WO1999055025A2 - Peak to average power ratio reduction in multicarrier modulation system - Google Patents

Abstract

The present inventions provide methods and systems for reducing the peak to average power ratio of a multi-carrier signal. Reducing the peak to average power ratio of a signal ensures that amplifiers and transmitters are not saturated, causing loss of data. Further, reducing peak to average power ratios reduces the consumption of power during transmission.

Description

Peak to Average Power Ratio Reduction

BACKGROUND OF THE INVENTION

This invention relates generally to communication systems. The present invention

relates more specifically to reducing peak to average power ratios in single carrier and multi-

carrier communication systems.

In recent years multi-carrier communication systems have received more attention.

Multi-carrier communication systems offer the promise of increased bandwidth combined with

two-way communications. However, several problems still remain to be solved to ensure the

widespread use of multi-carrier communication systems. One concern is how to reduce the

peak to average power ratio of a multi-carrier transmission.

Referring to figure 1 , a multi-carrier transmission is composed of a number of

independent signals. Figure 1 is a frequency domain plot of several signals 10(l)-10(n). Each

signal 10(l)-(n) is centered a different frequency f(l)-f(n). Often times the frequencies are

equally spaced apart. The frequencies are commonly referred to as carrier frequencies.

In most multi-carrier communication systems the signals 10(l)-(n) are combined

together as a vector. An inverse fast fourier transform (IFFT) is usually performed on the

vector to produce a discrete time domain signal which is converted to a continuous time domain

signal and transmitted. Figure 2 illustrates a continuous time domain representation of a typical

output signal 30 of a multi-carrier transmitter.

Signal 30 contains a number of peaks 31-34. A problem with the output signal is that

the peaks 31-34 often times exceeds the output capabilities of the transmitter. If the transmitter is only capable of transmitting at amplitudes of up to +/- 10 dB, the peaks saturate the

transmitter and the peaks are cutoff in the transmitted signal. Saturation causes the transmitted

signal to lose a significant amount of information, which may or may not be corrected for by the

receiver. Thus, it is important to reduce the peaks in order to maintain the integrity of the

transmitted signal.

Reducing the peak to average power ratio of a signal requires that the number and

magnitude of the peaks are reduced. There have been several attempts to reduce peak to

average power ratios, although they are only successful to a certain extent.

The placement of the different signals 10(l)-(n) at different carrier frequencies f(l)-f(n)

affects the shape of the output signal 30. One method randomly shuffles the phase of the

signals 10(l)-10(n) at each carrier frequency f(l)-f(n). Random shuffling does not completely

eliminate the problem, although randomizing has been shown to somewhat reduce the peak to

average power ratio to an extent. Random shuffling also requires performing an additional

IFFT. In addition to not completely reducing the peak to average power ratio to a practical

point, that particular method also requires that additional information, side information, be sent

along with the transmitted signal. In order for the receiver to be able to decode the transmitted

signal the receiver must also know how the signals 10(l)-10(n) were randomized. Thus, the

randomization scheme requires extra bandwidth to transmit the side information and does not

effectively reduce the peak to average power ratio.

Another method has been applied to multi-carrier communication systems that use a

small number of carrier frequencies. In that method all the different possible outputs of each

signal 10(l)-10(n) are simulated. For example, if each signal 10(l)-(n) is a 4-ary quadrature

amplitude modulated signal, each signal would be one of four different waveforms. If there are ten carrier frequencies, then over a million combinations are simulated. Those combinations of

the outputs of signals 10(l)-(n) that exhibit peak to power ratios that exceed a specified limit are

not used in actual transmissions. Typically, a channel must be simulated periodically because

of changes in the channel's characteristics.

The elimination of some of the possible combinations of the outputs of the signals,

however, reduces the bandwidth of the communication scheme. Further, the method can only

be applied to communication systems that use a few carriers since the number of simulations

required increases exponentially with an increase in the number of carriers. That is, if M-ary

QAM and N frequencies are used, N^M combinations must be simulated. M can be as high as

1024 and N even larger. Thus, this method becomes impractical when even a moderate number

of carriers are used.

A third method involves performing inverse fast fourier transforms on subsets of the

signals 10(l)-(n). For example, an IFFT may be performed on the first one fourth signals,

another IFFT for the second one fourth, and etc. The four output signals may then be linearly

combined to provide one output signal. Reducing the number of carriers within a single IFFT

output reduces the peak to average power ratio for that output signal since there are fewer signal

components. The linear combinations are compared to determine which combination has the

best PAR.

As the number of signals and carriers increase the number of IFFTs that must be

performed on the subsets of the signals increase, according to the number of signals

incoφorated within a single IFFT. The complexity of the transmitter thereby increases by the

number of IFFTs that must be performed, compared to a single IFFT. Further, information

about the linear combination of the transmitted signal must also be passed along to the receiver. This information is even more vital, and usually requires additional bandwidth to ensure proper

reception and decoding of the information.

In yet another method of reducing peak to average power ratio, the output signal of an

IFFT of all the signal components is scaled to bring the peaks below the maximum level. A

problem with this solution is that the signal to noise ratio is reduced proportionally with the

scaled factor. Reducing the signal to noise creates a great number of other problems which

makes this method unattractive. For example, as the signal to noise ratio decreases more errors

occur during transmission.

What is desired is a method of reducing the peak to average power ratio of a

transmission within a multi-carrier communication system. A method without a significant

decrease in the amount of usable bandwidth, and with low complexity such that reduction of the

peak to average power ratio may be performed in real time, is also desirable.

SUMMARY OF THE INVENTION

The present inventions provide methods and systems for reducing the peak to average

power ratio of communication signals. Reducing the peak to average power ratio of a signal

ensures that amplifiers and transmitters are not saturated, causing loss of data. Further,

reducing peak to average power ratios reduces the consumption of power during transmission.

In one embodiment, peak to average power ratios are reduced by selecting a subset of a

plurality of frequencies that make up a multi-carrier symbol. Peak reduction signals, carried at

the subset of frequencies, are computed to reduce the PAR of the symbol. In another embodiment, a kernel is generated that has components in the subset of

frequencies. The kernel is adjusted to negate one or more peaks in the multi-carrier symbol.

The adjustment of the kernel creates a subset of signals of a plurality of signals centered at the

plurality of frequencies. Negation of the peaks may be performed iteratively to remove any

peaks produced during prior peak reduction operations. In some embodiments, the subset of

frequencies are chosen prior to transmission. In alternate embodiments, the subset of

frequencies may be reselected during communication.

The subset of frequencies may be chosen to obtain a kernel that may better negate the

peaks of the multi-carrier symbol. In one embodiment the subset of frequencies may be chosen

based upon the characteristics of the channel. In other embodiments, the subset of signals may

be chosen randomly, pseudo-randomly, or combinations thereof.

In another aspect of the invention methods and systems are provided for estimating the

distortion of a received signal. Generally, any type of distortion may be estimated in order to

better decode the received signal.

In one embodiment the distortion is intentionally introduced into the signal in order to

reduce the peak to average power ratio of a single carrier or multi-carrier signal. Reducing the

peak to average power ratio of a signal ensures that amplifiers and transmitters are not saturated,

causing loss of data. Further, reducing peak to average power ratios reduces the consumption of

power during transmission.

In another embodiment of the present inventions the distorted signal is transmitted

without further attempts to embed information about the distortion back into the signals.

Rather, a receiver receives the distorted signal and estimates the missing information about the signal. The receiver reconstructs the original signal based upon the estimates of the distortion

of the signal.

In one particular embodiment, the signal is clipped as the form of distortion. The

receiver estimates the clipped portions of the signal in order to estimate and reconstruct the

original signal. In alternate embodiments some information about the clipping of the signal is

sent, such as the number of clips, magnitude of the clipping or information about the largest

clip. In alternate embodiments, side information is also provided to the receiver about the

distortion.

In another aspect of the invention methods and systems for reducing the peak to average

power ratio of a single carrier or multi-carrier signal are described. In one embodiment, peak to

average power ratios are reduced by selecting a subset of a plurality of frequencies that make up

a multi-carrier symbol. Peak reduction signals, carried at the subset of frequencies, are

computed to reduce the PAR of the symbol.

In one embodiment, a kernel is generated that has components in the subset of

frequencies. The kernel is adjusted to negate one or more peaks in the multi-carrier symbol.

The adjustment of the kernel creates a subset of signals of a plurality of signals centered at the

plurality of frequencies. Negation of the peaks may be performed iteratively to remove any

peaks produced during prior peak reduction operations.

In one embodiment, the subset of frequencies are chosen prior to transmission. In

alternate embodiments, the subset of frequencies may be reselected during communication.

The subset of frequencies may be chosen to obtain a kernel that may better negate the

peaks of the multi-carrier symbol. In one embodiment the subset of frequencies may be chosen based upon the characteristics of the channel. In other embodiments, the subset of signals may

be chosen randomly, pseudo-randomly, or combinations thereof.

In another embodiment the peak to average power ratios of a signal that may be a single

carrier or multi-carrier signal may be performed by applying a basis function to the signal. For

multi-carrier signals, a peak reduction signal is applied to one or more of the information

signals that make up the multi-carrier signal. The peak reduction signal is composed of one or

more kernels. The kernel is a basis function of the communication system. The application of

the basis function maps the information signal from an original constellation point to a duplicate

constellation point. A receiver decodes the information signal by mapping the duplicate

constellation point back to the original constellation point. The method of decoding may be

accomplished by performing a modulo operation on the received modified information signal.

These and other advantages of the present invention will become apparent to those

skilled in the art upon a reading of the following descriptions of the invention and a study of the

several figures of the drawing.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 illustrates a frequency domain plot of several signals of a multi-carrier

communication system.

Figure 2 illustrates a continuous time domain representation of a typical output signal of

a multi-carrier transmitter. Figure 3 illustrates a frequency domain plot of a DMT symbol prior to applying an

inverse fast fourier transform.

Figure 4 illustrates a signal constellation of a signal that is 4-ary quadrature amplitude

modulated.

Figure 5 illustrates a frequency domain representation of X in accordance with an

embodiment of the present inventions.

Figure 6 illustrates the frequency domain representation of C in accordance with an

embodiment of the present inventions.

Figure 7 illustrates the frequency domain representation of X+C in accordance with one

embodiment of the present inventions.

Figure 8 illustrates the continuous time domain representation of a symbol signal x(t) of

a multi-carrier communication system in accordance with an embodiment of the present

inventions.

Figure 9 illustrates a time domain representation of a desired symbol signal x^chp(t)=x(t) +

c(t) in accordance with an embodiment of the present inventions.

Figures 10A-C illustrate several approximate impulse functions p(t) in accordance with

an embodiment of the present inventions.

Figure 11 illustrates a multi-carrier transmitter in accordance with an embodiment of the

present inventions.

Figure 12 illustrates block diagrams of the modulator and the kernel applicator of figure

11 in accordance with an embodiment of the present inventions. Figure 13 illustrates a receiver in accordance with an embodiment of the present

inventions.

Figure 14 illustrates the preliminary process of determining the peak reduction channels

in accordance with an embodiment of the present inventions.

Figure 15 illustrates a flow chart of the operation of the kernel engine of figure 12 in

accordance with an embodiment of the present inventions.

Figure 16 illustrates a constellation of an information signal in accordance with an

embodiment of the present inventions.

Figure 17A illustrates a continuous time signal x₀(t) in accordance with an embodiment

of the present inventions.

Figure 17B illustrates a continuous time signal x_k(t) 500(t), which corresponds to signal

X_k 500, in accordance with an embodiment of the present inventions.

Figure 17C illustrates a continuous time signal x_N-1(t) in accordance with an embodiment

of the present inventions.

Figure 17D illustrates a combined continuous time signal x(t) in accordance with an

embodiment of the present inventions.

Figure 18 illustrates an original constellation with duplicate constellations in accordance

with an embodiment of the present inventions.

Figure 19A illustrates an original continuous component time signal x_k(t) 500(t) in

accordance with an embodiment of the present inventions. Figure 19B illustrates a sinusoid s(t) which would reduce the peak of x_k(t) at t = r^ in

accordance with an embodiment of the present inventions.

Figure 19C illustrates a modified continuous time signal x (t) , which corresponds to

modified signal X_k , in accordance with an embodiment of the present inventions.

Figure 19D illustrates the modified time signal x(t) , which is the sum of all the

component time signals x₀(t) through x_N.,(t), including modified time signal x_k (t) in

accordance with an embodiment of the present inventions.

Figure 20 is a constellation with duplicate constellations of a 16 QAM signal in

accordance with one embodiment of the present inventions.

Figure 21 illustrates a constellation map in accordance with an embodiment of the

present inventions.

Figure 22 illustrates a flow chart of the operation of the kernel engine of figure 12 in

accordance with another embodiment of the present invention.

Figure 23 illustrates a multi-carrier time domain signal, x(t).

Figure 24 illustrates a clipped signal, x^clιp(t), in accordance with an embodiment of the

present inventions.

Figure 25 illustrates a received signal y^cl,p(t) in accordance with an embodiment of the

present inventions.

Figure 26 illustrates a reconstructed signal x⁽¹⁾ (t) that is the first estimate of x(t) in

accordance with an embodiment of the present inventions. Figure 27 illustrates a reconstructed signal x^(q)(t) after q iterations of clip estimation in

accordance with an embodiment of the present inventions.

Figure 28A illustrates a block diagram of a transmitter in accordance with an

embodiment of the present inventions.

Figure 28B illustrates a block diagram of a transmitter in accordance with an another

embodiment of the present inventions.

Figure 28C illustrates a block diagram of a channel in accordance with an embodiment

of the present inventions.

Figure 29 illustrates a block diagram of a receiver in accordance with an embodiment of

the present inventions.

Figure 30 illustrates a multi-carrier time domain signal x(t) with a number of peaks.

Figure 31 illustrates a clipped signal x^cllp(t) in accordance with another embodiment of

the present inventions.

Figure 32 illustrates the clipped signal of figure 31 with reconstructed peaks in

accordance with an embodiment of the present inventions.

Figure 33 illustrates a reconstructed signal x^(q)(t) after q iterations of clip estimation in

accordance with an embodiment of the present inventions. DETAILED DESCRIPTION OF THE PRESENT INVENTIONS

The present inventions provide apparatuses and methods of reducing peak to average

power ratios in single carrier and multi-carrier communication systems without significantly

reducing the amount of bandwidth. The present inventions may also be implemented with a

low amount of complexity such that they may be implemented in real time. Additionally, no

significant amount of side information is required, which would reduce bandwidth, nor is there

a reduction in the signal to noise ratio or quality of service.

The present inventions apply to any type of communication systems utilizing multiple

carriers. By way of example, the present inventions apply to Discrete Multi-Tone (DMT),

Orthogonal Frequency Division Multiplexing (OFDM), Discrete Wavelet Multi-Tone (DWMT)

communication systems, Vector Coding Modulation. Alternate embodiments of the present

inventions apply to single carrier communication systems, such as Carrier-less Amplitude Phase

(CAPs), vestigial side band, amplitude modulation and the like.

Referring to figure 3, a multi-carrier communication system takes advantage of a

channel by sending several signals over a wide band of frequencies. Figure 3 is a frequency

domain plot of a DMT symbol 100 prior to applying an inverse fast fourier transform. The

DMT symbol is a function of a number of signals 110(0)- 110(N-1), each centered at a different

frequency 120(0)-(N-1). While details of the present inventions are discussed in terms of a

DMT communication system, the advantages of the present inventions apply readily to other

types of multi-carrier communication systems, and the present inventions are not restricted to

only DMT systems. Each signal 110(0)-(N-1) may carry any number of bits of information in a digital

system. By way of example, each signal may be modulated by M-ary quadrature amplitude

modulation, M-ary phase shift key, frequency modulation, amplitude modulation, continuous

phase modulation or any other type of suitable modulation scheme. The illustrated signals are

M-ary quadrature amplitude modulated. Thus, each signal 110(0)-(N-1) has a magnitude and a

phase in addition to its frequency.

Figure 4 is a signal constellation of signal 110(3) that is 4-ary quadrature amplitude

modulated. Signal 110(3) has an amplitude, A, and a phase, ø. Depending upon the amplitude

and phase, signal 110(3) may represent one of four binary values, 00, 01, 10 and 11, as

illustrated.

Each signal 110(0)-(N-1) are all quadrature amplitude modulated, but may have

different constellations. The number of constellation points that a signal represents depends

upon the characteristic of the channel for that particular frequency. That is, if frequency 120(4)

is less noisy than frequency 120(3), then signal 110(4) may have an 8-ary QAM constellation or

greater. Thus, by looking at the characteristics of the channel less noisy frequencies may carry

signals that represent a greater number of bits.

In one embodiment of the present inventions, those frequencies that have a lot of noise

and are capable of only carrying low bit rate signals are used as peak reduction frequencies.

The peak reduction frequencies may carry no signal at all. It has been found that having peak

reduction frequencies that carry no signal may sometimes marginally help to reduce the peak to

average power ratio of a transmission.

In another embodiment, the peak reduction frequencies carry peak reduction signals.

Peak reduction signals, like regular signals, have an amplitude and a phase. However, in one embodiment, the peak reduction signals generally do not carry any data. Rather, the peak

reduction signals are scaled and shifted such that the peaks of the output signal are dramatically

reduced.

In alternate embodiments of the present inventions, the peak reduction frequencies may

be chosen by any suitable method. Frequencies that are noisy are utilized as peak reduction

frequencies since the decrease in data rate of the output symbol is minimized. However, a

different selection of peak reduction frequencies may provide better peak to average power ratio

reduction with fewer peak reduction frequencies. It has been found that randomly selected peak

reduction frequencies provides good peak to average power ratio attenuation . Selection of peak

reduction frequencies is discussed further below.

Because of the properties of an inverse fourier transform changing the attributes of one

or more of the components of a signal before it is inverse fourier transformed effects the

transformed signal. In the case of DMT a discrete time signal x is generated from a number of

complex valued QAM modulated signals 110(0)-(N-1), or X. Where

x = [ x . X ]

N-1 ^J and

X = [ ^LX 0 ... X n ... X N-l ]

The elements of X are complex values that represent the amplitude and phase of the signals X₀-

X_N.„ where the frequencies f₀-f_N-1 are of equal bandwidth and separated by 1/T, where T is the

time duration of a DMT symbol. Each element of x is a symbol derived from X defined by:

1

. = -₇= ∑X_ke'²*"^N , k = 0,..., N - \

V N A=0 which can be written as x = QX, where Q is the IFFT matrix and the elements of Q are

l lπkn l N

The peak to average power ratio (PAR) of x is then:

PAR = ε [\\ x \\² ₂]/ N

where ||v||_∞ is the co-norm of the vector v, or the maximum absolute value, ||v||₂ is the 2-norm of

the vector v, or the root mean square, and ε[f(v)] is the expected value of the function f(v).

The peak reduction frequencies, once chosen, can be assigned arbitrary amplitudes and

phases. In one embodiment, the peak reduction frequencies may be initialized with zero

amplitude and zero phase. The values for the peak reduction signals are represented as the

vector c in the time domain, and C in the frequency domain, where.

x + c = Q(X +C)

The possible values for c are chosen such that

min,. II x + c l

PAR(c*) = c I I «

£ [|| X ||₂ J7.V ε [\\ x \\² ₂]/ N

where c* is the optimal solution for c. The value of the right side of the inequality is the PAR

of the signal generated from the vector x, and the left side of the inequality is the PAR of the

peak reduced signal generated from the vector x + c. The values for C at the peak reduction frequencies may be any suitable value that helps

to reduce the peaks in the transmitted multi-carrier symbol. However, the values for C at-the

non-peak reduction frequencies are always zero, such that the values of C do not interfere with

X. Thus,

Initially, C may be set to zero, and the values for C_k changed later to reduce the PAR. L is the

number of peak reduction frequencies that are utilized to reduce the PAR of x. If N frequencies

are available, then the ratio of peak reduction frequencies to the overall number of frequencies

is L/N. However, the actual bandwidth loss is the number of bits that the peak reduction

frequencies were capable of carrying over the total number of bits that all N frequencies are

capable of carrying. By selecting peak reduction frequencies that are capable of carrying few,

or zero, bits per symbol, bandwidth loss is minimized. The non-zero values, C for k e {i„ . . . ,

i_L} or c , are called the peak reduction signals, or peak reduction tones in the case of DMT, or

more generally dummy signals.

The values for X are zeroed at the peak reduction frequencies. Figures 5 and 6 show the

frequency domain representations of X and C, respectively, according to one embodiment of the

present inventions. The frequencies f„ f₅, f₆, f₇ and f_N.₂ are chosen as peak reduction

frequencies. Accordingly, the values for X,, X₅, X₆, X₇ and X_N.₂ are zero. The other values for

X correspond to the amplitude and phase of those signals.

In alternate embodiments, only one component of the values of X may be zeroed out and

used for peak reduction purposes. By way of example, the real part of the values of X,, X₅, X₆,

X₇ and X_N_, may be zeroed out and the imaginary part of the components used to carry information. Analogously, one of the phase or amplitude components of the values of X may

be zeroed out and used for peak reduction while the other is used to carry information.

The values for C correspond to the peak reduction frequencies. The index i conforms to

the peak reduction frequencies, e.g., i, is the index for the first peak reduction frequency f,, i₂ is

the index for the second peak reduction frequency f₅, and etc.

Figure 7 illustrates the frequency domain representation of X+C, in accordance with one

embodiment of the present inventions. In the combined signal all the frequencies contain a

signal. The non-zero values of peak reduction signals C are located at the peak reduction

frequencies, while the actual signals X are located at the non-peak reduction frequencies.

Initially, the peak reduction signals C may have any arbitrary values. However, it is useful to

initialize the values of C at zero.

The first set of values of C may then be represented as the initial values C(0). If C(0)

are zeroes, then X+C(0) = X, and x+c(0)=x. The time domain representation of x+c(0) is

equivalent to the unmodified signal x(t), as illustrated in figure 8. However, the values for C

should be chosen to provide a signal (x+c) that does not have peaks that exceed a

predetermined magnitude. Figure 9 is a time domain representation of a desired signal

x^cl,p(t)=x(t) + c(t) generated by the vector x + c.

The continuous time domain waveforms depicted in figures 8, 9 and other figures are

representative of analogous discrete time domain waveforms. A majority of the algorithms

used in the present inventions are predominantly performed in discrete time due to practical

considerations. The continuous time domain waveforms are used for purposes of illustration.

However, the scope of the present inventions includes analogous algorithms performed in

continuous time and frequency domains. The values for C* and c*, the optimal solution that would provide an x^cl,p(t) with the

smallest PAR, may be obtained by solving the following equation:

min x + c mm x + QC

Qis the sub-matrix of Q constructed from the columns i,, . . ., i_L, and c represents the non-zero

values of C. c* can actually be solved through linear programming. Solutions may also be

found separately for the real and the imaginary parts of x or X.

The above equation may be rewritten in the following form:

min t c subject to x + QC ≤_N t \_N, x + Q ≥_N -t l_N

Moving all the unknowns to the left hand side, the equations may be rewritten as:

min t c subject to QC - t 1 _N ≤_N - x,

QC + t l„ >„ - x

or

min t c subject to

The linear program has 2L+1 unknowns {Real(C ), Imag(C ), t} and 2N inequalities written in

the standard linear program form: min c^τx subject to Ax ≤_N b

Linear programming algorithms exist to solve for c*. The linear programming solutions

provide the ideal solution c*. Currently, the exact solution approach is most practical in

communication systems operating at data rates of approximately 500 kbps or lower because of

the amount of computations required to compute the exact solution for c*. However, good

approximations of c* may be obtained such that the PAR of x can be satisfactorily reduced in

real time for higher data rate systems. However, as processing power becomes more readily

available in the future the linear programming solution may be utilized in multi-carrier

communication systems operating at higher speeds in accordance with the present inventions.

Approximating c, C

As seen in figure 8, the time domain signal x(t) has several peaks 130-133. The peaks

130-133 can be reduced by adding or subtracting an appropriately scaled impulse function δ(t)

at those peak time values. The impulse function, however, must be constructed from the peak

reduction frequencies,! i„ i₂, . . . , i_L}. Since a true impulse function cannot be created by less

than all the frequency components, i.e., when L < N, an approximate impulse must be used, p.

Figures lOa-c illustrate several approximate impulse functions p(t), generated from

different values of p, in accordance with one embodiment of the present inventions. Since only

the L peak reduction frequencies can be used to create the approximate impulse function p(t), or

kernel, p(t) is not ideal. One useful constraint that may be placed upon p(t) is that the value for

p(0) is equal to one. This allows p(t) to be scaled more readily. Figure 10a may be a first approximation of an impulse. The lobes around the impulse

should however be reduced in magnitude. The side lobes should be reduced to ensure that when

the impulse is applied to x(t) to clip a particular peak no other portion of x(t) exceed the

maximum value. Another approximation of an impulse may look like the approximation in

figure 10b. Obviously, the secondary peaks of figure 10b poses a problem when applied to x(t).

Ideally, p(t) should resemble the waveform depicted in figure 10c.

Solving for the mean square error between p = Q P and an ideal discrete time impulse

e₀= [1 0 . . . 0]^τ provides the solution for an approximation of p that is the mean square error.

The mean square error minimizes the sum of all the peaks of the kernel, or power, other than the

peak at p(O).

P₂ ^b = arg min ||QP - e₀

The solution becomes:

P₂ ^b = (Q^TQr'Q^T o = Q^Te₀ = [! ...!]

N

Since the value for p₀ should be equal to one we can scale the result to obtain the mean square

error optimal solution for p, p*.

N p

^~ L ^L

Since P has non-zero values only at the peak reduction frequencies, C may be

represented as any suitable linear combination of P. The linear combinations of P correspond to

the scaled and shifted versions of the kernel, p, such that the scaled and shifted versions of p

negate the peaks of x. For example, if p(t) of figure 10c were to be applied to x(t) of figure 8,

p(t) would be inverted and shifted to t-2 in order to cancel out the first peak 130. Also, if the

first peak 130 exceeded the maximum value by some factor α, p(t-2) would be scaled by a value

greater than α, such as (1.2α). When x(t) and (1.2 )p(t-2) are added the value at t=2 would be

the maximum value + α - 1.2α, which gives us a value less than the maximum value (maximum

value -0.2α). The scaling and time shifting of p merely scales and phase shifts the values of P,

and therefore C . C , which is a linear combination of P, will have zero values at the non-peak

reduction frequencies.

Any number of peaks may be clipped in this fashion in one iteration. However,

reducing one or more peaks may cause the resulting waveform to exceed the maximum value at

other positions. Therefore, the process may be repeated with the resulting x^{c p}+c to achieve a

new x^chp with a PAR that is satisfactory.

In order to minimize the second highest peak of p(t), thereby reducing all the peaks other

than the peak at p(0), a linear program may be used to solve for the infinite norm equation.

P^ = arg mjn |[/>, p₂ ... p_NΛ f_∞ , subj. to p_Q = 1

p. = QP.

PL, provides the optimal solution, producing a p(t) that resembles the waveform illustrated in

figure 10c. The solution of p regardless of its order may be computed in advance, or off-line, since

only the peak reduction frequencies need to be known. Thus, p may be predetermined once the

peak reduction frequencies have been chosen. Once p is known, p may be linearly combined in

any fashion to produce the necessary values for c and C. The resulting c is a good

approximation of c* depending upon the number of iterations performed.

In one embodiment, the choice of the peak reduction frequencies may be based upon

obtaining a good kernel, p. Once the number of peak reduction frequencies, L, has been

determined, the location of the peak reduction frequencies may be determined based upon

deriving a good, or the best, kernel, p. Certain quality factors may be imposed before accepting

a p as a valid kernel. By way of example, a p with secondary peaks greater than a

predetermined magnitude may be rejected. That set of peak reduction frequencies may then be

rejected and a new set of peak reduction frequencies selected to provide a better p.

It has been found that randomly selected peak reduction frequencies will often times

provide a good kernel. If a first set of peak reduction frequencies chosen randomly does not

provide a good kernel, a new selection of peak reduction frequencies that swap a subset of the

first randomly chosen peak reduction frequencies sometimes provides a better kernel. The

combinations of peak reduction frequencies may be iteratively evaluated until a kernel with the

appropriate characteristics is obtained.

In another embodiment, peak reduction frequencies may be chosen based upon the bit

rates of the frequencies. In one instance, a pseudo-random selection of the peak reduction

frequencies may be performed with weights applied to those frequencies that have low bit rates

that make the selection of those frequencies more likely. If after several iterations a proper kernel, p, cannot be obtained the weights may be adjusted since the weighted frequencies may

not be good candidates for constructing a proper kernel.

After the peak reduction frequencies have been chosen the optimal, or a good

approximation of the optimal kernel is computed. Using the resulting kernel, p, the peak

reduction vector c, containing the peak reduction signals, or peak reduction signals, may be

constructed. Initially, the vector x + c(0), where the values of c(0) is all zeroes, is computed by

taking the IFFT of the vector X, containing zero values in the peak reduction frequencies. If

only one peak is negated during a single iteration of applying the kernel, p, is performed x^chp(l)

= x + c(l), where c(l) = A,p[(n-Δ_!)]_N in the discrete time domain, where is A a scaling factor

and Δ is a time shift. If two peaks are canceled in one iteration

x^clιp(l) = x + c(l), where c(l) = A,p(n-Δ,) + A₂p(n-Δ₂), and so on.

Any number of peaks may be canceled in a single iteration. Obviously, canceling more

peaks requires more computations per iteration without being able to readily determine if the

multiple application of several scaled and/or shifted kernels have not introduced newly created

peaks. Thus, in one embodiment, it may be advantageous to limit the number of peaks per

iteration. Once an iteration is complete the kernels may be linearly combined to produce c(j),

where j is the current iteration. After computing c(j) and adding it to x, the new x^cl,p, x^{c p}(j), can

be reevaluated to determine if further peaks require cancellation.

Further iterations may be performed by taking the previous x^cl,p and adding another set

of values for c, i.e., x^clιp(j) = x^clιp(j-l) + c(j). Since the values of x remain the same because p

and P are only functions of the peak reduction frequencies this sum expands to x^cl,p(j) = x + c(0) + c(l) + ... + c(j -l) + c(j), or x^cl,p(j) = x + ∑c(m), and m=0

I c* = c(m) as j — > ∞ m=0

The sum of c's is equal to a number of scaled and/or shifted kernels, p. If only one peak is

corrected (only one peak is canceled) per iteration then the equation becomes:

x ^,p(j) = x + A₁p[(n - Δ₁ )] + A₂p[(n - Δ₂)]_N + ... + A^pKn - Δ,_.,)],, + A _}p[(n - &,)] , oτ

x^clιp G) = x + ∑A_mp[(n - Δ_m)]_N m=0

Thus, c is computed simply by performing multiplies and adds, and does not require any

additional transforms, which are significantly more computationally intensive. Thus, the

present inventions require significantly fewer computational resources than other methods that

have been used to reduce the PAR of a multi-carrier signal.

The process may be repeated indefinitely until the summation of c approaches the

optimal peak reduction signal vector c*. But a good approximation of c* may be obtained in as

little as one or two iterations. The quality of c depends upon the quality of the kernel p, which

depends upon the number and location of the peak reduction frequencies. Thus, as L, the

number of peak reduction frequencies increases towards N, the total number of frequencies,

better approximations of c* are obtained in fewer iterations.

By way of example, four iterations at one kernel application per iteration when the ratio

of L N is 5% has produced good results. Application of the present inventions with higher L N

ratios produce better results with fewer iterations. In alternative embodiments, discussed further below, it be helpful to know the values of

C once c has been computed. In those cases a fourier transform of c provides the values for C.

Since c does not contain any frequency components in the non-peak reduction frequencies the

fourier transform of the entire signal x + c need not be computed. Further, if operations are

performed on C in order to provide better performance or added functionality the inverse fourier

transform of C may be taken to obtain a new c. The new c can be added to x to provide the new

x^clιp. Again, the inverse transform of X is not needed. Thus, even when additional transforms

are utilized the transformation operations are simpler than transforming the entire signal.

Once x^chp is determined it is transmitted to a receiver. The receiver, or demodulator,

decodes x^c!,p. A fourier transform is performed on the decoded signal. The values of the peak

reduction signals at the peak reduction frequencies are discarded since they typically do not

carry any information. The values of X_recened are then further decoded to extract the information

carried by those multiple carriers.

In alternate embodiments, the peak reduction signals may include some type of

additional information. In those embodiments the peak reduction signals, C_receιved, are also

decoded.

Figure 11 illustrates a multi-carrier transmitter in accordance with an embodiment of the

present inventions. Transmitter 200 includes an encoder 202, modulator 204, kernel applicator

206 and a digital to analog converter 208. Encoder 202 receives a stream of digital data and

encodes the data such that it can be transmitted over several different carriers. The encoder 202

provides the segmented data to modulator 204. Modulator 204 modulates the segmented data

using an appropriate modulation scheme, such as QAM. The individually modulated signals are combined together as a vector to produce a single frequency domain signal, X. Certain

predetermined frequencies, peak reduction frequencies, are not used.

Modulator 204 provides the frequency domain signal, X, to kernel applicator 206.

Kernel applicator 206 performs an inverse fourier transform to X to obtain x, which also

modulates the signals to the frequencies f₀-f_N-1. Kernel applicator 206 adds peak reduction

signals, c, to x in order to reduce the PAR of x. Initially, the peak reduction frequencies and a

kernel are predetermined, as discussed above. The choice of peak reduction frequencies, in one

embodiment, may be based upon the characteristics of the channel. In alternate embodiments,

the frequencies are chosen purely randomly, randomly with weights applied to frequencies with

low bit rates, according to channels that are not utilized by the particular communication

system, or any other suitable method.

Once kernel applicator 206 has finished reducing the peak to average power ratio of the

signal x, it provides x as another symbol of the discrete time sequence, x^chp(n) to digital to

analog converter (DAC) 208. DAC 208 converts the discrete time signal to a continuous time

domain signal x^clιp(t). The DAC may also include filters or other signal processing components.

The waveform of x^{c p}(t) has peaks that predominantly does not exceed a predetermined

maximum magnitude. Currently, it is desirable to limit the peaks of x^{c p}(t) to below 8-12 dB.

However, the present inventions may provide better PAR reduction depending upon the number

of peak reduction frequencies and iterations. By way of example, with a L N ratio of 20% the

PAR of a signal may be reduced to about 6 dB or lower within a finite number of iterations.

With proper peak to average power ratio reduction x^clιp(t) resembles the waveform illustrated in

figure 9 as opposed to the waveform illustrated in figure 8, which represents x(t) without the

application of a kernel. Figure 12 illustrates block diagrams of modulator 204 and kernel applicator 206 of

figure 11 in accordance with an embodiment of the present inventions. Encoder 202 segments

the data and provides the data to modulator 204. Modulator 204 includes a number of

modulators 210(0)-(N-1). Modulator 204 modulates the separate data streams with modulators

210(0)-(N-1). Modulators 210(0)-(N-1) modulate the individual data streams by the appropriate

modulation scheme.

In the illustrated embodiment the data streams are modulated by an M-ary QAM

scheme. However, any suitable type of modulation scheme may be utilized in accordance with

the present inventions. The output of modulators 210(0)-(N-1) provide the components of X,

X₀-X_N.,.

In an alternate embodiment, modulator 204 may also modulate the data segments to the

frequencies f₀-f_N_, . The modulated signals may be summed to produce x. This type of

modulation does not require an inverse fourier transform to obtain x, and x is directly fed to the

kernel applicator.

Selection of the peak reduction frequencies are made in advance. The modulators

210(0)-(N-1) corresponding to the peak reduction frequencies do not receive data from encoder

202. Rather, the peak reduction modulators are set to an initial value, such as zero amplitude

and phase.

Inverse fast fourier transformer (IFFT) 220 transforms X to provide the discrete time

equivalent x. IFFT 220 passes x to kernel engine 222, which applies a kernel to discrete time

equivalent x. The particular kernel is also computed beforehand based upon the selection of the

peak reduction frequencies. The kernel engine 222 analyzes x to determine how the kernel

should be scaled and time delayed to remove the peaks in x. The scaled and delayed kernel is added to x resulting in x^chp = x + c. c is a linear combination of one or more kernels that have

been scaled and time delayed to negate one or more peaks in x. Kernel engine 222 outputs x^{c p}

as part of the overall discrete time data stream x(n).

The value of c may result from one iteration of applying one or more kernels to x.

Alternatively, c may be accumulated over several iterations of applying the kernel to x.

Iteration is useful because the first iteration may negate the original peaks of x, but may also

create other peaks due to the imperfection of the kernel.

In the illustrated embodiment, more than one iteration of applying a kernel to x is

performed. Kernel engine 222 provides the values of c(j), the newest linear combination of the

kernel. In one embodiment c(j) may be the accumulated linear combination including past

iterations of applying the kernel. If no further iterations are necessary x + c is provided to DAC

208.

Once DAC 208 converts the discrete time signal into a continuous time signal, the

continuous time signal may be transmitted to a receiver through a channel. Again, DAC 208

may perform additional filtering and signal processing.

Figure 13 illustrates a receiver in accordance with an embodiment of the present

inventions. Receiver 300 includes a FFT 302, a demodulator 304 and a decoder 306. Before

FFT 302 receives the received signal x_r(t), the received signal may have been passed through

filtering and/or other signal processing. The received signal may also be converted from analog

to digital, providing a discrete time domain received signal x_r(n).

FFT 302 applies a fourier transform to the received signal to produce X_, which is

provided to demodulator 304, and C_r. ^ provides the values of the data signals centered at the non-peak reduction frequencies of f₀-f_N.,. The elements of X. are further decoded to extract the

data carried by those signals.

C_r is typically discarded if the peak reduction signals do not carry any information and

are not further decoded. However, in alternate embodiments where C_r does carry some type of

information those components of the received signal may be decoded as well.

In one embodiment, a number of band pass filters centered at frequencies f₀-f_N_, are

applied to X,. to extract the different frequency components of X,.. Individual demodulators then

demodulate the band passed signals to extract the separate data streams. The data streams are

recombined to reproduce the original data stream.

Figures 14 and 15 illustrate flow charts describing the process applying a kernel. Figure

14 illustrates the preliminary process of determining the peak reduction channels. Flowchart

400 begins at block 402 and proceeds to block 404. In block 404 the peak reduction frequencies

are chosen. The peak reduction frequencies may be chosen based upon the characteristics of the

channel. As described, frequencies that are capable of handling low bit rates, or no

communication at all, may be chosen as peak reduction frequencies.

In an alternative embodiment, the peak reduction frequencies may be chosen randomly.

Alternatively, the frequencies may be chosen pseudo-randomly with weights applied to the low

bit rate frequencies to make their selection more likely. Higher frequencies tend to be noisier

frequencies in many applications and the peak reduction channels may be chosen primarily in

the higher frequencies. But, in many cases peak reduction frequencies that are sequentially

grouped may provide less PAR reduction than randomly selected peak reduction frequencies.

The choice of peak reduction frequencies should, however, be made in light of obtaining a

sufficient kernel to perform adequate PAR reduction. The number of peak reduction frequencies compared to the number of overall

frequencies is also determined. A greater number of peak reduction frequencies provides better

performance. However, as the number of peak reduction frequencies increases more bandwidth

is lost to the peak reduction signals. Thus, a tradeoff must be made between performance and

bandwidth. A ratio of peak reduction frequencies to overall frequencies of about 5% has been

found to provide good performance while minimizing the loss of bandwidth. However, any

suitable ratio may be used depending upon the needs of the system.

Proceeding to block 406 a kernel is computed from the chosen peak reduction

frequencies. The above described algorithm may be used to compute a best approximation of

an impulse. The computation of the kernel may also be performed by linear programming. In

block 408 the chosen peak reduction frequencies and the computed kernel are applied to the

relevant parts of the transmitter. By way of example, the encoder and modulator are configured

to modulate data at the non-peak reduction frequencies. The kernel information is supplied to

the kernel engine. The flow chart ends in block 410.

Figure 15 illustrates a flow chart 450 of the operation of kernel engine 222 of figure 12.

The flow chart 450 begins in block 452 and proceeds to block 454. In block 454 x is received

from IFFT 220. Initially, IFFT 220 provides a peak reduction component, c(0), that is zeroed

out.

In block 456 the kernel engine analyzes x + c(j) and applies one or more kernels to x +

c(j) to reduce any peaks. In the first pass x^clιp(j) = x +c(j); j=0. The kernel engine may negate

one, two, or as many peaks as desired in one iteration. However, the more peaks that are

canceled in a single iteration the more computation that is required. A tradeoff may be made

based upon the available computational resources and the need for better performance. In block 457 the index j is incremented. Proceeding to block 458 the kernel engine

translates the scaling and shifting of the kernel into values for c(j). In block 459 the new peak

reduction components are accumulated by adding the previous peak reduction components; c(j)

= c(j) + cG-l).

The kernel engine determines whether more iterations are required in block 460. If no

other iterations are required the current x^clιp0) = x + c(j), is passed on to DAC 208 in block 464,

where c(j) is the accumulated sum of all the iterations of applying the kernel. When further

iterations are required, flow proceeds to back to block 456.

The operations of flow chart 400 of figure 14 may be performed before any

transmissions occur. The operations may also be performed periodically during transmission as

well. Whenever the characteristics of the channel changes new peak reduction frequencies may

be chosen, and a new kernel calculated. The transmitter may be updated on the fly, without

significantly interrupting communications.

Of course, the receiver must know which frequencies are peak reduction frequencies.

That information is transmitted to the receiver before communications with a new set of peak

reduction frequencies begin. The information about the identity of the peak reduction

frequencies is small and does not significantly affect the bandwidth of communications. The

peak reduction frequencies information is also intermittent, occurring rarely. By way of

example, peak reduction frequencies may be chosen in increments of minutes, hours, days,

weeks, months or years, depending upon the stability of the channel. Even if re-selection of the

peak reduction frequencies occurs every few minutes, the data would not prohibitively burden

the bandwidth of the communication system. In many applications the selection of peak reduction frequencies and a corresponding kernel need only be computed once, during

initialization of a communication system.

The operations of the transmitter may be performed by discrete components or more

general purpose devices. By way of example, a digital signal processor may perform any or all

of the functions of the encoder, modulator, and the kernel applicator. However, more

specialized devices may provide better performance.

In certain situations the average distribution of energy may be higher in the peak

reduction frequencies than the non-peak reduction frequencies. To alleviate this potential

concern a repeating pattern of peak reduction frequencies and kernels may be used for success

symbols transmitted. A first symbol would use one set of peak reduction frequencies, a second

symbol would use another set of peak reduction frequencies, and repeating after the last set of

peak reduction frequencies has been used. The receiver would also be informed in advance of

the different sets of peak reduction frequencies and synchronized. In this alternate embodiment

average energy is more evenly distributed over all the frequencies. Switching between different

sets of peak reduction frequencies may also be performed for other reasons besides energy

distribution.

The PAR is a time-varying quality and fluctuates per symbol that is transmitted, which

depends upon various factors. At times when the PAR of a particular symbol is low the PAR

reduction may be turned off for that symbol. This frees up the peak reduction frequencies to

carry data. For example, when the PAR is below 10 dB PAR reduction is turned off for that

symbol. When the PAR becomes a problem the peak reduction frequencies may then be used

for peak reduction. In addition, the number of peak reduction frequencies may be varied depending upon the conditions. Informing the receiver requires very little additional

information and does not take up a significant amount of the overall bandwidth of the system.

In a further embodiment, different sets of peak reduction frequencies, and corresponding

kernels, are precalculated. During the analysis of a symbol a selection of one of the sets of peak

reduction frequencies may be made based upon which set provides the best PAR reduction. In

one embodiment, the selection of one of the sets must be transmitted to the symbol, however the

bandwidth required for sending the information is low in comparison to other PAR reduction

schemes. In other embodiments, the receiver may be able to detect from the transmitted symbol

which peak reduction frequencies are being used.

Combined Information and PAR Reduction Signals

As mentioned the peak reduction signals may be used in alternate ways. By way of

example, the peak reduction signals may be used for peak reduction and to carry information.

In embodiments where the peak reduction and data signals include more than one component,

e.g., an amplitude and a phase value, or a real and an imaginary value, one of the two values

may be used specifically for peak reduction while the other may be used to carry information.

In such embodiments a set of kernels may be computed for increment of delay, rather than one

kernel that is shifted. This removes one dimension of variability in the peak reduction signals

such that a single component of the peak reduction frequencies may be used for peak reduction

and the other component used for other purposes. In one embodiment of the present inventions, all the frequencies carry information

signals. However, a subset of the information signals are augmented to act also as peak

reduction signals.

The information signals are modified by adding a basis function of the communication

scheme to the information signal. The addition of the basis function maps the original

information signal constellation to one or more duplicate constellations. The addition of the

basis function also reduces the contribution of the original information signal to the peak to

average power ratio of the transmitted symbol. One or more basis functions may be added to

the original information signal in order to reduce the peak to average power ratio. The use of

basis functions also facilitates decoding of the original information signal by the receiver. The

receiver may simply perform a modulo operation on the received modified signal to obtain the

original information signal.

For different types of communication systems different basis functions are used. By

way of example, Discrete Wavelet Multi-Tone communication systems use a wavelet as the

basis function. The present inventions may be applied to any suitable type of communication

system that utilize a basis function for encoding information into a signal.

Figure 16 illustrates a constellation of an information signal 500 in accordance with an

embodiment of the present inventions. The illustrated constellation is a 4 QAM constellation,

including four constellation points 501-504. The signal 500 carries 2b bits of information,

where b is the number of bits per dimension. In the illustrated constellation b=l . Thus, the

number of potential values is equal to 2^2b, which is equal to four in the illustrated constellation. Generally, the constellation points are separated by a distance, d, from its nearest

neighbor. The dimensions of the constellation is dM by dM, where M = 2^b and is the number of

levels per dimension. In the illustrated example, d = 2 and M = 2.

Signal 500 is composed of a real part and an imaginary part, designated by the in-phase

(i) axis and the quadrature axis (q), respectively. Signal 500, or X_k, may be written as X = R_k +

jl_k. The values for R_k and I_k can take the values {+/- d/2, +/-3d/2 . . . +/-(M-l)d/2}. The real

and the imaginary components determine the amplitude and phase of the signal. Where A =

sqrt(R_k ² + I_k ²) and ø = tan 'CVRJ.

When a receiver receives signal 500 some noise may be included with the signal. The

constellation for an uncoded signal is segmented according to the possible constellation points

for purposes of decoding. For example, the shaded region 506 is a constellation region

corresponding to the constellation point 504. If the receiver receives any signal falling within

region 506 the receiver decodes the signal as carrying the value of constellation point 504.

The illustrated constellation is arranged in a square constellation. However, other forms

of constellation packing may be used in accordance with the present inventions. By way of

example, hexagonally packed constellations, rectangular constellations, circular constellations,

cross-constellations or any other suitable constellation configuration may be used. The

following discussion of one embodiment of the present inventions focuses on a square packed

constellation, but any of the aforementioned constellation types may be utilized with slight

modifications. The dimensions of alternate configurations of constellations may have different

dimensions than the illustrated square packed embodiment. For example, a rectangular

constellation would have two values of M, M_; and M_q. A vector X is composed of QAM constellation values, X = [X₀ . . . X_N.,]. The

coreesponding time sequence x = IFFT(X) = [x₀ . . . x_N.,]. Referring now to figures 17A-17D,

the continuous time function x(t) is the sum of all the continuous time functions x₀(t) through

x_N.,(t). Figure 17A illustrates a continuous time signal x₀(t), which corresponds to signal X₀;

figure 17B illustrates continuous time signal x_k(t) 500(f), which corresponds to signal X_k 500;

and figure 17C illustrates continuous time signal x_N-,(t), which corresponds to signal X_N.,.

Figure 17D illustrates the combined continuous time signal x(t). Again, the signals are

represented in the continuous time domain for ease of illustration. The same principles of the

present inventions apply to operations in the discrete time domain.

The signal x(t) is the sum of the component time signals x₀(t) through x_N.,(t). x(t)

includes a number of peaks 610-613 that exceed the maximum values allowed. By observing

the component time signals x₀(t) through x_N.,(t) it can be determined which of the component

time signals contribute to the peaks. After determining which of the component time signals

contributes to the peak, those component time signals can be modified appropriately.

In the illustrated embodiment the component time signal that contributes most to the

peak is corrected. In an alternate embodiment it may be easier to modify several time

component signals that individually do not contribute as much, but as a whole significantly

contributes to a peak. Those time component signals may be modified rather than the largest

contributing component time signal. For example, a time component signal may contribute to

more than one peak. If the separation of the peaks of the time component signal are aligned a

single sinusoid added to that time component signal may be able to cancel more than one of the

peaks of the time component signal, thereby reducing more than one peak of the entire symbol. Other algorithms and methods for reducing one or more peaks by modifying the component

information signals of the symbol may be applied in accordance with the present inventions.

In some of the embodiments of the present inventions discussed above, peak reduction

signals were added to peak reduction frequencies. The peak reduction frequencies are generally

reserved for carrying peak reduction signals. In the illustrated embodiment, a peak reduction

signal is added to the component information. Thus, there is no reduction in the bandwidth of

the communication system.

While the illustrated embodiment depicts a basis function being added to a component

time information signal, it will be appreciated that the frequency domain equivalent of the basis

function may be added to the component frequency information signal. The present inventions

may be implemented in any suitable domain of the communication scheme.

In one embodiment the peak reduction signal is a sinusoid, which is the basis function of

the exemplary communication system. A sinusoid is applied to the component time signal that

contributes most to a particular peak in order to cancel the effects of that component time signal

and produce a modified information signal. The addition of a sinusoidal peak reduction signal

is easily mapped on an expanded constellation. The receiver can also easily decode the

modified information signal by accounting for the addition of the sinusoid, as discussed further

below.

For example, referring back to figures 17A-17D, it may be desired to reduce peak 612 of

time signal x(t) of figure 17D. Reviewing the signals x₀(t) through x_N-1(t), it may be determined

that component time signal x_k(t) (500(f)) contributes most to peak 612 at time n,,. A peak

reduction signal is then applied to component time signal x_k(t) in order to reduce its contribution

to peak 612. The peak reduction signal applied to component time signal x_k(t) is desirably designed

to be easily added to the information signal and easily decoded by the receiver. This maybe

achieved by duplicating the original constellation. The duplicated constellations are spaced

around the original constellation at certain intervals.

Figure 18 illustrates an original constellation with duplicate constellations in accordance

with an embodiment of the present inventions. The constellation points 501-504 of the original

signal 500, or X_k, remain in their original position. In addition, constellation points 511-514,

521-524, 531-534, 541-544, 551-554, 561-564, 571-574 and 581-584 are added around the

original constellation. The additional constellation points map directly to the original

constellation points. For example, constellations points 511, 521, 531, 541, 551, 561, 571 and

581 map to the original constellation point 501, and represent the same piece of information.

The duplicate constellation points are spaced at multiples of a distance D from the

original constellation point along the real or imaginary axes. For example, the distance between

original constellation point 503 and duplicate constellation point 553 is D along the real axis (i).

The distance between original constellation point 503 and duplicate constellation point 533 is

the magnitude of the vector (D + jD), which is equal to sqrt(2) • D.

Thus, if the original signal X_k 500, pointed to original constellation point 503, a

modified signal, X_k is created by adding a vector composed of orthogonal vectors of

magnitude D along the axes. That is, X_k = X_k + (p_kD + jq_kD). p_k and q_k are integers which

determine which of the duplicate constellations is being used. In the illustrated embodiment p_k

and q_k can take the values of 0 or +/-1. If further rings of duplicate constellations are used the

range of p_k and q_k would be accordingly larger. The values of p_k and q_k determine a number of

potential modified signals, X_k , 620, 622, 624, 626, 628, 630, 632 and 634. When different constellations are used the values for D may be different for each

dimension (e.g., a rectangular constellation would have dimensions D, and D_q). Additionally, D

varies with each component information signal since the constellations of each component

information signal may be different.

Referring back to figure 16, signal X_k 500 carried information indicating that the

constellation point 503 is selected. The corresponding component time signal x_k(t) (500(t) of

figure 17B) also showed that X_k contributed most to peak 612 of figure 17D at time = n₀. In

order to reduce peak 612 x_k(t) is modified by modifying X_k.

The values of p and q are chosen to define a sinusoid that is added to the original signal

X_k 500 that reduces the magnitude of x_k(t) at n„. In one embodiment, all the potential modified

signals are compared to the original signal to determine which of the modified signals, X_k ,

620, 622, 624, 626, 628, 630, 632 and 634 provides the best modification of the original signal

X_k. This approach is convenient in embodiments where the number of duplicate constellations

is low.

Referring to figures 18 and 19A through 19D, one iteration of modifying a signal is

depicted according to one embodiment. Figure 19A illustrates the original continuous

component time signal x_k(t) 500(t). x_k(t) 500(f) peaks at time t = n₀. Figure 19B illustrates a

sinusoid s(t) which would reduce the peak of x_k(t) at t = i

In the illustrated example, sinusoid s(t) of figure 19B corresponds to modifying X_k in the

frequency domain by a distance D in the real domain. Referring back to figure 18, modified

signal X_k is chosen at duplicate constellation point 553. Thus, X_k = X_k + (1)D + j(0)D, or p_k = 1 and q = 0. If the original values for the real and imaginary components is such that X =

R_k +jI_k = l + j, then X_k = (1 +j) + D +j(0) = (1+D) +j.

Figure 19C illustrates a modified continuous time signal x_k (t) , which corresponds to

modified signal X . The addition of sinusoid s(t) to x_k(t) to reduce the peak of x (t) produces

modified time signal x _k (t) . The value of x_k(t) is reduced at time r^, however, other peaks may

appear in x_k (t) . These additional peaks may or may not affect the peaks of the overall

modified time signal.

Figure 19D illustrates the modified time signal x(t) , which is the sum of all the

component time signals x₀(t) through x_N__](f), including modified time signal x_k(t) . Peak 612 of

x(f) is decreased due to the modification of x_k(t) to x_k(t) . However, the other peaks 610, 611

and 613 may be inadvertently increased. The process of modifying a component time signal is

repeated in the illustrated embodiment. In the next iteration a new component may be selected

that contributes to another peak that is to be reduced.

In another embodiment, where the number of duplicate constellations is great, the best

value for X_k may be computed. The computation of the appropriate sinusoid to be added to a

component signal also applies to the aforementioned embodiment.

The discrete time signal representation of a real baseband discrete multi-tone symbol

may be represented as:

x[n] = 0,...,N - l

where X = R_k + I_k, and g_k is the scaling factor for frequency k. Scanning the values of the symbol x[n] determines at which values of n peaks exist. For

example, a peak may be found at n = n₀. The value for x[nj is:

Assuming that the peak at ^ is a positive peak, the components of x[n₀] may be scanned to

determine which of the components contributes the most to the peak. A positive contributor

may be found at frequency k₀. If all the components, X_k, are 16 QAM, and the values for X_k

= 3 + j, or R_k = 3 and I_k = 1, and the value for cos(2πkn, N) is positive then the real part of the

component X_k may be reduced by D. The new peak x[n₀] is:

where / is a threshold factor, 0 < / < 1. The term / limits the use of sinusoids that are too small

at the peak location to be effective in canceling out a peak of component information time

signal. Values of / ranging from about 0.6 to about 0.8 have been found to provide good results.

The middle term of the inequality corresponds to the general case where D > dM. The right

most term is the specific case where D = dM.

The new transmit symbol x[n] can be computed without repeating the IFFT since the

new transmit symbol contains only one modified tone.

2 x[n] = x[n] - -= (g_koD_ko )cos(2πk₀n/N), n = 0,..., N - 1

Thus, the real portion of X_k is reduced. Should the imaginary portion of X_k also contribute to

the peak a similar algorithm may be performed on the sin() portion of that component. Similarly, X_k may be modified at other points in time to corcect the same or other peaks.

Further, other component information signals of X, may be similarly modified.

More generically, the modified symbol with one tone modified may be written as:

[n] = x[n] + (g_koq_k D_ko)sin(2 ιk₀n/N), n = 0,...,N - l

p_k and q_k are chosen to provide the best peak reduction. The equation is expanded as more

peaks are reduced by modifying different information signals at different frequencies. The

range of p and q varies with the number of duplicate constellations that are used. Varying the

values of p _k and q _k aligns the phase of the sinusoid such that the sinusoid effectively cancels

out one or more peaks at the given points in time. Again, in an alternate embodiment, if two

peaks are appropriately spaced a single sinusoid (or basis function) may cancel more than one

peak at a single time.

Independent of which method of determining the values for p and q the method for

decoding the modified symbols is of low complexity. The receiver performs a modulo

operation based upon the values of M and D. The bit rates and modulation scheme (and,

therefore, the constellation size) of each frequency is typically transmitted to the receiver during

initialization. Information about duplicate constellations may also be transmitted to the receiver

at the same time.

Once the receiver knows the values of D and M for each frequency, the receiver may

readily decode the duplicate constellation points and map them to the original constellation

points. In the case where D > dM, the mapping algorithm is:

When D = dM, the algorithm reduces to:

X_k = mod_D{ X_k }

In some embodiments it may be preferced to have D > dM, referring now to figure 20.

Figure 20 is a constellation with duplicate constellations of a 16 QAM signal in accordance with

one embodiment of the present inventions. The constellation map includes an original

constellation 700 with constellation points 700a-700p, and duplicate constellation s 701-708

with duplicate constellation points a-p.

If D = dM then the duplicate constellations 701-708 directly border original

constellation 700. As discussed, the nearest neighbors of each constellation point in the original

constellation 700 are separated by a distance d. When D = dM, neighboring duplicate

constellation points also become nearest neighbors to the outer ring of original constellation

points.

For example, when D > dM the nearest neighbors for original constellation point 700h

are neighboring original constellation points 700(d, g and 1), separated by a distance d. The

nearest duplicate constellation point to original constellation point 700h is 705e. The distance

between points 700h and 705e is d + (D-dM). When D = dM, the distance between points 700h

and 705e reduces to d, and 705e becomes a nearest neighbor.

Having duplicate constellation point 705 e as a nearest neighbor presents even a greater

problem than just having another nearest neighbor. Usually original constellation point 700h is

only one bit different from original constellation points 700d, g and 1. Typically, error correction coding may be implemented to correct the incorrect decoding of original

constellation point 700h if one of constellation points 700d, g or 1 is received since there is only

one bit of error. When duplicate constellation point 705e is received it is mapped to original

constellation point 700e, which may be more than one bit different from original constellation

point 700h. Enor correcting codes may not be able to compensate for the difference in that

case. The problem increases as the size of the constellation increases.

In an alternate embodiment, the problem may be alleviated by increasing the complexity

of the receiver. If the receiver has knowledge of the duplicate constellations, i.e., through the

initialization process, and the receiver performs error conection decoding before mapping the

received signal to the original constellation, the problem is significantly avoided. However, this

adds a bit more complexity to the receiver since it has to perform more than a simple modulo

operation on the data. Depending upon the channel it may be desirable to have D = dM

When duplicate constellation point 705 e is separated from point 700h only by the

distance d the probability of incorrectly decoding X_k increases unless the receiver is made

more complex. The bit enor rate, therefore, increases when D = dM. For large enough D > dM

the bit enor rate does not suffer with the addition of duplicate constellations, however, the

power of the transmitted symbol may be increased.

Thus, a potential concern with the illustrated embodiment is that the overall power of the

transmitted symbol may be inadvertently increased by adding peak reduction signals to

information signals. Through various methods an increase in the overall power of the

transmitted symbol may be minimized. One method is to minimize the value of D, or the separation between the original

constellation and the duplicate constellation. If power considerations are greater than bit enor

rate considerations then D may be minimized to dM. Otherwise, a value of D may be chosen to

be greater than dM without significantly increasing the overall power of the transmitted discrete

multi-tone symbol.

In such cases, other methods may be employed to minimize any increase in transmit

power. One method is to choose those signals X_k that have values that are outermost original

constellation points in the constellation. Referring again to figure 20, X_k is chosen for

modification if it's value is 700(a, b, c, d, e, h, i, 1, m, n, o or p). X_k is not chosen if its value is

700(f, g, j or k). By choosing to add increments of D to the outer original constellation points

the amount of added energy is less than the energy required to modify X_k if it's value is one of

the inner original constellation points.

In addition to choosing outer original constellation points, the method in which X_k is

modified may also be designed to minimize an increase in transmit power. If X_k has a value of

original constellation point 700d the value for X_k is R_k = 3 and I_k - 3. Duplicate constellation

points 701d-708d may be chosen to modify X_k. But, some of the duplicate constellation points

increases the power of the component to a lesser extent than others. For example, duplicate

constellation points 70 Id, 702d, 703d, 705d and 708d require significantly more power than

original constellation point 700d. Duplicate constellation points 704d, 706d and 707d require

only marginally more power than original constellation point 700d. Therefore, if power is a

consideration, original constellation point 700d is limited to being mapped to duplicate

constellation points 704d, 706d and 707d. Generally, values for p and q may be limited to the following constraints to minimize

the transmitter power.

sign(p) = -sign(R_k); and

sign(q) = -signflj

This relationship can be used for any of the original constellation points to minimize the

increase in transmit power. Use of the algorithm on the original inner constellation points may

only provide minimal savings in power. But, if applied to all the modified component signals

the combined savings may be significant. Also, the savings increases with the increase in size

of the original constellation.

In another method only partial duplicate constellations may be employed, referring now

to figure 21. Figure 21 illustrates a constellation map in accordance with an embodiment of the

present inventions. The constellation map depicts original constellation 700 and duplicate

constellations 701-708 of figure 20. However, duplicate constellations 702, 704, 705 and 707

include partial duplicate constellations 702', 704', 705' and 707'.

Partial duplicate constellations 702', 704', 705' and 707' represent alternative duplicate

constellations. Rather than mapping the original constellation points to all of the duplicate

constellation points, mapping may be confined to partial duplicate constellations 702', 704',

705' and 707'. The use of partial duplicate constellation points reduces the number of duplicate

constellation points, and also reduces the maximum amount of power increased involved in the

modification of X_k.

Alternately, partial duplicate constellations 702', 704', 705' and 707' may be weighted

during the determination of the values of p and q. In that embodiment, all of the duplicate constellation points are used, but preference is given to the constellation points that lie within

partial duplicate constellations 702', 704', 705' and 707'. Further, the partial duplicate

constellations may take any shape neighboring the original constellation 700. By way of

example, the constellation points 701(k, 1, o and p) may form another partial duplicate

constellation, or constellation points 701(1, o and p) may be used, along with conesponding

constellation points in the other duplicate constellation.

It is appreciated that any type of constellation mapping may be utilized within the scope

of the present inventions. For example, the duplicate constellation need not be configured

identically to the original constellation. It may be prefened to locate duplicate constellation

points near their original constellation point counterpart. In figure 21 constellation 704' may

include the points 704(a, b, e, f, i, j, m and n) rather than the illustrated points. Thus, the

duplicate constellation points are closer to their original constellation point counterparts in

original constellation 700.

In another example, the inner points (e.g., 700f, g, j or k) of the original constellation are

not modified. However, if the information signal is one of the outer points (e.g., 700 a, b, c, d,

e, h, i, 1, m, n, o or p) the basis function may be of magnitude D/2 rather than D. This requires

less energy to be added to the signal while still providing adequate mapping of the original

constellation point. Thus, there are a myriad number of ways to map the original constellation.

Any one method may be used depending upon the needs of the communication system.

The illustrated embodiments of figures 11-13 may utilize modified signals as discussed

above. The kernel applicator 206 of figure 11 would apply a basis kernel rather than a kernel

based upon peak reduction frequencies that do not carry any information signals. Any suitable

linear combinations of the basis kernel may be used to modify the information signal. The basis kernel is added to the information signal of frequencies that add to one or more peaks of a

symbol. Similarly, kernel engine 222 of figure 12 applies a basis kernel (or linear combinations

thereof), in this case a sinusoidal kernel, to a discrete time signal vector x provided by inverse

fourier transformer 220. In addition to using a basis kernel or a kernel that is a linear

combination of the basis function, appropriate kernels constructed from linear combinations of

basis functions may be precomputed to be added to the signal for PAR reduction.

Decoder 306 of figure 13 decodes all the frequencies used in the multi-carrier

communication system. Rather than decoding only frequencies that contain information signals

and ignoring the peak reduction frequencies, all frequencies are utilized with the use of

modified signals. Decoder 306 need only perform a modulo operation on those frequencies that

carry a modified signal rather than an unmodified information signal.

Figure 22 illustrates a flow chart 750 of the operation of the kernel engine of figure 12 in

accordance with another embodiment of the present invention. Initially, the bit rates of all the

available frequencies are determined. The bit rates of the frequencies dictate the type of

constellation used for each frequency. The number and location of duplicate constellations may

be determined based upon power, bit enor rate, peak values and locations, actual and desired

peak to average power ratios or any other suitable considerations. The original constellations

and the number of duplicate constellations may be sent to a receiver before transmission of

actual signals. This initialization process requires little bandwidth and typically need not be

performed more than periodically. The information may be as little as sending the values for D

for each information signal to the receiver. Alternatively, D may be fixed to equal dM + c,

where c is a constant that is known to the receiver. Thus, the receiver can derive D by knowing

the dimensions of the original constellation. In alternate embodiments, the values of D and constellations sizes and numbers may

vary per tone, or even per symbol. In which case, further information may be sent to the ^•

receiver. While some bandwidth may be required, optimization of the usage of the channel and

the reduction in the complexity of the receiver necessary to decode the symbol may make up for

the difference.

Referring to figure 22 and figure 11, the flow chart 750 begins in block 752 and

proceeds to block 754. In block 754 x is received from IFFT 220. Initially, IFFT 220 provides

a peak reduction component, c(0), that is zeroed out.

In block 756 the kernel engine analyzes x + c(j) and applies one or more kernels to x

+ c(j) to reduce any peaks. In the instant embodiment the kernel is a sinusoid, or a sum of

sinusoids. In the first pass x^{c p}(j) = x +c(j); j=0. The kernel engine may negate one, two, or as

many peaks as desired in one iteration by adding one or more sinusoidal kernels. The number

of sinusoidal kernels that are applied to x may be a significantly large number per iteration since

adding a sinusoid to x is a simple operation. The number of sinusoidal kernels that are applied

may be limited in order to verify that no new peaks have been created.

In block 757 the index j is incremented. Proceeding to block 758 the kernel engine

generates the sinusoidal kernel values for c(j). In block 759 the new peak reduction components

are accumulated by adding the previous peak reduction components; c(j) = c(j) + c(j-l).

The kernel engine determines whether more iterations are required in block 760 based

upon the number and size of peaks remaining in the signal x + c(j). If no other iterations are

required the current x^cl,p(j) = x + c(j), is passed on to DAC 208 in block 764, where c(j) is the

accumulated sum of all the iterations of applying the kernel. When further iterations are

required, flow proceeds to back to block 756. A single frequency may, thereby, carry an information signal component and a peak

reduction signal component. The information signal typically contributes to one or more peaks

in the discrete multi-tone symbol. The modification of the information signal to incorporate the

peak reduction signal may be as simple as adding a basis function, such as a sinusoid, to the

information signal. The modified signal contributes less to the peak than the original

information signal component. A basis function dummy signal may be added to the

information signal component by mapping the original constellation of the information signal to

one of a number of duplicate constellations. The use of duplicate constellations also provides

for simple decoding of the modified signal by the receiver. The receiver need only perform a

simple modulo operation to decode the modified signal.

Alternate embodiments of the present inventions may be applied to a single carrier

communication system. Particularly, the discussion of different embodiments of the present

inventions in reference to figures 14-22 may also be applied to single carrier communication

systems. The peak to average power ratio of each symbol in time is reduced in relation to the

other symbols preceding and succeeding the symbol. That is the overall transmitted signal is

comprised of a number of symbols transmitted at different time intervals (rather than a number

of different signals comprising a single symbol in the previously discussed embodiments). The

PAR of the overall signal may be reduced by individually modifying the symbols that make up

the signal.

The embodiments discussed in reference to figures 14-22 are readily applicable to

reducing PAR on a per symbol basis and may be utilized to reduce the effect of a symbol on the

PAR of that symbol and the overall signal. Buffering of preceding symbols may be required,

but the operations necessary to reduce the PAR may still be performed in real time. In a single carrier embodiment, the basis function of the communication system becomes the filter impulse

response.

In still another embodiment of the present inventions, the basis kernel may be

precomputed to optimize its peak canceling effect. The computation of the basis kernel may be

implemented in parallel to the discussion in reference to figures 3-15. An optimized basis

kernel may be implemented rather than an ordinary basis kernel.

In one embodiment, the optimized basis kernel may be optimized to cancel peaks at a

particular instance of time. While the optimized basis kernel may not approach an impulse

function in the way that the previously discussed embodiment does due to the constraint on the

peak reduction signal that it should be a multiple of D along the real and imaginary axes, the

optimized basis kernel may still be optimized to adequately reduce a single (or multiple) peaks.

The optimized basis kernel may be comprised of a linear combination of basis functions

precomputed for reducing a peak at a given point in time. For example, a precomputed

optimized basis kernel for a symbol that has ten frequency components may have the values

[0, 0, D, 0, (D - jD), 0, 0, jD, -D, (D+J2D)]. The optimized basis kernel only has peak reduction

signal components in the 3rd, 5th, 8th, 9th and 10th frequencies rather than in all the

frequencies. This begins to look similar to the previously described embodiment which uses

dedicated dummy frequencies. However, in this case no bandwidth is lost.

The components of the optimized basis kernel in the time domain are, thereby, modified

to coincide in time with the peaks. The constraint on the optimized basis kernel allows the

component basis functions to act as vectors in the frequency domain that may be used to map

the original constellation points to duplicate constellation points depending on the position of

the peaks in the time domain. In this manner PAR is reduced without sacrificing bandwidth. Merging the difference between the use of dedicated peak reduction frequencies and

peak reduction signal components, a hybrid embodiment may be utilized to maximize peak

reduction with minimal loss of bandwidth. Optimized basis kernels may be generated by

allowing peak reduction signal components in only selected frequencies. Those frequencies

may be dedicated to peak reduction and not carry an information signal, as in the previously

discussed embodiment. However, the peak reduction frequencies vary per symbol. In this

manner the best peak reduction frequencies may be used for each symbol, rather than having to

use the same peak reduction frequencies for each symbol. One constraint is that the peak

reduction signal component be larger than the size of the constellation established for those

frequencies when they do carry information, e.g., |Re(C_k)| > D/2 and llm C^j > D/2.

The advantage of the hybrid embodiment is that the receiver need not be informed of the

selection of the peak reduction frequencies. The receiver will decode all the frequencies of each

symbol. Those frequencies that carry signals that are larger than the constellation designated

for those frequencies are determined to be peak reduction frequencies for that particular symbol.

The peak reduction signals in those frequencies may be disregarded and the remaining

information signals may be properly decoded for that symbol. In the next symbol the receiver

detects a new set of peak reduction frequencies.

In a pure dedicated peak reduction embodiment the peak reduction frequencies could not

be changed on a per symbol basis without having to send side information to the receiver. This

caused the selection of peak reduction frequencies to be performed in view of reducing the PAR

for a wide variety of symbols. Thus, in some embodiments, the number of dedicated peak

reduction frequencies could be as high as 10% of the total number of frequencies to ensure

proper PAR reduction. In the hybrid embodiment a fewer number of peak reduction frequencies are required for each symbol since the peak reduction frequencies are optimally

selected for each symbol. Also, no side information is required since the receiver is capable of

automatically detecting the peak reduction frequencies. Thus, the amount of bandwidth lost is

minimized with the same, if not better, reduction of the PAR.

While the discussion has focused on reducing the peak to average power ratio as

measured at the transmitter, the present inventions apply equally to reducing the PAR as

measured at the receiver. Especially if the characteristics of the channel are known then the

transmitted symbols may be modified in order to reduce the PAR as measured at the receiver.

This is useful for channels with long impulse responses where the PAR of a symbol increases as

the symbol is transmitted to the receiver. By anticipating this increase in PAR the transmitted

symbol may be appropriately modified prior to transmission.

Of course, side information may also be used to facilitate the reduction of the peak to

average power ratios in a communication system. Side information may be sent to the receiver

concerning the clipping of the symbol, duplicate constellations or any suitable type of

information. The more information provided to the receiver, the easier it is for the receiver to

decode the transmitted symbol and the easier it is to reduce the PAR at the transmitter. The side

information may be sent to on a per symbol basis, e.g., if the PAR reduction scheme is based on

a per symbol algorithm, or less frequently depending upon the PAR reduction scheme. While a

focus of some of the embodiments of the present inventions is to reduce PAR without a

significant loss of bandwidth, there are cases where reducing the PAR of a signal is preferable

over the loss of bandwidth. In those cases alternate embodiments of the present inventions that

utilize side information may be employed. The present inventions apply to any type of communication systems utilizing single or

multiple carriers. By way of example, the present inventions apply to Discrete Multi-Tone

(DMT), Orthogonal Frequency Division Multiplexing (OFDM), Discrete Wavelet Multi-Tone

(DWMT) communication systems, Vector Coding Modulation. The basis functions for such

systems may include sinusoids, complex exponentials, singular vectors of channel matrix,

wavelet filters or any other suitable basis function for a multi-carrier communication system.

Alternate embodiments of the present inventions apply to single carrier communication

systems, such as Carrier-less Amplitude Phase (CAPs), vestigial side band, amplitude

modulation and the like. The basis functions for single carrier (or no carrier systems) are

generally the delayed versions of the transmit pulse or transmit filter.

Distortion Estimation

While the previously described embodiments have generally focused on using the

transmitter to reduce the peak to average power ratio of a signal, methods and apparatuses for

reducing the peak to average power ratio of a signal at the receiver may also be employed in

accordance with the present inventions. As discussed, the transmitter may always send side

information to the receiver and have the receiver perform a number of different functions to

perform decoding and peak to average power ratio reduction on the received signal or symbol.

The receiver, in another embodiment of the present inventions, may be configured to properly

decode a PAR reduced transmitted signal or symbol without any side information.

In one embodiment, the transmitted signal or symbol is distorted at the transmitter. The

transmitter simply distorts the signal without regard to retaining the integrity of the information

contained in the signal. Instead, the distorted signal is received by the receiver and the receiver performs an algorithm to estimate the distorted portions of the signal. Iterative estimation and

reconstruction of the distorted portions of the signal may be performed to obtain an

approximation of the original signal. The operations may be performed on multi-carrier signals

as well as single carrier signals.

In a particular embodiment, a transmitter introduces a distortion in the time domain

representation of an original signal of a multi-carrier system. The time domain may be the

discrete or continuous time domain. The distorted signal is sent to a receiver.

The receiver is informed of the type distortion that is applied to the original signal. The

receiver transforms the received distorted signal to provide the individual frequency domain

components of the distorted signal. The receiver decodes the individual frequency domain

components of the received distorted signal to generate a first estimate of the original signal.

Obviously, the first estimate of the original signal will contain enors due to the distortion. To

conect the enors, the first estimate of the original signal is distorted in the same manner as the

distortion of the original signal.

Distortion of the first estimate of the original signal provides a first estimate of the

distortion of the original signal. The first estimate of the distortion is extracted and combined

with the received signal. This combined signal is then decoded, which provides a next (and

typically better) estimated of the original signal. The next estimate of the original signal may be

further distorted to provide a next estimate of the distortion. The process may be repeated until

an acceptable estimate of the original signal is obtained.

In this general manner a receiver is able to receive and decode a distorted signal so long

as the type of distortion is known. Sending the type of distortion generally requires little

bandwidth since the type of distortion changes infrequently. However, the receiver need only know the general distortion function and not the specific details of distortion of a particular

signal.

The receiver estimates the distortion, in effect estimating the original non-distorted

signal. Thus, the transmitted signal may be distorted for any number of reasons, including for

peak to average power ratio reduction. The signal may be distorted for other reasons, for

example encryption. Regardless of the reason, the receiver is able to effectively decode a

distorted signal without requiring side information.

Any type of distortion may be utilized in accordance with the present inventions.

Clipping is one particular type of distortion that has been found to satisfactorily reduce the PAR

of a signal. A signal may be clipped, which reduces the PAR of a signal, and the clipped

portions of the signal discarded. The clipped portions may then be estimated by the receiver in

an attempt to reconstruct the original signal.

The distortion may be intentionally introduced to a signal in order to reduce the PAR,

which the receiver corrects through distortion estimation and reconstruction. However, the

receiver may also estimate and rehabilitate a signal that is inadvertantly distorted. For example,

non-linear effects and abenations that exist in analog components may be conected by the

receiver. Further, a channel may include intermediate amplifiers or repeaters (such as passing a

signal through a satellite) which add further distortion to the signal. The receiver may be

trained to conect the distortion introduced by these and other sources of distortion. However, it

may be necessary in some cases to inform the receiver about the types and/or magnitudes of the

distortion introduced by such intermediate devices.

Figure 23 illustrates a multi-carrier time domain signal, x(t). The signal includes a

number of peaks 791-794. In order to reduce the peak to average power ratio of x(t) the signal should be reduced to an amplitude A. Amplitude A may be equal to or less than the magnitude

necessary for sufficiently reducing the peak to average power ratio of the signal. Generally, it

may be preferred to set the value of A at less than the maximum necessary magnitude to allow

for deviations due to noise and the linear or non-linear characteristics of analog components.

The peaks have differing magnitudes which requires that the peaks be clipped by

different amounts. For example, peak 791 must be reduced by an amount α„ peak 792 by α₂,

peak 793 by α₃ and peak 794 by α₄. The transmitter clips peaks 791-794 by the appropriate

amounts.

Figure 24 illustrates a clipped signal, x^chp(t), in accordance with an embodiment of the

present inventions. The former peaks 791-794 have been reduced appropriately to bring the

signal within the magnitudes A and -A.

Again, the illustrated examples are depicted in continuous time, but the operations may

be performed in the discrete time or discrete frequency domain equivalents of the signal. Thus,

in the discrete time domain the ideal limiter of the signal x(t) would be:

The PAR for the clipped signal x^cl,p(n) is:

A

ΛM_« =101og₁₀ ε r n ιι² i

where, ε [ *_B|| ] is the average power of the transmitted signal. The limiting function outputs

the following discrete time domain signal: β"» = χ, - c ^X-^A)

where c[^x'^A) is the distortion at the transmitter, which is a function of the data vector X and the

clipping level A. In the frequency domain the signal is represented as:

X?» = FFT(x^{c φ}) = X_k - C ^{, A)}

Figure 25 illustrates the received signal y^chp(t) in accordance with an embodiment of the

present inventions. The received signal y^{c p}(t) equals the transmitted clipped signal x^chp(t)

convolved with the impulse response of the channel plus some noise n(t). In the discrete

frequency domain the received signal is represented as:

Y k^c,φ = H k (X k - C k^(X'^A)) ' + N k

where H_k is the frequency domain response of the channel.

The receiver receives the signal y^clιp(t) and attempts to accurately decode the information

contained within y^cl,p(t). In the frequency domain a maximum likelihood decoder may be

utilized to attempt to find the best estimate of the transmitted information. The estimate of the

signal is:

X = arg m ViXn Yi (H k (X k - C k⁽*'^A)) - Y k^c"^p)² k=0

In vector form the equation is:

X = arg min H o (X -C^(X'^A)) - Y^clip x where the operation v.wis the element by element vector product [v,w,, v₂w₂ V₃W₃ v_Nw_N]^T.

Substituting the value of Y^cl,p from the above equation provides:

X = argmin|Ho(X-C^(X'^A))-Ho(X-C^(X-^A))-N

This estimate X can be solved to obtain an estimate of the transmitted information.

However, the solution may be too complex to be solved in real time by today's systems. The

solutions for the vector functions C^(XΛ) and C^(XA) can be complicated functions of X and X,

respectively. Thus, finding X may require an exhaustive search over all the possible transmit

symbol vectors X and all conesponding C^(XA) . The solution is exponentially complicated, but

estimating or knowing C^(XΛ) would greatly simply the equation and provide a good estimate

X . In the future processing advances may provide the necessary power to solve the exact

solution in real time.

Currently, finding an estimation of C^(XA) may provide the necessary information to

adequately reconstruct the original signal. Once an approximate solution is found for C^(XA) the

C^(XA) term is not needed since the maximum likelihood receiver reduces as follows:

l|2

X = argmjn HoX -Ho(X-C^(X'^A) + C^,X'^A' )-N l|2

X = argmin H0X-H0X-N

where C^(X' ^A) is an estimate of C^(X' ^A) . The vector maximum likelihood receiver may be

decomposed into N scalar maximum likelihood receivers for each tone or frequency. Thus for

each constellation of a particular frequency the maximum likelihood receiver obtains an

estimate of the transmitted constellation point as follows: X k = arg °m Xin(H k k - H k X k - N)²

The maximum likelihood receiver for a single frequency breaks down to choosing the closest

constellation value nearest the received signal of that frequency, or

where the operation (f) denotes selecting the closest constellation point to the value f. While the

maximum likelihood choice provides good estimates of the transmitted constellation point, it

may be improved upon with enor conection coding. Enor conection coding may be used in

another embodiment of the present inventions to increase the accuracy of the estimation

process.

Knowing the values for C^{(X A)} would therefore greatly reduce the complexity of the

reconstruction of the received signal. The values for C^{(X A)} may be transmitted along with the

clipped signal in any number of the methods described above. Additionally, side channels may

be utilized to transmit the clipping information.

In an embodiment of the present inventions the value for C^(X'^A) is estimated without any

side information provided by the transmitter. Initially the receiver derives a first estimate X ⁽ⁱ⁾.

The receiver uses the first estimate to determine a first estimate, C^{(X A)} , and uses it to obtain a

second estimate X ⁽²⁾. The process is reiterated a number of times until a good estimate of

C^{(X A)} is obtained. Correspondingly, a good estimate of C^{(X A)}, C^(X" ^A) , also provides a good

estimate of X, X ^(q).

In one embodiment, the first estimate of X may be obtained by using:

Performing an IFFT on X ^m provides x „(⁽D '. From x⁽ ( ^υ1) and information about the type of

distortion, in the illustrated embodiment the clipping level A, c^{(x A)} and C^(x '^A) is computed.

Figure 26 illustrates a reconstructed signal x⁽ '(f) that is the first estimate of x(t). Signal

x ^} (t) includes estimated peaks 801-804. Estimated peaks 801-804 are derived from the

estimate C^(x '^A) . The first estimate, as illustrated, may not perfectly estimate the original peaks

791-793. However, the estimated peaks 801-804, or estimate C^(X ' ^A) , may be used to obtain a

better estimate of x(t), or X, i.e., C^(X ' ^A) . The following equation may be used to obtain a next

estimate of X.

or generally the iterative algorithm may be written

= 0;

C^(x<q,'^A)= FFT(X^(q)- X⁽ c^qli⁾p),q>l;

χ^(q ₊» ₌((Y"*O1/H) + C (X^(q),A)

which may be written as:

X^(q+1)=argmin HoX -Ho(X-C^(λ,A'+C (X^(q).A) )-N where X and X^<q) are taken from the cunent estimate. The cunent iteration is indicated by the

index q. The q+1 estimate of X is obtained using the values for C^{(x A)} obtained from the

cunent, q^th, iteration.

The q^Λ iteration C^{(X A)}is obtained by taking the IFFT of X^(q) . The conesponding time

domain signal x^(q)(t) obtained by the IFFT is clipped through a similar clipping procedure

performed at the transmitter. An FFT is performed on the clipped portions of the peaks and the

frequency domain representation of the clipped portions of the peaks are used as the new

estimate C^{(X q} ' ^A) . The new estimates are used in the above equations to generate a new

estimate, X^(q+1) .

The IFFT of the previous estimate X^(q) need not be performed on the entire signal. Only

the differential between the previous estimate X^(q) and the cunent estimate X^(q+1) need be

computed. If X^(q) and χ^(q+1) are very similar than the difference between the two is mostly

zeroes, which is easier to compute. Thus, only a portion of X^(q) and χ^(q+1) need be transformed

during each iteration of the estimation process. Additionally, c^{(X Λ)} only contains information

in those portions where clipping, or more generally distortion, occuned. Thus, the FFT of

c^(x '^A) need only be performed on those portions of c^{(X A)} where clipping occuned. Selective

transformation during the iterative process helps to increase the speed of the estimation process.

The number of iterations of the estimation procedure may be performed a predetermined

number of times. As few as two iterations of the estimation process has shown to provide

satisfactory symbol enor rates for multi-carrier systems. In one embodiment two to five iterations have been sufficient to provide enor rates in a multi-carrier signal that approaches the

ideal minimum enor rates of conventional systems.

In another embodiment, when X^(q) and χ^(q+1) are very similar, or exactly the same, then

no further iterations are necessary. Typically successive iterations that are very similar means

that the estimation process has achieved the best possible estimation.

Figure 27 illustrates a signal x^(q)(t) reconstructed after q iterations of clip estimation in

accordance with an embodiment of the present inventions. Signal x^(q)(t) includes peaks 821-

824 which more closely resemble the original peaks 791-794.

The described embodiment of the present inventions, however, may be applied to single

carrier systems as well. Instead of employing the concept over a number of carriers comprising

a single symbol, the described procedures may be applied to a number of time sequential

symbols of a single carrier signal.

Figure 28 A illustrates a block diagram of a transmitter 830 in accordance with an

embodiment of the present inventions. Transmitter 830 includes an encoder 832, an inverse

fourier transformer 834, a digital to analog converter 836 and a distorter 838.

Encoder 832 and IFFT 834 modifies a set of data for transmission, similar to the

operations of the encoders and transformers/modulators discussed in reference to figures 11-13.

The IFFT 834 outputs a discrete time signal x(n). Digital to analog converter converts the

signal x(n) to a continuous time signal x(t). Distorter 838 then distorts the continuous time

signal x(t) and transmits x^d,st(t) to a receiver.

Generally the appropriate signals may be represented as: , dist (X.d) .

X - c dist _ y _ (X,d) k k k

where d represents the type of distortion performed on the signal. In an alternate embodiment

the distortion function d is a non-memory-less function. That is, the distortion function may

take into consideration the values of the previous symbols. The receiver simply needs to store

the estimations of the distortions of previously received symbols to obtain estimations of the

present symbols. Thus, the present inventions may be apply a broader range of distortion

functions to the signals or symbols. Extra complexity is added in such a scheme, but better

estimations may be achieved if previously estimated symbols were accurately reconstructed.

In another embodiment, distorter 838 may perform clipping on the discrete time

sequence x(n) rather than on the continuous time signal x(t). Thus, distorter 838 may be

embodied in a digital signal processor, a central processing unit or other type of computational

device. Digital to analog converter 836 may then be used to convert the distorted discrete time

sequence into a continuous time waveform.

Performing the clipping operation on the discrete time sequence x(n) has the advantage

of providing the ability to store the clipped information. The clipped information may be sent

to the receiver as side information through a variety of methods, as discussed and as may be

well known in the art.

Figure 28B illustrates a block diagram of a transmitter 830' in accordance with an

embodiment of the present inventions. Transmitter 830' includes an amplifier 839 in addition

to the previously described elements of transmitter 830. An amplifier is described, but

amplifier 839 is exemplary of any device with non-linear characteristics. That is, an type of

analog device that is not perfectly linear, which most analog devices are not, introduce distortion. Thus, x^d,st(t) includes the distortion added by distorter 838 and the distortion

introduced by amplifier 839. Thus, the distortion function includes a term for each type of

distortion. If the distortion added by amplifier 839 is known the receiver may be ananged to

estimate that type of distortion as well.

Figure 28C illustrates a block diagram of a channel in accordance with an embodiment

of the present inventions. The channel includes a first channel 833, an intermediate transceiver

835 and a second channel 837. In many instances direct point to point communications is not

feasible or even possible. Typically, a signal is passed from transmitter through a number of

transceivers to a receiver, the final destination. Each transceiver between the source and the

destination may add additional distortion. Also, components within the receiver may also add

distortion to the signal.

Channel 833 has a channel response h'(t) and channel 837 has a channel response h²(t).

Transceiver 835 introduces some distortion d¹. The distortion introduced by the transmitter is d'

and the distortion introduced by the receiver is d^r. The general formula may be written as:

X = arg mjn (H o X - C (X, //d', ff₂d', d^r ) ) - (H ° X- X, //d' , .tf₂d' , d^rK N x

where Η = Η,Η₂, the frequency responses of channels 833 and 837. Note that the distortion is a

function of all the sources of distortion. The distortions of the transmitter and transceiver 835

are affected by one or more of the channel responses of the channels 833 and 837. The

distortion from the transmitter is affected by the channel response of all the channels since

distortion d^l traveled through both channels 833 and 837. The distortion of transceiver 835,

however, is affected only by the frequency response of channel 837. And, the distortion of the receiver is not affected by any of the frequency responses of the channels since the distortion d^r

is introduced at the receiver. The general algorithm may be written as:

_D(.) _(χO) _{) = FFT (i}(.) . _χd,s,(.) _{) ;}

D'⁰ [TLX⁰' - H.CDJ⁰ (X⁽¹⁾))] = D₂ ¹⁾ (Z⁽¹⁾) = FFT(z^(,) - z^d,st(,)); and

D⁽ _r° [H₂(H,X⁽¹⁾ -

- H. ,¹' (Z⁽¹⁾))] = D^< ₂ ¹⁾ (W⁽¹⁾) = FFT (w^{(, )} - w^d,s,(1));

The channels themselves may also add distortion. The above equations may be modified

appropriately to account for the distortion introduced by the channels.

Figure 29 illustrates a receiver 840 in accordance with an embodiment of the present

inventions. Receiver 840 includes an analog to digital converter 842, a fourier transformer 844,

a decoder 846, and a distortion estimator 855. Estimator 855 includes an inverse fourier

transformer 856, a distorter 858 and a fourier transformer 860.

Analog to digital converter 842 receives the transmitted signal y^dlst(t) and converts the

signal into a discrete time sequence y^dιst(n). Fourier transformer 844 performs a fourier

transform on the discrete time sequence to obtain the frequency domain representation of the

received signal, Y^dlst. Decoder 846 decodes the frequency domain signal. Decoder includes the

initial estimation of the clipped portions of the signal. Decoder 846 provides an estimate, X^(q) ,

q=l . Frequency equalization is also performed on the received signal. Frequency equalization

may be performed by decoder 846 during the first estimation process. Or, frequency

equalization may be performed right after transformer 844 has transformed the received signal. IFFT 856 performs an inverse fourier transform on X^(q) to provide an estimated discrete

time sequence x^(q)(n) . Distorter 858 distorts the time signal x(n) and estimates the distorted

portions c^(q)(n) . FFT 860 performs a fourier transform on c^(q)(n)to obtain the frequency

domain estimation of the clipped portions, C^{(χ q} '^Λ) . That information is fed back to decoder

846.

Decoder 846 uses C^(x '^A) to perform a new estimation of X based upon C^(X ' ^A) . After

a predetermined number of estimations have been performed decoder 854 decodes the last

estimate X^(q) , q = last iteration. Decoder 854 provides a set of data that should be equivalent to

the original set of data provided to the transmitter.

The components of the illustrated transmitters and receivers may be embodied in

hardware, software operating on general purpose processing device, or a combination of both.

The operations described can be performed in any suitable domain. While the illustrated

embodiment has described a signal based upon the fourier transform, other types of signals may

be utilized in accordance with the present inventions. By way of example, DWMT variants of

the present inventions apply wavelet transforms and vector coding applies transmit matrices.

For example, discrete cosine transforms, Hartley transforms and suitable type of multiple-input

multiple-output matrix operation may be utilized. Similarly, any suitable type of transmission

methodology may be utilized in accordance with the present inventions.

Additionally, in alternate embodiments, the functions of decoder 846 may be

incoφorated into the estimator 855. Generally, an estimator includes any mechanism used to

estimate the distorted portions of the received signal. Any suitable clipping methodology may be used in accordance with the present

inventions. In another embodiment, the peaks of a signal x(t) may be clipped by a standard

amount rather than clipping the peaks by variable amounts to a predetermined magnitude.

Figure 30 illustrates a signal x(t) with a number of peaks. Rather than clipping the peaks

by a variable amount to reduce the peaks to the same amplitude, A, the peaks are reduced by a

set value. In the illustrated example the peaks are reduced by α_A. The value of _A is set such

that the highest peak is reduced to below a maximum value necessary for satisfactory PAR

reduction.

Figure 31 illustrates a clipped signal x^clιp(t) in accordance with another embodiment of

the present inventions. Figure 32 illustrates the clipped signal of figure 31 with estimated peaks

901-904. In the illustrated embodiment the estimations of the clipped portions of the received

signal is made easier since one constraint is known. It is known that the magnitudes of the

clipped portions are all α_A. Knowing the magnitudes of the clipped portions reduces the

complexity of the estimation process. Figure 33 illustrates a reconstructed signal x^(q)(t) after q

iterations of estimation. Reconstructed signal x^(q)(t) includes estimated peaks 911-914 that

more closely resemble the original peaks.

In another embodiment of the present inventions, an incremental value α_Δ may be

utilized. Instead of the clipping the peaks by a standard value, such as α_A, the peaks may be

clipped by increments of α_Δ. Depending upon the overshoot of the peak the peak is clipped by a

multiple of α_Δ. Using an incremental value minimizes the distortion of the signal. In still a

further embodiment, a set of predetermined clipping increments may be utilized. The values of

the clipping increments may be varied according to the size of the peaks, or other criteria. For example, the values of the clipping increments may range logarithmically from the smallest

clipping increment to the largest clipping increment.

Again, sending information about the clipping, or any type of distortion, improves the

receivers ability to estimate the distorted portions of the signal. The side information may

include the number of distorted peaks, the magnitude of the distortion, the characteristics of the

type of distortion (e.g., in clipping the clipping level), location of some or all of the peaks,

values for the clipping increments and other suitable pieces of side information. Sending side

information also decreases the complexity of the receiver. Additionally, oversampling the data

at the transmitter may also allow the receiver to provide better estimates of the distorted signal.

Clipping the signal is just one type of distortion that is applied to the signal in order to

reduce the peak to average ratio. Any suitable type of distortion other than clipping may be

used to reduce the peak to average power ratio as long as the receiver is capable of estimating

the distortion. The receiver determines the distortion of the signal in order to reconstruct the

original signal. Determination of the distortion may be performed through direct computation,

or through estimation and iteration.

Generally, the present inventions provides methods and apparatuses for reducing the

peak to average power ratio of a signal without losing a significant amount of bandwidth.

Additionally, the complexity of some embodiments allow peak reduction to be performed in

real time.

Also, predistortion techniques have been utilized to minimize the distortion introduced

by devices. Predistortion techniques may also be used in conjunction with the present

inventions. Smaller distortions are generally easier to estimate and predistortion techniques

provide better speed and accuracy to the estimation process. While these inventions have been described in terms of several preferred embodiments,

it is contemplated that alternatives, modifications, permutations and equivalents thereof will

become apparent to those skilled in the art upon a reading of the specification and study of the

drawings. It is therefore intended that the following appended claims include all such

alternatives, modifications, permutations and equivalents as fall within the true spirit and scope

of the present invention.

Claims

C L A I M S

1. A receiver for use in a communication system, the receiver receiving a received signal, the received signal being a function of an original signal and an original distortion, the receiver comprising: an estimator, the estimator estimating the distortion of the received signal to provide a final estimate of the original distortion, wherein the receiver uses the final estimate of the original distortion to remove the original distortion from the received signal to provide a final estimate of the original signal, whereby the final estimate of the original signal is an estimate of the received signal without the original distortion.

2. A receiver as recited in claim 1 , the receiver further comprising: a decoder coupled to the estimator, the decoder decoding the received signal, wherein in the process of decoding the received signal the decoder makes a first estimate of the original signal, the decoder providing the first estimate of the original signal to the estimator; and a transformer coupled to the decoder, the transformer performing a transform on the received signal, the transformer providing the transformed received signal to the decoder.

3. A receiver as recited in claim 2, wherein the estimator receives the first estimate of the original signal from the decoder and estimates a first estimate of the original distortion based upon the first estimate of the original signal, the estimator providing the first estimate of the original distortion to the decoder, the first estimate of the original distortion contributing to the determination of the final estimate of the original distortion and the final estimate of the original signal.

4. A receiver as recited in claim 3, wherein the decoder receives the first estimate of the original distortion from the estimator, the decoder decoding the received signal in conjunction with the first estimate of the distortion and providing a next estimate of the original signal.

5. A receiver as recited in claim 4, wherein the estimator receives the next estimate of the original signal and estimates a next estimate of the original distortion based upon the next estimate of the original signal, the estimator providing the next estimate of the original distortion to the decoder, whereby the next estimate of the original distortion contributes to the determination of the final estimate of the original distortion.

6. A receiver as recited in claim 4 or 5, wherein the transform is chosen from a group of transforms consisting of an inverse fourier transform, a discrete wavelet transform, a vector matrix, a discrete cosine transform and a Hartley transform.

7. A receiver as recited in any of the preceding claims, wherein the received signal is a function of the original signal and the original distortion, the original signal including a peak and the distortion being a clip of the peak, wherein the clipping of the peak by the distortion reduces a peak to average power ratio of the received signal.

8. A receiver as recited in any of the preceding claims, wherein the original distortion function is applied in a manner chosen from a group consisting of: applying the original distortion function to the original signal at a transmitter, wherein the received signal originates from the transmitter; applying the original distortion function to the original signal at a transceiver, wherein the received signal passes through the transceiver before being received by the receiver; applying the original distortion function to the original signal at a channel, wherein the received signal passes through the channel before being received by the receiver; and applying the original distortion function to the original signal at the receiver.

9. A receiver as recited in any of claims 2-8, wherein the original distortion that is part of the received signal is applied to the original signal by an original distortion function, and the estimator includes a distorter, wherein the distorter distorts the first estimate of the original signal according to a first distortion function, wherein the application of the first distortion function to the first estimate of the original signal introduces a first distortion, wherein the first distortion provides the first estimate of the distortion.

10. A receiver as recited in claim 9, wherein the estimator further includes: an inverse transformer coupled to the decoder and the distorter, the inverse transformer performing an inverse transform on the first estimate of the original signal, the inverse transform being the inverse of the transform performed by the transformer and providing the transformed first estimate of the original signal to the distorter; and another transformer coupled to the distorter and the decoder, the another transformer receiving the first estimate of the original signal including the first estimate of the original distortion and performing the transform on the first estimate of the original signal including the first estimate of the original distortion, and extracting the first estimate of the original distortion and providing the first estimate of the distortion to the decoder.

11. A receiver as recited in any of the preceding claims, wherein the signals are multi-carrier signals.

12. A method of conecting an original distortion in a received signal in a communication system, wherein an original distortion function is applied to an original signal to introduce the original distortion into the received signal, the method comprising: estimating the original distortion of the received signal, thereby providing a final estimate of the original distortion; and applying the final estimate of the original distortion to approximately remove the original distortion from the received signal to obtain a final estimate of the original signal.

13. A method as recited in claim 12, further comprising: performing a transform on the received signal prior to decoding the received signal; decoding the received signal, the process of decoding providing a first estimate of the original signal; and estimating a first estimate of the original distortion based upon the first estimate of the original signal.

14. A method as recited in claim 13, further comprising: a next step of decoding the received signal in conjunction with the first estimate of the original distortion, providing a next estimate of the original signal; a next step of estimating a next estimate of the original distortion based upon the next estimate of the original signal; and decoding the received signal in conjunction with the next estimate of the original distortion and repeating the step of estimating the next estimate of the original distortion and the step of decoding the received signal in conjunction with the next estimate of the original distortion a predetermined number of iterations, whereby the final estimate of the original distortion is determined.

15. A transmitter that transmits a multi-carrier symbol in a multi-carrier communication system, wherein the transmitted multi-carrier symbol has a peak to average power ratio and is a function of a plurality of information signals, each of the plurality of information signals being centered at one of a plurality of frequencies, the plurality of information signals including a modified signal, the modified signal including an information component and a peak reduction component, wherein the peak reduction component of the modified signal is ananged to reduce the peak to average power ratio of the transmitted multi-carrier symbol.

16. A transmitter as recited in claim 15, wherein the peak reduction component of the modified signal is a basis function of the multi-carrier communication system.

17. A transmitter as recited in claim 16, wherein the basis function is selected from the group consisting of a sinusoid, a wavelet and a complex exponential.

18. A transmitter as recited in any one of claims 15-17, wherein each one of the plurality of information signals is a quadrature amplitude modulated signal that is based at least in part upon an associated information signal constellation that includes a plurality of information signal constellation points, wherein the modified signal has at least one associated duplicate constellation, each duplicate constellation including a plurality of duplicate constellation points, wherein the plurality of information signal constellation points of the information signal constellation associated with the modified signal are mapped to the plurality of duplicated constellations points in each duplicate constellation.

19. A transmitter for use in a multi-carrier communication system, the transmitter transmitting a multi-carrier symbol, the multi-carrier symbol having a peak to average power ratio and being a function of a plurality of information signals, the transmitter comprising: a kernel applicator, wherein the kernel applicator reduces the peak to average power ratio of the multi-carrier symbol by modifying a selected information signal of the plurality of information signals, wherein the modified signal includes an information component and a peak reduction component.

20. A transmitter as recited in claim 19, the transmitter further comprising: an encoder, wherein the encoder encodes a first set of data into a plurality of sets of data; and a modulator, coupled to the encoder, that receives the plurality of sets of data and modulates each set of data of the plurality of sets of data to produce the plurality of information signals, which are combined, the modulator also coupled to the kernel applicator, the modulator providing the plurality of information signals to the kernel applicator.

21. A transmitter as recited in claims 19 or 20, wherein the kernel applicator includes: a transformer, coupled to the modulator, the transformer performing a transform on the plurality of information signals producing a transformed signal; and a kernel engine, coupled to the transformer, wherein the kernel engine analyzes the transformed signal to detect at least one peak in the transformed signal, wherein when at least one peak is detected the kernel engine applies a kernel to the at least one peak of the transformed signal, wherein the kernel is a basis function of the transmitter, the basis function being applied to the information component of the modified signal, whereby the basis function is the peak reduction component of the modified signal.

22. A transmitter as recited in any of claims 19-21 , wherein the kernel applicator retains information about the reduction of the peak to average power ratio of the multi-carrier symbol such that the transmitter also transmits the information to a receiver to facilitate decoding of the transmitted multi-carrier symbol.

23. A multi-carrier communication system for transmitting a multi-carrier symbol having a peak to average power ratio and being a function of a plurality of information signals, the multi- carrier communication system comprising a transmitter as recited in any of claims 15-22 and a receiver that receives the multi-carrier symbol transmitted by the transmitter, wherein the transmitted multi-carrier symbol is modified by the addition of the basis function to the information component of the modified signal by the transmitter such that the receiver receives the transmitted multi-carrier symbol and the modified signal, the receiver mapping the modified signal from the selected duplicate constellation point to the associated information signal constellation to obtain the information component.

24. A multi-carrier communication system as recited in claim 23, wherein the transmitter transmits the multi-carrier symbol to the receiver through a channel such that the kernel applicator takes into account a characteristic of the channel to reduce the peak to average power ratio of the multi-carrier symbol as measured at the receiver.

25. A method of reducing a peak to average power ratio of a multi-carrier symbol of a multi- carrier communication system, wherein the multi-carrier symbol is a function of a plurality of signals, each of the plurality of signals centered at each one of a plurality of frequencies, the method comprising: analyzing the multi-carrier symbol to detect a peak in the multi-carrier symbol; determining a first signal of the plurality of signals that contributes to the peak; and modifying the first signal by applying a peak reduction component to the first signal, which reduces the peak to average power ratio of the multi-carrier symbol.

26. A method as recited in claim 25, the method further comprising: repeating the steps of analyzing, determining and modifying until the peak to average power ratio is reduced to a predetermined level.

27. A method as recited in claim 25 or 26, the method further comprising: transmitting the multi-carrier symbol to a receiver; and decoding the multi-carrier symbol when the receiver receives the transmitted multi- carrier symbol including, decoding the modified first signal.

28. A method as recited in claim 27, wherein decoding the modified first signal includes performing a modulo operation on the modified first signal.

29. A method as recited in claim 27 or 28, the method further comprising: transmitting information about the reduction of the peak to average power ratio of the multi-carrier symbol, wherein the information facilitates the decoding of the multi-carrier symbol.

30. A transmitter for use in a communication system, the transmitter transmitting a signal wherein the transmitted signal has a peak to average power ratio and is a function of a plurality of information symbols, each symbol being transmitted at each one of a plurality of intervals of time, a selected information symbol of the plurality information symbols including an information component and a peak reduction component, wherein the peak reduction component modifies the information component and reduces the peak to average power ratio of the transmitted signal.

31. A transmitter as recited in claim 30, wherein the peak reduction component is a basis function of the transmitter the communication system has an impulse response such that the basis function is the impulse response of the communication system.

32. A transmitter for use in a multi-carrier communication system, the transmitter transmitting a multi-carrier symbol, the multi-carrier symbol having a peak to average power ratio and being a function of a plurality of information signals, the transmitter comprising: a means for reducing the peak to average power ratio of the multi-carrier symbol.

33. A transmitter for use in a multi-carrier communication system, wherein a symbol transmitted by the transmitter has a peak to average power ratio and is a function of a plurality of signals, each one of the plurality of signals being centered at one of a plurality of frequencies, wherein a subset of the plurality of signals are configured to reduce the peak to average power ratio before the symbol is transmitted.

34. The transmitter of claim 33, wherein the subset of the plurality of signals carry data and configured to reduce the peak to average power ratio.

35. The transmitter of claim 33, wherein the subset of the plurality of signals do not carry data and the signals are comprised of information about the reduction of the peak to average power ratio of the symbol.

36. The transmitter of any of claims 33-35, wherein the configuration of the subset of the plurality of signals includes scaling a one or more of the signals of the subset of the plurality of signals.

37. The transmitter of any of claims 33-36, wherein the configuration of the subset of the plurality includes phase shifting one or more of the signals of the subset of the plurality of signals.

38. The transmitter of any of claims 33-37, wherein the configuration of the subset of the plurality includes time shifting a one or more of the signals of the subset of the plurality of signals.

39. The transmitter of any of claims 33-38, wherein the plurality of signals are modulated by a method of modulation chosen from the group of modulation schemes consisting of quadrature amplitude modulation, phase shift key modulation, frequency modulation, amplitude modulation and continuous phase modulation.

40. The transmitter of any of claims 33-39, wherein the subset of the plurality of signals are configured by applying a kernel to the symbol.

41. The transmitter of claim 40, wherein the kernel has a primary and at least one secondary peak each peak having an amplitude, the amplitude of the at least one secondary peak being smaller than the amplitude of the primary peak, such that the subset of the plurality of frequencies are chosen from the plurality of frequencies such that the amplitude of the at least one secondary peak is minimized with respect to the amplitude of the primary peak.

42. The transmitter of any of claims 33-41, wherein each of the signals has an associated bit rate and the subset is chosen in part based upon the bit rate of the signals.

43. The transmitter of any of claims 33-42, wherein the subset of the plurality of signals are configured by a linear program based upon the subset of the plurality of frequencies.

44. The transmitter of any of claims 33-43, wherein the subset of the plurality of signals are configured by modifying the subset of the plurality of signals and applying a fourier transform to the subset of the plurality of signals, which are then incorporated into the plurality of signals and applying an inverse fourier transform to the plurality of signals and determining if the subset of the plurality of signals require more modification, and repeating if the subset of the plurality of signals require more modification.

45. The transmitter of any of claims 33-45 wherein the transmitter includes: an encoder, wherein the encoder encodes a first set of data into a plurality of sets of data; a modulator, coupled to the encoder, that receives the plurality of sets of data and modulates each set of data of the plurality of sets of data to produce the plurality of signals, which are combined; and a kernel applicator, coupled to the modulator, wherein the kernel applicator reduces the peak to average power ratio of the combined plurality of signals by modifying a one or more signal of the subset of the plurality of signals, and producing the symbol, whereby the peak to average power ratio of the symbol is reduced.

46. The multi-carrier communication system of claim 45, wherein the kernel applicator includes: an inverse fourier transformer, coupled to the modulator, the inverse fourier transformer performing an inverse fourier transform on the combined plurality of signals producing a transformed signal; and a kernel engine, coupled to the inverse fourier transformer, wherein the kernel engine analyzes the transformed signal and detects any peaks in the transformed signal, if a peak is detected the kernel engine applies a kernel to the peak of the transformed signal by adjusting the kernel, wherein the kernel is an approximation of an impulse generated from the subset of the plurality of signals, such that the kernel is adjusted by modifying the subset of the plurality of signals, whereby the peak to average power ratio of the symbol is reduced.

47. A multi-carrier communication system comprising a transmitter as recited in any of claims 33-46 and a receiver, wherein the receiver receives the symbol and extracts the plurality of signals from the symbol.

48. The multi-carrier communication system of claim 47, wherein the receiver discards the subset of the plurality of signals and extracts information from the plurality of signals that are not the subset of the plurality of signals.

49. A method of reducing a peak to average power ratio of a signal of a multi-carrier communication system, wherein the signal is a function of a plurality of signals, an each of the plurality of signals centered at an each one of a plurality of frequencies, the method comprising: choosing a subset of the plurality of frequencies, whereby a conesponding subset of the plurality of signals are also selected, wherein the subset of the plurality of signals are peak reduction signals; and modifying the subset of the plurality of signals such that the peak to average power ratio of the signal of the multi-carrier communication system is reduced.

50. The method of claim 49, wherein modifying the subset of the plurality of signals includes one selected from the group of: scaling a one or more signal of the subset of the plurality of signals; phase shifting a one or more signal of the subset of the plurality of signals; and time shifting a one or more signal of the subset of the plurality of signals.

51. The method of claim 49 or 50, wherein choosing the subset of the plurality of frequencies is based upon optimizing a kernel to approximate an impulse function, the kernel being a function of a plurality of components, an each one of the plurality of components centered at an each one of the subset of the plurality of frequencies, such that the plurality of components are a part of the subset of the plurality of signals, whereby optimizing the kernel facilitates reducing the peak to average power ratio.

52. The method of claim 51 , wherein choosing the subset of the plurality of frequencies is further based in part upon at least one selected from the group consisting of: a random process; a channel which the multi-carrier communication system uses as a medium of communication; bit rates of the plurality of frequencies.

53. The method of any one of claim 49-52, wherein modifying the subset of the plurality of signals includes: applying the kernel to the symbol, the kernel being a function of the subset of the plurality of frequencies, such that the subset of the plurality of signals are determined by the application of the kernel to the symbol, wherein if the symbol has a peak, the kernel is scaled and applied to the symbol to reduce the peak, whereby a one or more signals of the subset of the plurality of signals is modified.

54. The method of claim 53, wherein the symbol has a plurality of peaks, and the step of applying the kernel to the symbol is repeated such that the kernel is repeatedly adjusted and applied to the plurality of peaks.

55. The method of claim 53, wherein the kernel has a primary peak having an amplitude and at least one secondary peak having an amplitude, the amplitudes of the secondary peaks being smaller than the amplitude of the primary peak, such that the subset of the plurality of frequencies are chosen from the plurality of frequencies such that the amplitudes of the secondary peaks are minimized with respect to the amplitude of the primary peak.

56. The method of claim 49, wherein modifying the subset of the plurality of signals, wherein the determination includes, modifying the subset of the plurality of signals, applying a fourier transform to the subset of the plurality of signals, incorporating the fourier transformed subset of the plurality of signals into the plurality of signals, applying an inverse fourier transform to the plurality of signals, determining if the subset of the plurality of signals require more modification, and repeating modifying the subset of the plurality of signals if the subset of the plurality of signals require more modification.