CN101442297B - Digital decimation filter - Google Patents

Abstract

The invention provides a digital decimation filter. The digital decimation filter comprises a cascade integral comb filter, a cascaded Lagrange half-band filter and an FIR filter, which are orderly connected. The digital decimation filter has the advantages of less pass band dropping, small quantized noise power after filtering, and less power consumption.

Description

A kind of decimation filter of digital

Technical field

The present invention relates to filter, relate in particular to a kind of decimation filter of digital.

Background technology

In the analog to digital converter (ADC) that decimation filter of digital (Digital Decimation Filter) is applied to modulating based on ∑-Δ.Analog to digital converter based on ∑-Δ modulation uses over-sampling, and noise shaped, decimation filter of digital obtains higher signal to noise ratio.Use over-sampling to be because: along with the increase of multiple R, the overlapping component of quantization noise spectrum and signal spectra is fewer and feweri.The noise shaped noise energy that makes focuses on HFS.Decimation filter of digital frequency band π/R＜| ω | remove quantization noise in the≤π, and signal component does not change, and has therefore improved signal to noise ratio.Decimation filter of digital not only can be used among the ADC that modulates based on ∑-Δ, can also be used in digital down converter (DDC) and the digital up converter (DUC).

The purpose of decimation filter of digital is to be f in order to make over-sampling speed _sDigital signal return to Nyquist speed f by filtering and down-sampling _NSignal, and keep simultaneously as far as possible only | ω | the signal in≤π/R, and leach out-of-band noise.It in fact be with regard to the cut-off frequency that serves as after the noise moulding | ω | the role of the low pass filter of≤π/R.R is called sample rate again and changes the factor, R=f _s/ f _NBy the Nyquist sampling thheorem, our interested analog signal peak frequency f then _c≤ f _N/ 2=f _s/ 2R is as normalization f _s/ 2 is π, then ω _c≤ π/R, wherein ω _c=2 π f _c

Because cascaded integrator-comb (Cascaded Integrator-Comb is abbreviated as CIC) filter has characteristic of simple structure (when using the structure of integration-extraction-differential, not having multiplying), therefore generally use the cic filter of low order at front end.Behind the process cic filter, speed decline R1 doubly.Than low rate the time, can use the interior fluctuation of passband little, the FIR filter that stopband attenuation is very fast.The tap number of such FIR filter is more, but owing to be operated under the low rate, therefore the increase of the complexity of bringing is little.

The cic filter on N rank is H (z)=((1-z ^-R)/(1-z ^-1)) ^N, wherein N is an exponent number.Using the reason of cic filter at the front end of decimation filter of digital is to realize simply, but does not have multiplication when using the IIR structure, and as N greatly the time, its side lobe attenuation is bigger.Generally speaking, the extraction factor R is 2 L power, i.e. R=2 ^L, then the transfer function of cic filter can be written as $H\left(z\right)={(1+z^{-1})}^N{(1+z^{-2})}^N...{(1+z^{{-2}^{L-1}})}^N.$ Therefore cic filter can be implemented as the filtering-2 times extraction form of cascade.

General tolerance decimation filter performance has two indexs: (1) passband landing: the maximum attenuation of the interior filter of useful signal bandwidth (band of interest).(2) aliasing error: fold into the attenuation in the useful signal bandwidth.Aliasing error is of comparison key.Analog signal can be with sampling frequency f after being sampled _sFor periodically repeating frequency spectrum in the center, when sampling rate drops to f _s/ R then can be with f _s/ R is that frequency spectrum is periodically repeated at the center.Therefore after extracting R times, at [kf _s/ R-f _c, kf _s/ R+f _c] frequency band can be aliased in the interested frequency band, k=1 wherein, 2 ...,

R/2

As normalization f _s/ 2 is π, then [2k/R-ω _c, 2k/R+ ω _c] frequency band can be aliased in the interested frequency band, these frequency bands are called as the aliasing band.

The compensating filter shortcoming of CIC is: the passband of cic filter descends bigger, and the increase that passband descends has caused reducing of bandwidth; The stopband attenuation of the cic filter of high-order is bigger, but the landing of passband is also bigger, causes that bandwidth reduces, and can make passband more smooth after compensating, but can amplify stopband simultaneously.Exponent number is high more, and it is many more that passband need compensate, and cause that the amplification of stopband is big more, and the passband fluctuation of the compensating filter of the cic filter of high-order is very big, causes that the fluctuation of passband of the cic filter after the compensation is also very big.

Existing decimation filter of digital comprises half-band filter, compensating filter and the FIR filter of cic filter, cascade.Satisfy condition:

$h\left[\mathrm{Ln}\right]=\left\{\begin{array}{cc}\mathrm{\α}& n=0\\ 0& \mathrm{otherwise}\end{array}\right.$ Filter h, be called Nyquist filter or L band filter.

Especially, the L band filter during L=2 is called half-band filter.The transfer function of half-band filter is: H (z)=α+z ^-1E ₁(z ²), E wherein ₁(z) be any filter.This formula is equivalent to H (z)+H (z)=1, when α=1/2.If H (z) has real coefficient, then H (e ^{J ω})=H (e ^{J (π-ω)}), and H (e is arranged ^{J ω})+H (e ^{J (π-ω)})=1, i.e. H (e ^{J (pi/2-θ)}) and H (e ^{J (pi/2+-θ)}) add up to 1, H (e in other words ^{J ω}) about being symmetrical with the frequency pi/2 partly, this is the origin of " half-band filter " just.This also can derive two character of half-band filter: (1) passband and stopband fluctuation equate δ _p=δ _s(2) passband and stopband edge are about pi/2 symmetry, ω _p+ ω _s=π.So behind the down-sampling, [pi/2, ω _s] in frequency component only can advance [ω by aliasing _p, pi/2] and in the frequency band, and [ω _s, π] in component enough little, therefore to passband, promptly [0, ω _p] in frequency band influence very little.But the decay of the half-band filter of existing decimation filter of digital in the aliasing band is big inadequately, the tap coefficient complexity, and the passband landing that existing decimation filter of digital brings is bigger, needs compensating filter to compensate.

Summary of the invention

The object of the present invention is to provide a kind of decimation filter of digital, the decay of half-band filter in the aliasing band that is intended to solve existing decimation filter of digital is big inadequately, tap coefficient complexity, and the bigger problem of passband landing brought of existing decimation filter of digital.

Technical scheme of the present invention is to realize like this, a kind of decimation filter of digital, the Lagrange half-band filter and the FIR filter that comprise cascade integral comb filter, cascade, the Lagrange half-band filter of described cascade integral comb filter, cascade links to each other successively with the FIR filter, the tap coefficient of described Lagrange half-band filter is to be the end with 2 power, and the transfer function of the Lagrange half-band filter of described cascade is $H_{\mathrm{CLag}}\left(z\right)=H\left(z\right)H\left(z^2\right)...H\left(z^{2^{L-1}}\right),$ Wherein H (z) is the transfer function of Lagrange half-band filter, and 2 ^L=R, described R are that sample rate changes the factor.

The technical scheme that the present invention takes also comprises: described decimation filter of digital comprises that 8 rank Lagrange half-band filters that the 4 classes connection integral comb filter of 22 times of extractions adds 22 times of extractions add 4 times again and extract the cascade of FIR filter.

Beneficial effect of the present invention is: decimation filter of digital of the present invention comprises the Lagrange half-band filter of cascade, the passband of Lagrange half-band filter is smooth, and it is very big at the aliasing band attenuation, little, the filtered noise power of passband landing that decimation filter of digital of the present invention brings is little, its power consumption is less, and add that for the cascade integral comb filter that adopts 4 rank 8 rank Lagrange half-band filters add the mode of FIR filter, can be without compensating filter.

Feature of the present invention and advantage will be elaborated in conjunction with the accompanying drawings by embodiment.

Fig. 1 is the structural representation of existing decimation filter of digital;

Fig. 2 is the structural representation of decimation filter of digital of the present invention;

Fig. 3 is the structural representation of the Lagrange half-band filter of cascade of the present invention;

Description of drawings

Fig. 4 is the logarithm amplitude spectrum of CIC+FIR 4:1 mode and noise;

Fig. 5 is the logarithm amplitude spectrum of CIC+L8-half+FIR 4:1 mode and noise;

Fig. 6 is the logarithm amplitude spectrum of CIC+L12-half+FIR 4:1 mode and noise;

Fig. 7 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+FIR 4:1;

Fig. 8 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+L8-half+FIR 4:1;

Fig. 9 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+L12-half+FIR 4:1.

In order to make purpose of the present invention, technical scheme and advantage clearer, below in conjunction with drawings and Examples, the present invention is further elaborated.

See also Fig. 2, be the structural representation of digital decimation filter of the present invention. Digital decimation filter of the present invention comprises: Lagrange half band filter and the FIR wave filter of cascade integration pectination (CIC) wave filter, cascade. Lagrange half band filter of cascade integration comb filter, cascade links to each other successively with the FIR wave filter.

For the FIR wave filter, F (z)=z^N-1(1+z ^-1) ^NR (z), wherein R (z) is the multinomial (N is even number) that a number of times is N-2. F (z) can generate one and half band filters, by suitable selection R (z) so that the frequency response F (e of wave filter F (z)^jω) be maximally-flat in ω=0 and ω=π place. Half band filter of this class is called as binomial (binomial) or maximally-flat (maxflat) wave filter. The maximally-flat here refers to Dⁿ(F(e ^jω))| _ω＝0＝δ _0，n，D ⁿ(F(e ^jω))| _ω＝π=0, D whereinⁿN rank differential operators, n=0 ... N. The same with the transfer function of single-stage cic filter, it also has (1+z^-1) ^NThis, thus guaranteed that its amplitude response is at the flatness at ω=π place. And its 1 to N order derivative in ω=0 is 0, has therefore guaranteed that its amplitude response is at the flatness at ω=0 place.

Embodiment

Daubechies has constructed the F (z) with minimum length in the process of structure orthogonal wavelet.If note H ₀(z)=((1+z)/2) ^pR ₀(z), F (z)=((2+z+z then ^-1)/4) ^pP (y), 2p=N wherein, y=sin ²(ω/2)=(2-z-z ^-1)/4, P (y)=R ₀(z) R ₀(z ^-1).Because R ₀(z) R ₀(z ^-1) be the multinomial of cos ω, so can be write as

Multinomial.Again because R ₀(z) be real coefficient, so R ₀The R of (-z) ₀(z ^-1) be

Multinomial (coefficient is identical with P (y) coefficient), be designated as P (1-y)=R ₀The R of (-z) ₀(z ^-1).Therefore condition F (z)+F (z)=1, changes into

(1-y) ^pP(y)+y ^pP(1-y)＝1 (1)

The lowest-order multinomial that satisfies the P (y) of this formula is

$P\left(y\right)=\underset{k=0}{\overset{p-1}{\mathrm{\Σ}}}\left(\begin{array}{c}p-1+k\\ k\end{array}\right)y^k---\left(2\right)$

Obtain accordingly

$F\left(z\right)={\left(\frac{2+z+z^{-1}}{4}\right)}^p(\underset{k=0}{\overset{p-1}{\mathrm{\Σ}}}\left(\begin{array}{c}p-1+k\\ k\end{array}\right){\left(\frac{2-z-z^{-1}}{4}\right)}^k)---\left(3\right)$

This filter also can be generated by the Lagrange interpolation, therefore is also referred to as the Lagrange half-band filter.

The tap coefficient of Lagrange half-band filter all is to be the end with 2 power, does not need to round when using fixed-point number to calculate, therefore can loss of accuracy.

The amplitude response of Lagrange half-band filter has flatness at ω=π place, has brought near its logarithm amplitude response decay ω=π very big.

See also Fig. 3, the transfer function of the Lagrange half-band filter of cascade is $H_{\mathrm{CLag}}\left(z\right)=H\left(z\right)H\left(z^2\right)...H\left(z^{2^{L-1}}\right),$ Wherein H (z) is the transfer function of Lagrange half-band filter, and 2 ^L=R.Notice H (z ²) at ω=pi/2 place there be N zero point, so H (z) H (z ²) at ω=π and ω=pi/2 place all there be N zero point, H (z) H (z like this ²) the logarithm amplitude response more smooth near ω=π and ω=pi/2, the signal energy that therefore is aliased in the band of interest is less.Furtherly, transfer function is H _CLagThe Lagrange half-band filter of cascade (z) all has N zero point at the 2k/R place, wherein

So H _CLag(z) logarithm amplitude response is all more smooth near these zero points, and near the band attenuation corresponding zero point is bigger, and therefore the aliasing error that causes is less, makes that the output quantization noise power is less, and this is the not available advantage of general half-band filter.Suppose total extraction factor R=64, normalization f _s/ 2 is π, and then band of interest is at 0～π/R _ΔInterior (R _Δ≤ R), then the aliasing band is [2 π k/R-π/R _Δ, 2 π k/R+ π/R _Δ], wherein

R in general the application _Δ=R, so the aliasing band is in the whole interval of [0, π].Therefore in 0 to the π scope of decimation filter all the noise power to output exert an influence.

Compare CIC+FIR 4:1 mode below respectively, CIC+L8-half+FIR 4:1 mode, CIC+L12-half+FIR 4:1 mode, wherein CIC+FIR 4:1 represents that 4 rank cic filters of 42 times of extractions add 4 times and extract the cascade of FIR filter, and CIC+Lx-half+FIR 4:1 represents that x rank Lagrange half-band filter that 4 rank cic filters of 22 times of extractions add 22 times of extractions adds 4 times again and extracts the cascade of FIR filter.When adopting the mode of band Lagrange half-band filter, have only the cic filter cascade on two 4 rank, the passband that brings landing is little, so do not adopt compensating filter, can reduce complexity like this.

See also Fig. 4, Fig. 5 and Fig. 6, in Fig. 4, Fig. 5 and Fig. 6, empty vertical line is the starting point of aliasing band, and real vertical line is the terminal point of aliasing band.If H is (f _d) be the frequency response of whole decimation filter, signal and the total power spectral density of noise after then extracting are:

S _z(f _d)＝|H(f _d)| ²S _x(f _d)+|H(f _d)| ²S _N(f _d) (4)

S wherein _x(f _d) be the power spectral density of input signal, S _N(f _d) be the power spectral density of quantizing noise.Here we only discuss the power spectral density of quantizing noise.For the sigma-delta modulator on p rank, the power spectral density of quantizing noise is:

S _N(f _d)＝E(f _d)·[2sin(πf _d)] ^p (5)

Wherein $E\left(f_d\right)=({\mathrm{\Δ}}^2/12)\sqrt{2/f_s}$ For supposing that quantizing noise is the sampling noiset spectrum density of white noise, f _dNormalized frequency, corresponding angles frequency are ω=2 π f _d, f _d∈ [0,0.5], Δ=(2V _r/ 2 ^N-1) is the quantization step of N bit resolution quantizer, 2V _rDynamic range for the quantizer input.The transfer function of supposing quantizing noise is (1-z ^-1) ³, so the power spectral density of quantizing noise is:

$S_N\left(f_d\right)={\left|{(1-e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="S2007101247815E00021.png,S2007101247815E00031.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{f}_d})}^3\right|}^2---\left(6\right)$

This is equivalent to E (f _d)=1, the situation during p=6.Below what carry out performance relatively is exactly in this case comparison.

The noise power spectral density of note output is below:

S _No(f _d)＝|H(f _d)| ²S _N(f _d) (7)

The noise power of output is (only calculating the power in the aliasing band):

$P_{\mathrm{xo}}=\underset{k}{\mathrm{\&Sigma;}}{\&Integral;}_{k/R-f_c}^{k/R+f_c}{S}_{\mathrm{No}}\left(f_d\right){\mathrm{df}}_d---\left(8\right)$

Wherein For R is even number.

See also Fig. 7, Fig. 8 and Fig. 9, as can be seen when adopting the mode of band Lagrange half-band filter, have only the cic filter cascade on two 4 rank from Fig. 7, Fig. 8 and Fig. 9, even using compensation filter not, the passband that brings landing is also little.

Following table is the noise power that several modes are estimated.

Table 1: noise power relatively

?	Pn(dB)
		CIC+FIR?4:1	-100.33
CIC+L8-half+FIR?4:1	-112.64

Power consumption is estimated as following formula:

$P=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}\frac{{\mathrm{NP}}_i*W_i}{{\mathrm{\&Pi;}}_{j=1}^i{M}_j}---\left(4\right)$

NP wherein _iBe quantity in the partial product of i in the stage, W _iBe the input word length in i stage, M _jBe the extraction factor in i stage, l is the extraction stage sum.Here the partial product of saying is meant and the product of 2 power time, when multiplying each other with 2 power, just can be converted into integer and move to left when hardware is realized.For example when N=5, the FIR filter transfer function is (1+z ^-1) ⁵=1+5z ^-1+ 10z ^-2+ 10z ^-3+ 5z ^-4+ z ^-5, the impulse response of corresponding filter be [1 5 10 10 5 1], 2 corresponding systems are expressed as [0,001 0,101 1,010 1,010 0,101 0001], therefore have 10 times with the inferior product of 2 power, promptly have 10 times integer to move to left, corresponding NP _i=10.Because move to left each time with regard to a corresponding sub-addition, thus the actual number that moves to left with addition of having represented of the number of partial product, the complexity when it has well represented the hardware realization.Divided by ∏ _J=1 ⁱM _jBe because the multiple that on behalf of speed, i stage total extraction number descend, and the default rate power consumption of one times of computing of being carried out one times of this fact that just descends that descends.

Compare CIC+FIR 4:1, the power consumption of CIC+L8-half+FIR 4:1 dual mode below.Suppose the realization of FIR filter is all realized with the leggy form.

Table 2: the comparison of power consumption

?	P (power consumption)	Relative power consumption
			CIC+FIR?4:1	?1731	100％
CIC+L8-half+FIR?4:1	?1609	93％

Power consumption minimum when adopting 8 rank Lagrange half-band filters, and its filtered noise power is more much smaller than CIC+FIR 4:1 mode.

The above only is preferred embodiment of the present invention, not in order to restriction the present invention, all any modifications of being done within the spirit and principles in the present invention, is equal to and replaces and improvement etc., all should be included within protection scope of the present invention.

Claims (2)

1. decimation filter of digital, comprise cascade integral comb filter and FIR filter, it is characterized in that: the Lagrange half-band filter that also comprises cascade, the Lagrange half-band filter of described cascade integral comb filter, cascade links to each other successively with the FIR filter, the tap coefficient of described Lagrange half-band filter is to be the end with 2 power, and the transfer function of the Lagrange half-band filter of described cascade is $H_{\mathrm{CLag}}\left(z\right)=H\left(z\right)H\left(z^2\right)...H\left(z^{2^{L-1}}\right),$ Wherein H (z) is the transfer function of Lagrange half-band filter, and 2 ^L=R, described R are that sample rate changes the factor.

2. decimation filter of digital as claimed in claim 1, it is characterized in that described decimation filter of digital comprises that 8 rank Lagrange half-band filters that the 4 classes connection integral comb filter of 22 times of extractions adds 22 times of extractions add 4 times again and extract the cascade of FIR filter.

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