WO2006056486A1

WO2006056486A1 - Method and device for analogue digital conversion with asymmetry correction

Info

Publication number: WO2006056486A1
Application number: PCT/EP2005/013538
Authority: WO
Inventors: Michel Robbe
Original assignee: Eads Secure Networks
Priority date: 2004-11-29
Filing date: 2005-11-29
Publication date: 2006-06-01
Also published as: FR2878667B1; FR2878667A1

Abstract

In an analogue-digital converter having two quadrature channels I and Q each comprising an associated input (301, 301') and an associated output (303, 303'), a complex analogue input signal is converted into a complex digital output signal. The converter comprises a complex filter having a first and a second stage, each comprising a respective input (301, 301', 320, 320') and a respective output (308, 308', 309, 309') on each of the I and Q channels. Each output of the converter is coupled, on the one hand, to the respective input of the first stage assigned with a first correction coefficient via a first feedback loop (305, 305'), and on the other hand, to the respective input of the second stage assigned with a second feedback coefficient via a second feedback loop (304, 304'). A correction of asymmetry of the transfer function of the converter is furthermore performed by coupling the I and Q channels upstream or downstream between the first stage and the second stage of the complex filter.

Description

METHOD AND DEVICE FOR ANALOGUE DIGITAL CONVERSION WITH ASYMMETRY CORRECTION

The present invention relates to analogue-digital converters of complex signals, having two quadrature channels. More particularly the invention addresses the problem of the asymmetry of the transfer function which affects certain complex bandpass converters. Known among analogue-digital converters are Sigma-Delta analogue- digital converters. These converters exhibit a much greater oversampling frequency than the Nyquist frequency of the input signal. Therefore, they make it possible to obtain significant resolution for relatively low cost.

The converters of this type also make it possible to reject quantization noise outside the useful frequency band of the output signal, which band possesses a central frequency fo. Specifically, in such a converter, the quantization noise is controlled by combined action of the signals of the feedback loops and of a complex filter.

Bandpass Sigma-Delta converters can convert complex analogue signals into complex digital signals. They then exhibit two quadrature channels, conventionally referenced I (for "In phase") and Q (for "Quadrature"). They have advantages over converters of real signals, in terms of stability and width of passband.

Figure 1 illustrates a complex bandpass Sigma-Delta converter of this type. Each of the I and Q quadrature channels comprises an input 301 , 301'. The signal entering through the input 301 , 301' is added by an adder 103, 104 to the signal of the feedback loop 114, 115. Then the signal passes through the complex filter 105. It is thereafter processed by a quantizer 106, 107 before being delivered on outputs 303, 303'. The feedback loops each comprise at least one digital-analogue converter 112, 113 for providing the feedback signal.

Figure 2 illustrates a curve exhibiting two noise shaping shoulders 202 and 203 for a converter of complex signals. Thus, the noise is rejected out of the useful frequency band 201 of central frequency fo, thereby making it possible to improve the signal/noise ratio and therefore to boost the performance of the converter. In a second-order bandpass filter, an asymmetry of processing of the signal may occur between the first stage and the second stage of the filter. In the case of asymmetry, the shoulders 202 and 203 of the noise shaping curve are not symmetric. When a transfer function of a converter has an asymmetry of processing between various stages, the noise on the image frequency is not correctly rejected.

The present invention aims to reduce the noise in such a way as to improve the performance of such a converter, by addressing the problem of asymmetry of the transfer function.

A first aspect of the invention proposes a method of analogue digital conversion using an analogue-digital converter having two quadrature I and Q channels each comprising an associated input and an associated output, for converting a complex analogue input signal into a complex digital output signal. The converter comprises a complex filter having at least one pair of stages with a first stage and a second stage, each comprising a respective input and a respective output on each of the I and Q channels. A respective direct link links the output of the first stage to the respective input of the second stage on the I channel, respectively on the Q channel. Each of the outputs of the converter is coupled, on the one hand, to the respective input of the first stage assigned with a first feedback coefficient, denoted a, via a first feedback loop, and on the other hand, to the respective input of the second stage assigned with a second feedback coefficient, denoted b, via a second feedback loop. A correction of asymmetry of the transfer function of the converter, corresponding to the pair of stages of the filter, is performed by coupling, per pair of stages, the I and Q channels upstream of the first stage of the pair of stages or between the first and second stages of the pair of stages.

By virtue of these provisions it is possible to boost the performance of an analogue-digital converter such as this.

The correction may be performed by coupling the I and Q channels downstream of the first stage of the complex filter. The method then comprises the following steps: - coupling the output of the Q channel of the first stage to the output of the I channel of the first stage via a coupling link assigned with a first correction coefficient;

- coupling the output of the I channel of the first stage to the output of the Q channel of the first stage via a coupling link assigned with a second correction coefficient;

- assigning a third correction coefficient to the respective direct links of the I and Q channels between the first and the second stage.

In an embodiment of the present invention, the first correction coefficient has a value equal to sin θ_c, the second correction coefficient has a value equal to -sin θ_c and the third coefficient has a value equal to cos θ_Cl where θ_c substantially satisfies one of the following equations according to the relative value of the zero of each stage ; _n bsinθz θ_r = arete + θz for fzl < fz2 a + (b + 2)(1- costfz) ^J θ_c =-arctg ^^ θz for fzl > fz2

for θz equal to 2πf_z/f_s, where f_z is the frequency corresponding to the zero of a transfer function of the filter, f_s is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

In another embodiment of the present invention, the correction is performed by coupling the I and Q channels upstream of the first stage of the complex filter. The method then comprises the following steps:

- coupling the output of the I channel of the converter, on the one hand, at the input of the first stage of the I channel, via the first feedback loop assigned with a first correction coefficient, and, on the other hand, at the input of the first stage of the Q channel via a coupling link assigned with a second correction coefficient;

- coupling the output of the Q channel of the converter, on the one hand, at the input of the first stage of the Q channel via the first feedback loop assigned with a third correction coefficient and, on the other hand, at the input of the first stage of the I channel via a coupling link assigned with a fourth correction coefficient. In such a case the first correction coefficient may have a value equal to a^*cosθ₀, the second correction coefficient may have a value equal to -a*sinθ_c, the third correction coefficient may have a value equal to a*cosθ_c, and the fourth correction coefficient may have a value equal to a^*sinθ_c; where θ_c substantially satisfies one of the following equations ;

_ _v bsinθ∑ . . θ. - arctg + θz for fzl < fz2 a+ (b+ 2)(l- cosθz) ^J _{θ =} -arctg ^^ θz for fz\ > fzl

for θz equal to 2πf_z/f_s, where fz is the frequency corresponding to the zero of a transfer function of the filter, f_s is the frequency of sampling of the input signal, fzl corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

In an embodiment of the present invention, the complex filter comprises a number of stages greater than two successively forming one or more pairs of stages, each comprising successive first and second stages. For each of the pairs of stages, each output of the converter is coupled, on the one hand, to the respective input of the first stage assigned with a first feedback coefficient, and on the other hand, on the respective input of the second stage assigned with a second feedback coefficient. Each of the pairs has a transfer function exhibiting an asymmetry. A correction of asymmetry of the transfer function or functions corresponding to each pair of stages of the complex filter is performed upstream of the first stage or downstream of the first stage.

When the number of stages of the complex filter is an odd number greater than or equal to three, the correction of asymmetry is performed on the pair or pairs of stages which follow the first stage. A second aspect of the present invention proposes an analogue-digital converter having two quadrature I and Q channels each comprising an associated input and an associated output, for converting a complex analogue input signal into a complex digital output signal.

The converter comprises a complex filter having at least a first stage and a second stage, each comprising a respective input and a respective output on each of the I and Q channels. Each of the outputs of the converter is coupled, on the one hand, to the respective input of the first stage assigned with a first feedback coefficient via a first feedback loop, and on the other hand, on the respective input of the second stage assigned with a second feedback coefficient via a second feedback loop.

The analogue-digital converter comprises a complex coupler suitable for coupling the I and Q channels upstream or downstream of the first stage of the complex filter and thus for applying a correction of asymmetry of the transfer function of the converter. In an embodiment of the present invention, the complex coupler is placed downstream of the first stage of the complex filter and comprises ;

- a first coupling link linking the output of the first stage of the Q channel on the output of the first stage of the I channel, the first coupling link being assigned with a first correction coefficient; - a second coupling link linking the output of the first stage of the I channel on the output of the first stage of the Q channel, the second coupling link being assigned with a second correction coefficient. In this case, the first correction coefficient may be equal to sin θ_c and the second correction coefficient may be equal to -sin θ_c, the outputs of the first stage of the I and Q channels each being assigned with a coefficient of value cos θ_c; where θ_c substantially satisfies one of the following equations ; θ = arctg bsmθz_ ₊ ^ fzl < fz2 a+ ψ+ 2)(l- cosθz) ^J θ, = - arctg ^-^ θz for fή > fz2 a+ (b + 2)(l~ cosθz) for θz equal to 2ττfz/fs_> where f_z is the frequency corresponding to the zero of a transfer function of the filter, f_s is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

Such a converter can furthermore comprise in the first stage:

- on the I channel, an internal return link from the output of the first stage of the I channel, the link comprising a substantial delay and being assigned with a coefficient of value cos G₁; - on the Q channel, an internal return link from the output of the first stage of the Q channel, the link comprising an appreciable delay and being assigned with a coefficient of value cos θ-i;

- on the I channel, an adder receiving: - the input of the I channel of the filter;

- the first feedback loop of the I channel, assigned with the first feedback coefficient;

- the internal return link of the I channel; and

- a return link from the internal return link of the Q channel, assigned with a coefficient of value -sin θi ;

- on the Q channel, an adder receiving:

- the input of the Q channel of the filter;

- the first feedback loop of the Q channel, assigned with the first feedback coefficient; - the internal return link of the Q channel; and

- a return link from the internal return link of the I channel, assigned with a coefficient of value sin θi; where G₁ is equal to 2πf_z/f_s, where fz is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal; in the second stage:

- on the I channel, an adder or several adders receiving:

- the input of the I channel of the second stage;

- the second feedback loop of the I channel assigned with a coefficient of value b*cos θ₂; where b is the second feedback coefficient;

- a return link from the feedback loop of the Q channel assigned with a coefficient of value equal to -b*sin Θ₂;

- an internal return link from the output of the I channel of the second stage assigned with a coefficient of value cos θ₂;

- a return link from the output of the Q channel of the second stage assigned with a coefficient of value -sin θ₂;

- on the Q channel, an adder or several adders receiving: - the input of the Q channel of the second stage;

- the second feedback loop of the Q channel assigned with a coefficient of value b*cos θ₂; where b is the second feedback coefficient; - a return link from the feedback loop of the I channel assigned with a coefficient of value equal to b*sin θ₂;

- an internal return link from the output of the Q channel of the second stage assigned with a coefficient of value cos θ₂;

- a return link from the output of the I channel of the second stage assigned with a coefficient of value sin θ₂; where θ₂ is equal to 2πf_z/f_s, where fz is a frequency corresponding to a zero of the transfer function of the filter and f_s is the frequency of sampling of the signal; the converter comprising between the first stage and the second stage, the complex coupler having the first correction coefficient of value equal to

-sin θ₃ and the second correction coefficient of value equal to sin θ₃; the outputs of the first stage of the I and Q channels each being assigned with a coefficient of value cos θ₃; where θ₃ substantially satisfies the following equation ; Θ₃ = θ₂- Θ_c; where θ_c substantially satisfies one of the following equations ; bsinθz . . . , . _ θ_r = arctg + θz for fή < fzl α+ (δ + 2)(l- cos£z) ^J _θc = -_arctg ^bJ^£l _θ2 f_or f_{A >} fe₂

for θz equal to 2τrf_z/f_s, where f_z is the frequency corresponding to the zero of a transfer function of the filter, f_s is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

In an embodiment of the present invention, the complex coupler is placed upstream of the first stage and comprises :

- a return loop linking the output of the Q channel of the converter to the input of the first stage of the I channel assigned with a first correction coefficient;

- a return loop linking the output of the second stage of the I channel to the input of the first stage of the Q channel assigned with a second correction coefficient. In this case, the first feedback loop on the I channel is assigned with a third correction coefficient and on the Q channel is assigned with a fourth correction coefficient.

The first correction coefficient may be equal to a*sinθ_G; the second correction coefficient may be equal to -a*sin θ_c; the third and the fourth correction coefficients may be equal to a*cos θ_c; where θ_c substantially satisfies one of the following equations: bsmθz _{+ θz} f_z\ < fz1 a + (b + 2)(l-cosθz) bsinθz _n , _ ., _ _ rctg θz for fz\ > fz2 a + (b + 2)(l-cosθz) for θz equal to 2πf_z/fs, where fz is the frequency corresponding to the zero of a transfer function of the filter, f_s is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

In another variant, the first correction coefficient has a value equal to

-tan θ_c; the second correction coefficient has a value equal to tan θ_c; the third and the fourth correction coefficients have a value equal to -1. The direct links between the first and the second stage on each of the I and Q channels are assigned with a coefficient d whose value is -a.

In an embodiment, the complex filter comprises a number of stages greater than two and a respective coupler per pair of stages, each pair comprising successive first and second stages. Each of the couplers is placed upstream or downstream of the first respective stages and is assigned with a correction coefficient.

Other aspects, aims and advantages of the invention will become apparent on reading the description of one of its embodiments.

The invention will also be better understood with the aid of the drawings, in which: - Figure 1 is a complex bandpass Sigma-Delta analogue-digital converter according to the prior art, as detailed above;

- Figure 2 is a noise shaping curve for a bandpass Sigma-Delta analogue-digital converter of complex signals, as detailed hereinabove;

- Figure 3 illustrates a conventional modelling of a converter comprising a second-order complex filter;

- Figure 4 illustrates a common modelling of a filter such as illustrated in Figure 3 in which are detailed complex couplings between the two I and Q channels;

- Figure 5 represents another modelling, closer to an implementation diagram, of the same complex converter as that modelled in Figure 4;

- Figure 6 illustrates a modelling of an analogue-digital converter comprising a complex correction coupler downstream of the first stage according to an embodiment of the invention;

- Figure 7 illustrates a modelling closer to an implementation diagram of an analogue-digital converter such as illustrated in Figure 6 according to an embodiment of the invention; - Figure 8 illustrates a modelling of an analogue-digital converter comprising a complex correction coupler upstream of the first stage according to an embodiment of the invention;

- Figure 9 illustrates a modelling closer to an implementation diagram of an analogue-digital converter as illustrated in Figure 8 according to an embodiment of the invention;

- Figure 10 is an implementation diagram of a filter according to an embodiment of the present invention;

- Figure 11 represents a noise shaping spectrum curve for a converter of the prior art; - Figure 12 represents a noise shaping spectrum curve for a converter according to an embodiment of the present invention. For the sake of clarity, an embodiment of the present invention in which the complex filter of the analogue-digital converter introduces an appreciable delay 307, denoted Z₂ ^"1 on the main I and Q channels, is more precisely described. A modelling of such a filter is illustrated by Figure 3.

However, such a configuration is not limiting on the present invention which covers all types of converters, whether they introduce one or more appreciable delays.

The following sections pertain to the converter illustrated in Figure 3.

In Figure 3, the two I and Q channels are shown diagrammatically by two continuous lines. The converter thus illustrated comprises an input 301 and an output 303. It comprises a first stage 314 and a second stage 315. The output is looped, on the one hand, back to the input of the first stage via a first feedback loop 305 with a first feedback coefficient, denoted a, and, on the other hand, to the input of the second stage via a second feedback loop 304 with a second feedback coefficient, denoted b.

The first stage has an input 301 and an output 308. The output 308 is looped back to the input 301 via a return loop 311 comprising a delay 306, denoted Z-f¹. At the input of this first stage, an adder 312 receives and adds the signal of the input 301 , the signal of the return loop 311 and the signal of the feedback loop 305.

The second stage has an input 320, corresponding to the output of the first stage, and an output 309. This stage introduces an appreciable delay 307, denoted Z₂ ^"1, into the processing of the signal. The output 309 is looped back to the input 320 via an internal return link 310. At the input of this second stage, an adder 313 receives and adds the output signal of the first stage, the signal of the return link 310 and the signal of the feedback loop 304. For such a converter, the transfer function exhibits an asymmetry. Such an asymmetry is illustrated in Figure 11 representing a noise shaping spectrum curve pertaining to such a converter. It is noted that the shoulders 111 and 112 of this curve are not symmetric.

A correction of such an asymmetry may be determined according to two different approaches.

A first approach is based on an estimation Group Propagation Time (GPT) of the transfer function of the high-pass filter. The GPT is defined as being a derivative of the phase φ, denoted dφ, with respect to the variation in angular frequency ω, denoted dω, and therefore satisfying the following equation:

GPT = -^- dω

For a second-order filter, the high-pass transfer function may be written in the following biquadratic form:

_ \ + k_lZ^~1 + k₂Z-² ^{z ~} 1 + aZ^~x + βZ^'1

When the attenuated band is narrow, the coefficient k₂ is substantially equal to 1. It may therefore be replaced by 1. An approximation of the first- order coefficient of the numerator may then be provided by the following equation:

The high-pass transfer function may then be written:

with Z = cos6> - j sin<9 and θ = 2π β θz substantially satisfies the following equation:

fz being the frequency corresponding to the zero of the transfer function of the filter and f_s being the frequency of sampling of the signal. The coefficients α and β correspond to the poles of the transfer function while the coefficient δ corresponds to the zero of the transfer function.

A phase calculation is then performed and the following equation is obtained:

We thus note that a literal calculation of GPT is relatively complex. In this type of approach, one can advantageously perform numerical applications of the phase φ for θz and for two values around θz. Then by linear regression, it is thus possible to calculate an approximation to the tangent at the point θz. We calculate the value of a correction arc φ_COr expressed in degrees through the equation: φ_cor = -GPT*fz*360

In a second approach, an analytic procedure of the transfer function of the filter is applied so as to determine the correction to be applied so as to correct the asymmetry of the transfer function of the complex filter. The filter is preferably a prototype high-pass filter. This second approach makes it possible to obtain substantially the same results as those that can be obtained in the first approach.

Referring to Figure 3, the first stage is assigned with Z-f¹ and the second stage is assigned with Z2^"1.

Moreover, the delay Zo^"1 corresponding to the central frequency f₀ of the converter may be written:

Z₀ ^"1 = cos x -j sin x where x is equal to -f/f_s.

For a frequency fi satisfying the following equation:

we may write:

Z₁ ^'1 = Z₀ ^"1 * (cos θz + j sin θz) where θz is equal to 2πf_z/f_S) where fz is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal.

Next, for a frequency f₂ satisfying the following equation:

If₂I > |fo| we may write:

Z₂ ^"1 = Z₀ ^"1 * (cos θz - j sin θz) We then obtain the following equations: Z₁-U z₂ ^"1 = 2 Z₀ ^"1 * cos θz and y -1 * v -1 _ ₇ -2

An equation for the transfer function, denoted T, of such a filter may then be written:

Next, the equation can then easily be written in the following form:

The denominator is approximated to that of the high-pass transfer function of a second-order real Sigma-Delta modulator such that:

We note that in the equation for T, more precisely in the denominator, the coefficient a, assigned to the feedback loop linked to the input of the first stage of the filter, is associated with the delay Z₂ ^"1.

The expression a* Z₂ ^"1 is advantageously replaced with the expression a^*Z_∞r^"1, where Z_cor ^'1 corresponds to a symmetry correction vector for the transfer function. We then obtain the following equation:

1 - aZcor^"1 - (b+1)Z₂ ^"1 - Z-f¹ + (b+1 )Z₀ ^"2 = 1 - (a+b+2)Z₀ ^"1 + (b+1 )Z₀-².

Thus, we can calculate a value of the correction vector as a function of the coefficients a and b as well as the delays Z₀ ^'1, Z-f¹, and Z₂ ^"1. The value of the correction vector Z_∞r satisfies the following identity:

1 7^"1

Z^'l = -((α + b + 2)Z^"1 - Z₁ ^""1 - (b + I)Z₂ ^"1) = -^((α + b + 2) - (2 + b)cosflz + jbsinθz) a a

We obtain in succession the real part and then the imaginary part as well as the phase of the correction vector Z₁ cor ^{"1 ■} j re{Z;l) = -(a + (b + 2)(1 - cosøz)) a bsinθz

*^«(O = a

,__{l s} . bsinθz φ(Z2.) = arctg a + (b + 2)(l - cosθz) Consequently, the following equation may be written:

This last equation may advantageously be applied to various numerical cases and thus provide an asymmetry correction value for various values of coefficients a and b of the feedback loops and various values of θz.

Thus, for example, in a numerical application where a is equal to -1.210019, b is equal to -0.723772 and θz is equal to 3.923 degrees, we obtain a phase value θ_cor of the correction vector with respect to the vector of the central frequency equal to -2.349 degrees. Such a correction may advantageously be applied before the first stage or else after the first stage of such a filter.

In an embodiment of the present invention, such a correction is applied via a complex signal coupler for which values of particular coefficients which make it possible to correct the asymmetry of the transfer function of such a filter are determined.

Figure 4 illustrates a modelling of such a second-order filter in which are detailed complex couplings of the two I and Q channels.

A description of the Q channel being readily deducible from a description of the I channel by symmetry, only the I channel is detailed hereinbelow. In this figure and the subsequent ones, the I and Q channels are represented. For the sake of clarity and simplicity, each element referenced on the Q channel bears the same reference as the element symmetric to it on the I channel, the reference then being associated with a prime.

Thus, the signal of the return loop 311¹ of the Q channel is injected into the input 301 of the I channel of the first stage of the filter by the adder 312, via a return link 403 assigned with a coefficient -sin θ, the internal return loop 311 being itself assigned with the coefficient cos G₁. By symmetry, a link 403' makes it possible to inject the signal of the return loop 311 into the input of the first stage of the Q channel, this link is assigned with a coefficient sin θi, the internal return loop 311 ' being itself assigned with the coefficient cos G₁. It is noted that θi is equal to 2πfz/fs_> where fz is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal.

Next, the output signal 309' of the second stage of the Q channel is injected at the output 309 of the second stage of the I channel by the adder 302 via a return link 401 which is assigned with a coefficient -sin θ₂. By symmetry, a link 401' makes it possible to inject the output signal 309 of the second stage of the I channel at the output 309' of the second stage of the Q channel, this link is assigned with a coefficient sin Θ₂. The signal of each of the outputs 309 and 309' of the second stage is itself assigned with a coefficient cos θ₂.

It is noted that Q₂ is equal to 2πfz/fs, where f_z is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal.

The I and Q channels each comprise an analogue-digital converter for providing the output signal 303 and 303' with the converter in a digital form and a digital analogue converter for converting the digital output signal into an analogue signal intended to be injected into the feedback loops 304, 304' and 305, 305'. These converters are shown diagrammatically by a single entity 402 and 402' on each of the channels. Figure 5 represents another modelling of one and the same complex converter as that modelled in Figure 4. One of the benefits of this modelling is that it is akin to an implementation diagram for such a converter.

This modelling depicts a complex coupler 500 between the first stage and the second stage. It is noted that the adders 302 and 302' are no longer used.

A complex coupler 500 such as this comprises a first coupling link 501 which makes it possible to inject an output signal 308' of the first stage of the Q channel at the output 308 of the first stage of the I channel. This coupling link 501 is assigned with a coefficient -sin θ₂. It also comprises a second coupling link 501' which makes it possible to inject an output signal 308 of the first stage of the I channel at the output 308' of the first stage of the Q channel. This coupling link 501' is assigned with a coefficient sin θ₂. An adder 502, 502' on each of the channels of the output of the first stage makes it possible to couple such signals.

It is noted that in this modelling, the feedback loop 304, 304' is assigned with a coefficient equal to b*cos G₂. Moreover, the adder 313 of the I channel furthermore receives the output signal 303' of the Q channel of the filter via a link 503 assigned with a coefficient -b*sin Θ₂. It also receives the output signal 309 of the second stage of the I channel via the internal return link 310 assigned with a coefficient cos G₂. It also receives the output signal 309' of the second stage of the Q channel via a return link 505 assigned with a coefficient -sin G₂.

By symmetry, the adder 313' of the Q channel furthermore receives the output signal 303 of the I channel of the filter via a link 503' assigned with a coefficient b*sin G₂. It also receives the output signal 309' of the second stage of the Q channel via the return link 310" assigned with a coefficient cos G₂. It again receives the output signal 309 of the second stage of the I channel via a return link 505" assigned with a coefficient sin G₂.

Figure 6 illustrates a modelling, similar to that illustrated in Figure 4, in which an analogue-digital converter comprises a complex correction coupler 600 according to an embodiment of the invention. In an embodiment of the present invention, a correction of asymmetry of the transfer function is applied via the complex correction coupler 600 placed between the first stage and the second stage of a complex filter of a converter such as modelled in Figure 4.

This complex correction coupler 600 comprises a coupling link 601 between the Q channel and the I channel making it possible to inject the output signal 308' of the Q channel of the first stage at the output 308 of the I channel of the first stage. It also comprises a coupling link 601 ' making it possible to inject the output signal 308 of the I channel of the first stage at the output 308' of the Q channel of the first stage. An adder 602, 602' on each channel adds together the output signal of the respective channel and the signal of the coupling link of the quadrature channel for this complex coupler 600.

The asymmetry correction is applied via the various coefficients assigned with to the links of the correction coupler. Thus, the link 601 is assigned with a coefficient sin θ_c and the link 601' is assigned with a coefficient -sin θ_c, where θ_c substantially satisfies one of the following equations:

θ = arctg ^bJ^l + θz for fA < fzl α+ (δ + 2)(l- cos0z) ^J θ = -arctg ^bJ^l _θz f_Qr f_{Λ >} β₂ ^c a+ (b + 2)(l- cosθz) ^J for θz equal to 2πf_z/fs, where fz is the frequency corresponding to the zero of a transfer function of the prototype high-pass biquadratic filter, f_s is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

The output signal of the I channel and that of the Q channel of the first stage are assigned with a coefficient of value θ_c.

These particular coefficients each have a value which advantageously makes it possible to balance the transfer function corresponding to the filter comprising the first and the second stage.

With a central frequency f₀ of value f_s/8, to which there corresponds a vector in the z plan placed at 45 degrees, one obtains an application of the following numerical values. For a value of a equal to -1.210019, for a value of b equal to -0.723772 and for a value of θz equal to 3.923 degrees, the following values are successively obtained: θi = 45-3.923=41.077 degrees; Q₂ = 45+3.923=48.923 degrees; θ_c= 2.349+3.923=6.272 degrees. Then, therefore: sin θi = 0.65707 cos B₁ = 0.75383 sin Θ₂ = 0.75383 cos Θ₂ = 0.65707 sin θ_c = 0.10925 cos θ_c = 0.994014. For a complex analogue-digital converter comprising a complex coupler between the first and the last stage, such as that illustrated in Figure 5, it is advantageous to determine particular coefficients for this complex coupler in such a way that they make it possible to correct the asymmetry of the transfer function of the second-order filter.

Thus, Figure 7 represents another modelling of one and the same filter as that modelled in Figure 6 according to an embodiment of the present invention. Such a modelling is much like an implementation diagram. The filter modelled in Figure 7 may correspond to the filter modelled in Figure 5 in which a correction of asymmetry of the transfer function is applied.

It is noted that the addition of the signals carried out by the adders 313 and 313' in Figure 5, is carried out by the adders 313, 701 , 313' and 701' in Figure 7. The adders 313 and 701 and respectively the adders 313' and 701' may readily be implemented in the form of a single adder. Respectively for each of the channels it is also possible to group together the additions carried out by the adder 602, respectively 602', in this single adder.

Advantageously, in this embodiment, the various coefficients of the coupler 600 are determined in such a way as to correct the asymmetry of the transfer function of the filter. Thus, advantageously, the link 601 is assigned with a coefficient -sin θ₃ and the link 601' is assigned with a coefficient sin θ_3ι where θ₃ substantially satisfies the following equation:

Θ₃ = θ₂ - θc.

The output signal 308 of the I channel and 308' of the Q channel of the first stage are assigned with a coefficient of value cos θ₃. The various coefficients may thus be determined, in the case where the coefficient a is equal to -1.210019, the coefficient b is equal to -0.723772 and the arc θz is equal to 3.923 degrees.

We obtain: θi = 45-3.923 = 41.077 degrees and hence: sin O₁ = 0.65707 and cos G₁ = 0.75383

Then we obtain: θ₂ = 48.923 degrees. and hence: sin Q₂ = 0.75383 and cos θ₂ = 0.65707; then b*sin θ₂ = 0.54560 et b*cos G₂ = 0.47557 And finally, we obtain θ₃ = 42.651 degrees and therefore sin θ₃ = 0.67753 and cos θ₃ = 0.73549

The embodiments of the present invention that are detailed in the above sections correspond to an implementation of a complex coupler for correcting a symmetry placed downstream of the first stage of the complex filter.

The following sections detail embodiments of the present invention in which a complex coupler for correcting asymmetry is introduced upstream of the first stage of the complex filter.

Figure 8 illustrates a modelling of a filter with two stages according to an embodiment of the present invention.

Numerous similarities with figures described previously are noted. The sections hereinbelow detail the essential differences with the previous figures. Thus, an asymmetry correction is applied by a complex correction coupler 800. Such a coupler is placed upstream of the first stage of the filter. It comprises a coupling link 801' between the output 303 of the I channel of the complex converter and the input 301 ' of the Q channel of the filter. It also comprises a coupling link 801 between the output 303' of the Q channel of the filter and the input 301 of the I channel of the filter.

Thus a coupling of the I and Q channels is effected before the first stage of the filter.

Each of these coupling links is advantageously assigned with a coefficient making it possible to correct the asymmetry of the transfer function of this filter.

Thus, one determines that the coupling link 801' is assigned with a coefficient of a value -a*sin θ_c and that the coupling link 801 is assigned with a coefficient of a value a*sin θ_c, when each of the respective coefficients of the feedback loops 305 and 305' has a value equal to a*cos Θ_G.

On the I channel, the adder 312 of the input of the filter receives and adds to the input signal 301 , the signal of the feedback loop 305 assigned with the coefficient a*cos θ_c, the output signal of the first stage of the I channel via the internal return link 311 assigned with the coefficient cos θ-i, as well as the signal of the link 403 assigned with the coefficient -sin θi and the signal of the link 801 of the complex correction coupler 800, assigned with the coefficient a*sin θ_c.

By symmetry, the Q channel is readily obtained. A modelling of an easier implementation of filter such as this may readily be obtained, in particular for an implementation as switched capacitances.

Thus, Figure 9 illustrates a modelling in which a value of the coefficient a satisfying the following equation has been determined: a*cos θ_c = -1

Such a modelling facilitates an implementation of such a filter. In this case a coefficient "d" is introduced between the two stages which compensates for this change.

Consequently, a modelling is obtained in which a complex coupler 800 comprises a coupling link 801' assigned with a coefficient of value tan θ_c for injecting the output signal 303 of the I channel at the input 301 ' of the Q channel. It comprises a coupling link 801 assigned with a coefficient of value - tan θ_cfor injecting the output signal 303' of the Q channel at the input 301 of the filter of the I channel. Taking a previous example of a filter whose frequency is centred on fs/8, to which there corresponds a vector in the z plane placed at 45 degrees, we obtain for a equal to -1.210019, b equal to -0.723772, and θz equal to 3.923 degrees, the following numerical values:

G₁ = 45-3.923= 41.077 degrees; Q₂ = 45+3.923= 48.923 degrees; θ_c = 2.349+3.923= 6.272 degrees. Then, therefore: sin G₁ = 0.65707 and cos G₁ = 0.75383; sin θ₂ = 0.75383 and cos θ₂ = 0.65707; sin θ_c= 0.10925 and cos θ_c = 0.994014; a*sin θ_c= -0.132193 and a*cos θ_c= -1.202776. Figure 10 illustrates an implementation diagram for such a filter according to an embodiment of the present invention.

Such an implementation is based on switched capacitances. It is noted that this diagram comprises capacitances with negative values. In a differential embodiment, such capacitances are connected on complementary signals and therefore have positive physical values. Not all the elements of this implementation diagram for a filter according to an embodiment of the present invention are detailed. They stem from the modelling represented in Figure 7 and detailed with reference to Figure 7.

An integrator with switched capacitances operates conventionally over a period comprising two phases driven by one of the control signals without overlap Φ1 , Φ2.

A first phase corresponds to a half period during which the signal Φ1 is active and the signal Φ2 is not active (switches 1 closed and switches 2 open). Next, a second phase corresponds to the other half period during which the signal Φ2 is active and the signal Φ1 is not active (switches 1 open and switches 2 closed). According to the layout of the switched-capacitance units, input signals charge capacitors in the course of one of the operating phases.

It is noted that four switch units 1005 operate on control signals Φ1', Φ2'. These units are placed at the input of each stage of the filter on each channel. They make it possible to take account of a time shift with the control signals Φ1 , Φ2.

The adders are implemented in the form of operational amplifiers 1001-1004.

Thus, for example, the adder 312 of the input of the filter on the I channel is implemented by the operational amplifier 1001. It receives the input signal 301 of the I channel, the signal of the link 403 comprising a capacitor 1006, of value Cd satisfying the equation: Cd = Ce1*sin(τr/4 - θz); where Ce1 is the capacitance of the input of the filter on the I channel and θz is equal to 2πf_z/f_s, where fz is a frequency corresponding to the zero of the transfer function of the prototype high-pass filter and fs is the frequency of sampling of the signal. The operational amplifier 1001 also receives the signal via the link 305, as well as the signal via the link 311 which comprises a capacitance Ci1 satisfying the following equation:

CM = Ce1/G; where G is the gain of the first stage. It is also noted that the complex correction coupler 600, placed between the first stage and the second stage, is implemented via the links 601 and 601' each comprising a switched-capacitance unit 1007 and a switch- capacitance unit 1008. The switched-capacitance unit 1007 is based on a capacitor Cx2 of value equal to Ci2*d*cos(π/4 +/- Θ_cor)/G. The switched- capacitance unit 1008 is based on a capacitor Cy2 of value equal to Ci2*d*sin(τr/4 +/- Θ_cor)/G. The sign associated with the arc θ_cor depends on the relative position of the frequency of the zero of each of the two stages.

The various capacitances satisfy the following equations for the frequency of the zero of the first stage lower than of the zero of the second stage:

CiI = CeIIG Ccl = Celsin(π/4~θz) Cdl = Ce\(\ -cos(πl 4 - θz))lG Cx2 = Ci2 * dcos(π/4 + θcor)/G Cy2 = Ci2 * dsin(π/4 + θcor)/G Cb2 = Ci2 *bcos(π/4 + θz) Cf2 = Ci2 *bsin(π/4 + θz) Cc2 = Ci2sin(π/4 + θz) Cd2 = Ci2(l - cos(π/4 + θz)) for the frequency of the zero of the first stage higher than that of the zero of the second stage:

CiI = CeIIG CcI = CeI sin (πl 4 + θz) Cdl = Cel(l - ∞s(π/4 + θz))lG Cx2 = Ci2 * dcos(π/4 - θcor)IG Cy2 = Ci2 * dsin(π/4 - θcor)/G Cb2 = Ci2 *b∞s(πlA - θz) Cf2 = Ci2 *bsm(π/4- θz) Cc2 = Ci2sin{πl4 - θz) Cd2 = C/2(l - cos (^r /4 - θz))

_ bsinθz θcor = -arctg - — r with % ₊ (& + 2)(l-co8fe) The coefficient d corresponds to the parameter "-a" of the NTF.

The coefficient b corresponds to the parameter "b" of the NTF.

The previous sections have described embodiments of the present invention in the case where a complex filter of an analogue digital complex converter comprises two stages.

The present invention covers the case where the complex filter of such a converter comprises more than two stages.

In an embodiment of the present invention, an asymmetry correction such as described in the previous sections is advantageously determined and applied to each pair of successive stages of the complex filter. Thus, the asymmetry of the transfer functions of each of the pairs of successive stages is corrected. Each pair is then considered to comprise a first and a second stage and the principles detailed in the previous sections may thus readily be extended to each of the pairs of stages. A correction may be applied either upstream or downstream of the first stage of each pair.

Preferably, an asymmetry correction is applied to each of the pairs of successive stages. However, the invention makes it possible to boost the performance of a converter even in the case where an asymmetry correction is applied only to some of the pairs of successive stages. In the case where the complex filter comprises an odd number of stages, no correction is applied to the stage which does not form part of a pair, such a stage not exhibiting an asymmetric transfer function since it is advantageously placed on the central frequency.

In such a case, the pairs of stages are formed from the second stage, the first stage therefore not forming part of a pair.

Figure 11 is a curve representing a shaping spectrum of the quantization noise in a converter in which no asymmetry correction according to an embodiment of the present invention is applied.

It is noted that a first shoulder 111 and a second shoulder 112 of the curve are not symmetric.

Figure 12 is a curve representing a shaping spectrum of the quantization noise in a converter in which an asymmetry correction is applied according to an embodiment of the present invention. It is noted that a first shoulder 121 and a second shoulder 122 of the curve are substantially symmetric.

The curves are obtained by simulation carried out under scilab of INRIA. The FFTs of 104000 points are smoothed by windows of 4 Blackman- Harris -92dB terms, having 1000 points each and 50% interleaved. The equivalent noise band equals 26.3 kHz.

An embodiment of the present invention therefore makes it possible to boost the performance of a complex analogue-digital converter based on a complex filter of order higher than or equal to two.

Claims

1. Method of analogue digital conversion using an analogue-digital converter having two quadrature channels I and Q each comprising an associated input (301 ,301') and an associated output (303,303'), for converting a complex analogue input signal into a complex digital output signal, said converter comprising a complex filter having at least one pair of stages with a first stage and a second stage, each comprising a respective input (301 , 301', 320, 320') and a respective output (308, 308', 309, 309') on each of the I and Q channels, a respective direct link linking the output of the first stage of the pair of stages to the respective input of the second stage of the pair of stages on the I channel, respectively on the Q channel; each of the outputs of the converter being coupled, on the one hand, to the respective input of the first stage of the pair of stages, assigned with a first feedback coefficient (a) via a first feedback loop (305,305'), and on the other hand, to the respective input of the second stage of the pair of stages, assigned with a second feedback coefficient (b) via a second feedback loop (304,304'); wherein a correction of asymmetry of the transfer function of the converter, corresponding to the pair of stages of the filter, is performed by coupling, per pair of stages, the I and Q channels upstream of the first stage of the pair of stages or between the first and second stages of the pair of stages.

2. Method according to Claim 1 , according to which the correction is performed by coupling the I and Q channels between the first and second stages of the pair of stages of the complex filter; said method comprising, for the pair of stages, the following steps;

- coupling the output (308') of the Q channel of the first stage to the output (308) of the I channel of the first stage via a coupling link (601 ) assigned with a first correction coefficient;

- coupling the output (308) of the I channel of the first stage to the output (308') of the Q channel of the first stage via a coupling link (601') assigned with a second correction coefficient;

3. Method according to Claim 2, wherein the first correction coefficient has a value equal to sin Θ_G; wherein the second correction coefficient has a value equal to -sin θ_c; and wherein the third correction coefficient has a value equal to cos θ_c; where θ_c substantially satisfies one of the following equations;

θ = arctg + θz for fzl < fzl a+ (h + 2)(1- cos θz) ^J θ_r - -arctg θz for fzl > fzl ^c a+ (b + I)(I- cosθz) ^J for θz equal to 2πf_z/f_S) where f_z is the frequency corresponding to the zero of a transfer function of the filter, fs is the frequency of sampling of the input signal, fzl corresponds to the frequency of the zero of the first stage of the pair of stages and fz2 corresponds to the frequency of the zero of the second stage of the pair of stages.

4. Method according to Claim 1 , wherein the correction is performed by coupling the I and Q channels upstream of the first stage of the pair of stages of the complex filter; said method comprising, for the pair of stages, the following steps ;

- coupling the output (303) of the I channel of the converter, on the one hand, at the input of the first stage of the I channel, via the first feedback loop (305) assigned with a first correction coefficient, and, on the other hand, at the input (301') of the first stage of the Q channel via a coupling link (801') assigned with a second correction coefficient;

- coupling the output (303') of the Q channel of the converter, on the one hand, at the input of the first stage of the Q channel via the first feedback loop (305¹) assigned with a third correction coefficient and, on the other hand, at the input (301 ) of the first stage of the I channel via a coupling link (801 ) assigned with a fourth correction coefficient.

5. Method according to Claim 4, wherein the first correction coefficient has a value equal to a^*cosθ_c; wherein the second correction coefficient has a value equal to -a*sinθ_c; wherein the third correction coefficient has a value equal to a*cosθ_c; and wherein the fourth correction coefficient has a value equal to a*sinθ_c; where θ_c substantially satisfies one of the following equations ; θ = arctg ^^ + θz for fzl < fz2 a+ (b + 2)(1- co&θz) ^J θ_r = -arctg θz for fzl > fzl a+ φ + 2)(1- oosθz) ^J for θz equal to 2πf_z/fs, where fz is the frequency corresponding to the zero of a transfer function of the filter, fs is the frequency of sampling of the input signal, fzl corresponds to the frequency of the zero of the first stage of the pair of stages and fz2 corresponds to the frequency of the zero of the second stage of the pair of stages.

6. Method according to any one of the preceding claims, wherein the complex filter comprises a number of stages greater than two successively forming one or more pairs of stages, each comprising successive first and second stages, for each of said pairs of stages, each output (303, 303') of the converter being coupled, on the one hand, to the respective input of the first stage assigned with a first feedback coefficient, and on the other hand, on the respective input of the second stage assigned with a second feedback coefficient; each of said pairs having a transfer function having an asymmetry; and wherein a correction of asymmetry of the transfer function or functions corresponding to each pair of stages of the complex filter is performed between the first and second stages of the pair or of the pairs of stages according to Claim 2 or 3 or upstream of the first respective stage of the pair or of the pairs of stages according to Claim 4 or 5.

7. Method according to Claim 6, wherein the number of stages of the complex filter is an odd number greater than or equal to three; and wherein the correction of asymmetry is performed on the pair or pairs of stages which follow the first stage of the converter.

8. Analogue-digital converter having two quadrature channels I and Q each comprising an associated input (301 ,301') and an associated output (303,303'), for converting a complex analogue input signal into a complex digital output signal, said converter comprising a complex filter (105) having at least one pair of stages with a first stage and a second stage, each comprising a respective input (301 , 301', 320, 320') and a respective output (308, 308', 309, 309') on each of the I and Q channels; each of the outputs of the converter being coupled, on the one hand, to the respective input of the first stage of the pair of stages, assigned with a first feedback coefficient (a) via a first feedback loop (305,305'), and on the other hand, on the respective input of the second stage of the pair of stages, assigned with a second feedback coefficient (b) via a second feedback loop (304, 304'); said analogue-digital converter comprising a complex coupler suitable for coupling the I and Q channels, per pair of stages, upstream of the first stage of the pair of stages or between the first and second stages of the pair of stages of the complex filter and thus for applying a correction of asymmetry of the transfer function of the converter corresponding to the pair of stages.

9. Converter according to Claim 8, wherein the complex coupler is placed between the first and second stages of the pair of stages of the complex filter and comprises for the pair of stages ;

- a first coupling link (601 ) linking the output (308') of the first stage of the Q channel on the output (308) of the first stage of the I channel, said first coupling link being assigned with a first correction coefficient;

- a second coupling link (601 ') linking the output of the first stage of the I channel on the output (308¹) of the first stage of the Q channel, said second coupling link being assigned with a second correction coefficient.

10. Converter according to Claim 9, wherein the first correction coefficient has a value equal to sin θ_c; and wherein the second correction coefficient has a value equal to -sin θ_c; the outputs (308, 308') of the first stage of the I and Q channels each being assigned with a coefficient of value cos θ_c; where θ_c substantially satisfies one of the following equations ; θ. - arctg + θz for fz\ < fz2 a+ (b + 2)(l- cosθz) ^J θ. = -arctg θz for fzl > fz2 a+ (b + 2)(l- cosθz) ^J for θz equal to 2πfz/fs, where f_z is the frequency corresponding to the zero of a transfer function of the filter, fs is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage and fz2 corresponds to the frequency of the zero of the second stage.

11. Converter according to Claim 9, comprising furthermore in the first stage of the pair of stages ;

. on the I channel, an internal return link (311 ) from the output (308) of the first stage of the I channel, said link comprising substantial delay

(306) and being assigned with a coefficient of value cos θ-i; . on the Q channel, an internal return link (311') from the output (308') of the first stage of the Q channel, said link comprising an appreciable delay (306¹) and being assigned with a coefficient of value cos θi;

- on the I channel, an adder (312) receiving ;

- the input of the I channel of the filter (301 );

- the first feedback loop (305) of the I channel, assigned with the first feedback coefficient; - said internal return link of the I channel (311 ); and

- a return link (403) from said internal return link (311 ') of the Q channel, assigned with a coefficient of value -sin θ-i; - on the Q channel, an adder (312') receiving ;

- the input of the Q channel of the filter (301 ');

- the first feedback loop (305') of the Q channel, assigned with the first feedback coefficient;

- said internal return link of the Q channel (311 '); and

- a return link (403') from said internal return link (311 ) of the I channel, assigned with a coefficient of value sin θ-i; where θi is equal to 2πf_z/fs, where fz is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal; in the second stage of the pair of stages:

- on the I channel, an adder or several adders (313, 701 ) receiving

- the input (320) of the I channel of the second stage;

- the second feedback loop (304) of the I channel assigned with a coefficient of value b*cos G₂; where b is the second feedback coefficient;

- a return link (503) from the feedback loop of the Q channel assigned with a coefficient of value equal to -b*sin θ₂;

- an internal return link (310) from the output (309) of the I channel of the second stage assigned with a coefficient of value cos θ₂;

- a return link (505) from the output (309¹) of the Q channel of the second stage assigned with a coefficient of value -sin θ₂; - on the Q channel, an adder or several adders (313', 701') receiving :

- the input (320^*) of the Q channel of the second stage;

- the second feedback loop (304') of the Q channel assigned with a coefficient of value b*cos θ₂; where b is the second feedback coefficient;

- a return link (503') from the feedback loop of the I channel assigned with a coefficient of value equal to b*sin θ₂;

- an internal return link (310') from the output (309¹) of the Q channel of the second stage assigned with a coefficient of value cos θ₂; - a return link (505') from the output (309) of the I channel of the second stage assigned with a coefficient of value sin θ₂; where θ₂ is equal to 2πf_z/f_s, where fz is a frequency corresponding to a zero of the transfer function of the filter and fs is the frequency of sampling of the signal; said converter comprising between the first stage and the second stage, the complex coupler (600) having the first correction coefficient of value equal to -sin Θ₃ and the second correction coefficient of value equal to sin θ₃; the outputs (308, 308') of the first stage of the I and Q channels each being assigned with a coefficient of value cos Θ₃; where θ₃ substantially satisfies the following equation ;

Θ3 = Θ₂ - Θ_G; where θ_c substantially satisfies one of the following equations ; _n bήnθz . . ,_Λ . _ θ = arctg + θz for fzl < fz2 a+ (b + 2)(l- ∞sθz) ^J θ = -arctg θz for fz\> fzl a+ (b + 2)(l- cosθz) ^J for θz equal to 2πfz/fs, where f_z is the frequency corresponding to the zero of a transfer function of the filter, % is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage of the pair of stages and fz2 corresponds to the frequency of the zero of the second stage of the pair of stages.

12. Converter according to Claim 8, wherein the complex coupler is placed upstream of the first stage of the pair of stages and comprises for the pair of stages ;

- a return loop (801 ) linking the output (303¹) of the Q channel of the converter to the input (301) of the first stage of the I channel assigned with a first correction coefficient;

- a return loop (801') linking the output (303) of the second stage of the I channel to the input (301') of the first stage of the Q channel assigned with a second correction coefficient; and wherein the first feedback loop (305) on the I channel is assigned with a third correction coefficient and on the Q channel is assigned with a fourth correction coefficient.

13. Converter according to Claim 12, wherein the first correction coefficient has a value equal to a*sinθ_c; wherein the second correction coefficient has a value equal to -a*sin θ_c; wherein the third and the fourth correction coefficients have a value equal to a*cos θ_c; where θ_c substantially satisfies the following equation : θ_c = arctg ^^ + θz for fz\ < fz2 a + (b + 2)(l- cosθz) θ_c = arctg ^?Ξ θz for fz\ > fz2 a + (b + 2)(l- cosθz) for θz equal to 2πf_z/fs, where f_z is the frequency corresponding to the zero of a transfer function of the filter, fs is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage of the pair of stages and fz2 corresponds to the frequency of the zero of the second stage of the pair of stages.

14. Converter according to Claim 12, in which the first correction coefficient has a value equal to -tan Θ_G; in which the second correction coefficient has a value equal to tan θ_c; in which the third and the fourth correction coefficients have a value equal to

-1 ; where θ_c substantially satisfies one of the following equations : θ_c = arctg ^bJE?l + Qz for fz\ < fz2 a + φ + 2)(l- cosθz) _n hsinθz . . ._n . _

6> = arctg θz for fz\ > fz2 α + (έ> + 2)(l- cos0z) for θz equal to 2πfz/fs, where fz is the frequency corresponding to the zero of a transfer function of the filter, fs is the frequency of sampling of the input signal, fz1 corresponds to the frequency of the zero of the first stage of the pair of stages and fz2 corresponds to the frequency of the zero of the second stage of the pair of stages.

15. Converter according to any one of the preceding claims, the complex filter comprising a number of stages greater than two and a respective coupler per pair of stages, each pair comprising successive first and second stages; wherein each of said couplers is placed upstream of the first stage of at least one pair of stages or between the first and second stages of at least one pair of stages and is assigned with a correction coefficient according to any one of Claims 8 to 14.

16. Converter according to Claim 15, wherein the number of stages of the complex filter is an odd number; and wherein the couplers are associated solely with the pairs of stages which follow the first stage of the converter.