FR2835071A1 - METHOD FOR FILTERING A DIGITAL SIGNAL, FREQUENCY SYNTHESIZER AND CORRESPONDING COMPUTER PROGRAM - Google Patents

Abstract

The invention relates to a method of filtering an original digital signal, represented by a sequence of samples if, i varying from 1 to N, where N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on parasitic signals, said method making it possible to reduce the level of parasitic signals and to obtain a filtered digital signal having good spectral quality. According to the invention, the method comprises a digital filtering step based on a simplification of a matrix Σ appearing in the singular value decomposition (SVD) of the Hankel matrix of the original digital signal.

Description

of the dimension of variation of the opacity: r

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  Method for filtering a digital signal, frequency synthesizer and

  corresponding computer program.

  The field of the invention is that of filtering digital signals.

  The filtering method according to the invention has many applications. It can in particular, but not exclusively, be used in frequency synthesizers, used in electronic devices such as receivers (radiocommunication or other), signal generators, modulators, demodulators, mixers, etc. More generally, it can be applied in all devices or systems which

  use sinusoidal signals which must have good spectral purity.

  It is assumed that the original signal to be filtered is a digital sinusoidal signal, comprising a dominant sinusoidal component on which are superimposed

spurious signals.

  This original digital signal to be filtered is for example obtained with a direct digital frequency synthesis system (SND system). This SND system can optionally be combined with a digital / analog converter (DAC), if

  want to produce an analog signal.

  It is known that, during direct digital synthesis (SND), spurious signals (lines) are generated due to the different truncations carried out on the number of bits used (phase accumulator, addressing of the sine table and bus width data), as well as by non-integer values of the frequency ratio

  clock on frequency of the synthesized signal.

  Known solutions for reducing the level of parasitic signals consist either in limiting the use of the synthesizer to frequencies where the parasites have levels

  weak, or to increase the number of bits of the various elements used in the SND.

  The first solution mentioned above is unacceptable in applications where the frequency must

  be variable. The second aforementioned solution is expensive and limits the processing speed.

  According to yet another known filtering technique, a pseudo-random sequence is added to the signal phase (so-called "dither method"). This other known technique is presented for example in the document ("data sheet") of the company GRAYCHIP INC entitled: "Multistandard quad DDC chip" rev 0.3, April 20,

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  2000. It is shown there that, in certain cases, one can reduce the level of the parasitic lines

about 14 dB.

  However, the potential for improvement of this other known technique is intrinsically limited since it consists in distributing the energy of the parasites throughout the frequency band. The object of the invention is in particular to overcome these various drawbacks of

the state of the art.

  More precisely, one of the objectives of the present invention is to provide a method of filtering a digital signal which is simple to implement and inexpensive in number of operations, so as not to require computing times high

  and be able to be implemented in real time.

  Another object of the invention is to provide such a method making it possible to obtain

  an improvement of more than 14 dB, and up to 60 dB or even more.

  These various objectives, as well as others which will appear subsequently, are achieved according to the invention using a method of filtering an original digital signal, represented by a sequence of samples if, i varying from 1 to N. o N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on parasitic signals, said method making it possible to reduce the level of the parasitic signals and obtain a filtered digital signal exhibiting good spectral quality, the method comprising a digital filtering step based on a simplification of a matrix appearing in the decomposition into singular values (SVD) of the Hankel matrix of the signal

original digital.

  It will be noted that it was not obvious to use the SVD decomposition of the Hankel matrix of a signal, in the present context of filtering of a sinusoidal digital signal. Indeed, it is a known filtering method in the field of signal processing, to extract several compos antes sinus odales from noise. The major drawback of this method is that it is greedy from the point of view of the number of operations to be performed. It was therefore difficult to envisage its use when a bet

  in real time is desired.

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  The present invention is based on a simplification of the decomposition into singular values of the Hankel matrix of the signal, and more precisely on a simplification of a matrix appearing in this decomposition. This simplification is based on the fact that, in the present context, the original signal comprises, in addition to the noise, only a dominant sinusoidal component. As explained in detail below, the matrix l; obtained after simplification, Émodifée, makes it possible to obtain a filtered Hankel matrix, Hfi, rée, whose first and last lines (which put end to end constitute the sequence of N

  "filtered" samples filtered signal) can be calculated quickly and easily.

  Advantageously, the matrix I: is a matrix n x p written:

, O. O O

0 â2 0. O

É- O 0 â3 À

_ ..

O O .. O

  o the elements 6j are singular values classified in descending order and correspond to the powers of the components present in the original digital signal. The simplification of the matrix consists in canceling all the values of c ,; except the largest a "corresponding to the dominant sinusodal component of the signal

original digital.

  Thus, the matrix obtained after simplification, Emodified, is very simple with a

only non-zero element (a,).

  Advantageously, the simplification of the matrix also consists in posing a, =

  1. This is a simple question of standardization.

  In a preferred embodiment of the invention, the method comprises the following steps: - obtaining a vector v, whose components Vk, k varying from 1 to N / 2, are the N / 2 values of the function d 'translated autocorrelation of the sample sequence s; the original digital signal; - calculation of scalars u, and u: such that:

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  N / 2 * U1 = Évksk k = 1 N12 * U2 = ÉVkSk + NI2 k = 1 - calculation of a sequence of samples sfiltréj, i varying from 1 to N. representing the filtered digital signal, according to the following equations: * sfiltrék = u vk k = 1, 2, 3 N / 2 * sfiltrék + N, 2 = u2 vk k = 1, 2, 3 N / 2 In this way, the digital filtering applied to the original digital signal is

  fast and inexpensive in computing power.

  The scalar calculation equations u, and u2, as well as the calculation equations for the N samples of the filtered signal, result from a simplification, as discussed above and presented in detail below, of a matrix appearing in the decomposition into singular values (SVD) of the Hankel matrix of the signal

original digital.

  In the whole of this description (and as it appears in the light of

  equation (e) (see below) of the components vk of the vector v), the expression “translated autocorrelation function” means the translated autocorrelation function of a unit and

  such that the zero offset corresponds to k = 1 (instead of k = 0 in the classic case).

  Advantageously, the step of obtaining the vector v consists in calculating the components Vk, k varying from 1 to N / 2, according to the following equation:

(N12) 1

  Vk = SiS,; k-1 k = 1, 2, 3 ... N / 2 i = l

  o if * is the conjugate complex of s ;.

  This calculation can be applied in all cases, whether or not the frequency of the original digital signal is known, since it is carried out from the samples

of the original digital signal.

  According to an advantageous variant, in which the frequency of the original digital signal is known, the step of obtaining the vector v consists in calculating N / 2 theoretical values of the translated autocorrelation function of the sequence of samples s; of

original digital signal.

  Thus, this variant applies when the frequency of the original digital signal is known. This is particularly the case when the original digital signal results from a direct digital frequency synthesis (SND). This variant allows

  to obtain better results in terms of spectral quality.

  Preferably, the step of obtaining the vector v is carried out once

  for all, for a given frequency of the original digital signal.

  In other words, the vector v should only be recalculated in the event of a change in

  frequency of the original digital signal.

  In a particular embodiment of the invention, the step of obtaining the vector v is carried out beforehand several times, for different possible frequencies of the original digital signal, so as to have a set of vectors v each associated with a separate frequency. The method comprises a step of storing the set of vectors v, and, during the step of digital filtering of the original digital signal having a given frequency, obtaining the vector v consists in reading from

  the means for storing the vector v associated with said given frequency.

  This variant is particularly interesting if the frequency changes are frequent (for example synthesis of a "chirp" signal). Indeed, it saves time during the filtering of the original digital signal, the vectors v having been

previously calculated.

  The invention also relates to a frequency synthesizer comprising

  means for implementing the above process.

  The invention also relates to a computer program, comprising sequences of instructions suitable for implementing a method as mentioned above.

  when said program is executed on a computer.

  The invention also relates to a computer program product for filtering an original digital signal, represented by a sequence of samples s ;, i varying from 1 to N. o N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on parasitic signals, said method

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  allowing the level of the parasitic signals to be reduced and a filtered digital signal having good spectral quality to be obtained, said computer program product comprising program code instructions recorded on a medium usable in a computer comprising computer-readable programming means for perform a digital filtering step based on a simplification of a matrix appearing in the decomposition into singular values (SVD) of the Hankel matrix of the signal

original digital.

  The invention also relates to a computer program product for filtering an original digital signal, represented by a sequence of samples if, i varying from 1 to N. o N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on spurious signals, said method making it possible to reduce the level of spurious signals and obtaining a filtered digital signal having good spectral quality, said computer program product comprising code code instructions program recorded on a support usable in a computer comprising: - computer-readable programming means for performing a step of obtaining a vector v, the components Vk, k varying from 1 to N / 2, are the N / 2 values of the translated autocorrelation function of the sample sequence if of the original digital signal; - computer-readable programming means for performing a step of calculating scalars u, and U2 such that: N / 2 * U = EVkSk k =] N / 2 * U2 = iVkSk + N / 2 k = l - means computer-readable programming steps to perform a step of calculating a sequence of sfiltered samples ;, i varying from 1 to N. representing the filtered digital signal, according to the following equations: * sfilteredk = u, vk k = 1, 2 , 3 N / 2 * sfiltrék + N, 2 = u2 vk k = 1, 2, 3 ... N / 2

(computer program product).

  Other characteristics and advantages of the invention will appear on reading

  the following description of a preferred embodiment of the invention, given as

  of indicative and nonlimiting example, and of the appended drawings, in which: - Figure 1 presents a simplified flowchart of a particular embodiment of the digital signal filtering method according to the invention; - Figure 2 is a block diagram illustrating the implementation of a particular embodiment of the filtering method according to the invention; - Figure 3 shows the spectrum of an example of original digital signal, generated by direct digital synthesis; - Figures 4 to 6 each show the spectrum of the original digital signal of Figure 3, after filtering according to a particular embodiment of the present invention, by blocks of 512 points (fig. 4), 2048

  points (fig. 5) and 4096 points (fig. 6) respectively.

  The invention therefore relates to a method of filtering an original digital signal. As illustrated in the simplified flowchart of Figure 1, we assume in the

  continuation of the description that the original digital signal results from a step (1) of

  synthesis allowing to obtain an original digital signal (two channels I and Q quadrature), represented by a sequence of samples s ;, i varying from 1 to N. where N is the

number of samples to be processed.

  This synthesis step (1) consists for example in the implementation of a direct digital frequency synthesis (SND). The SND technique is well known to those skilled in the art and will not be described in more detail below. It will be noted that with this technique, the frequency of the synthesized signal is known. It is clear, however, that the present invention applies whether or not the frequency of the signal is known.

original digital.

  Optionally, if it is desired to obtain an analog signal having high spectral purity, the filtering according to the method of the invention (2) is followed by a

  digital / analog conversion step (3).

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  In the particular embodiment illustrated in FIG. 1, the digital filtering (2) according to the method of the invention comprises: - a step (21) of obtaining a vector v, whose components Vk, k varying from 1 to N / 2, are the N / 2 values of the translated autocorrelation function of the sequence of samples s; the original digital signal; - a step (22) of calculating scalars u, and U2 such that: N / 2 * lll = EVkSk (a) k = l N / 2 2 VkSk + N / 2 (b) k = l - a step (23 ) calculating a sequence of samples sfiltréi, i varying from 1 to N. representing the filtered digital signal, according to the following equations: * sfiltrék = u, vk k = 1, 2, 3 ... N / 2 ( c) * sfiltrck + N, 2 = u2 vk k = 1, 2, 3 ... N / 2 (d) The steps (22) of calculating scalars u, and u2 and (23) of calculating a sequence filtered samples; require 2N complex multiplications and N-2 complex additions. Thus, to treat N = 512 points for example, it takes about 1024 multiplications and 510 additions. The processing time of a sequence of N points depends linearly on N and the method according to the invention is therefore particularly

  interesting from the point of view of computation time.

  The step (21) of obtaining the vector v can be carried out in two different ways. In the general case, applicable in particular if the frequency of the synthesized signal is not known (and a fortiori if it is known), step (21) of obtaining the vector v consists in calculating the components Vk, k varying from 1 to N / 2, according to the following equation:

(N / 2) +1

  Vk = SiS, + k-1 k = 1, 2, 3 ... N / 2 (e) i =]

  o sj * is the conjugate complex of s ;.

  In this general case, the calculation of the vector v requires (N2 / 4 + N / 2) complex multiplications and N2 / 4 complex additions. Thus, for N = 512 for example, it takes about 66,103 multiplications and 66,103 additions. As shown below, this calculation

  can be done once and for all.

  A variant can be envisaged in the particular case where the frequency of the synthesized signal is known. This is for example the case if the synthesis step (1) consists in the implementation of a direct digital frequency synthesis (SND). According to this variant, step (21) of obtaining the vector v consists in calculating N / 2 theoretical values of the translated autocorrelation function of the sequence of samples s; of the original digital signal. We then obtain better results in terms of spectral quality. Recall that the translated auto-correlation function of the synthesized signal, which is a sinusoidal signal, is given by: vk = exp [j2F (k-l)] o F is the frequency of the synthesized signal

  and k varying from 1 to N / 2, the N / 2 sampling instants.

  We now present, in relation to the block diagram of Figure 2, the implementation

  _work of a particular embodiment of the filtering method according to the invention.

  We observe that equations (a) and (b) are calculated in the same way, and that it is the same for equations (c) and (d). From an implementation point of view, it is therefore sufficient to design a circuit calculating the filtered signal for k = 1 to N / 2, and the same circuit will be used for k = (N / 2) +1 to N. In the assembly presented in figure 2, it is assumed that a counter 8 is clocked by a clock H and has a capacity N / 2. The outputs of the counter 8 make it possible to address a RAM memory 9 which contains the N / 2 components of the vector v, which are assumed to have been calculated beforehand (see detailed discussion below). The number u, is calculated by the association of a first multiplier 4 and an accumulator 5. More precisely, at time i, the value v; is presented at the input of the first multiplier 4 and it is multiplied by if. The operation is performed for i

  varying from 1 to N / 2. For i = N / 2, the accumulator 5 contains the value u (equation (a)).

  This value is stored in a memory 6 by the output signal from the decoder 7, when the counter 8 has reached its maximum value. The output signal from the decoder 7, delayed by an element referenced 11, then causes the accumulator to be reset to zero.

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  5. The value of u, is then used to calculate the filtered signal "sfiltrck", for k = 1 to N / 2 (equation (c)), in the second multiplier 10. During this phase, the value u2 is calculated by the first multiplier 4 (equation (b). When the value of u2 is available, it is used in the second multiplier 10 to generate the following N / 2 elements of the filtered signal, according to equation (d). As indicated below above, the N / 2 components of the vector v can be calculated beforehand, once and for all, for a given frequency of the original digital signal. In this case, the vector v is stored and it must only be recalculated

  frequency change of the synthesized signal.

  If the frequency changes are frequent (for example synthesis of a "chirp" signal), one can envisage a variant, to save time, consisting in calculating beforehand several vectors v, each associated with a particular possible frequency of the synthesized signal , and to store this set of vectors v in a memory (RAM, ROM, EPROM, ...). Thus, during digital filtering (2), obtaining the vector v consists in reading in this memory the vector v, among the plurality of stored vectors,

  associated with the frequency of the original digital signal to be filtered.

  The performances of the method according to the invention, in the embodiment presented above in relation to FIGS. 1 and 2, were evaluated using a

  program simulating an SND signal generator.

  FIG. 3 illustrates the spectrum P1 l of an example of signal generated at the frequency F = 11.33 MHz, for a clock frequency FH = 64 MHz. The analyzed sequence has 16,348 points and is apodized by a Kaiser window. The strongest parasitic line is at -60 dBc and the average noise level is between -70 and -80 dBc. Figures 4 to 6 each show the spectrum of the original digital signal of Figure 3, after filtering according to the invention, in blocks of 512 points (fg.4) '2048 points (fig. 5) and 4096 points (fig . 6) respectively. We observe that the level of the lines

  parasites gradually decreases when the frequency moves away from that of the carrier.

  We also observe that the signal becomes more and more pure when the number of

treated points increases.

  The degree of spectrum improvement slightly depends on the frequency chosen, but the simulations show that the processing on 2048 points makes it possible to obtain a

  spectral purity close to 100-110 dB whatever the frequency synthesized.

  The intellectual journey of the inventor is presented below to arrive at the above equations. We assume that the signal includes N values s = {s, S2 S3 S4 ... SN}. The Hankel matrix of this signal is an n x p matrix, with p = N / 2 and n = (N / 2) + 1 = p + 1, formed as follows: s, s2 À To sp

S2 S3. AT

H = S3

  Sn À À À SN This writing highlights the fact that we find the initial sequence in

  taking the left line of H. followed by the last line.

  The SVD decomposition (singular value decomposition) of this Hankel matrix of the signal is written:

H = VI; VH

  where U and V are matrices n x n and p x p respectively, formed of singular vectors, VH indicates the Hermitian transpose of V and is a matrix n x p composed of the singular values classified in descending order:

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cy, O. O O

O (T2 0. O

O 0 3 =

TO TO P

O O .. O

  Ul l U12 À à À Uln Vl I V12 .. Vlp

U21 U22 À à À à V21 V22 ...

U =. V =.

  . U ',, Un2 À à À Unn _ V plV p2. V pp The values 6j correspond to the powers of the components present in the

  Ssignal. The lowest values correspond to noise.

  The general principle of this type of filtering consists in putting the values which correspond to noise to zero, in order to eliminate it, then in performing the product of the matrices to obtain a filtered Hankel matrix: Hfi "= U dii V This method filtering, known in itself, gives good results.

  drawback is that it is heavy from the point of view of computation time.

  In the context of the present invention, the matrix is simplified by canceling all the values of 6; except the largest 6 "corresponding to the component

  1 Dominant sine wave of the original digital signal, and we set 6, = 1.

  Under these conditions the Emodified matrix becomes simple with a single non-zero element equal to 1, the calculations are simplified and we obtain:

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* * u, lv "u" v2 ,. . u "vp,

* * *

U2IVIl u2, v2, AT â21VPI

Hfltrée = .. ..

  . .. UnlVII UnlV2. . UnlVp To find the filtered signal, it suffices to calculate the first and the last line of Hfi, rée. One notes on the preceding matrix that only the two components u "and one of the matrix U must be known and that only the first line v" *, v2, *, ... vp, * of

the VH matrix must be known.

  The components v ,, v2 "... vp can be calculated using the following matrix product: v = HHH (:, 1) where HH is the Hermitian transpose of H and H (:, 1) represents the first column of H. This equation corresponds to the above equation (e), used during step (21)

obtaining the vector v.

  For reasons of speed of computation, it was necessary to find a trick to compute u, and one quickly, without having to compute the whole matrix U. After study, the inventor found that u, and Un could be obtained by: N12 Uri = ÉVkSk k = 1 N12 k = 1,2,3 N / 2 Unl = ÉVkSk + N12 k = l Thus, u "and un correspond respectively to the scalars u, and u2 above. These two equations correspond to equations (a) and (b) used during the step referenced

22 in Figure 1.

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  We now have all the elements to go to the filtering step (23) proper, which consists in calculating the first and last Hams line to obtain the signal filtered by the equation: Sf / tré {V UniV} This equation corresponds to equations (c) and (d) used during the step

S referenced 23 in FIG. 1.

Claims (10)

  1. A method of filtering an original digital signal, represented by a sequence of samples s ;, i varying from 1 to N. o N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on spurious signals, said method making it possible to reduce the level of spurious signals and obtaining a filtered digital signal exhibiting good spectral quality, characterized in that the method comprises a digital filtering step based on a simplification of a matrix l; appearing in the decomposition into singular values   (SVD) of the Hankel matrix of the original digital signal.   2. Method according to claim 1, characterized in that the matrix is an n x p matrix written: a] O. O O O (T2 0. O É- O 0 â3 À _ .. O O .. O   o the elements c '; are singular values classified in descending order and correspond to the powers of the components present in the original digital signal, and in that the simplification of the matrix consists in canceling all the values of s; except the largest a "corresponding to the dominant sine component of the signal original digital.   3. Method according to claim 2, characterized in that the simplification of the   matrix l; further consists in posing a, = 1.   4. Method for filtering an original digital signal, represented by a sequence of samples s ;, i varying from 1 to N. o N is the number of samples to be processed, the original digital signal comprising a dominant sinusoidal component superimposed on spurious signals, said method making it possible to reduce the level of spurious signals and obtaining a filtered digital signal having good spectral quality, characterized in that the method comprises the following steps: 16 2835071   obtaining a vector v, whose components Vk, k varying from 1 to N / 2, are the N / 2 values of the translated autocorrelation function of the sequence of samples s; the original digital signal; computation of scalars u, and U2 such that: N / 2 * U1 = EVkSk k = 1 N / 2 * U2 = 2.iVkSk + N, 2 k = l - calculation of a sequence of sfiltrated samples ;, i variant from 1 to N. representing the filtered digital signal, according to the following equations: * sfltrék = u1 vk k = 1, 2, 3 N / 2 sfiltrék + N, 2 = u2 vk k = 1, 2, 3 N / 2 5 Method according to claim 4, characterized in that the step of obtaining the vector v consists in calculating the components Vk, k varying from 1 to N / 2, according to the following equation: (N / 2) +1 Vk SiS.; K-1 k = 1, 2, 3 ... N / 2 i = 1   1S o sj * is the conjugate complex of s ;.   6. Method according to claim 4, the frequency of the original digital sig n being known, characterized in that the step of obtaining the vector v consists in calculating N / 2 theoretical values of the auto function - translated sequence correlation   samples; of the original digital signal.   7. Method according to any one of claims 4 to 6, characterized in that   the step of obtaining the vector v is carried out once and for all, for a frequency   given the original digital signal.   8. Method according to any one of claims 4 to 7, characterized in that   the step of obtaining the vector v is carried out beforehand several times, for different possible frequencies of the original digital signal, so as to have a set of vectors v each associated with a distinct frequency, in that the method includes a step of storing the set of vectors v, 17 2835071   and in that, during the digital filtering step of the original digital signal having a given frequency, obtaining the vector v consists in reading from the means   the vector v associated with said given frequency.   9. Frequency synthesizer, characterized in that it comprises means   allowing the implementation of the method according to any one of claims 1 to   8. 10. Computer program, characterized in that said program comprises sequences of instructions adapted to the implementation of a method according to one   any of claims 1 to 8 when said program is executed on a   computer. 11. Product computer program for filtering an original digital signal, represented by a sequence of samples s ;, i varying from 1 to N. o N is the number of samples to be processed, the digital signal d origin comprising a dominant sinusoidal component superimposed on spurious signals, said method making it possible to reduce the level of spurious signals and obtaining a filtered digital signal exhibiting good spectral quality, said computer program product comprising program code instructions recorded on a medium usable in a computer comprising computer-readable programming means for performing a digital filtering step based on a simplification of a matrix l; appearing in the decomposition into singular values (SVD) of the Hankel matrix of the signal original digital.   12. Product computer program for filtering an original digital signal, represented by a sequence of samples s ;, i varying from 1 to N. o N is the number of samples to be processed, the digital signal d origin comprising a dominant sine component superimposed on spurious signals, said method making it possible to reduce the level of spurious signals and obtaining a filtered digital signal having good spectral quality, said computer program product comprising program code instructions recorded on a support usable in a computer comprising: 18 2835071   computer-readable programming means for performing a step of obtaining a vector v, whose components Vk, k varying from 1 to N / 2, are the N / 2 values of the translated autocorrelation function of the sequence of samples if of the original digital signal; - computer-readable programming means for performing a step of calculating scalars u, and U2 such as: N / 2 * ui = EVkSk k = l N / 2 2 VkSk + N / 2 k = l - readable programming means by computer to perform a step of calculating a sequence of samples sfiltré ;, i varying from 1 to N. representing the filtered digital signal, according to the following equations: * sfiltrék = u, vk k = 1, 2, 3. .. N / 2