WO2003063350A2

WO2003063350A2 - Digital signal filtering method, a frequency synthesiser and the corresponding computer programs

Info

Publication number: WO2003063350A2
Application number: PCT/FR2003/000180
Authority: WO
Inventors: Alain Bourdillon
Original assignee: Universite De Rennes 1
Priority date: 2002-01-22
Filing date: 2003-01-20
Publication date: 2003-07-31
Also published as: WO2003063350A3; FR2835071A1

Abstract

The invention relates to a digital signal filtering method, a frequency synthesiser and the corresponding computer programs. More specifically, the invention relates to a method of filtering an original digital signal which is represented by a sequence of samples si, wherein i varies between 1 and N and N is the number of samples to be processed, said original digital signal comprising a dominant sinusoidal component which superimposes clutter signals. The inventive method can be used to reduce the level of clutter and to obtain a filtered digital signal having a good spectral quality. According to the invention, the method comprises a step involving digital filtering based on a simplification of a matrix Σ that appears in the singular value decomposition (SVD) of the Hankel matrix of the original digital signal.

Description

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 sinusoidal digital signal, comprising a dominant sinusoidal component on which parasitic signals are superimposed.

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

It is known that, during direct digital synthesis (SND), signals

(lines) parasites are generated due to the different truncations performed on the number of bits used (phase accumulator, addressing of the sine table and width of the data bus), as well as by the non-integer values of the clock frequency ratio on frequency of the synthesized signal.

Known solutions for reducing the level of the parasitic signals consist either in limiting the use of the synthesizer to frequencies where the parasites have low levels, or in increasing the number of the 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 solution mentioned above 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: "Multi-standard quad DDC chip" rev 0.3, April 20, 2000. It is shown there that, in certain cases, the level of the parasitic lines can be reduced by 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 invention particularly aims to overcome these various drawbacks of the state of the art.

More specifically, 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 can be implemented in real time.

Another objective of the invention is to provide such a method making it possible to obtain an improvement greater than 14 dB, and which can range up to 60 dB, or even beyond.

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 s, 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 the parasitic signals and obtain a filtered digital signal having good spectral quality, the method comprising a digital filtering step based on a simplification of a matrix Σ appearing in the decomposition into singular values (SVD) of the Hankel matrix of the original digital signal.

It will be noted that it was not obvious to use the SND decomposition of the Hankel matrix of a signal, in the present context of filtering of a digital sinusoidal signal. Indeed, it is a known filtering method in the field of signal processing, to extract several sinusoidal components of a 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 real-time implementation is desired. 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 Σ 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 Σ obtained after simplification, ∑ _modified , makes it possible to obtain a filtered Hankel matrix, H _flUrée , the first and last lines of which (placed end to end constitute the sequence of N samples "sfiltered;"of the filtered signal) can be calculated quickly and easily.

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

0 0 0

0 σ 2 0 0 0 0 σ

Σ =

0 0 0 where the elements σ _; 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 Σ consists in canceling all the values of σ _; except the largest σ „corresponding to the dominant sinusoidal component of the original digital signal.

Thus, the matrix Σ obtained after simplification, ∑ _modified , is very simple with only one nonzero element (σ,).

Advantageously, the simplification of the matrix Σ also consists in setting σ _l = 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 v _k , k varying from 1 to N / 2, are the

N / 2 values of the translated autocorrelation function of the sample sequence s _; the original digital signal; calculation of scalars u _λ and u ₂ such that: N / 2

»1 = Σ V *

* = 1

JV / 2

* ^W 2 ^_ 2 ^V k ^S k + N / 2

Jt = l calculation of a sequence of samples sfiltré ;, i varying from 1 to N, representing the filtered digital signal, according to the following equations: * sfiltré _k = u, v _k k = l, 2, 3 ... N / 2

* sfιltré _{k + N / 2} = u ₂ v _k 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 Uj and u ₂ , 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 Σ appearing in the decomposition into singular values (SND) of the Hankel matrix of the original digital signal. Throughout this description (and as appears from the equation (e) (see below) of the components v _k of the vector v), the expression “translated autocorrelation function” means the autocorrelation function translated by one unit and such that the zero offset corresponds to k = l (instead of k = 0 in the classic case).

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

(JV / 2) + l ^v k = ∑ + * - ι k = 1, 2, 3 ... N / 2 where Si ^* 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, the method according to claim 4, 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 the 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 makes it possible to obtain better results in terms of spectral quality.

Preferably, the step of obtaining the vector v is carried out once and for all, for a given frequency of the original digital signal.

In other words, the vector v should only be recalculated if the frequency of the original digital signal changes.

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 the vector from the storage means 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 calculated previously.

The invention also relates to one comprising means allowing the implementation of the above-mentioned method.

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 _i5 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 allowing the level of the spurious 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 Σ appearing in the decomposition into singular values (SND) of the Hankel matrix of the original digital signal.

The invention also relates to a computer program product for filtering an original digital signal, represented by a sequence of samples s _i; i varying from 1 to

Ν, where Ν 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 having good quality spectral, said computer program product comprising program code instructions recorded on a medium usable in a computer comprising: computer-readable programming means for performing a step of obtaining a vector v, the components of which v _k , k varying from 1 to Ν / 2, are the N / 2 values of the translated autocorrelation function of the sequence of samples s _; the original digital signal; computer-readable programming means for performing a step of calculating scalars u _} and u ₂ such as:

N / 2

* ^w ι = ∑vΛ k = \

- computer-readable programming means for performing 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: * sfiltered _k = u, v _k k = 1, 2, 3 ... N / 2 * sfiltered _{k + N / 2} = u ₂ v _k k = 1, 2, 3 ... N / 2

(computer program product).

Other features and advantages of the invention will appear on reading the following description of a preferred embodiment of the invention, given by way of non-limiting example, and the accompanying drawings, in which: Figure 1 shows a simplified flowchart of a particular embodiment of the digital signal filtering method according to the invention; FIG. 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 flow diagram of FIG. 1, it is assumed in the following description that the original digital signal results from a synthesis step (1) making it possible 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 original digital signal is known.

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 step (3) of digital / analog conversion. 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 v _k , 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 Uj and u ₂ such as:

NI2 ^w ι = ∑ * (a)

4 = 1

NI 2

^U 2 = Σ ^V k ^S k ₊ N, 2 Φ) k = \ a step (23) of calculating a sequence of samples sfiltré ;, i varying from 1 to N, representing the filtered digital signal, according to the equations following:

* sfiltered _k = Uj v _k k = 1, 2, 3 ... N / 2 (c)

* sfiltered _{k + N / 2} = u ₂ v _k k = l, 2, 3 ... N / 2 (d)

The steps (22) of calculating scalars U [and u ₂ and (23) of calculating a sequence of samples filtered; 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 advantageous from the point of view of the 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 v _k , k varying from 1 to N / 2, according to the following equation:

(W / 2) + l ^v * = ∑ ^s ι ^s k-ι k = l, 2, 3 ... N / 2 (e)

1 = 1 where S; ^* is the conjugate complex of S ;.

In this general case, the calculation of the vector v requires (N ^2/4 + N / 2) complex multiplications and N ^2/4 complex additions. So, for N = 512 for example, it takes about 66 10 ³ multiplications and 66 10 ³ 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: v _k = exp £ j2πF (kl)] where 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 FIG. 2, the implementation 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 = l at N / 2, and the same circuit will be used for k = (N / 2) + l at N.

In the assembly presented in FIG. 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 Uj 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 S ;. The operation is performed for i varying from 1 to N / 2. For i = N / 2, the accumulator 5 contains the value u _x (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. 5. The value of U _j is then used to calculate the filtered signal "sfiltré _k ", for k = l at N / 2 (equation (c)), in the second multiplier 10. During this phase, the value u ₂ is calculated by the first multiplier 4 (equation (b). When the value of u ₂ is available, it is used in the second multiplier 10 to generate the next N / 2 elements of the filtered signal, according to equation (d).

As indicated 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 should not be recalculated unless the frequency of the synthesized signal changes. 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 which it is desired to filter.

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. Figure 3 illustrates the FFT spectrum of an example signal generated at the frequency

F = ll, 33 MHz, for a clock frequency F _H = 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 (fig.4), 2048 points (fig.5) and 4096 points (fig .6) respectively. It is observed that the level of the parasitic lines gradually decreases when the frequency moves away from that of the carrier. It is also observed that the signal becomes increasingly pure when the number of points processed increases. The degree of improvement of the spectrum depends slightly 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≈fsj s ₂ s ₃ s ₄ ... s _N }. The Hankel matrix of this signal is an nxp matrix, with p = N / 2 and n = (N / 2) + l = p + l, formed as follows:

This writing highlights the fact that we find the initial sequence by taking the ^1st line of H, followed by the last line.

The SND (decomposition into singular values) decomposition of this Hankel matrix of the signal is written:

H = UΣV ^H

where U and N are matrices n x n and p x p respectively, formed of singular vectors,

N ^H indicates the Hermitian transpose of V and Σ is a matrix nxp composed of the singular values classified in descending order: 0. 0 0 0. 0

0 0

Σ =

0 0

The σ values _; correspond to the powers of the components present in the signal. 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:

"filtered ~ U ^ ¹ modified ^

This filtering method, known in itself, gives good results. Its disadvantage is that it is heavy from the point of view of computation time.

In the context of the present invention, the matrix Σ is simplified by canceling all the values of σ _{ except the largest σ ,, corresponding to the dominant sinusoidal component of the original digital signal, and we set σj = 1.

Under these conditions the _modified matrix ∑ becomes simple with a single non-zero element equal to 1, the calculations are simplified and we obtain: "i.Vn

"llVpl

* _*

^M 21 ^V 11 ^M 21 ^V 21 "21 ^V „ 1

H filtered

^M "ι ^v π" Λ U „l ^V pl

To find the filtered signal, it suffices to calculate the first and the last line of Η _filtered . One notes on the preceding matrix that only the two components u _u and u _nl of the matrix U must be known and that only the first line v _π ^* , v ₂₁ ^* , ... v _pl ^* of the matrix N ^H must be known.

The components v _u , v ₂₁ , ... v _pl can be calculated using the following matrix product:

v = H ^H H {:, \) where H ^H is the Hermitian transpose of H and H (:, l) represents the first column of H.

This equation corresponds to the above-mentioned equation (e), used during step (21) of obtaining the vector v.

For reasons of speed of computation, it was necessary to find a trick to compute u _π and u _nl quickly, without having to compute the whole matrix U. After study, the inventor found that u _π and u _nl could be obtained by:

NI2 ^w π = ∑ ^V Λ k = \ NI2 k = l, 2,3 ... N / 2

"„ 1 = Σ ^V k ^S k + N / 2 k = \

Thus, u _u and u _nl correspond respectively to the scalars

and u ₂ above. These two equations correspond to equations (a) and (b) used during the step referenced 22 in FIG. 1. We now have all the elements to go to the filtering step (23) proper, which consists in calculating the first and last line of _filtered H to obtain the signal filtered by the equation:

s _β , t _r é = n ^v "> ^u „ ι ^v "} This equation corresponds to equations (c) and (d) used during the step referenced 23 in FIG. 1.

Claims

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

2. Method according to claim 1, characterized in that the matrix Σ is an n x p matrix written as: σ> 0 • 0 0

0 σ ₂ 0 . 0

0 0 σ,

Σ = σ.

0 0 . . 0 where the elements σ 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 Σ consists of canceling all the values of σ, except the largest σ corresponding to the dominant sinusoidal component of the original digital signal.

3. Method according to claim 2, characterized in that the simplification of the matrix Σ further consists of setting σ ₁ = l.

4. Method for filtering an original digital signal, represented by a sequence of samples s„ i varying from 1 to Ν, where Ν is the number of samples to be processed, the original digital signal comprising a component dominant sinusoidal superimposed on parasitic signals, said method making it possible to reduce the level of parasitic signals and obtain a filtered digital signal having good spectral quality, characterized in that the method comprises the following steps:

obtaining a vector v, whose components v _k , k varying from 1 to N/2, are the N/2 values of the auto-correlation function translated from the sequence of samples Si of the original digital signal ; calculation of scalars Uj and u ₂ such that:

NI2

* ^U ₂ = Σ V _W/2 k=\ calculation of a sequence of sfiltered samples,, i varying from 1 to N, representing the filtered digital signal, according to the following equations: * sfiltered _k = u, v _k k = 1, 2, 3 ... N/2 * sfiltered _k+N/2 = u ₂ v _k k = 1, 2, 3 ... N/2

5. Method according to claim 4, characterized in that the step of obtaining the vector v consists of calculating the components v _k , k varying from 1 to N/2, according to the following equation:

(Λf/2)+lv* = ∑ *A ₊ *-ι k = l, 2, 3 ... N/2 ι=l where s ^* is the conjugate complex of s,.

6. Method according to claim 4, the frequency of the original digital signal being known, characterized in that the step of obtaining the vector v consists of calculating N/2 theoretical values of the auto-correlation function translated from the sequence of samples s, 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 given frequency of 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 several times beforehand, 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 comprises a step of storing the set of vectors v, and in that, during the step of digital filtering of the original digital signal having a given frequency, obtaining the vector v consists of reading in the storage 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 any one of claims 1 to 8 when said program is executed on a computer.

11. Computer program product for filtering an original digital signal, represented by a sequence of samples S;, i varying from 1 to N, where N is the number of samples to be processed, the digital signal d origin comprising a dominant sinusoidal component superimposed on parasitic signals, said method making it possible to reduce the level of parasitic signals and obtain a filtered digital signal having 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 carrying out a digital filtering step based on a simplification of a matrix Σ appearing in the singular value decomposition (SND) of the Hankel matrix of the digital signal of origin.

12. Computer program product for filtering an original digital signal, represented by a sequence of samples s _i; i varying from 1 to Ν, where Ν 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 digital signal filtered having 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 carrying out a step of obtaining a vector v, whose components v _k , k varying from 1 to N/2, are the N/2 values of the translated auto-correlation function of the sample sequence S; of the original digital signal; computer-readable programming means for carrying out a step of calculating scalars u _t and u ₂ such that:

NI2

U, =Σ V*

4=1

computer-readable programming means for carrying out a step of calculating a sequence of sfiltered samples _i5 i varying from 1 to N, representing the filtered digital signal, according to the following equations:

* sfiltered _k = Ui v _k k = 1, 2, 3 ... N/2

* sfiltered _k+N/2 = u ₂ v _k k = 1, 2, 3 ... N/2