METHOD FOR BANDPASS SAMPLING BY POSITION MODULATED WAVELETS

Abstract

The present invention relates to a wavelet bandpass sampling method, with low aliasing and a corresponding device. The analogue signal to sample is correlated with a sequence of wavelets succeeding each other with a rate f.sub.p of which the positions in the sequence are temporally modulated from arguments of a CAZAC sequence, notably a Zadoff-Chu sequence. The correlation results are next sampled at a frequency f.sub.sf.sub.p and digitally converted to provide a compressed representation of the signal. The temporal modulation of the positions of the wavelets makes it possible to obtain incoherent aliasing of the correlation signal in the sampling band and thereby to reduce aliasing.

Claims

1. Method for bandpass sampling of an analogue signal, the spectrum of said signal having a set of sub-bands, said signal being correlated with a sequence of wavelets succeeding each other with a rate f.sub.p equal to the sum of the widths of said sub-bands, the central frequencies of the wavelets belonging to said sub-bands, characterised in that the respective positions of the wavelets in the sequence are temporally modulated by means of arguments of successive elements of a CAZAC sequence, the correlation results with the sequence of wavelets thereby position modulated being sampled with a frequency f.sub.sf.sub.p and digitally converted.

2. Bandpass sampling method according to claim 1, characterised in that the sequence of wavelets is formed of a plurality of base sequences, each base sequence being of duration T.sub.p with T.sub.p=1/f.sub.p, the CAZAC sequence being of length , the respective positions of the wavelets in a base sequence being temporally modulated by means of arguments of successive elements of the CAZAC sequence.

3. Bandpass sampling method according to claim 2, characterised in that the CAZAC sequence is formed by the complex elements .sub.k,k=1, . . . , where the temporal positions of the wavelets of the sequence are the instants kT.sub.p+.sub.k with T.sub.p=1/f.sub.p and ${\mathrm{.Math.}}_k=\frac{\arg \left({\mathrm{.Math.}}_k\right)}{2.Math..Math..Math.j_c^v},$ where f.sub.c.sup.v is the central frequency of the wavelets.

4. Bandpass sampling method according to claim 3, characterised in that the CAZAC sequence is a Zadoff-Chu sequence of which the elements are defined by: ZC.sub.R(k)=e.sup.jRk(k1)/, if the length of the sequence is odd, and ZC.sub.R(k)=e.sup.jR(k1)(K1)/, if the length of the sequence is even and where R, are positive integers that are coprime, R representing the excursion of the instantaneous frequency, expressed as a multiple of f.sub.p, along the sequence and k[1,].

5. Bandpass sampling method according to claim 4, characterised in that the sequence of wavelets is formed of at least a first and a second base sequence, the first base sequence being constituted of wavelets of which the positions are temporally modulated by means of arguments of successive elements of a first CAZAC sequence and the second base sequence being constituted of wavelets of which the positions are temporally modulated by means of arguments of successive elements of a second CAZAC sequence, the first and second CAZAC sequences being of same length but of distinct excursions of instantaneous frequency.

6. Bandpass sampling method according to claim 4, characterised in that the spectrum of the analogue signal includes a first and a second sub-band, and that the sequence of wavelets is formed of at least a first and a second base sequence, the first being constituted of wavelets of which the central frequency belongs to the first sub-band and the second being constituted of wavelets of which the central frequency belongs to the second sub-band.

7. Bandpass sampling method according to claim 1, characterised in that the wavelets are Gabor functions.

8. Bandpass sampling method according to claim 1, characterised in that the wavelets are Morlet wavelets.

9. Device for bandpass sampling of an analogue signal by bandpass sampling, the spectrum of said signal having a set of sub-bands, the device including at least one sampling branch comprising: a wavelet sequence generator, the wavelets of said sequence succeeding each other with a rate f.sub.p equal to the sum of the widths of said sub-bands; a correlator for correlating the analogue signal with said sequence of wavelets and supplying correlation results at the rate f.sub.p; a sampler for sampling the correlation results at the rate f.sub.p; an analogue digital converter for decimating and digitally converting the correlation results thereby sampled; characterised in that it further includes: a temporal position modulator intended to modulate the temporal positions of the wavelets of the sequence of wavelets generated by the generator, before correlation with the analogue signal, the temporal modulation being carried out by means of arguments of successive elements of a CAZAC sequence.

10. Bandpass sampling device according to claim 9, characterised in that the CAZAC sequence is a Zadoff-Chu sequence of which the elements are defined by: ZC.sub.R(k)=e.sup.jRk(k1)/, if the length of the sequence is odd, and ZC.sub.R(k)=e.sup.jR(k1)(K1)/, if the length of the sequence is even and where R, are positive integers that are coprime, R representing the excursion of the instantaneous frequency, expressed as a multiple of f.sub.p, along the sequence and k[1,].

11. Bandpass sampling device according to claim 10, characterised in that the wavelet sequence generator includes a VCO oscillator switched by a switching signal, the VCO oscillator including: an oscillating circuit, the frequency of which is controlled to be equal to the central frequency of the wavelets of the sequence; a pair of crossed transistors, mounted between the terminals of the oscillating circuit; a common current source, the current source being switched by the switching signal; a clock generator, for generating a clock signal at the frequency f.sub.p, a frequency modulator, for linearly modulating the frequency of the clock signal in such a way that it varies linearly from the start to the end of the sequence of wavelets with an excursion of Rf.sub.p.

12. Bandpass sampling device according to claim 10, characterised in that the wavelet sequence generator further includes a pulse shaping module, transforming each pulse of the clock signal into a pulse having the waveform of a wavelet.

13. Bandpass sampling device according to claim 9, characterised in that the wavelets are Gabor functions.

14. Bandpass sampling device according to claim 9, characterised in that the wavelets are Morlet wavelets.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Other characteristics and advantages of the invention will become clear on reading a preferential embodiment of the invention, described with reference to the appended figures, among which:

[0040] FIG. 1, already described, schematically represents a sampler using a WBS sampling method known from the prior art;

[0041] FIG. 2represents the spectrum of the sequence of wavelets used in the sampler of FIG. 1 in the case of uniform sampling (UWBS);

[0042] FIG. 3Aand

[0043] FIG. 3Brepresent the evolution of the instantaneous phase, at each start of pulse, as well as the instantaneous frequency along a Zadoff-Chu sequence;

[0044] FIG. 4Aand

[0045] FIG. 4Brepresent respectively phase components and frequency components of a sequence of wavelets temporal position modulated by a Zadoff-Chu sequence;

[0046] FIG. 5represents the spectrum of a sequence of wavelets temporal position modulated by a Zadoff-Chu sequence;

[0047] FIG. 6schematically represents an example of sensing matrix used within the scope of a WBS sampling method according to the present invention;

[0048] FIG. 7schematically represents a WBS sampler according to a first embodiment of the present invention;

[0049] FIG. 8schematically represents a WBS sampler according to a second embodiment of the present invention;

[0050] FIG. 9represents an example of sequence of wavelets position modulated by a CAZAC sequence;

[0051] FIG. 10schematically represents a wavelet sequence generator PPM modulated by means of a Zadoff-Chu sequence for the AWBS sampler of FIG. 7 or FIG. 8;

[0052] FIG. 11represents signals at different points of the generator of FIG. 10.

DETAILED DESCRIPTION OF PARTICULAR EMBODIMENTS

[0053] A WBS (wavelet bandpass sampling) sampler, of uniform or non-uniform type, will be considered hereafter. It will be recalled that the wavelets used by this sampler are elementary functions of which the temporal support is bound and of which the central frequency is adjustable, these functions forming a base or even an overcomplete set of L.sup.2(). We will also assume, with the aim of illustration and without loss of generality, that the wavelets in question are Gabor functions, such as defined above. Alternatively, they could be for example Morlet wavelets or even Haar wavelets. The input signal is correlated with a sequence of wavelets succeeding each other at the rate f.sub.p and the correlation results thereby obtained for the different wavelets are sampled at a sampling frequency f.sub.s which may be equal to f.sub.p (uniform sampling) or instead be less than it (non-uniform sampling).

[0054] The basic idea of the present invention is to carry out a modulation of the temporal positions of the wavelets, also called PPM (Pulse Position Modulation) by means of a modulation sequence derived from a CAZAC (Constant Amplitude Zero Auto Correlation) sequence so as to break the phase coherence of the spectral rays of the signal formed by the sequence of wavelets, before aliasing of the spectrum due to sampling.

[0055] It will be recalled that a CAZAC sequence is characterised by the fact that all its elements are of same module and that its cyclic autocorrelation is a zero centred Dirac. In other words, the cyclic correlation of a CAZAC sequence with this same shifted sequence is zero except when this shift is itself zero in which case the result of the correlation is equal to 1.

[0056] More precisely, if one considers a given central frequency f.sub.v.sup.c and a CAZAC sequence .sub.1, . . . , .sub.(where .sub.1, . . . , .sub.are complex values) the sequence of PPM modulation, .sub.1, . . . , .sub., is obtained by means of:

[00009]$\begin{array}{cc}{\mathrm{.Math.}}_k=\frac{1}{2.Math..Math..Math..Math.f_v^c}.Math.\arg \left({}_k\right),k=1,\mathrm{.Math.}.Math.,.Math.& \left(8\right)\end{array}$

[0057] If one considers now a sequence of wavelets

[00010]$p\left(t\right)=\underset{k=1}{\overset{.Math.}{.Math.}}.Math.{}_{f_v^c,{}_k}\left(t\right),$

the sequence obtained after PPM modulation may be expressed by:

[00011]$\begin{array}{cc}\tilde{p}\left(t\right)=\underset{k=1}{\overset{.Math.}{.Math.}}.Math.{}_{f_v^c,{}_k}\left(t-{\mathrm{.Math.}}_k\right).Math..Math.\mathrm{where}.Math..Math.{}_{f_v^c,{}_k}\left(t\right)={\left(\frac{2}{}\right)}^{\frac{1}{4}}.Math.\frac{1}{\sqrt{}}.Math.e^{2.Math..Math..Math..Math.{\mathrm{if}}_v^c\left(t-{}_k\right)}.Math.e^{-{\left(\frac{t-{}_k}{}\right)}^2}& \left(9\right)\end{array}$

in the case where Gabor functions are used.

[0058] The duration of the sequence {tilde over (p)}(t) is of T.sub.acq=T.sub.p where f.sub.p=1/T.sub.p is the rate at which the wavelets succeed each other. The sequence {tilde over (p)}(t) may itself be repeated at the rate f.sub.acq=1/T.sub.acq.

[0059] If one assumes that the wavelets of the sequence are of same central frequency f.sub.v.sup.c and noting (t) the common waveform c, that is to say .sub.f.sub.v.sub.c.sub.,.sub.k(t)=(t.sub.k), the sequence of wavelets after PPM modulation is written:

[00012]$\begin{array}{cc}\tilde{p}\left(t\right)=\underset{k=1}{\overset{.Math.}{.Math.}}.Math.\left(t-k.Math..Math.T_p-{\mathrm{.Math.}}_k\right)=\left(t\right).Math.\underset{k=1}{\overset{.Math.}{.Math.}}.Math.\left(t-k.Math..Math.T_p-{\mathrm{.Math.}}_k\right)& \left(10\right)\end{array}$

where .Math. represents the convolution product.

[0060] As an example of CAZAC sequences, Zadoff-Chu (ZC) sequences will advantageously be chosen, defined by:

ZC.sub.R(k)=e.sup.jRk(k1)/, if the length of the sequence is odd, and ZC.sub.R(k)=e.sup.jR(k1)(k1)/, if the length of the sequence is even and where R, being positive integers that are coprime (that is to say pgcd(R,)=1), R/ being the slope of the instantaneous frequency along the sequence and k[1,].

[0061] If a ZC sequence of even length is chosen, the elements of the sequence may be written:

[00013]$\begin{array}{cc}{\mathrm{ZC}}_R\left(k\right)=e^{-j.Math..Math..Math..Math.{\mathrm{Rk}}^2.Math.\text{/}.Math.}=e^{j.Math..Math.{}_R\left(k\right)}.Math..Math.\mathrm{with}.Math..Math.{}_R\left(k\right)=-\frac{.Math..Math.{\mathrm{Rk}}^2}{.Math.}.& \left(11\right)\end{array}$

The ZC sequence may be considered as a chirp of which the instantaneous frequency takes the discrete values:

[00014]$\begin{array}{cc}{f}_R=\left(k\right)=\frac{1}{2.Math.}.Math.\frac{d.Math..Math.{}_R.Math.\left(k\right)}{\mathrm{dk}}=-\frac{R}{.Math.}.Math.k& \left(12\right)\end{array}$

[0062] In the case of a ZC sequence, after PPM modulation of the sequence of wavelets the expression (10) then becomes:

[00015]$\begin{array}{cc}{\mathrm{.Math.}}_k=\frac{{}_R\left(k\right)}{2.Math..Math..Math.f_v^c}=-\frac{R}{2.Math..Math..Math.f_v^c}.Math.k^2& \left(13\right)\end{array}$

[0063] By using a narrow band approximation of the wave function, (t), it will be understood that this PPM modulation is equivalent to a phase modulation of the wavelets using the continuous phase law, .sub.R(t), of period T.sub.acq=T.sub.p, defined over [0,T.sub.acq] by:

[00016]$\begin{array}{cc}{}_R\left(t\right)=-\frac{R.Math..Math.}{.Math..Math.T_p^2}.Math.t^2={}_R.Math.t^2.Math..Math.\mathrm{with}.Math..Math.{}_R=-\frac{R}{.Math.T_p^2}=-\frac{{\mathrm{Rf}}_p}{T_{a.Math.c.Math.q}}.& \left(14\right)\end{array}$

The result is that the instantaneous frequency

[00017]$\frac{1}{2.Math.}.Math.\frac{d.Math..Math.{}_R}{\mathrm{dt}}$

undergoes a linear variation of slope .sub.R.

[0064] As an example, in FIGS. 3A and 3B are respectively represented the evolution of the phase and the instantaneous frequency along a Zadoff-Chu sequence. More precisely, the curves represented correspond to a sequence of length =64, of root R=1 and a rate of f.sub.p=125 MHz. It may notably be remarked that the instantaneous frequency varies linearly along the sequence, with a slope .sub.R.

[0065] Given the scale change properties of the Dirac function and the narrow band character of the spectrum of the wavelet, it is possible to show that:

[00018]$\begin{array}{cc}\tilde{H}\left(f\right)\left(f\right).Math.2.Math..Math..Math.f_p.Math.\mathrm{FT}\left(\underset{k=1}{\overset{.Math.}{.Math.}}.Math..Math.e^{-\mathrm{ik}.Math..Math.\left(t\right)}\right)& \left(15\right)\end{array}$

where {tilde over (H)}(f) and (f) are respectively the spectra of {tilde over (p)}(t) and (t), FT(.) signifies the Fourier transform and (t) is a period phase law T.sub.acq, verifying over [0,T.sub.acq]that (kT.sub.p)=arg(.sub.k)=2f.sub.v.sup.c.sub.k.

[0066] The expression (15) may be rewritten in the following form:

[00019]$\begin{array}{cc}\tilde{H}\left(f\right)\underset{k=1}{\overset{.Math.}{.Math.}}.Math.{\tilde{H}}_k\left(f-{\mathrm{kf}}_p\right)& \left(16\right)\end{array}$

where {tilde over (H)}.sub.k(f)=2f.sub.p(kf.sub.p)FT(e.sup.jk(t)) is the frequency response of a filter {tilde over (H)}.sub.k associated with the kth aliasing band. By comparing with the expression (6), it may be seen that the spectrum {tilde over (H)}(f) is no longer a spectrum of lines but has elementary bands of width f.sub.p. It may further be remarked that, compared to a conventional WBS sampler, the term FT(e.sup.ik(t)) introduces a phase modulation linked to the CAZAC sequence.

[0067] FIG. 4A represents the evolution of the real part of (e.sup.ik(t)) over time, for different aliasing indices k and FIG. 4B represents the evolution of the real part (FT(e.sup.ik(t))) as a function of the frequency for these same indices.

[0068] This representation has been given as an example for a Zadoff-Chu sequence of length =64 and of root r=1, a rate of f.sub.p=125 MHz and a wavelet central frequency f.sub.v.sup.c=4 GHz.

[0069] FIGS. 4A and 4B are consistent with the fact that the Fourier transform of a Zadoff-Chu sequence is still a Zadoff-Chu sequence. The spectral band {tilde over (H)}.sub.k(f) appearing in the expression (16) is the result of a modulation (in the space of the frequencies) with a Zadoff-Chu sequence of root kR.

[0070] Given that two Zadoff-Chu sequences of same length and different roots (here kR, kR with kk) have a low inter-correlation value (it is possible moreover to show that it is a minimum value for sequences having a perfect autocorrelation function), the different spectral bands are decorrelated.

[0071] In a more general manner, PPM modulation using a CAZAC sequence leads to a weak result of intercorrelation between two spectral bands, kk (the intercorrelation between {tilde over (H)}.sub.k(f) and {tilde over (H)}.sub.k(f) is weak except for k=k).

[0072] The result is that, despite the aliasing of these different spectral bands in the base band

[00020]$\left[-\frac{f_p}{2},+\frac{f_p}{2}\right]$

on account of the bandpass sampling, it will be necessary to discriminate the original sub-bands and to recover the information of the signal by means of a suitable filtering.

[0073] FIG. 5 represents the spectrum of a sequence of wavelets position modulated by the Zadoff-Chu sequence illustrated previously.

[0074] As a comparison, the spectrum of the same sequence of wavelets, in the absence of modulation, is also shown.

[0075] One verifies that, for a spectrum of rays at intervals f.sub.p, a continuous spectrum has been substituted of which the sub-band to sub-band intercorrelation properties are optimal. As mentioned previously, the rays of the spectrum of the non-modulated signal as well as the zeros of the modulated signal coincide with the multiples of the repetition frequency f.sub.p.

[0076] Another manner of appreciating the decoherence between bands introduced by PPM modulation derived from a ZC sequence is to calculate the correlation between two columns of the measurement matrix.

[0077] If the same formalism as that used in equation 2 is again taken, the bandpass sub-sampling by sequence of wavelets (WBS) PPM modulated may be represented by:

y=R.sub.TDW.sup.HF.sup.1s=R.sub.TDAs (17)

with A=(FW).sup.H, D the matrix of size N of which the elements are D(k,n)=exp(j2.sub.knf) where f=1/T.sub.acq and is the Hadamard product and N=T.sub.acqf.sub.Nyq.

[0078] The sensing matrix representing the bandpass sub-sampling by sequence of PPM modulated wavelets, designated hereafter WBS may thus be written:

A.sub.=D(FW).sup.H (18)

[0079] In FIG. 6 is represented the relationship (17) in the equivalent form y=R.sub.TDW.sup.Hx where x is the vector of size N representative of the signal sampled at the Nyquist frequency.

[0080] The matrix DW.sup.H contains on each line the samples (at the frequency f.sub.Nyq) of a wavelet of frequency f.sub.v.sup.c centred on the time .sub.k+.sub.k. The matrix R.sub.T represents the selection of M samples among correlation results (in the case of a non-uniform sampling or NU-WBS) in the time interval T.sub.acq.

[0081] In the case where the CAZAC sequence used for the PPM modulation is a ZC sequence, the elements of the matrix A.sub..sup.norm obtained by normalising the vector columns of the matrix A.sub.in such a way as to avoid the need for the amplitude of the wavelet (Gaussian term in the case of Gabor wavelets), may be written in the form:

[00021]$\begin{array}{cc}{A}_{}^{\mathrm{norm}}\left(k,n\right)=\exp \left(-j.Math..Math.2.Math..Math.\left({}_k+{\mathrm{.Math.}}_k\right).Math.n.Math..Math..Math..Math.f\right)=\exp \left(-j.Math..Math.2.Math..Math.\left(\frac{k}{f_p}-\frac{{\mathrm{Rk}}^2}{2.Math.f_v^c.Math.}\right).Math.n.Math..Math..Math..Math.f\right)& \left(19\right)\end{array}$

[0082] If one chooses f=1/T.sub.acq, the correlation between two components of the signal separated by f.sub.p=.Math.f, in other words between two columns of the matrix A.sub..sup.norm separated by may be expressed in the form:

[00022]$\begin{array}{cc}{}_{n,n+.Math.}=\underset{k=1}{\overset{.Math.}{.Math.}}.Math.\exp \left[j.Math..Math.2.Math..Math.\left(\frac{{\mathrm{Rk}}^2}{2.Math..Math.f_v^c}\right).Math.f_p\right]=\underset{k=1}{\overset{.Math.}{.Math.}}.Math.\exp \left[j.Math..Math.2.Math..Math.\left(\frac{{\mathrm{Rk}}^2}{2}\right).Math.\frac{.Math..Math.f}{f_v^c}\right]& \left(20\right)\end{array}$

[0083] It may be seen that the correlation result is no longer periodic in f.sub.p as in the non-modulated case (corresponding to R=0, .sub.n,n==) but is a sum of complex terms on the unit circle of which the module is weak.

[0084] FIG. 7 schematically represents a AWBS sampler according to a first embodiment of the present invention.

[0085] This first embodiment corresponds to a series configuration, that is to say to a single sampling branch. In other words, the different sub-bands of the input signal are sampled sequentially.

[0086] The input signal, x(t), is firstly multiplied (Hermitian product) thanks to the mixer 710 with a sequence of wavelets position modulated by a CAZAC sequence such as a ZC sequence. Obviously, instead of carrying out a Hermitian product in the mixer 710, one could simply generate conjugated wavelets and carry out a conventional multiplication in the mixer 710.

[0087] The wavelet sequence generator 750 supplies a base sequence of duration T.sub.acq, repeating every T.sub.acq. The base sequence comprises wavelets succeeding each other at the rate f.sub.p, where is the length of the CAZAC sequence. The period of the sequence of wavelets, designated sensing period, is thus T.sub.acq=T.sub.p. The frequency f.sub.p is chosen equal to the bandpass sub-sampling frequency, that is to say to

[00023]$\underset{i=1}{\overset{N_b}{.Math.}}.Math.B_i^w$

where B.sub.i.sup.w, i=1, . . . , N.sub.b are the widths of the sub-bands of the input signal.

[0088] The positions of the wavelets in the sequence of wavelets are modulated by the time modulator 760. More precisely, the wavelets of a base sequence are position modulated around the times .sub.k=kT.sub.p with

[00024]${\mathrm{.Math.}}_k=\frac{\arg \left({}_k\right)}{2.Math..Math.f_v^c}$

where .sub.k, k=1, . . . , are the elements of the CAZAC sequence.

[0089] The Hermitian product at the output of the mixer is integrated by the integrator 720 over a time interval corresponding to the temporal support of the wavefunction. The correlation results at the output of the integrator succeed each other at the rate f.sub.p. These correlation results are sampled in 730 with a frequency f.sub.p.

[0090] The sequence of samples is, if needs be, decimated in a non-uniform manner, and the samples thereby obtained are next digitally converted in the ADC converter 640. For example, all the samples of the sequence could be conserved (f.sub.s=f.sub.p) or instead only M samples among could be retained over a sensing period T.sub.acq (one then has f.sub.sf.sub.p), which corresponds to a decimation factor /M.

[0091] It should be noted that all of the elements 710-630, 750, 760 operate on analogue signals. The generator and the modulator could further be realised by a same analogue circuit as described hereafter. Thus, the WBS sampler is not subjected to the passband/sampling rate constraints inherent in samplers at the Nyquist frequency.

[0092] FIG. 8 schematically represents a WBS sampler according to a second embodiment of the present invention.

[0093] This second embodiment corresponds to a configuration with a plurality P of sampling branches operating in parallel.

[0094] For example, each sampling branch may correspond to a band of the input signal. In other words, the different sub-bands of the input signal are then sampled in parallel and the sampling within a band is carried out sequentially.

[0095] Generally speaking, the different branches may correspond to different CAZACs with sensing periods T.sub.acq.sup.i, identical or distinct. For example, the CAZAC sequences of the different branches may be ZC sequences of same length but with different indexes R thus excursions of frequency Rf.sub.p that are different over a same sensing period T.sub.acq. Preferably, ZC sequences will be used with indexes R that are coprime. The fact of using distinct CAZAC sequences (for example of distinct index) makes it possible to improve the diversity of the correlation results and leads to better rejection of aliasing.

[0096] The input signal x(t) is distributed over a plurality P of sampling branches in parallel, each branch P having the same structure as that represented in FIG. 7, namely comprising a wavelet sequence generator 850.sub.p, a temporal position modulator 860.sub.p, a mixer 810.sub.p, an integrator 820.sub.p, a sampler at the frequency f.sub.s, 830.sub.p, and an analogue digital converter, 840.sub.p.

[0097] As in the first embodiment, the sampling may be uniform or non-uniform.

[0098] Whatever the embodiment, the signal at the output of the WBS sampler, noted y(t), makes it possible to reconstruct the input signal, x(t). To do so, the signal y(t) is firstly the subject of a transform into z:

[00025]$\begin{array}{cc}{Y}_d\left(z\right)=\underset{n=0}{\overset{+}{.Math.}}.Math..Math.y\left(n\right).Math.z^{-n}& \left(21\right)\end{array}$

with y(n)=y(nT.sub.p), it being understood that in the case of non-uniform sampling with

[00026]$f_s=\frac{M}{}.Math.f_p$

the missing samples are replaced by zeros.

[0099] The result Y.sub.d(z) is next correlated with the responses of the filters {tilde over (H)}.sub.k, in other words is the subject of a suitable filtering with these filters in the frequency domain. The spectra obtained, i.e.

[00027]${}_k\left(f\right)={\left[Y_d\left(z\right)*H_k\left(z\right)\right]}_{z=e^{\mathrm{j2}.Math..Math..Math..Math.f_p}}$

(where * represents the correlation) are next frequency transposed then summated to obtain the spectrum of bands:

[00028]$\begin{array}{cc}\left(f\right)=\underset{k=0}{\overset{-1}{.Math.}}.Math..Math.{}_k\left(f\right).Math..Math.\mathrm{where}.Math..Math.=\frac{f_{N.Math.y.Math.q}}{f_p}.& \left(22\right)\end{array}$

The spectrum (f) is that of a signal stemming from a conventional WBS sampling, the signal x(t) then being able to be deduced therefrom as described in the aforesaid patent EP-A-3319236.

[0100] FIG. 9 represents an example of sequence of wavelets position modulated by a CAZAC sequence, such as used in the sampler of FIG. 7 or in a sampling branch of FIG. 8.

[0101] As indicated previously, the sequence of wavelets has a period T.sub.acq=T.sub.p. The line (A) represents the non-modulated sequence, such as supplied by the generator 710 or one of the generators 850.sub.p. The line (B) represents the time shifts, .sub.k stemming from the CAZAC sequence .sub.k, k=1, . . . ,. Finally, the line (C) shows the sequence of wavelets position modulated by the time shifts stemming from the CAZAC sequence. The wavelets are henceforth centred on the instants kT.sub.p+.sub.k.

[0102] FIG. 10 schematically represents a wavelet sequence generator PPM modulated for the WBS sampler of FIG. 8 or FIG. 9.

[0103] This wavelet sequence generator is particularly advantageous in so far as it combines the function of generation of wavelets and that of temporal position modulation.

[0104] The generator 1000 is based on a VCO oscillator switched to the repetition frequency f.sub.p. The oscillator includes an oscillating circuit LC, 1010, using a Varicap controlled by a voltage V.sub.f so as to be able to control the oscillation frequency of the oscillating circuit. The voltage V.sub.f is chosen equal in such a way that the oscillating circuit oscillates at the central frequency of the wavelets, f.sub.v.sup.c. The voltage V.sub.f may thus vary at each new period T.sub.p. The generator further includes in a conventional manner a pair of crossed transistors, 1021, 1022, mounted between the terminals of the oscillating circuit LC and a common current source, 1030. The current source is switched using a switching signal, {tilde over ()}.sub.(t), obtained by modulating the temporal positions of the rise edges of a clock Clk at the frequency f.sub.p.

[0105] The clock Clk has low phase jitter. It may be generated by means of the clock circuit 1040, directly by a quartz resonator, or even by a VCO oscillator locked by means of a phase locking loop on a low frequency signal, stable in frequency and in phase. The frequency of the clock signal is modulated by the frequency modulation circuit, 1050. Said circuit periodically modulates the frequency of the clock according to a linear law of slope .sub.R over the duration T.sub.acq of the base sequence. In other words, the instantaneous frequency of the clock thereby modulated travels over an excursion of Rf.sub.p over the duration T.sub.acq=T.sub.p. The clock signal thereby frequency modulated, {tilde over ()}(t), is digitally shaped in a pulse shaping module, 960. This shaping module restricts the duration of the clock pulses to the duration . The signal thereby shaped is used as switching signal {tilde over ()}.sub.T(t) of the current source.

[0106] The initial phase and the polarity of the initial pulse generated by the generator 900 are set by forcing the oscillating circuit to start in one direction. This may be done by introducing an unbalance between the 2 branches, for example by providing MOS transistors having different geometric ratios or instead by adding a capacitive charge in one of the branches.

[0107] The signal at the output of the switched oscillator is next filtered by an analogue shaping filter, 970, in such a way that each train of sinusoidal oscillations has the envelope of the wavefunction. Alternatively, the form of the wavefunction may be imposed in a digital manner by modulating the amplitude of the signal {tilde over ()}(t) in the shaping module 960.

[0108] Two generators 900 could be provided, operating with clock signals Clk in quadrature so as to be able to have available signals of complex wavelets and to carry out the Hermitian product in the input mixer. The quadrature clock could be inversed so as to obtain simply the conjugated signals and use a conventional mixer on the paths I and Q.

[0109] FIG. 10 gives as illustrative example a chronogram of the signals at the different points of the wavelet sequence generator of FIG. 9.

[0110] Line (A) represents the clock signal Clk generated by the clock circuit 950. Line (B) represents the clock switching signal (t), at the output of the frequency modulation circuit. It may be remarked that the rise edges of the switching clock are shifted by .sub.k with respect to the rise edges of the clock Clk. Line (C) represents the switching signal .sub.T(t) obtained by digital shaping of the clock switching signal by reducing the clock pulses to the width of the wavelet. At line (D) is represented the signal at the output of the oscillator when it is not switched (sinusoidal at the central frequency of the wavelet f.sub.v.sup.c). The signal at the output of the switched oscillator is represented at line (E): this is constituted of wave trains at the frequency f.sub.v.sup.c starting at the instants kT.sub.p+.sub.k. This signal is shaped in the shaping filter 960 to supply the sequence of wavelets {tilde over (p)}(t).