WO2010072963A1 - Polyphase decomposition of a filter bench for oversampled ofdm - Google Patents

Abstract

The invention relates to a device for demodulating an oversampled multi-carrier signal. According to the invention, the device includes Δ stages, Δ being an integer higher than or equal to 1, supplied with received data representative of source data, each of said stages including an analysis filter bench having N ₀ branches and a decimation factor equal to M ₀ , N ₀ and M ₀ being prime numbers relative to each other and M ₀ > 1, wherein an analysis filter bench includes a decomposition matrix V ⁽ⁱ⁾ (z) and a Fourier matrix F _N0 . The decomposition matrix is a para-unit matrix having some coefficients forced to zero according to a predetermined criterion that takes into account said decimation factor M ₀ and number of branches N ₀ in order to enable a perfect reconstruction of the source data.

Description

 POLYPHASE DECOMPOSITION OF A FILTER BENCH FOR SURFACEFUL OFDM

FIELD OF THE INVENTION The field of the invention is that of the transmission of digital signals based on multicarrier modulations.

More specifically, the invention relates to the modulation and demodulation of Oversampled Orthogonal Frequency Division Multiplexing (Quadrature Amplitude Modulation) signals.

The invention applies to wireless or wired communications. In the case of wired transmission, oversampled OFDM is still called FMT (Filtered MultiTone). The invention finds particular applications in DVB-type broadcasting systems or in VDSL type transmissions (in English "Very-high-bit-rate Digital Subscriber Line", in French "digital subscriber line very high speed ").

2. Prior Art

Multi-carrier transmission techniques, for example of the OFDM type, are used for wireless (DVB-T, WiFi, WiMax) or wired (xDSL, PLC) communications. They make it possible in particular to improve the spectral efficiency in so-called frequency selective transmission channels. Indeed, with these techniques, the spectrum is divided into sub-channels and the signal is transmitted on several subcarriers. These techniques can in particular be combined with multiple access techniques, which increase the number of users who can access a resource at any time.

Conventionally, the OFDM signal is written in baseband in the following form:

with: or :

M is the number of carriers on which we want to transmit complex symbols of data c _mn , with and

n ε Z;

- / (O is an integrable square function, also called a prototype function;

- T ₀ is the symbol duration;

F ₀ is the spacing between two successive carriers; and - j ² = -1.

Appendix A, which is an integral part of the description, recalls the main characteristics of OFDM modulations.

Over-sampled OFDM, compared to conventional FOFDM, presents the possibility of introducing an optimized waveform with appropriate criteria for a given transmission channel. This specific waveform is produced by an exponential modulation of a filter, called a prototype filter.

Appendix B, which is also an integral part of the description, recalls the main features of oversampled TOFDM.

More precisely, the discrete signal of oversampled OFDM, making it possible to transmit M data symbols on JV samples, with N greater than M, can be expressed in the form:

with:

or :

- k is the index of the samples of the discrete signal;

- J [k] represents the prototype filter that will be assumed to have a finite impulse response (FIR) of length L;

- D is a parameter related to the delay of treatment introduced by the system of transmission; - ψ _mn is a phase term.

Equations (4) and (5) can be rewritten by revealing a bank of synthesis filters with M sub-bands having an expansion factor equal to N consisting of m filters F _m (z) of impulse responses f _m [ k].

Such a system, also called a transmultiplexer, is illustrated in FIG. 1, and in particular described in the European patent EP 1 354 453 (reference [3] in Appendix G) in the name of the same applicant. Appendix C, which is an integral part of the description, recalls its main features. It should be remembered that to realize in the form of a transmultiplexer a so-called oversampled OFDM system or FMT, a synthesis bank is placed on the transmission (modulation side) and an analysis bench on the reception side (demodulation side), including a necessary delay. to the realization of a causal system. In a dual manner, to produce a so-called sub-band coding system, an input signal is processed by a modulation-side analysis bank and the signal x (n) is reconstructed on reception (demodulation side) by a bank. of synthesis.

The most significant advances on the theory of oversampled modulated filter banks concerned devices for subband coding. For example, Z. CVETKOVIC, "Oversampled modulated filter banks and tight Gabor frames in I ² (Z)". However, the authors present only "toy" examples, unusable in a transmission application, and do not provide any generic construction rules.

The perfect reconstruction conditions, as well as the associated optimization criteria, are also described in document [3] cited above. In addition, the document "Perfect reconstruction conditions and design of oversampled DFT modulated transmultiplexers" (reference [4] in Appendix G) specifies the conditions for perfect reconstruction in the orthogonal case, and the information needed to obtain efficient implementation schemes. and fast Fourier Transform Fast (FFT). Annex D, which is an integral part of the description, recalls the main aspects.

Moreover, the above-mentioned document [4] proposes a decomposition theorem that splits the initial transmission system of the perfect reconstruction equations into small equation independent subsystems, which can be solved by means of appropriate angular parameterizations. The technique presented in this document [4] provides, in the end, an algorithm for obtaining oversampled transmultiplexers for rational oversampling ratios 3/2, 4/3 and 5/4 and arbitrarily large prototype filter lengths.

Thus, the technique described in this document [4] guarantees the obtaining of perfect reconstruction (RP) solutions, that is to say perfectly orthogonal solutions, and allows, thanks to the use of a parametric representation in angular form. that this orthogonality is structurally verified. It nevertheless has some disadvantages, and in particular:

- parametric representations do not show regular realization structures. This does not allow to deduce solutions for polyphase components of a high degree prototype filter;

Some solutions with perfect reconstruction can be contained in other solutions. In other words, the described technique can lead to parametric representations which are a priori different, but which after analysis prove to be equivalent;

The complexity of the technique does not allow to consider oversampling ratios of less than 5/4.

The term "solution" is understood to mean a technique for implementing the transmultiplexer that will guarantee that the coefficients of the prototype filter and the overall implementation of the modulator and the demodulator will make it possible to verify the orthogonality conditions. 3. Presentation of the invention

The invention proposes a new solution that does not have all of these disadvantages of the prior art, in the form of a device for demodulating an oversampled multicarrier signal. According to the invention, such a device comprises Δ stages, Δ being an integer greater than or equal to 1, powered by received data representative of source data, each of said stages comprising an analysis filter bank having N ₀ branches and a factor of decimation equal to M _Q , with N ₀ and M _Q prime between them and M ₀ > 1.

Specifically, an analysis filter bank includes a decomposition matrix

, and the decomposition matrix is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account the decimation factor M _Q and the number of branches N ₀ , so as to allow a perfect reconstruction of the source data. .

The invention thus proposes a new technique of demodulation with perfect reconstruction, making it possible to ensure the orthogonality of the carriers of an oversampled multicarrier signal (M ₀ <N ₀ ), and in particular of an oversampled OFDM / QAM signal.

In particular, the prototype filter of the demodulator thus constructed allows multicarrier transmissions using up-to-date oversampling ratios, and in particular ratios close to unity. Indeed, it is recalled that over oversampling is low and close to unity, plus the spectral efficiency of the multicarrier signal is important, which is desirable.

The proposed technique makes it possible to obtain oversampled multicarrier systems for linear or nonlinear phase prototype filters, because of the use of a number of stages that can be arbitrary.

The use of a non-linear phase prototype filter has several advantages, such as obtaining regular structures with a minimum of rotation angles for the implementation of the prototype filter as described below, or a higher attenuation for the prototype filter for a given prototype length and oversampling ratio.

Annex E, paragraph 5.8, recalls the definition of para- unitary matrices. For example, the predetermined criterion forces the coefficients of the decomposition matrix V (z) of size N _Q XM _Q to zero, according to the following equations:

with:

M is an integer greater than or equal to 1;

'integer such that UM ₀ - vN ₀ = 1, with | u | <N ₀

with O ≤ k ≤ N ₀ - 1 and 0 ≤ l ≤ M ₀ - 1 and τem (a, b) an operator defining the remainder of the division of the integers a by b. The demonstrations associated with the determination of this criterion are presented in more detail in the Annexes, including Annex E, paragraphs

5.2 and 5.8.1.

According to a particular characteristic, the decomposition matrix consists of m identical matrices B, each matrix B consisting of at least one rotation and at least one offset, m being greater than or equal to 1.

It may be noted that the different rotations of a matrix B are independent. In other words, a rotation being defined by an angle of rotation, there are no dependencies between the different angles of rotation. According to another particular aspect, the number of branches N ₀ is equal to M ₀ + 1.

The oversampling ratio is then close to unity.

In particular, the number of rotations of a matrix B is equal to M _Q. In this way, if the decomposition matrix V ^ '(z) consists of m identical matrices B, the total number of rotations is TΠM _Q , which ensures a minimum dimension of the solution.

According to another particular characteristic, the number of stages Δ is an even integer, greater than or equal to 2. Indeed, if the prototype filter is a linear phase filter, it is necessary that the number of stages Δ be an integer pair for symmetry reasons.

According to another particular aspect, the stages supply first permutation means PT comprising ΔN ₀ inputs and AN _Q outputs, feeding N ₀ Fourier matrices F _Δ , which themselves feed second permutation means P.

It is thus possible to decompose the Fourier transform operation performed at the demodulator.

In another embodiment, the invention relates to a method for demodulating an oversampled multicarrier signal. According to the invention, such a method implements Δ stages fed by received data representative of source data, each of said stages implementing an analysis filter having No branches and a decimation factor equal to M ₀ , with N ₀ and M ₀ prime between them and M _Q > 1, comprising the following steps: - multiplication by a decomposition matrix V ^l '(z) and multiplication by a Fourier matrix F ^ ₀ .

In particular, the decomposition matrix is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account the decimation factor M _Q and the number of branches N ₀ , so as to allow a perfect reconstruction of the data. source. Such a method can be implemented by a demodulation device as described above. This method may of course include the various characteristics relating to the demodulation device according to the invention.

In particular, the decomposition matrix consists of m identical matrices B, each matrix B consisting of at least one rotation and at least one offset, m being greater than or equal to 1, and the method comprises a preliminary phase of determination of the matrices B, so that the matrices B are written in the following form:

where Z ^ ₀ is a diagonal square matrix providing an offset and R _{1 n + 1} a rotation matrix.

The invention furthermore relates to a device for modulating an oversampled multicarrier signal.

According to the invention, such a device comprises Δ stages, fed by data source, each stage comprising a synthesis filterbank having N ₀ branches and an expansion factor equal to M _0, N is ₀ and M ₀ first between them and M ₀ > 1, a synthesis filter bank comprising a conjugate Fourier matrix FyV ₀ and a conjugated decomposition matrix

≠ ' ^{} *} (z). In particular, the conjugate decomposition matrix is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account the expansion factor M ₀ and the number of branches N ₀ , so as to allow a perfect reconstruction. source data.

For example, the predetermined criterion forces the coefficients of the conjugated decomposition matrix V ^ ^* (z) of size M ₀ x N _{0 to zero} , according to the following equations:

, where T corresponds to the transposition operation, with: - m an integer greater than or equal to 1;

- (M, V) a pair of integers such that wM ₀ - VN ₀ = 1, with | M | <N ₀ and | v | <M ₀ ; with 0 </ c <N ₀ - 1 and 0 </ ≤ M ₀ - 1 and rem (α, / 3) an operator defining the remainder of the division of a by b.

Such a modulation device is able to modulate a signal intended to be received by the demodulation device described above.

The characteristics and advantages of this modulation device are the same as those of the demodulation device. Therefore, they are not detailed further. Indeed, it is noted that the modulator and the demodulator operate symmetrically, to allow a perfect reconstruction of the source data. Thus, the decomposition and Fourier matrices used on the modulation side correspond to the conjugates of the decomposition and Fourier matrices used on the demodulation side.

In another embodiment, the invention relates to a method of modulating an oversampled multicarrier signal. According to the invention, such a method implements Δ stages powered by the source data, each stage employing a synthesis filter having N branches ₀ and an expansion factor equal to M _0, N is ₀ and M ₀ between themselves and M _Q > 1, comprising the following steps: multiplication by a conjugate Fourier matrix F ^ ₀ , multiplication by a conjugated decomposition matrix V ^ (z).

In particular, the conjugated decomposition matrix is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account the expansion factor M _Q and the number of branches N ₀ , so as to allow perfect reconstruction. source data.

The invention also relates to a computer program comprising instructions for implementing the modulation method and / or the demodulation method as described above when the program is executed by a processor. The invention also relates to a system for transmitting an oversampled multicarrier signal, comprising a modulation device and a demodulation device as described above, so as to allow a perfect reconstruction of said source data.

4. List of Figures Other features and advantages of the invention will appear more clearly on reading the following description of a particular embodiment, given as a simple illustrative and non-limiting example, and the accompanying drawings, among which: FIG. 1 presents a conventional transmultiplexer, associated with OFDM modulations oversampled with N> M; FIGS. 2A and 2B illustrate an analysis bench and its equivalent diagram for an oversampled system of N / M ratio, with N> M; Figure 3 shows a modulated bench for oversampled report analysis

N / M, with N ≥ M, demodulator side, in decomposed form; FIG. 4 shows an oversampled synthesis modulated bench of N / M ratio, with N ≥ M, modulator side, in decomposed form; Figure 5 shows a rotation, with N _Q = 6, k - 2, I = 5; Figures 6A and 6B provide two equivalent representations of a matrix operation; FIG. 7 illustrates an embodiment diagram of a matrix of decomposition V ^ Cz); FIGS. 8A and 8B illustrate an example of a prototype filter, with its impulse response and its frequency response.

5. Description of an embodiment of the invention 5.1 General principle

The invention proposes a new modulation / demodulation technique making it possible to determine different structures for producing the prototype filter used to shape the carriers of an oversampled multicarrier signal, in a transmultiplexer type system or, in a dual mode, in a subband system.

The use of the prototype filter thus determined makes it possible to ensure orthogonality of the carriers, that is to say to maintain the distances between the multicarrier symbols transmitted at the output of the modulator, which makes it possible to optimize the transmission performance. In other words, if the transmission channel is the ideal channel, the data reconstructed at the output of the demodulator will be identical to the source data input of the modulator. In the case of a white Gaussian noise (AWGN) channel, because of the perfect orthogonality of the multicarrier modulation system, the symbol error probabilities (or bit) will be minimal. In particular, the invention proposes to determine the most appropriate implementation structures for the implementation of the prototype filter P (z) in its so-called polyphase form, by decomposing the matrix H (z) representative of the polyphase components of the prototype filter.

The solutions are written in the form of matrix products, which translate for implantation in the modulator or the demodulator by cascades of rotation and shift cells. It will be recalled that a solution here corresponds to obtaining a perfectly orthogonal filter bank because of its production structure.

In particular, the proposed technique makes it possible to preserve the orthogonality of the carriers because of the independence of the parameters, even if the angles of the rotations are changed.

5.2 Particular embodiment

As illustrated in FIGS. 2A and 2B, and demonstrated in appendix E, paragraph 5.1, which is an integral part of the description, the prototype filter can be implemented in its polyphase form, by means of a polyphase matrix H (z).

The invention proposes, according to this particular embodiment, to break down the polyphase matrix in the form of matrix products.

The decomposition of the polyphase matrix H (z) on the demodulator side is illustrated in FIG.

More precisely, as explained in Appendix E, the polyphase matrix H (z) can be factored into the form:

H (z) = F _N V (z) (31) where F _N is a Fourier matrix of dimension N x N and the matrix V (z) of dimension N x M is the product:

By denoting Δ the greatest common divisor between M and N, with M = ΔM ₀ , N ₀ > M ₀ > 1, N = ΔN ₀ , and Δ> 1, we can decompose the Fourier matrix F _N as described in the appendix. E, paragraph 5.3, in the following form:

D is the diagonal matrix whose line element and column i + 1 is ofl ^r if i = qA + r, we denote the root Ν-th of the unit. The operator X of FIG. 3 corresponds to the matrix D.

More precisely, a demodulator according to this decomposition comprises Δ stages powered by received data. Received data feeding the first stage undergo a decimation according to a decimation factor Δ, the received data supplying the second stage undergo a delay z and a decimation according to the factor Δ, the received data supplying the Δth stage suffer (Δ - 1) delays z ^"1 and a decimation according to the factor Δ Each stage comprises an analysis filter bank having N ₀ subbands or branches and a decimation factor equal to M _Q , with N ₀ and M _Q prime between them. analysis includes a decomposition matrix

F ^ ₀ in size

N ₀ x N ₀ , with / ranging from 0 to Δ - 1. According to FIG. 3, the permutation matrices P, P take the form

T of an interleaver π, II respectively. The first interleaving is provided by the outputs of the decomposition matrices

FIG. 4 illustrates the decomposition of the polyphase matrix H (z) on the modulator side, implemented symmetrically with the demodulator. More precisely, a modulator according to this decomposition comprises Δ stages fed by source data after an inverse Fourier transform. Each stage comprises a synthesis filterbank having N branches ₀ and an expansion factor equal to M _Q, where N is ₀ and M ₀ coprime ₀ and N> ₀ M> 1. A synthesis filterbank comprises a conjugated Fourier matrix Fχ _Q , of size N ₀ XN _Q , and a conjugated decomposition matrix V ^ (z), of size M ₀ x No, with / ranging from 0 to Δ - 1.

We then try to define the different structures of realization of the matrices of decomposition

(z), depending on the values of the parameters No and M ₀ . To do this, one considers decomposition matrices, V ^ ¹ '(z) demodulation side and V ^''(z) modulation side, para-unitary, some coefficients of which are forced to zero according to a predetermined criterion taking into account the factor decimation M _Q and the number of branches N ₀ , so as to allow a perfect reconstruction of the source data. Such a criterion is more precisely defined in Annex E, paragraphs 5.2. 5.8.1.

In this way we define motifs, also called "pattems", defining the set of possible solutions for the decomposition matrices, again denoted S _{MQtNQ> m} . The set S ^ ₀ ^ _{0 m} thus corresponds to the union of a set of solutions, each solution being a set of para- unitary decomposition matrices.

, for / ranging from 0 to Δ-1. Para-unitary matrices

correspond to a product of at least one rotation and / or shift.

The value m is defined according to the length L of the prototype filter: L = mAM ₀ N ₀ .

For example, in the case where N ₀ = M ₀ + 1, as illustrated in FIGS. 5, 6A, 6B and 7 for M ₀ = 5, a decomposition matrix

is obtained by the product of m identical elementary para-unitary matrices B {z). As shown in Appendix F, which is an integral part of the description, one can indeed write V ^{t) (z) = B ^m (z) U ₀ .

Elementary matrices B are written for example in the following form:

_NCJ wherein Z = Z _{+ 1} ^ ₀ is a diagonal square matrix providing a shift and ^ _i, A i _f o + ^{a my} sorting ^this rotation, as described in Appendix F. It is noted that the rotations of the rotation matrix R ^ M _n + i ^are independent of each other, which means that modifying one of the rotations in the matrix R _1, M _{Q +} 1 does not lead to a modification of the other rotations in the matrix

^R ι, Mo + l ^• As described in Annex E, Section 5.4, it is possible to use a parametric representation of para-unitary matrices, allowing these matrices to be represented by means of angles and offsets. Due to the independence of the angles and parameters, the modulator and the demodulator thus constructed allow a perfect reconstruction of the source data (x (n) = x (n)), including included in the case where the rotation angles are quantized. In other words, the modulator and the demodulator thus constructed make it possible to preserve the orthogonality of the carriers, even if the angles of the rotations are modified.

FIGS. 8A and 8B illustrate an example of a finite impulse response prototype filter obtained according to this embodiment, optimized to minimize out-of-band energy, for a system of size M = 60 and N = 72, with M the number of carriers and N the expansion factor, and the following parameters

M ₀ = 5, N ₀ = 6, Δ = 12, and m = 4.

FIG. 8A illustrates the impulse response h (n) of the prototype filter, making it possible to determine the value of its coefficients, and FIG. 8B illustrates the amplitude of the filter (in dB) as a function of the frequency (frequency response).

Finally, Appendix E, paragraph 5.8, presents the number of solutions obtained, for different triplets of parameters (M ₀ , N ₀ , m).

For example, for the triplet (4,5,2), we obtain a set of 61 solutions for the implementation of the decomposition matrices, of which seven solutions have a dimension equal to 12, and four solutions have a dimension equal to 4.

Thus, Appendix E presents a complete set of solutions such that no solution is contained in another and each solution is defined with a minimum number of angular parameters (called dimension of this solution), for some values of the triplet ( M ₀ , N _Q , m).

"Minimum number of parameters" therefore means that in each embodiment structure the necessary number of rotations used can no longer be reduced. 5.3 Other features and advantages of the invention

According to at least one embodiment, the invention makes it possible: to obtain the parametric angular solutions allowing the structural realization, at the level of the modulator and the demodulator, of the perfect reconstruction condition; - the adequacy of the proposed solutions to the use of the method of compact representation (as recalled in Annex E, paragraph 5.6) to determine the angular parameters of the prototype structure or function, and thus to the optimization of arbitrary-sized modulation / demodulation systems, even for a high Δ number;

obtaining regular structures that provide solutions for all the values of the parameters (M _Q , N _Q , m);

to obtain an OFDM-type multicarrier transmission system having a spectral efficiency comparable to that of a cyclic prefixed OFDM system (CP) of reduced value, that is to say less than 25%;

to work in discrete time, which allows the direct application of the filters thus constructed in the modulators / demodulators, in particular in digital form;

- a matrix interpretation of perfect reconstruction systems allowing direct deduction of generic and regular realization structures;

to obtain, for any value of the triplet (Mo, N _Q , m), families of solutions of minimum or maximum dimension;

to obtain a set of solutions verifying the following properties: each solution is described using a minimum number of independent angular parameters;

no solution is contained in another, and in particular two different solutions are not equivalent; - any para-unitary matrix, solution of the problem posed, belongs to at least one solution.

Thus, it is possible to obtain, according to one embodiment of the invention, a modulator and / or a demodulator in which the Δ stages (or sub-blocks of polyphase filtering of the modulated banks DFT are made using a minimum number of independent angular parameters. For example, for N ₀ = M _Q + 1, it is possible to obtain all the solutions for prototype filter lengths L = IΠAN _Q M _Q including the following cases:

- m = 1, ..., 5 for M ₀ = 2; - m = 1, ..., 4 for M ₀ = 3;

- m = 1, ..., 3 for M ₀ = 4;

m = 1.2 for M ₀ = 5;

- m = 1 for M ₀ = 6.

In particular, for parameter values M ₀ = 2, N ₀ = 3 and a prototype filter of length L = 6mΔ, we obtain a set of 3m - 1 solutions.

For odd M ₀ , N ₀ = M ₀ + 1 and a prototype filter of length L = MAN _Q M _Q , we obtain the solution of maximum dimension. For M ₀ any par,

N ₀ = M ₀ + 1 and a prototype filter of length L = mΔN ₀ M ₀ , we obtain the m - 1 solutions of maximum dimension. For any M ₀ , N ₀ = M ₀ + 1 and any m, we build a solution with minimum dimension mM ₀ .

According to another embodiment, by duality, the invention also allows the realization of a uniform filter bank with N sub-bands and rational oversampling

1 APPENDIX A: OFDM Modulation

For OFDM modulation on M carriers we want to transmit complex data symbols C ^ _n (m € I = {0, • • •, M - 1} and n € Z). The usual representations for these symbols correspond to the alphabets used for amplitude modulations such as QAM (Quadrature Amplitude Modulation) or PSK (Phase Shift Keying) and its differential coding variant DPSK. . The OFDM signal is written in baseband:

0 n O no with

and 10 - f (t) an integrable square function generally called a prototype function;

- To symbol duration;

The spacing between two successive carriers; - J ² = -1.

Thus defined, (/ m, _n ) _(m , _n) ei _x z constitutes a family of Gabor. The goal is to find the symbols C ^ _n from the signal s (t). This is possible if and only if (/ _m , _n ) (m, n ₎ eiχz constitutes a free family. Thus, the density d = ψγ ~ must be less than or equal to 1 to be able to demodulate the signal. Now the spectral efficiency η, which measures the bit rate per unit of time and frequency, is related to the density by the relation η ≈ bd.

It is deduced that the spectral efficiency is maximal when d = 1, i.e. Fo = 1 / TQ. In this case, the Balian-Low theorem indicates that f (t) is a function necessarily badly localized in the time-frequency plane, that is why it may be necessary to reduce the spectral efficiency by taking d <1. The system is said to be subcritical density.

To use the write convention of [1], referenced in Appendix G, we can represent an OFDM system by the triple (/, T ₀ , FQ) and we say that an OFDM system is orthogonal if

The condition of orthogonality is not essential to find exactly the symbols of data c _mιTl after demodulation. Thus in the case of a so-called biorthogonal system a family of Gabor functions different from that of the emission can be used to

M the reception. In this case it is necessary to determine a pair of functions of Gabor, for example f _m , _n and g _mtn which satisfies condition (3). On the other hand, in the presence of Gaussian additive white noise (AWGN) it is the orthogonal solution that ensures that the error rate will be minimal.

By the Balian-Low theorem we know that it is impossible to find a triplet (/, T ₀ , Fo), or a quadruplet (f, g, T ₀ , F ₀ ) in the biorthogonal case, which simultaneously verifies three conditions:

1. The functions f _m , n are orthogonal (or the pair f _mηn , 9 _m , _n is biorthogonal);

2. The function / is well localized in time and in frequency (or the functions / and g are both well localized in time and in frequency);

3. T ₀ F ₀ = 1.

Thus in the case of the basic OFDM, ToFo = 1 but the prototype function is then a function poorly localized in frequency.

As a result, the only two OFDM alternatives are

- The OFDM / OQAM which, by releasing the orthogonality constraint by limiting it to the real body by the Offset-QAM technique (OQAM), allows the introduction of waveforms well localized in time and frequency, all maintaining maximum spectral efficiency;

- The subcritical density OFDM which introduces a loss of proratal spectral efficiency to 1 / d.

2 APPENDIX B: Oversampling OFDM

To obtain a numerical modulation system from the mathematical expressions of the signal (1, 2), one can either proceed by a simple sampling or directly use the Weyl-Heisenberg family theory. The link between time-continuous and time-discrete formulations can be made by choosing a sampling period such that the density is written d = jψ. Hence N> M so that to transmit M data symbols N samples are necessary, we will therefore speak of oversampled OFDM system. In the end, the discrete signal of oversampled OFDM is expressed in the form

with

In this last expression

- f [k] represents the prototype filter which will be assumed to be the finite impulse response (FIR) of length L;

- D is a parameter related to the processing delay introduced by the system. In the "orthogonal case D = L - I.

~ Ψ _{m, n} is a phase term that will also allow us to describe the detail of a causal realization of an oversampled OFDM modem (or modulator-modulator).

The condition of orthogonality is then written in the form

20 If this condition is realized at the demodulation we get

Note at this stage that the phase term φ _{m <n} has no influence on the orthogonality condition (6). Two remarks are also needed regarding the terminology.

Referring to [1], referenced in Appendix G, any system modulated by Gabor functions defined by a triplet (Z, T ₀ , FQ) is an OFDM system.

- In the time-discrete version, the notion of oversampling appears explicitly and this is referred to as oversampled OFDM. However a risk of confusion is still possible because the oversampled OFDM is also used for OFDM (or CP-OFDM) systems where the modulated signal is obtained for N = M (possibly with the addition of a cyclic prefix or CP) and where the operation oversampling is performed on the modulated signal or on the received signal.

3 ANNEX C: Over-sampled OFDM Transmultiplexer

3.1 Reminders

The equations (4) and (5) can be rewritten by revealing a bank of M synthesis filters sub-bands having an expansion factor equal to N consisting of the filters F _m (z) of impulse responses:

provided that φ _m , _n - - _jj m-nN, and applying the entries c _mtTl to the bench thus formed. We then obtain

+ oo MI

Now consider demodulation. It is given by equation (7). Since we have set φ _{m <n} = - _jj mnN, the demodulation is written

k oo

To handle the processing delay D, we define the integers a and b, a> 1 and O <b <N - 1 by

So

D. Assuming that the family {f _m , _n } is a basis of the space it generates, equation (10) is then rewritten

By bringing this equation closer to the input-output relationship of an analysis bank, it is concluded that the demodulation can be performed with a delay of a samples using an M-sub analysis bench. bands, with an equal decimation factor & N and consisting of analysis filters H _m (z) of impulse responses

provided that h [k] = f * [D - k] and apply the input s [k - b] to the bank thus M constituted. The discrete time OFDM / QAM modem can therefore be realized using a transmultiplexer as described in FIG. 1. Over-sampled OFDM transmultiplexers were introduced in [2], referenced in Appendix G, as dual systems of oversampled filterbanks. In the patent [3], on behalf of the same Applicant, the causality conditions, which result in the parameters a and b in FIG. 1, are also introduced.

3.2 Efficient realization scheme

Direct realization of the OFDM / QAM modem using the transmultiplexer of Figure 1 would be very expensive in terms of calculations. Indeed, during transmission and reception, it involves M filters of length L. Moreover, these filterings are performed at a rate TV times greater than the transmission rate of the symbols C _{77 n} , d where the idea of using polyphase filtering to reduce the operating cost of the modem. In the patent [3], it has been shown how to obtain fast transformation based schemes for BFDM. As this is a special case, we simply give here the main lines to achieve this type of scheme.

We first define the integers N ₀ and M ₀ by:

with lcm the smallest common multiple.

Type I polyphase decompositions of transmission and reception prototypes are also used:

with:

After some calculations, we obtain the diagrams of the modulator and the demodulator illustrated in the figures of the patent [3]. Several variants were then proposed using either FFT or IFFT on transmission and reception.

In these diagrams, inputs and outputs are noted in the domain transformed into z, while II corresponds to a permutation block. 4 APPENDIX D: Reconstruction

4.1 The perfect reconstruction conditions

The conditions for perfect reconstruction of the OFDM / QAM system have already been demonstrated in the patent [3] in the BFDM case. Then on the basis of a so-called decomposition theorem, they have been specified for the orthogonal case in [4]. To express them, we first define the expression

We then denote by Δ the GCD ("Most Great Common Divisor") of N and

M, ie N = AN _0, M = AMo- We note Gι _tp (z) = ₊ Gι _{& P} (z). For a prototype filter of

B length L = mN ₀ M = TnAN ₀ MB, we can then show, by a so-called decomposition theorem [4], that perfect reconstruction conditions can be reduced to Δ independent systems, such as

where Gι = _tP (z) and where T denotes the paraconjugué an expression z, which in the case of a polynomial function with real coefficients is reflected for example by the equality

4.2 The optimization criteria

The choice of the optimization criterion of the prototype filters generally depends on the type of application envisaged. In the case of a transmission system in a static channel such as that of the ADSL ("Asymmetric Digital Subscriber Line"), a common choice is to minimize the out-of-band energy. For radio-mobile type channels, the model of modeling corresponds to a filter varying in time, the channel is then dispersive in time and frequency. In this case, where the temporal and frequency dimension are equally important, the time-frequency localization criterion is currently the reference criterion. This is for example the case in [Ij, referenced in Appendix G.

4.2.1 Time-frequency localization

For a discrete signal x, we use a discrete localization measure, noted ξ such that

where the quantities π ^ z) and Tt ₂ (x) correspond respectively to the moments of order 2 temporal and frequency as defined by Doroslovacki [5].

4.2.2 Minimizing out-of-band energy

The second criterion that we consider is that of the minimization of the weighted norm of the error, denoted E (u), in frequency. For a weighting function, denoted W (u), the objective function is written

In this case we choose a constant weighting function in each interval defining the low-pass and low-pass bands of the prototype low-pass filter. For a prototype filter p, whose Fourier transform is P (v), the function to be minimized is then written

where a and p are two real numbers between 0 and 1, with a the weighting parameter and p the roll-off factor, which determines in part the limits of the bandwidths, f _p , and attenuated, / _s . So for a system with 2M carriers (or subbands), we have

Since the effect of the perfect reconstruction constraint promotes good bandwidth behavior, we choose to take a = 1.

5 APPENDIX E: Demonstrations

5.1 Principle of resolution

Note by P (z) the finite impulse response filter FIR called prototype filter. For exponential modulation, the N subband and decimation factor analysis array M, shown in FIG. 2A, comprises N analysis filters H _t (z), i = 0,. . . , N-1 such as

where ω is the root iV-th of the unit, ω = e ^~ ^, connected to N decimators of order M. The polyphase components of order MiVo of P (z) are denoted by V _t (z), i = 0,. . . , MNQ

and for i = 0, . . , N - 1 the polyphase components of order M of H _t (z) are denoted H _tJ (z) o with

Using equations (26, (27), (28) and the relation MN ₀ = JVM ₀ , we deduce that

The polyphase matrix of the analysis filter bank is a matrix of size N x M having the elements H _υ (z). As shown in FIG. 2B, for a signal x (n), which is transformed into z denoted X (z), the processing produced by the analysis bench corresponds to the matrix product H (z) X (z) where X (z) ) is a column matrix M x I such that

As Rima Hleiss [2] has already shown, the matrix H (z) factors in the form

where F _N is the Fourier matrix of dimensions N x N and the matrix V (z) of dimensions N x M is the product

V (z) (32)

where I _n denotes the identity matrix of size n. The matrix of I _N contains Mo copies concatenated horizontally and that of IM contains N ₀ vertically concatenated copies. The problem to be solved is to determine the most appropriate realization structures for the implementation of the matrix H (z) in the case of the exponential modulation and for different values of M ₀ , N ₀ , Δ and m. The principle of resolution is based on equivalences between the paraunitarity of polyphase matrices and the perfect reconstruction condition. The demonstration is therefore linked to the identification of specific properties which are then used to produce new implementation schemes for oversampled filter banks.

5.2 Matrices associated with the prototype filter for Δ = 1

The problem of perfect reconstruction (RP) can be reduced to Δ independent systems. We study more particularly an elementary system in the case where Δ = 1. Let M ₀ > 1, N ₀ > MQ and gcd (Mo, N ₀ ) = 1, for each a such that 0 <a <M ₀ N ₀ - 1, we can determine by Euclidean division unique integers k and p such that

or

(with gcd the "greatest common divisor" or greater common divisor). Since Mo and N ₀ are prime between them, according to Bezout's theorem, there exists a unique pair of integers (u, v) such that:

Again by Euclidean division, it is also possible to show that given k and / with 0 <k <N ₀ , 0 <I <M ₀ , there exist unique integers p (k, l), q (k, I ) and a {k, I) such that

where rem (", υ) corresponds to the rest of the division of u by v. We also define the integer β (k, I) such that

and the integer a (k) such that a (k) = 0 if Mo divides A; and 1 otherwise. We can thus introduce an integer ε (k, I) such that ε (k, I) = a (k) -β (k, I) takes its values in {0, 1}. In the case where N ₀ = M ₀ + 1, a particular case of important practical interest, it can be shown that:

with b any integer.

All these notations allow us to express the polyphase decomposition of the prototype P (z), corresponding to the case where Δ = 1, as follows. The parameters M ₀ , N ₀ are such that Mo <N ₀ , gcd (M ₀ , No) = 1, and the prototype filter P (z) is assumed FIR. We denote by P _a (z), a = 0,. . . , M ₀ Nc, - 1 the polyphase components of order M ₀ AO of P (z), ie

The matrix Vp (z) of size N ₀ x M ₀ is defined by

and the matrix Up (z) of size N ₀ x M ₀ is defined by

X5 We denote by T ₁ (^) the square diagonal matrix of size N ₀ whose (k + l) -th element is _z - ^{i (} kf ⁾ + ^{a (k) N} ₀ ^ Q <k <N ₀ and by T ₂ (z) the square diagonal matrix of size M ₀ with for (Z + 1) -th element

0 <I <M ₀ . From the relation (39) and the definition of the function ε (k, l), we can show that

5.3 Implementation structure using decomposition

M In the general case Δ> 1 and the prototype filter will be written in the form

with for all i = 0,. . . , Δ - 1,

The relation (44) can be generalized as follows

In order to make the lines and columns of V (z), the indices of which even remain in the Euclidean division by Δ, contiguous, we introduce the permutation σ _/ v on the set {0,. . . , N - 1} defined by

and the permutation σu on the set {0,. . . , M - 1} defined by

We then denote P the permutation matrix of dimension N corresponding to σ ^ defined by

Since N = ANo, the Fourier matrix F _N admits a decomposition taking into account the composite nature of TV.

where the diagonal matrix of blocks F _No contains Δ, that of blocks FA contains NQ and D is the diagonal matrix whose line element and column i + 1 is ω ^qr if i = qA + r. FIG. 2B then produces the diagram represented in FIG. 3, where each of the Δ-polyphase components of the prototype filter P (z) intervenes in this diagram to define a modulated sub-band at iV ₀ with a decimation factor equal to Note that in this diagram:

~ The matrix P is denoted II;

- The impact of the last matrix P in the expression (51) is implicitly taken into account in the re-ordering of the matrix blocks

As can be deduced the synthesis bench shown in Figure 4. 5.4 Some characteristics of the solutions obtained

Throughout this study, we consider real filters with finite impulse response (FIR), that is to say the real polynomials in the variable z ^{~ ι} .

If A (z) is a matrix with N rows and M columns whose elements are filters, we write  (z) the matrix with M rows and N columns whose elements are given by

A paraunitary matrix is a matrix A (z), with M <N, satisfying

where IM is the unit matrix of dimension M.

The theory of the super-enhanced oversampled modulated filter banks and the oversampled OFDM / QAM transmultiplexers having the perfect reconstruction property both lead to the use of paraunitary matrices with M and N prime between them. The decomposition theorem is the basic result in this reduction.

When it comes to using paraunitary matrices and for example to make an optimization on all these matrices, we can choose to represent these matrices by their

15 coefficients. But we must in this case force the coefficients to check the constraints resulting from the matrix equation (53). It is best to use a parametric representation of all paraunit matrices using a set of independent parameters, a representation that automatically implies that equation (53) is true. For example, a constant paraunitary matrix with M = N is simply

2o an orthogonal matrix and it is well known to represent these matrices using M (M-1) / 2 angles and a discrete parameter which is the sign of the determinant (generalization of Euler angles).

Reference [4] gives a set of parametric representations of the particular class of paraunitary matrices that occurs for modulated transmultiplexers

2s OFDM / QAM with perfect reconstruction. This method is based on an analysis of the system of algebraic equations obtained from equation (53).

5.5 Perfect reconstruction in matrix form

Compared to [4] the rewriting of perfect reconstruction equations (RP) with equivalences in matrix form is an important element in the development of the M solutions described later.

For a given prototype filter P (z), the following conditions are equivalent: (a) The matrix V [z) is paraunitary.

(b) The matrices V ^ (z) are paraunitary for i = 0,. . . , Δ - 1.

(c) The matrices

are paraunitary for i = 0,. . . , Δ - 1.

(d) The following relations are verified by the components MiV ₀ -polyphases V ₁ (z) s of the filter P (z)

d \ M ₀

1 = 0,. . . , M ₀ - I, A = 1,. . . , M ₀ (54) with n _{\ t \} =

and where d _mn is the symbol of congraence.

Note in these equivalences that the matrices

are deduced from the relations (42) and (43) described in the case Δ = 1 and that in the general case these matrices are linked by the relation (47). Also differing from [3, 4] the symbols U and V are used instead of K and G to record the polyphase components which explains the difference between (54) and (21).

5.6 Symmetry and Linear Phase Prototype Filter

Recall that the MΛO-polyphase components of the prototype filter are of degree m - 1 with m> 1. The filter P [z) is then of length L = MoNoAm and of degree L - I5 in z ^'1 .

The symmetry of the prototype filter P {z) for the parameters (Mo, N ₀ , Δ, m) is denoted by definition σχ, (P) (z). Its polyphase components of order MN ₀ , denoted W _t {z), i = 0,. . . , MN ₀ - 1, are connected to polyphase components V _t (z) of MΛO-polyphase order of P (z) by the equations

In the same way, the components Δ-polyphase Q ₁ [Z), i - 0,. . . , Δ - 1 of σι [P) [z) are given by

where P _t [z), i = 0,. . . , Δ - 1 are the Δ-polyphase components of P [z).

For n> 0, we introduce J _n as the nxn matrix defined by

= 1, k = 1,. . . , n and 0 elsewhere. We can then show that the matrices are connected by the equation

We can highlight a similar relationship between matrices

i = 0,. . . , Δ - 1, but this requires the introduction of two new diagonal matrices.

We define T $ (z) as the square diagonal matrix of size NQ with (A; + 1) - th element z ( β ( *, o ) + " (JV b- i - fc , o ) + " ) / α iV b- (*) - β (Λ b i - f c)) g ^ that \ a I t r ice T 4 (z), also 5 square diagonal but with size M 0 to (7 + l) -th element z (Φ i) + q (p, o M i - i) + i - u) / N Q) ^

The matrices are connected by the equation:

JM ₀ T ₄ (Z). (58)

From previous results (57), (58), as well as from the decomposition theorem (see Appendix, section IV.1), we can deduce that a prototype filter P (z) has the property PR for the parameters (Mo, NQ, Δ, m) if and only if the symmetric prototype filter σ ^ _i (P) (z) io has the same property.

When Δ is even, the same arguments make it possible to construct a symmetric prototype filter, ie P (z) = <7 _£ i (P) (z), by choosing the first Δ / 2 components Δ-polyphase such that P _t (z ), i = 0,. . . , _j - 1 with the PR property and finally obtain the remaining polyphase components by

"5.7 Angular parameterization and solution concept 5.7.1 Elementary rotations

For N> 2 and l <i, j <N, i φ j, we call elementary rotation of coordinates i, j and of angle θ, the rotation of H ^N whose matrix R = R _ι , _j (θ) is given by

The set of rotations [R ₁ ^ (O), θ € R} is denoted R ₁ ^.

This notation is extended to the product of elementary rotations by agreeing that the angles that appear in the various rotations are independent parameters. For example

The group of direct rotations in R ^ is designated by SON (R): its elements correspond to real matrices R of TV dimension satisfying RR ^T = I ^ and det (ft) ≈ 1. In R ³ , the representation of a direct rotation using Euler angles corresponds to the identity

This multiplicative factorization of a rotation of H ³ by means of elementary rotations is generalized to R

This makes it possible to express any direct rotation of Nl ^N as a product of ₂ ^~ 'elementary rotations.

In the following we use the notation R _ιlt ... _TTK with k <N for designating in SO _JV (R) the subgroup of rotations which leave invariant j ^'-th coordinates with

o A product of elementary rotations can sometimes be simplified. For example, the product -R 1, R 2, R 2, R 2, R 2, 3 is equal to R 1, R 2, R 2, R 2, since the latter product represents, using Euler angles. any direct rotation of R ³ .

For N = 2, the result is trivial thanks to the identity

For N = 3, the representation by Euler angles is generalized by the following result5. We have

5.7.2 Offsets

Let N> 2, I be a subset of indices between 1 and TV and Zi be the diagonal matrix0 of dimension N whose diagonal elements are given by [2T _/ ] _J , _J = z ^{~ ι} if i £ / and 0 if not. If A (z) is a paraunitary matrix with N rows and M columns with

M <N, then ZjA (z) also. We denote Zj the transformation defined by the matrix

Zi, which is called a shift.

5.7.3 Definitions Let M ₀ and NQ with Mo <No, M ₀ and No first between them. We denote by UM _O , N _O the set of paraunitary matrices at N ₀ rows and Mo columns. ^ _M0 , N ₀ , _m with m> 0 denotes the subset of UM _O , N _O matrices whose elements are of degree in z ^{~ x} less than or equal to m. In particular UM _O , N _O , _O is the set of paraunitary matrices of dimension JV ₀ x Mo constants.

A matrix A (z) of U _M0 , N ₀ , _m can be represented as a polynomial in z ^{~ ι} with constant matrix coefficients:

The matrices of U _M0 , N ₀ , _m satisfying the following property: for 0 <k <N ₀ - 1 and

0 </ <M ₀ - 1, if a (k, M ₀ ) - ε (k, l) = 1 then [A ₀ ] _f c _{+1) (+} i = 0, otherwise [A] _m ] _{k + lti + 1} = 0 form a subset denoted S _M0 , _N0 , _m - These paraunitary matrices have the form of matrices U ^ '(z) considered in section 5.2.

A matrix of UM _O , N _O constant whose elements are in the set {0, 1, -1} is said elementary. Such a matrix necessarily has Mo non-zero values, equal to 1 or -1, so that two non-zero elements are neither on the same column nor on the same line. Notation.- An elementary matrix U containing only 0s or 1s has an element equal to 1 in each column, without having two elements equal to 1 on the same line. For column I with 1 </ <Mo, we denote by k {l) the line index between

1 and N ₀ such that U _{k (I) 1I} = 1. It is then appropriate to note U in the more compact form U ₀ [Ar (O) k (1). . . AT (M ₀ - 1)]. For example the matrix

is denoted U ₀ [A 3 2 5], the number of rows of this matrix being set by the context.

A solution S of S _M0 , _N0 , _m is a subset of the form S = TiT ₂ . . . T _k U ₀ where

T, for i - 1,. . . k is either an offset Zi with / G {1,. . . , No}, or a set of elementary rotations of the form R _1J with 1 <i, j <N ₀ , ι φ j. An element of S is a function of n independent angular parameters where n is the number of elementary rotations in the sequence Ti,. . . , T ^.

Two solutions S = T ₁ T ₂ . . . T _k U ₀ and S '≈ T [Tl ₁ . . . T _k ', U' _o are equivalent if S = S 'setwise. S is said to be an injective solution if there is no solution equivalent with a strictly lower number of rotations in the course of its transformations. The notion of dimension that will be introduced later makes it possible to establish that a solution is injective if its number of angular parameters is equal to its dimension.

5.7.4 Symmetrical solutions

With the definition of the preceding paragraph, we say that a solution S is symmetrical if φ _m (S) = S.

For MQ> 2 and iV ₀ = Mo - + - 1, if No and m are odd, there is no symmetric solution.

5.8 General description of all solution families 5.8.1 General resolution algorithm

For Mo, N ₀ , m fixed, we give a general algorithm allowing to express the set S _M0 , _N0 , _m in the form of the union of a set of solutions, each solution being a set of paraunitary matrices obtained as produced by a sequence of rotations and shifts of an elementary paraunitary matrix.

Definition of patterns, also called "patterns":

The set S _MO , _NO , _TTI consists of paraunitary matrices A (z) of dimensions TVo x M ₀

where the constant matrices Ao and A _m of dimensions No x Mo have preassigned null elements:))

for 0 <k <N ₀ - 1, 0 <I <M ₀ - 1.

We then introduce the more general notion of patterns or pattern that corresponds to the set of paraunitary matrices of the form (69) for which the matrices AQ and A _m have preassigned null elements. Let A be a subset of S = [1,. . . , N ₀ ] x [1,. . . , Mo]. We denote by M. _A the set of real matrices M of dimensions No x Λ / o such that M _τj - 0 for (ι, j) $. AT. To every subset A of S is associated an array of dimensions No XM _O , denoted A, whose elements are in {0, x} such that A _{% J} = x if (i, j) GS and 0 elsewhere.

The same notation A is used for the set U _Ma , _N0 , _m HM _A which is called an orthogonal pattern, that is to say the set of orthogonal matrices No ^χ M> ^with pre-assigned zeros for the indices in HER.

For A and 5 ^ 0 two subsets of 5, we denote by i ooo β the subset of matrices M in U _MO , _NO such that Mo € Λ ^ _A and M _m € M _{B where} m is the degree of M with m> 0.

By convention A ooo 0 ≡ A ooo S_.

The subset of paraunitary matrices of degree m in AoooB_ is denoted Aooo _m B_ and consistent with the previous convention A ooo _m φ ≡ A ooo _m _ ₁ 5.

The sets A ooo _m H and i ooo β are called paraunitary patterns.

For M ₀ > 2, N ₀ > M ₀ , gcd (M ₀ , No) = 1, let us define

so

according to equations (70) at (71) and

is the set of paraunitary matrices with perfect construction.

Example.- For MQ = 4, No = 5 and with notation using tables

It is straightforward to verify that if A _\ CA ₂ and B _\ CB ₂ then ₁ 000 H ₁ c A ₂₀₀₀ H _2.

Action of the rotations on the patterns: Let R _ι , _j (θ) be a rotation of any angle θ as defined in paragraph 5.7.1 and A an orthogonal pattern. Then since the rotation acts on lines i and j, if M € A 000 H the constant element of line 1 or j and column /,! </ <M ₀ of the matrix R ₁ ^ (O) M is not zero if (i, j) GA or (j, I) G A. By noting  the set of (k, I) such that (k, I) € A if k φ i and k φ j or k = i or k = j with (i, k) GA or (j, k) GA, we thus have the set notation for the rotation R _tj AC A.

In the same way we define the action of a rotation R ₁₃ on a paraunitary pattern s A 000 B and we establish that R ₁ ^ A 000 B ci 000 B_.

It should be noted that the lines i and j of the patterns obtained after the action of a rotation R ₁₃ are identical.

Conversely, we say that a parity pattern A 000 B is rotatable if there are two indices i and j, i φ j such that the two lines i and j of the pattern are identical with at least one element non-zero (i, k) GA and a non-zero element (i, I) G B. In this case, there are two possibilities. If the columns A; of A and I of H have the only common elements of lines i and j non-zero, we put A '= A - {(j, k)}, B' = B - {(i, l)}, and if A '= A, B' ≈ B - {(i, l)}.

We then have A ooo B_ = R, _i3 A '000 B'. xδ Similarly, we say that an orthogonal pattern A is pivotable i and j, i φ j such that the two lines i and j of the pattern are identical with at least two non-zero elements (i, k) and (i, I) in A. In this case, we distinguish two cases. If the columns k and I of A have the only elements of lines i and j not common, we put A '= A - {(j, k), (i, l)}, and if A' = A - { (he)}.

We then have A = R _{1 3} A '.

Offset action on patterns:

Let I be a non-empty subset of indices between 1 and N ₀ and Z ₁ the offset defined in section 5.7.2. If MG AOOO _m _ B_ ₁ is a pattern paraunitaire m> 1, then Z ₁ MG TO ooo _m £? where the sets A and B are defined by _^ 4 = {(i, k), (i, k) GA, kg /} 25 B = {(i, k), (i, k) EB, ke I} .

In the same way, for an orthogonal pattern, we have ZiA C iooo ₀ β where i =

{(t, k), (z, k) e A, k ^ i}, E = {(1, k), (i, k) e A _y kei}.

Conversely, we say that a paraunitary pattern A ooo B_ is reducible in degree if for any index i between 1 and N ₀ the line i of A has all its null elements, or the line 30 i of B_. We note / the set of indices for which the line i of A has all its null elements. We can then write A ooo _m B_ = ZιAiooo _m _ _x B ^ where A '- {(i, j), (i, j) GA or i G /} and B' - {(ι, j), (i , j) GB or i $ /}, and it is said that it is a paraunitary shift. We also have A oOo ₀ B = Z ₁ AL where A '= {(i, j), (hj) £ ^A , ii 1} U {(t, j), (i, j) GB, i G /} and it is said that it is an orthogonal shift.

The splitting operation:

If the matrix given by equation (69) is paraunitary, then in particular we have the relation A ^ _n Ao = 0, which means that the column vectors of AQ are orthogonal to the column vectors of A _m . If a column of index k of Ao and a column I of A _m are such that [Ao] _1A ; = 0 or [A _m ], _/ = 0 except for i = i $ with 1 <io <NQ, then the orthogonality relation between the column vectors k of A ₀ and I of A _{n is} written [Ao], ^^] ^ = 0 and so either [A _o ] _ιok = 0, or [A _{m] % ol} = 0.

We then say that a paraunitary pattern A ooo B_ is cleavable if its matrices A and B_ io satisfy the previous property. We can then write

where A! and B 'are defined by A' = A - {(i ₀ , k)}, B '= B - {( _iQ , l)}.

However, the paraunitary patterns ^ ooo β or i ooo β ^ can themselves be further cleavable. So we can start the splitting operation again as long as we can, and we can show that whatever order is chosen for splits if

There are possibilities of choice, we obtain a unique decomposition of the cleavable pattern

A ooo B_ in the form

where K> 2, A _k CA, B _k CB, 1 <k <K and where each pattern A_ _k _ ooo B _j1 is no longer cleavable. This decomposition of the pattern A ooo B_ is called its total splitting and the A _k ooo B _k the splitting components.

M quasi-elementary orthogonal patterns:

The column vectors of an orthogonal matrix are orthogonal. A quasi-basic orthogonal pattern is called a pattern A such that for every line i, there exists at most one column k such that (i, k) E A. For 1 <k <MQ _1, we then write I _k the set of lines i such that (i, k) 6 A.

Since the column vectors of an orthogonal matrix are also unitary, two possibilities arise.

Either there exists a column index such that I _k is empty and in this case A = 0, ie all Ik, 1 <k <MQ are non-empty and we can denote / _& = {^, ₁ , ^, 2, - - •, ik, c _k }, where c ^> 1 is the number of elements of I _k - It is then easy to show that A can be written in the form

where * 2, i _> - • • '* M), i] ^es * ^an elementary orthogonal matrix as defined in 7.3. We then see that a quasi-elementary pattern is empty or is in the form of a solution.

s Definition of STAP:

We always assume in this section that M ₀ > 1 and N ₀ > M ₀ with gcd (Mo, N ₀ ) = 1 are given.

The acronym STAP designates a Suite of Transformations Applied to a Pattern. An STAP consists of a sequence Ti, T ₂ , T _k of transformations, possibly empty, each T _% being a rotation R ₁₀ with 1 <i <j <N ₀ or an offset Z ₁ with / C {1 ,. . . , N ₀ ] and a paraunitary pattern Aooo β or orthogonal A

Examples. An empty list of transformation and the perfectly reconstructed paraunitary pattern is called the perfect reconstruction STAP. Has a solution

S = T ₁ T ₂ . . . T _k U ₀ may be associated with the STAP consisting of the sequence [T _x , T ₂ ,. . . , T _k ) and χ5 orthogonal pattern A defined by A = {[i, j), [Uo] _t3 = 1} where U ₀ is an elementary paraunitary matrix. This second type of STAP is called an STAP solution.

If the pattern of an STAP is rotatable by the rotation R ₁ ^ then we say that the STAP is pivotable. We define the rotation on the STAP by

where A ', B' are defined in paragraph 5.8.1, and

Where A 'is defined in paragraph 5.8.1.

Similarly, if the pattern of an STAP is reducible in degree with a set of indices /, the STAP is said to be reducible in degree and one defines its shift with a paraunitary pattern by]

where A ', B' are defined in paragraph 5.8.1. Its offset with an orthogonal pattern ∞ is defined by

where A 'is defined in 5.8.1.

If the pattern of an STAP is cleavable, then we can write it as equivalent to a set of STAP whose patterns are the components of the cleavage, and we say that the STAP is cleavable and that we performed the splitting of STAP. Finally, if the pattern A of an STAP E is a quasi-elementary orthogonal pattern, which is empty or written in the form

is an elementary orthogonal matrix, then the STAP represents an empty set or equal to the STAP note QE [E)

defined by A '- {(i, j),

= 1} which is therefore an STAP solution.

First step of the general algorithm: For fixed values of MQ, NQ, m, one starts from the set I constituted by the STAP with perfect reconstruction and one constructs the set of STAP SQ by the following algorithm

ToDo = I

£ o = 0 as Todo φ 0 do {x = first (ToDo) ToDo = Todo - {x} if x then cleave ToDo - ToDo U split (x) otherwise if x pivotable then ToDo = ToDo U {rotation (a; }} else if x is reducible in degree then SQ = EQ U {X} else error "STAP not transformable"

} return SQ

In this algorithm x = fιτst (ToDo) means that we choose an element x in ToDo, the smallest for any order on ToDo, for example the first if ToDo is represented by a list. The SQ set consists of STAPs which are all reducible in degree. The set S _Q is the set of paraunit offsets of the elements of SQ. We then construct the set S _\ using the same algorithm where T is replaced by S _Q and SQ by S _\ and so on. The construction of S ₁ , i = 0,. . . , k is given by the following algorithm. from i = 0 to k make {if i = 0 then ToDo = 1 otherwise ToDo - {offset (: r), x G ^ _t _i}

as Todo φ 0 do {x = first (ToDo) ToDo = Todo - {x} if x then cleave ToDo = ToDo U split (x) else if x pivotable then ToDo = ToDoU {rotation (a;)} otherwise if x reducible in degree then E ₁ - E _% U {x} else error "STAP non transformable"

}} return {Eo,. . . , Sk)

The STAP sets O _n , n> 1 are the STAPs obtained from the STAPs of E _n - _\ by shifting their reducible patterns in degrees to orthogonal patterns. We then apply the following algorithm to transform O _n into a set of quasi-elementary STAPs, then a set S _n of STAP solutions.

ToDo = O _n

<S "= 0 as Todo φ 0 do {x = first {ToDo) ToDo ≈ Todo - {x} if x then cleave ToDo = ToDo U split (x) otherwise if x pivotable then ToDo = ToDo U {rotation (x }} else if x quasi-elementary then S _n = EQ U {QE (X)} else error "STAP not transformable"

} return S _n To the set S _n corresponds a set of solutions whose union is S _M0 , _{N0, m} , that is to say exhaustive. But, after this first step of the algorithm, this set of solutions has the following drawbacks:

One solution may be contained in another, that is, the set S _n is redundant.

- A solution may depend on a number of rotations strictly greater than its dimension.

The next step is to start from S _n to build a set of solutions no longer presenting these two disadvantages.

ιo Second step of the general algorithm:

Consider a solution S = Ti, T ₂ ,. . . , TkUo, depending on rotations R _{11 Jx,.} . . FL _1n ^ _n the application φs: (9χ, ..., θ _n ) → M (θχ, ..., θ _n ) where M (θχ, ..., θ _n ) is the matrix N ₀ x JW ₀ real obtained by affecting the angles θ _t , ι = 1,. . . , n to those transformations T ₁ which are rotations is called parametric representation of the solution. The application β βs can be considered as an application of H ⁿ in JR, ^ ^{0 M} ° and the maximum dimension of the tangent space in φ (θχ, ..., θ _n ) is reached almost everywhere on IR ". This maximum dimension is called the dimension of the solution and we denote dim (S '). We always have (Um (S) <n

It can then be shown that it is possible to keep the same solution by suppressing n - dim (S) rotations among the n contained in the initial solution. The algorithm for obtaining this reduction in the number of rotations is as follows: by calculating successively TkUo, Tk-χTkUo,. . . , Tχ. . . TkUo, one removes the transformations T ₁ which are rotations and do not increase of 1 dimension.

Given two solutions Sx: Ti. . . TkU ₀ and S ₂ : Ux. . . Ui VQ, with dim (Si) = k <25 I = dim (S2) and φs ₁ , φs ₂ their respective parametric representations, it is possible to demonstrate that one of the following cases is verified:

1. let φs ₁ (Εi ^k ) be contained in φs ₂ (1R ^ι ), in other words Sx CS ₂ ,

2. either φg * (φ _Sl (R ^k ) n φs ₂ BB!)) Is of zero measure in H *.

To show that 5i C 5 ₂ , we therefore choose a random value θ = (#i, ..., # _& ) of

M Sx parameters and we try to minimize the distance δ = min _α

when a - (ax, ..., ai) € H. In case 1, J = O while in case 2, δ> 0. A numerical global optimization software is used to calculate the value of δ and the optimization is started with several initial values for a. 5.8.2 Direct application to the case NQ = M ₀ + 1

The resolution algorithm, with its two steps, leads to the sets of solutions whose properties are described in Table 1 in the case of some values of (M ₀ , N ₀ , m) with N ₀ = M ₀ + 1.

The 2 solutions of the case Mo = 2, No - 3, m - 1 are, by way of example, provided in Table 2.

However, the general algorithm can be used also in cases where NQ> MQ + 1. For example for M ₀ = 3, No = 5, m = 1 we obtain that £ _3.5,1 consists of 4 given solutions in Table 3.

The algorithm presented here constitutes a considerable improvement of the one proposed in [4] since, for example, the set £ _4.5,2 is described by 61 solutions, a number which can not be reduced, while was using 13502 solutions.

5.8.3 Comprehensive enumeration of the solutions for the case Mo = 2 and TV ₀ = 3 The set of solutions obtained in this case can be characterized as follows.

Let m> 1. The m - 1 subsets A ₀₁ , a = 0,. . . , m - 2 of U ₂ , ₃ , _m defined by

are m - 1 distinct solutions of <S ₂ , ₃ , _m of dimension 2m and symmetric null support

{dl, 0, <74, m-1}.

They verify the relations of symmetry φ _m ^ _\ {A _a ) = _^ 4 _m _ ₂ - _α , a - 0,. . . , m - 2. If m is even, then ^ l _(m _ ₂₎ / ₂ ^ t is a symmetrical solution of £ ₂ , ₃ , ™ of dimension 2m. For m> 1 and a = 0,. . . , m - 2, we pose, for a <m - 3

TABLE 2 - Solutions for M ₀ = 2, N ₀ ≈ 3, m = 1.

TABLE 3 - Solutions for M _Q = 3, N ₀ ≈ 5, m = 1.

and

The are m - 1 distinct solutions of 2m dimension

and unsymmetrical zero support

The are m - 1 distinct solutions of <S ₂ , ₃ , "_> of dimension 2m and

unsymmetrical zero support

The following symmetry relationships are verified:

2.

We finally ask

and

Then D and E are two solutions of <S ₂ , ₃ , _m of dimension 2m, unsymmetrical null supports

5.8.4 Obtaining Maximum Dimension Solutions for Odd Mo and

For MQ <2, we define the elementary unitary matrix UM ₀ of dimensions M ₀ x (M ₀ + 1) by:

- If MQ is even, MQ = 2mo:

- If MQ is odd, M ₀ = 2mo + 1

Examples.

Is

Let the matrices NQ X NQ be defined, for odd MQ, by

The solution S _M0 defined by

is a solution of S _M0 , N ₀ , m- This solution is symmetrical and of dimension TΠ {NQ / 2) ² .

We conjecture that the solution S _M0 of the previous theorem is the unique solution of s S _M0 , _No , _m of maximum dimension m (No / 2) ² for odd Mo.

5.8.5 Obtaining Maximum Dimension Solutions for Pe M and N ₀ =

M ₀ + 1

Let them be

NQ X JV ₀ matrices defined, for peer Mo, by

10

For e

"Then

distinct solutions of

and likewise null support. The following symmetry relations are verified

We conjecture that the solutions A _no , _α>

of Theorem 2 are of maximum dimension

and when m is even,

is the only symmetric solution of maximal dimension in S _M0 , _N0 , _m - If m is odd, there is no symmetrical solution of maximum dimension. 5.8.6 Obtaining Minimal Dimension Solutions for Any Even Peer and

For Mo> 1 and No = Mo + 1, we define the unitary rectangular matrix UQ NQ X MO by [Uo] i _j = 1 if i = j and [UQ] ^ J = 0 $ ii φ j. The set of matrices B is defined by

So for m> 1, all the matrices

defined by

is a solution in S _{M0, N0, m} dimension mmole We conjecture that this dimension MMO is the minimum dimension of the solutions in "SM .M _{O O} + I, ™ when this assembly is described as a union set of irreducible solutions.

6 APPENDIX F: Examples

In what follows we detail the particular case of a filter bank system for the parameter values Mo = 5, JV ₀ = 6, Δ = 12, m = 4. This calculation corresponds to that required for an OFDM system. oversampled in a 6/5 s ratio and including 60 subcarriers, which translates to a prototype length filter

L = 1440.

In the case of this example, we need only apply the following method. For k, I with 1 <k <I <JV ₀ , we define the rotation matrix R _k i (θ) of dimension JV ₀ x JVo and angle θ in the plane (k, I) by

, _J io In the illustrated patterns, a R _k ι rotation (θ) will be represented as in Figure 5.

Finally Zjv ₀ is the diagonal square matrix of dimensions JVo x JV ₀ whose diagonal terms are equal to 1 except on the last line whose term is z ^{~ x} .

For MQ> 1 e £ JV ₀ = Mo + 1, we define the unit matrix UQ of size NQ X MQ 15 by [t7o] ', _j = 1 if i = j and [Uo] ij = 0 if i φ j. The set of matrices B is defined by

For this set equality the rotations receive all the possible values for the angles.

Then, according to 5.8.6, for m> 1, the set of matrices S ^ _{0 m} defined by

is an IUMQ dimension solution. M This means that for any value of Δ a prototype P (z) defined by

and if for all i = 0,. . . , Δ - 1,

where the matrices V ^w (z), i = 0,. . . , Δ - 1

check the equation (47)

where the matrices U ^ (z), i = 0,. . . , Δ - 1 belong to the solution S ^ _n , whereas P (z) is perfectly reconstructed.

Here Ti (Z) is the diagonal matrix 5 x 5, non-causal, having as diagonal elements 1, z, z ² , z ³ , z ^A , z ^{~ x} and T ₂ (z) the diagonal matrix 4x4 of diagonal elements 1, z ^{~ ι} , z ^{~ 2} , z ^{~ 3} . Since, for i fixed, £ / ^w (z) is in S $ _om , we have (111) and (105), where B denotes a matrix dand B for which the angles have been fixed.

The matrix B (z) defined by

wherein the angles of rotation matrices R _{I, M0 + I} are not indicated. The equalities

then allow to write

The equality (114) is shown in the diagrams of embodiments of FIGS. 6A and 6B and the diagram of embodiment of the blocks V 1 (z) of FIG. 3 is given in FIG. 7. The mM ₀ = 20 rotations present in FIG. the diagram of Figure 7 correspond to 20 different angles. Therefore in Figure 3 are present mM ₀ A = 240 angles.

An example of such a prototype filter optimized for out-of-band energy, in the case of a non-linear phase, is shown in FIGS. 8A and 8B. 7 APPENDIX G: Bibliography

References

[1] A. Paulraj and T. Strohmer. Method for drawing shape for ofdm. Patent WO 2006/004980, 2006.

[2] R. Hleiss, P. Duhamel, and M. Charbit. Oversampled OFDM Systems. In Proc. International Conference on Digital Signal Processing, Santorini, Greece, July 1997.

[3] C. Siclet and P. Siohan. A method of transmitting a multicarrier signal associated with synchronously modulated carriers, modulation and demodulation methods, and corresponding devices Patent Application PCT / FR 02/00329, January 2002.

[4] C. Siclet, P. Siohan, and D. Pinchon. Perfect reconstruction conditions and design of oversampled DFT modulated transmultiplexers. Eurasip Journal on Appled Signal Processing, 2006. Article ID 15756, 14 pages.

[5] Mr. I. Doroslovacki. Product of second moments in time and frequency for discrete time and uncertainty. Signal Processing, 67 (1), May 1998.

Claims

1. Device for demodulating an oversampled multicarrier signal, characterized in that it comprises Δ stages, Δ being an integer greater than or equal to 1, fed by received data representative of source data, each of said stages comprising a filter bank analysis with no branches and a decimation factor equal to M _Q , with N ₀ and M ₀ prime between them and M _Q > 1, an analysis filter bank comprising a decomposition matrix V ^ (z) and a Fourier matrix F ^ ₀ , and in that said decomposition matrix associated with a bank of analysis filters is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said decimation factor M ₀ and said number of branches N ₀ , so as to allow a perfect reconstruction of said source data, and in that said predetermined criterion forces to zero the coefficients of the decomposition matrix V ^ \ z) of size N ₀ x M ₀ , according to the following equations:

with:

m is an integer greater than or equal to 1;

() () β ()

- β (k, l) an integer such that:

- (u, v) a pair of integers such that UM _Q - VN ₀ ≈ 1, with | "| <N ₀ and | v | <M ₀ ; with 0 ≤ k ≤ N ₀ - 1 and 0 ≤ / ≤ M ₀ - 1 and τem (a, b) an operator defining the remainder of the division of the integers a by b.

2. Demodulation device according to claim 1 characterized in that said decomposition matrix consists of m identical matrices B, each matrix B consisting of at least one rotation and at least one offset, m being greater than or equal to 1.

3. Demodulation device according to claim 1, characterized in that N ₀ is equal to M ₀ + 1.

4. Demodulation device according to claim 3, characterized in that the number of rotations of a matrix B is equal to M _Q.

5. Demodulation device according to claim 1, characterized in that the number of stages Δ is an even integer, greater than or equal to 2.

6. Demodulation device according to claim 1, characterized in that

Said stages supply first permutation means P comprising ΔN _Q inputs and ΔNo outputs, supplying N ₀ Fourier matrices F i, which themselves feed second permutation means P.

7. A method for demodulating an oversampled multicarrier signal, characterized in that it implements Δ stages, where Δ is an integer greater than or equal to 1, fed by received data, representative of source data, each of said stages implementing implement an analysis filter having N branches ₀ and a decimation factor equal to M _Q, where N is ₀ and M first and M _Q them _Q> 1, comprising the following steps: multiplication by a decomposition matrix V (z) and multiplying by a Fourier matrix F ^, and in that said decomposition matrix associated with an analysis filter bank is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said factor of decomposition. decimation M _Q and said number of branches N ₀ , so as to allow a perfect reconstruction of said source data, and in that said predetermined criterion forces to zero coeffici ents of the decomposition matrix V ^ι \ z) of size N ₀ x M ₀ , according to the following equations:

with:

m is an integer greater than or equal to 1;

β - β (kj) an integer such that:

- (w, v) a pair of integers such that UM _Q - VN ₀ »1, with |« | <N ₀ and | v | <M ₀ ; with 0 ≤ k ≤ N ₀ - I and 0 ≤ / ≤ M ₀ - 1 and rem (α, è) an operator defining the remainder of the division of the integers a by b.

8. Demodulation method according to claim 7, characterized in that said decomposition matrix consists of m identical matrices B, each matrix B consisting of at least one rotation and at least one offset, m being greater than or equal to at 1, and in that said method comprises a preliminary phase for determining said matrices B, so that said matrices B are written in the following form:

where ZH ₀ is a diagonal square matrix providing an offset and / ?, Λ / O + I ^a rotation matrix.

9. Computer program comprising instructions for implementing the demodulation method according to claim 7 when said program is executed by a processor.

10. A device for modulating an oversampled multicarrier signal, characterized in that it comprises Δ stages, where Δ is an integer greater than or equal to 1, fed by source data, each of said stages comprising a synthesis filter bank having N ₀ branches and an expansion factor equal to M _Q , with N ₀ and M ₀ first between and M ₀ > 1, a synthesis filter bank comprising a conjugate Fourier matrix Fχ _Q and a conjugated decomposition matrix V ^ (z), and in that said conjugate decomposition matrix associated with a filter bank of analysis is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said expansion factor M ₀ and said number of branches N ₀ , so as to allow a perfect reconstruction of said source data, and in that that said predetermined criterion forces to zero the coefficients of the conjugate decomposition matrix V (z) of size M ₀ X No, according to the following equations:

with

where T corresponds to the transposition operation, with:

m is an integer greater than or equal to 1; f ^{w *} l

- β (k, l) an integer such that:

- (M, V) a pair of integers such that "M _o - VN ₀ = 1, with | w | <N ₀ and | v | <M ₀ ; with 0 ≤ £ ≤ N ₀ - 1 and 0 ≤ / ≤ M ₀ - 1 and τem (a, b) an operator defining the remainder of the division of a by b.

11. A method of modulation of an oversampled multicarrier signal, characterized in that it implements Δ stages, Δ being an integer greater than or equal to 1, powered by source data, each of said stages using a synthesis filter having N branches ₀ and an expansion factor equal to M _0, N is ₀ and M ₀ and M coprime _Q> 1, comprising the following steps: multiplication by a matrix Fourier conjugate F ^ ₀ , - multiplication by a conjugate decomposition matrix V ^ (z); and in that said conjugate decomposition matrix associated with an analysis filter bank is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said expansion factor M _Q and said number of branches. N _Q , so as to allow a perfect reconstruction of said source data, and in that said predetermined criterion forces to zero the coefficients of the conjugate decomposition matrix V ^ '' (z) of size M ₀ x No _" according to the following equations :

where T corresponds to the transposition operation, with:

m is an integer greater than or equal to 1;

- k) - β (k, l);

- β (k, l) an integer such that:;

- (M, V) a pair of integers such that κM ₀ - VN ₀ ≈ 1, with | w | <N ₀ and | v | <M _Q ; with 0 ≤ k ≤ N _Q - I and O ≤ l ≤ M _Q - I and rem (α, è) an operator defining the rest of the division of a by b.

Computer program comprising instructions for implementing the modulation method according to claim 11 when said program is executed by a processor.

13. System for transmitting an oversampled multicarrier signal, characterized in that it comprises:

a modulation device comprising Δ stages, Δ being an integer greater than or equal to 1, fed by source data, each of said stages comprising a synthesis filter bank having N ₀ branches and an expansion factor equal to M ₀ , with N ₀ and M ₀ prime between them and M ₀ > 1, a synthesis filter bank comprising a conjugate Fourier matrix F _{# 0} and a conjugate decomposition matrix V (z), said conjugate decomposition matrix associated with a bench of analysis filters is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said expansion factor M _Q and said number of branches N _Q , said predetermined criterion forces to zero the coefficients of the conjugated decomposition matrix V ^ ¹ '(z) of size M ₀ x No _> according to the following equations:

z ^{~ m} , where T corresponds to the transposition operation, with:

m is an integer greater than or equal to 1;

- (M, V) a pair of integers such that UM _Q - VN ₀ = 1, with | «| <N ₀ and | v | <M ₀ ; with 0 ≤ k ≤ N ₀ - 1 and 0 ≤ / ≤ M ₀ - 1 and τem (a, b) an operator defining the remainder of the division of a by b, and - a demodulation device comprising Δ stages powered by received data, representative of the data source, each of said stages comprising an analysis filterbank having ₀ N branches and a decimation factor equal to M _Q, an analysis filter bank comprising a decomposition matrix V ^ '' (z) and a Fourier matrix F ^, and in that said decomposition matrix associated with an analysis filter bank is a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said factor of decimation M ₀ and said number of branches N ₀ , said predetermined criterion forces the coefficients of the conjugated decomposition matrix V ^ (z) of size M ₀ x No _{> to zero} according to the following equations:

where T corresponds to the transposition operation, with: - m an integer greater than or equal to 1;

- β (k, l) an integer such that:

- (w, v) a pair of integers such that UM _Q - VN ₀ = • 1, with | «| <N ₀ and | v | <M ₀ ; with 0 ≤ k ≤ N _Q - 1 and O s / ≤ M _Q - I and τem (a, b) an operator defining the remainder of the division of a by b, so as to allow a perfect reconstruction of said source data.

14. Device for demodulating an oversampled multicarrier signal, implementing a prototype filter implemented in its polyphase form, characterized in that it comprises:

- Δ analysis filter banks, Δ being an integer greater than or equal to 1, fed by received data representative of source data after decimation according to a decimation factor Δ and application of a delay, each of said filter banks of analysis presenting N ₀ branches, a decimation factor equal to M _Q , with No and M _Q prime between them and M _Q > 1, a decomposition matrix

of size No x MQ and a Fourier matrix F ^ _n of size No x No. said decomposition matrix associated with a bank of analysis filters being a para-unitary matrix, some coefficients of which are forced to zero according to a predetermined criterion taking into account said decimation factor M _Q and said number of branches N ₀ ,

means for permutation of the data obtained at the output of said banks of analysis filters, delivering permuted data, No Fourier matrices F ^ of size Δ x Δ, transforming said permuted data into transformed data,

means for permutation of said transformed data, delivering demodulated data enabling perfect reconstruction of said source data.