WO2003015270A2 - Method for determining filtering coefficients of a modulated filter bank, prototype filter, modulated filter bank, terminal and corresponding uses - Google Patents

Abstract

The invention concerns a method for determining filtering coefficients of a modulated filter bank, based on a prototype filter of length L designed to produce a modulated system with M sub-bands, comprising the following steps: (a) determining, for at least two low values of M, based on a first parameter space P, of a set of first filtering coefficients, for at least a predetermined criterion to be optimised; (b) analysing said set(s) of first filtering coefficients, so as to determine a subset of corresponding filters, with which is associated a second parameter space R, enabling another representation of said filters; (c) optimising a second set of filtering coefficients, for at least two values of M and at least one value of L, using said second parameters space R; (e) storing at least one of said second sets of filtering coefficients.

Description

 Method for determining the filter coefficients of a modulated filter bank, prototype filter, modulated filter bank, terminal and corresponding application. Y. Field of the invention The field of the invention is that of digital filtering, and more precisely of modulated filter banks. In particular, the invention relates to the determination of the prototype function of such modulated filter banks.

Modulated filter banks constitute a particularly important family of the family of filter banks. Their two main advantages are: the possibility of implantation using fast algorithms using polyphase decompositions and fast transforms of DCT type ("Discrète Cosine Transform", in French: transformed into discrete cosine) for what concerns benches modulated in cosine, or DFT type ("Discrete Fourier Transform", in French: discrete Fourier transform) for benches modulated by complex exponentials; a design method which for a given type of transform (DCT or DFT) involves only one or two prototype filters. Their applications are potentially very numerous.

They are in fact suitable for multi-carrier reception, transmultiplexers, coding in sub-bands, ...

Thus, in the field of digital communications, the applicants of this patent application have highlighted the formalism of modulated filter bank for multicarrier modulation systems using for each carrier an amplitude modulation in quadrature with offset (OQAM). In the biorthogonal case [1, 2] (to simplify the reading, the various references cited are grouped in appendix 9), the BFDM / OQAM modulation, (BFDM: "Biorthogonal Frequency Division Multiplex") and in the orthogonal case [3] of OFDM / OQAM (OFDM: "Orthogonal Frequency Division Multiplex"). In these various works, it is also demonstrated that the problem of calculating the prototype filter (s) is identical to that encountered for orthogonal or biorthogonal sub-band coding systems. Another potential application therefore relates to bit rate reduction audio coding with possible possibilities of improvement, in orthogonal [4], or in biorthogonal [5], of a standard of MPEG1 type (Moving Picture Expert Group) [6] . 2. State of the art

The basic principles of modulated filter banks are described in particular in [7, Chapter 8]. It is shown in particular that the design of the prototype filter leads to a non-linear problem whose complexity increases with the length of the prototype.

In the examples presented in this reference, the maximum length treated is 110. However, it is well known that, for multicarrier transmission systems, the number of carriers can be more than a hundred, or even a few thousand, and that the length of the prototype is, in general, a multiple ranging from 1 to 4 of the number of carriers. It therefore appears necessary to have rapid and efficient calculation methods.

We now recall different categories of solutions that can be envisaged to solve this problem. We consider here the most frequent case which is that where the shaping of the signal is carried out by a prototype filter of type RIF (Finite Impulse Response).

2.1 Category 1: Use of continuous functions

In this case the prototype filters are obtained by truncation of the support in time then discretization of continuous functions which verify the condition of orthogonality in continuous time on a support of infinite duration.

A defect of this technique is that, in discrete, the orthogonality is checked only approximately, and a prototype considered as optimal in continuous time for a given criterion is no longer necessarily it in the discrete case. We have for example implemented this technique within the framework of the use of the IOTA function (Isotropic Orthogonal Transform Algorithm) [9] for the realization in digital form of an OFDM / OQAM modem. In this case, we rely on functions which form an orthogonal basis in continuous time and for which we conjecture that they constitute an optimum for time-frequency localization.

In the case of digital processing, the orthogonality is no longer verified and the location of the discrete prototype is not directly optimized.

22 Category 2: Optimization of prototypes with near-perfect reconstruction

We then make sure that the so-called “back-to-back” filter bank, for example the modem taken in isolation for a transmission system, checks approximately. perfect reconstruction condition.

According to this approach, the reconstruction is therefore not perfect and the cost of optimizing the coefficients can become prohibitive for long lengths.

For example, in audio coding, the MPEG1 filter bank, layers 1 and 2, has been calculated according to this principle [6]. It does not allow to obtain a perfect reconstruction. In more recent standards, the recommended filter banks must be perfectly orthogonal. 23 Category 3: Orthogonalization of RIF prototypes

The discrete counterpart of the continuous orthogonalisation as it is produced by the IOTA process is based on a so-called Zak transform. From an initial prototype of given length, we obtain a prototype of double length which checks the conditions of orthogonality. However, in this case, the calculation takes into account only the orthogonality constraints and does not really allow optimization of a precise criterion.

This method has been used for example by the ENST to synthesize prototypes of great length for OFDM / OQAM systems [10]. 2.4 Category 4: Optimization of orthogonal RIF prototypes and biorthogonal

It is both a question of optimizing a given criterion and of verifying exactly biorthogonality or orthogonality conditions. Most often, the filter banks calculated in this way use parametric representations, for example in the form of a scale (in biorthogonal) or lattice (in orthogonal), which by construction will guarantee that the reconstruction is perfect.

It is then necessary to solve a nonlinear optimization problem, with constraints if we consider the transverse representation of the prototype and without constraints for representations of the ladder or lattice type. The difficulty is linked to the large size of certain problems, for example systems

OFDM / OQAM or BFDM / OQAM with several hundred or thousands of carriers.

We are thus confronted with this problem during the design of biorthogonal and orthogonal prototypes of the BFDM / OQAM [11, 2] and OFDM / OQAM [3] systems. A more general parametrization than the trellis, because it also applies to oversampled orthogonal systems, is proposed in reference [10], but it does not in any way simplify the problem of synthesis in the case of so-called critical sampling.

A similar problem also arises in the context of sub-band coding [5], [1]. 3. Objectives of the invention

The invention relates more particularly to the last category of solution. An objective of the invention is to make it possible to synthesize prototypes with several hundreds or thousands of coefficients.

Compared to the categories of solutions 1 and 3 which already achieve this objective, it should be remembered that:

- Category 1 solutions cause losses of orthogonality which are all the more important when one tries to reduce the length of the prototype for a given number of carriers, or sub-bands.

- Category 3 solutions do not precisely optimize a given criterion. The invention particularly aims to overcome these drawbacks of the state of the art.

More specifically, an objective of the invention is to provide a method for calculating prototype filters, as well as prototype filters and corresponding modulated filter banks, which are quasi-optimal for predetermined optimization criteria.

An objective of the invention is thus to provide such prototype filters, the lengths of which are clearly greater than the lengths which can be processed hitherto by known optimization methods. For example, in OFDM / OQAM, it is necessary to calculate prototypes for systems comprising more than 2048 carriers. Similarly, in BFDM / OQAM (where the 2048 length cap for a system with 512 carriers was difficult to reach according to the prior art), the invention aims in particular to allow the processing of systems of length 8192 with 2048 carriers , for example in a reasonable calculation time.

Another objective of the invention is to ensure, for a given prototype length, incomparably shorter computation times than with known optimization methods.

The invention also aims to provide such prototype filters, which can be implemented easily and efficiently in terminals.

These objectives, as well as others which will appear subsequently, are achieved according to the invention using a method for determining the filter coefficients of a modulated filter bank, based on a prototype filter of length L intended to produce, by modulation, a modulated system with M sub-bands, comprising the following steps:

(a) determination, for at least two low values of M, as a function of a first space of parameters P, of a set of first filtering coefficients, corresponding to a global optimum of the corresponding problem, denoted {P, M, L}, for at least one predetermined criterion to be optimized; (b) analysis of said set or sets of first filter coefficients, so as to determining a subset of filters corresponding to said optimums, to which a second space of parameters R is associated, with card R ≤ card P, allowing another representation, called compact, of said filters; (c) optimization of a second set of filter coefficients, for at least two values of M and at least one value of L, using said second space of parameters R, corresponding to a global optimum of the corresponding problem, noted {R, M, L};

(e) storing at least one of said second sets of filter coefficients. In this way, as will be seen later, very long prototype filters (for example a few thousand) can be efficiently determined in a very short time, compared to known techniques.

Advantageously, the method for determining filter coefficients of the invention further comprises a step:

(f) rapid interpolation of a set of filter coefficients for any value of M, from the one or more said second sets of stored coefficients.

Preferably, in said step (c), said optimization is calculated for at least two values of M which are powers of 2.

According to a characteristic of the invention, said optimization of said step (c) is calculated by taking into account at least one fixed value for at least one annex parameter A.

The said additional parameter (s) A may in particular include a fixed deadline.

Advantageously, in said step (b), said analysis takes account of at least one of the properties belonging to the group comprising: the linearity of the phase; the continuity of the parameters from one polyphase component to another; belonging to a predetermined range of angular parameters; - continuity and / or derivability properties of limit curves, when the length tends to infinity. Preferably, the method of the invention comprises, after said step (c), a step of:

(d) validation of the results of the resolution of said second problem, for at least a low value of M.

The method for determining the filter coefficients of the invention also advantageously comprises, after said step (f), a step of: (g) checking and / or optimizing the interpolated values for the corresponding problem {P, M, L} or {R, M, L}. The one or more predetermined criteria to be optimized may in particular belong to the group comprising: location in time and / or in frequency; minimization of out-of-band energy. Advantageously, in said step (b), at least one function, called code, is implemented, associating with an optimal prototype filter P (z) an approximate representation P _x (z) of said optimal prototype filter P (z), expressed using said second set of parameters.

According to a preferred aspect of the invention, said modulated filter bank is used for the implementation of a multicarrier modulation system. Said multicarrier modulation is an OFDM / OQAM modulation or a BFDM / OQAM modulation.

In the case of an OFDM / OQAM modulation, said first space P is advantageously the space of the angular coefficients:

with card (P) = m * M / 2 where: 2M is the number of subcarriers of said modulation; L = 2mM is the initial length of said prototype filter. In this case, said second space preferably belongs to the group comprising: - a space R _κo generated by the functions generating the points _M (), 1 = 0, ..., (M / 2) -l, with u (l) a bijection from {0, (M / 2) -l} to {0, 1/2}; a space R _K1 corresponding to the subspace of polynomials of degree less than or equal to Kl, K>0; - a space R _K2 corresponding to the subspace of polynomials of degree less than or equal to Kl, K> 0, in the base of the Chebyshev polynomials. According to various advantageous embodiments, said codes are respectively: - for said space R _κo : code n ° 0:

for said space R _K1 : code n ° 1:

for said space R _κ2 : code n ° 2: cosθf = Φ (^ 7I (4 <M /) - l)).

In a particular embodiment, said code No. 2 can be used with degree 8.

Preferably, said interpolation step (f) implements a sub-step of searching for a polynomial of degree in log ₂ M approaching, in the least squares sense, all of the values already stored.

In the case where the modulation is a BFDM / OQAM modulation, said first space P is advantageously the space of the coefficients in scale such that

with card (P) = (2m + l) / 2 where: 2M is the number of subcarriers of said modulation;

L = 2mM is the initial length of said prototype filter. In this case, said second space advantageously belongs to the group comprising: a space R _κl consisting of polynomials of degree less than or equal to K1, K> 0, represented by their coefficients; a space R _K2 consisting of polynomials of degree less than or equal to Kl, K> 0, represented by their Chebyshev development; a space R _κ3 made up of polynomials of degree less than or equal to Kl, K> 0, satisfying

; a space R _κ4 made up of polynomials of degree less than or equal to Kl, K> 0, satisfying

represented by their development of Chebyshev. According to different embodiments, said codes can be one of the following codes: code n ° 0: the coefficients x ^l _j , l = ^), ..., -, j = 0, ... 2m are the coefficients a ^l _{J + 1} , l = 0, ....- ^ = 0, ... 2m for said space R _κl : code n ° 1:

k = 0 - for said space R _K2 : code n ° 2: a ^l _{J + 1} = ^ x * Tk (4φ _M (l) -l), j = 0, ... 2m

A = 0 for said space R _K3 : code n ° 3:

k = 0 for said space R _K4 : code n ° 4:

Advantageously, said optimization step (c) implements a local optimization algorithm and / or a genetic global minimization algorithm.

Preferably, said prototype function is described using lattice representations, or in scale, in the form of a function a (t) such that: for code n ° 1:

where d is the degree considered and a _t , i = 0, ..., dl the corresponding d coefficients; for code 2:

where the Tι are the Chebyshev polynomials; for code n ° 3:

According to a particular aspect of the invention, said filtering coefficients are advantageously obtained, for all> 0, using the equation: P _a (z) = z- ^2sM H (z) + ^ r ⁱ K _i (z ^2M ) where H (z is a filter of length M satisfying ξ _mod (H) = 1, ξ _mo being a function delivering a measure of discrete localization of a filter; and α is such that ^" ξ, _moά (P ^)> 1-ε.

The invention also relates to: - the prototype filters for modulated filter banks, the filter coefficients of which are obtained using the method of determining the filter coefficients described above; the modulated filter banks formed from a prototype filter, the filter coefficients of which are obtained using the method for determining the filter coefficients described above; the terminals for receiving a multicarrier signal, implementing a bank of modulated filters formed from a prototype filter, the filter coefficients of which are obtained using the method of determining filter coefficients described above.

The invention also relates to the applications of a modulated filter bank formed from a prototype filter, the filter coefficients of which are obtained using the method of determining the filter coefficients described above in at least one of the domains belonging to the group including: BFDM / OQAM type multicarrier modulation; OFDM / OQAM type multicarrier modulation; coding in sub-bands. 4. Preferential embodiments Other characteristics and advantages of the invention will appear more clearly on reading the following description of preferential embodiments of the invention, given by way of simple illustrative and nonlimiting examples, and of the figures. appended, among which: FIG. 1, commented on in appendix 1, illustrates the combined "lifting" and "dual lifting" stages, without increasing the delay, corresponding to a representation in scale for a BFDM / OQAM transmultiplexer; FIG. 2, also commented on in appendix 1, presents the stages of "lifting" and "lifting dual" combined, with increase in delay, also retaining the property of biorthogonality; FIGS. 3A to 3D show the coefficients Θ \ (as a function of

/ + 1), for ξ _mod , for M = 32 and respectively m = 1, ..., 4, and 0 ≤ i

≤ m-1 for the optimization of angular coefficients in the case of OFDM / OQAM; - Figures 4A to 4D show the coefficients θ \ (as a function of +1), for ξ _mod , for M = 128 and respectively m = 1, ..., 4, and 0 ≤ i

≤ m-1 for the optimization of angular coefficients in the case of OFDM / OQAM; FIGS. 5A to 5D present the coefficients θ \ (as a function of / + 1), optimal for out-of-band energy, for M = 32 and respectively m = 1, ..., 4, and 0 ≤ i ≤ m -1 for the optimization of angular coefficients in the case of OFDM / OQAM; FIGS. 6A and 6B illustrate the responses of a prototype filter w optimal filters for code 2 of degree 8; - figure 14, also commented in appendix 6, shows the error relative committed on the best out-of-band energy with the OFDM prototype filters of length L = 2M constructed from the parameters of the optimal filters for code 2 of degree 8; FIG. 15, also commented in appendix 6, is a representation with support in [-1, +1] of the 2M transverse coefficients of the optimal filters for the location for M = 32 (cross) and M = 1024 (solid line); FIG. 16, also commented in appendix 6, shows the relative error made on the best out-of-band energy with the OFDM prototype filters of length L = 2M constructed from θ _ε (f); FIG. 19, also commented in appendix 6, shows the relative error made on the best out-of-band energy with the OFDM prototype filters of length L = 4M (m = 2) constructed from the parameters of the optimal filters for code 2 of degree

8, for M being 128 and 1024; FIG. 20, also commented in appendix 6, shows the relative error made on the best out-of-band energy with the BFDM prototype filters of length L = 4M (m = 2) constructed from the parameters of the optimal filters for code 2 of degree

8, for M being 128 and 512; Figure 21, also commented in appendix 6, shows the relative error made on the best out-of-band energy with the BFDM prototype filters of length L = 6M (m = 3) constructed from the parameters of the optimal filters for code 2 of degree

8, for M being 128 and 512; Figure 22 is a simplified flowchart showing the general principle of the method of the invention. 4.1 General presentation The invention therefore relates in particular to a method for rapidly calculating a prototype filter with real coefficients, noted p (n), where for a filter of length L, n is an integer between 0 and Ll. To be admissible in the sought-after category of solutions, the coefficients p (n) must optimize a given criterion and verify biorthogonality or orthogonality conditions. In appendix 1 we present in detail two criteria (time-frequency localization and minimization of out-of-band energy) that can be retained, as well as the biorthogonality conditions which apply to different families of modulated filter banks.

To recall the application context of these preferred embodiments, we briefly present, in appendix 2, the principle of obtaining a BFDM / OQAM transmultiplexer based on the modulation of a prototype p (n).

42 The parameter reduction method

The method of the invention, called parameter reduction, applies to a very large set of modulated filter banks. It can be suitable, in particular, for any system for which equations (10) to (13) (appendix 1) constitute sufficient conditions for the property of perfect reconstruction: modulated banks of type DCT II and IV and of type MDFT I and II, oversampled BFDM / OQAM and BFDM / QAM modulators [17], etc.

One can also consider applying it in the case where the prototypes of the analysis and synthesis benches differ [5, 16]. First, we describe the process used in a very general context. We then explain how it applies to BFDM / OQAM and OFDM / OQAM, and we finally deduce restrictions which may possibly exist in contexts different from those treated. 4.2.1. General principle

We take as a starting hypothesis that the function to be optimized, for a given criterion, is a prototype filter of length L which by modulation will produce a system modulated with M sub-bands.

For such a prototype, the method of reducing parameters is validated by the succession of steps which follow, illustrated by the flowchart in FIG. 22. 1. Optimization 11 on a set of parameters, noted P, in principle in sufficient number, ie card P ≥ number of degrees of freedom, to obtain the global optimum of the problem, noted {P, M, L) , for the chosen criterion, and analysis of optimal solutions for low values of M. 2. Interpretation 12 of the properties of the solutions obtained as belonging to a subset of filters. In these subsets, we choose approximate subspaces of finite dimension where the filters can be represented with a reduced number of parameters.

This representation is called “compact” in this patent application. The properties considered of the optimal filters obtained during the first step can be for example:

- the linearity of the phase;

- the continuity of the parameters from one polyphase component to another; belonging to a limited range of angular parameters (for example [0, π]);

- the properties of continuity, of derivability of limit curves when the length tends towards infinity.

This results in a representation with a new set of parameters, noted R, deduced from P and which is said to be reduced when cardR <cardP. 3. Optimization 13 of the prototype with the new set of parameters R, for given values of L, M (for example the powers of 2 for the values of

M) and, optionally, values also fixed for a set of other additional parameters, denoted A (including for example the delay), making it possible thereafter to cover all the possible solutions of the problem. Verification 131, for the small values of M, that the resolution of this problem, noted {R, M, L), produces an undegraded optimum, compared to that obtained for the problem {P, M, L ^} .

4. Storage 14 of the best results obtained for a set of values of

M, for example the powers of 2, and for a set of fixed values of A, when solving the different problems (R, M, L). Interpolation fast 141, without optimization, of the results for all the values of M and verification 142 of the results with optimization of the corresponding problem of type iP, M, Ω or {P, M, L}.

A set R of parameters which makes it possible to validate steps 1 to 4 constitutes a compact representation which satisfies the conditions required for our method of reducing the space of parameters.

This method is now illustrated in the case of OFDM / OQAM and in that of BFDM / OQAM. The optimization criteria used to do this are those of time-frequency location and out-of-band energy described in appendix 1.

In addition, it will be noted that, the problems to be solved being non-linear, we used several optimization software, local and global, to validate these results.

4.2.2 Case of OFDM / OQAM We first present the case of OFDM / OQAM for which we have already validated the entire 4-step procedure in Figure 22, for prototypes of length L = 2m.

4.2.2.1 Optimization of the angular coefficients in the space P One chose a space of initial parameters which is that most usual for this type of problems, and which is introduced in appendix 1 in paragraph 2.2, that is to say that of the angular coefficients {θfp≤i≤m-1 P≤l≤ψ- 1}. In this case we have card () = mxM / 2 = number of degrees of freedom. The optimization problems to be solved are therefore without constraints and perfectly guarantee the orthogonality as well as the phase linearity of the prototype. The observation of the angular parameters obtained during a direct optimization shows, as illustrated by FIGS. 3 to 5, that these, after reduction by the period 2π, can always be chosen in the interval [0, π ]. On the other hand, the parameters which correspond, that is to say those associated with the same index i for a given set of polyphase components l, evolve continuously. 4.2.2.2 Representation in compact form

The objective is to obtain an approximate representation P _x (z) of the optimal prototype filter P (z) using a reduced set of parameters {xf ≤k≤Kp≤i≤ml}. The prototype filter P (z) is said, in the present patent application, coded by the (f) _k , ₀ , .. jc-i o _r ..jn-ι ^and the application

is called a code. K is called the degree of the code.

The OFDM prototypes thus coded are coded using mK parameters, and they are said to be represented in compact form. It is generally impossible to represent a prototype filter in compact form for a given code. However, the observation of the properties of the optimal filters, for a given criterion, shows that they belong to subsets of prototype filters

OFDM can be represented in compact form for a certain code.

In this case, the search for optimal filters, for longer lengths, can be done in the set of filters that can be represented in compact form for this code. The number of parameters for the optimization problem is considerably reduced. For a given subspace R _k , the choice of a base B = (p _} , ..., p _κ ) determines the choice of variables in the optimization problem.

The behavior of the optimization programs makes it possible to highlight a possible bad conditioning. It is then possible to change the base to try to improve the packaging: this therefore leads to a new code.

Appendix 3 presents three examples of codes that can advantageously be used. Of course, other codes can be considered. 4.2.2.3 Optimization of the compact form

The choice of a compact form given by the code 1 or 2, with <4 [- necessarily leads to a reduction of dimension of the space of the parameters and therefore to a significant reduction of the computation time. Many optimizations of this type are described in documents [14, 18]. Table 1 (all the tables in this patent application are grouped in the appendix 8) constitutes an illustration of a set of results obtained, for the location criterion, with code 2 and local optimization software called CFSQP (Feasible Sequential Quadratic Programming in C) [19].

In this table Test the execution time in seconds on a RISC6000 at 200 MHz, N _f is the number of calls to the objective function. It is clear that the gains in speed only exist for M ≥ 16, that is to say from the moment one begins to need it.

In comparison the results of table 2 obtained with the trivial code clearly show that, except for the low values of M where the smaller number of calls favors this code, the difference in time increases in an almost exponential manner with M, while l reduced space for parameters has very little impact on the quality of the results.

The examples of FIGS. 6 and 7 illustrate forms of responses typical of optimal solutions respectively for the location and out-of-band energy criteria.

4.2.2.4 Rapid interpolation as a function of M

The comparison of the relative efficiency of two codes available for the optimization of localization or energy shows that the best results are obtained globally for code number 2 with degree 8. We therefore retained these parameter values for generate a quasi-optimal set (R _κ , M, L) of results.

In this set, the values of m are the integer values between 1 and 4 and those of M correspond to all the powers of 2 of the integers between 3 and 12. Examination of the angular coefficients and the optimum reveals a very regular behavior of the parameters of the compact representation as a function of M when M describes a set of powers of 2 successive. We then look for a polynomial of degree fixed in log ₂ M which best approaches, in the sense of least squares, the set of points already obtained, for each parameter of the compact representation. Least squares optimization variables the coefficients of this polynomial.

Examples of such interpolation polynomials, of degree 5, are shown in FIGS. 8 and 9 in the case m = 1 for code number 2 of degree 8 and for the first four of the 8 parameters, for location and out-of-band energy respectively.

Tables 3 and 4 show the results of the interpolation for intermediate values of M. In general, the relative difference, noted ε, is very small between the interpolated solutions, which are obtained almost instantaneously, and the optimal solutions . Interpolation on an 11 M scale could significantly improve the interpolation results which are of lower quality. 4.2.3 Case of the BFDM / OQAM

We now move on to the situation detailed in paragraph 2.1. of appendix 1, i.e. where L = 2m and d = 2M - 1. 4.2.3.1 Optimization of the coefficients

The initial parameter space is then that of the scale coefficients such as {α) ^' = l, ... 2 + l, / = 0, ..., ^J ₂ -l}. We thus have card (P) = (2 + l) ^j ₂ ^L which is therefore the number of degrees of freedom. The problems of optimizing the coefficients a are without constraints and perfectly guarantee biorthogonality. The observation of the parameters obtained during a direct optimization shows, in FIGS. 10 and 11, that for the two optimization criteria considered these have continuity characters which are also usable for obtaining compact representations.

4.2.3.2 Representation in compact form By considering again the application defined by (54) (appendix 3), one introduces again subspaces R _κ of dimension K of the whole of the continuous functions defined on the interval [ 0, 0.5]. R _κ has a base B = (p _λ , ..., p ^) and we associate with JT-tuples of coordinates (x _r ..jcf), where j varies between 1 and 2m + l, coefficients) by the application: a = a φ _M (D) (59) with

Unlike the OFDM case, the coefficients of the scale representation are not angles. These are real parameters, without periodic behavior, and it is therefore not necessary to use a function like the function φ.

Codes that can be used are presented in appendix 4. 4.2.3.3 Optimization of the compact form BFDM

As for OFDM, the gains in computation time of the compact form BFDM appear as soon as K <M / 2. Furthermore, the results obtained in BFDM have a particularity when we consider the criterion of localization.

As an example, we can examine the results obtained with "Cfsqp".

"Cfsqp" is local optimization software. Thus with an initialization provided by a prototype with coefficients all zero, and almost whatever the bounds of variation authorized for, we obtain the results reported in Table 5.

Other tests were carried out with the "Dega" software (Differential Evolution Genetic Algorithm). "Dega" is a genetic algorithm for the global minimization of a function defined on a set of continuous variables. With this program available on the web

(htt: // http icsi .berkeley, ed u / sorn / code .html). the global optimization capacity makes it possible to obtain the optimal results obtained in table 6. It should be noted that in this table and in the one presented next (cf. table 8) the number of calls to the objective function (Nj) is fixed a priori. These two types of results are each of particular interest.

- the non-defective solution. The local optimization results (cfsqp) show that we can improve the results of the OFDM / OQAM, delay and / or localization value, without significantly modifying the frequency behavior. Thus in Figure 12, we can see that biorthogonality makes it possible to reduce the delay, here s = 0, compared to what we obtain in orthogonal (cf. Figure 6). Through against the frequency responses remain very close for a given criterion. - the defective solution. The global optimization results (Dega) lead to solutions close to the absolute optimum. As shown in a section to come (§ 4.3) this optimal solution can be formally constructed without optimization of any kind. However, the frequency behavior is then not as satisfactory as with the solutions which are not perfectly optimal.

As shown in reference [14], a single parameter allows a type distinction to be made by its values, either close to 0 for a non-defective solution or close to 1 for a defective solution.

This particular behavior linked to the location criterion does not concern the out-of-band minimization criterion [14] for which a set of results similar to that provided in Tables 5 and 6 is presented in Tables 7 and 8. 4.2.3.4 Interpolation fast as a function of M

The basic principle of this interpolation is the same as that described for OFDM / OQAM.

43 Case of degenerate BFDM / OQAM solutions with optimal location The case of the degeneration phenomenon, underlined in paragraph 1.1, of appendix 1, leads to a more precise examination of the relevance of the location criterion in the biorthogonal case. This aspect is detailed in Annex 5.

4.4 Asymptotic behavior of the compact representations Knowing the coefficients of the compact representations of optimal OFDM or BFDM prototype filters for different values of M, the interpolation method consisted in calculating the coefficients for the same compact representation of prototype filters for the intermediate values of M .

According to another approach, detailed in appendix 6, we can try to understand the behavior when M tends to infinity of the coefficients of the compact representation of optimal prototype filters. The number of coefficients for a given value of M does not depend on the value of M. We can therefore ask a first question: do these coefficients provide good prototype filters for a higher value of M which can be arbitrarily large ? The expected answer is that this is probably the case if the optimal coefficients tend towards a limit when M tends to infinity.

45. Prototype functions

The results established for the energy criterion assume in expression (8) (appendix 1) that ω _s = ω _p = -§-. Naturally, the methods of reduction of parameters and of interpolation can be implemented with other values of these two parameters, but this is nevertheless translated by computation times which become very high for the high values of M. This is then naturally all the more critical as m is also high.

An approach entitled "prototype functions" is proposed in appendix 7, which allows rapid optimization for any variant of the energy criterion.

In addition, in the case of the location criterion, the determination of prototype functions, also made it possible to establish theoretical limits, validating the previous results, but without allowing them to be improved [18]. 4.6. Conclusion We have therefore described a parameter reduction method which allows the rapid calculation of optimal orthogonal and biorthogonal modulated banks.

This method makes it possible to deal with problems of dimension, in number of sub-bands and length of the prototype, never reached. It is characterized in particular by: - the process of reducing parameters;

- the rapid interpolation process;

- the analytical formulas resulting from the calculations of the limit functions;

- the method of calculation by the prototype functions.

In addition, it has also been shown that a rapid interpolation method could make it possible to obtain results for a large set of values of the number of channels 2M. In some cases, the stability of the compact representation even makes it possible to obtain the result even more directly, by freeing oneself from the interpolation method, and by basing oneself on simple analytical expressions.

These methods have been illustrated with numerous examples which correspond to multicarrier modulation systems of the BFDM / OQAM and OFDM / OQAM type, which are also equivalent to the filter banks of the dual systems used in coding in sub-bands.

ANNEX

1 Optimization criteria

The choice of the optimization criteria for the prototype filters generally depends on the type of application envisaged. In the case of a transmission system in a static channel like that of ADSL (Asymmetric Digital Subscriber Line), a usual choice is to minimize the out-of-band energy. For radio-mobile type channels, the modeling model corresponds to a time-varying filter, the channel is then dispersive in time and frequency. In this case, where the time and frequency dimension are also important, M. Alard [8] has shown that the time-frequency location criterion is essential.

1.1 The time-frequency location

For continuous time signals, it is well known that the Gaussian function is the reference in terms of time-frequency localization where it reaches the limit fixed by the principle of uncertainty.

However, it has been shown recently that to measure the localization of discrete-time signals, it was preferable to use modified definitions of time and frequency moments of order 2 [12].

So for a discrete signal x whose standard is defined by

X Σ \ x (k) \ * (i) fc - = - oo the definitions proposed by Doroslovacki [12] for a real signal are the following:

- temporal center of gravity:

Σ (k - l / 2) \ x (k) + x (k - l) \ ²

T (x) = k = —oo

(2)

∑ \ x (k) + x (k - 1) \ ² k = - oo

- moment of order 2 in frequency:

1 + 0O t ₂ ( ^χ )% ∑ \ x (k) - x (k - l) \ ² (3)

\ χ k = - oo moment of order 2 in time:

We can then derive from these quantities a discrete localization measure which we denote by ξ _m0d to differentiate it from the measure ξ deduced in references [13], [3] from the continuous case [8]

For any discrete signal x of finite energy, i.e. such as x G / ₂ (Z), we have 0 <ξ _od (^) ≤ 1- The maximum location, _m0d (£) = L ^are reached for a function x _opt whose Fourier transform is expressed by X _opt (v) = C \ cos

with CGC and K> -. The function x _opt can therefore be considered as a discrete homologous function of the continuous Gaussian function. Its expression in time is such that a τt,, - ^Λ °) T (K + 1) _(r

where r (.) is the Gamma function, with [r ^ A;)] ^"-1 = 0 if A; <0, k GZ; ΛT _O pi (0) is a non-zero constant and r (x _opt ) is the temporal center of gravity deduced from (2) .To obtain a causal function in the interval [0, L - 1], we can set K = L - the T {x _{0 ≠} ) = ⁱ .

The location criterion has the particularity of being invariant by translation in time and frequency. This property has important consequences, which allows us in particular to identify a class of optimal solutions which is obtained directly from the function x _pi . This important particular case is dealt with in Annex 5.

1.2 The minimization of out-of-band energy

The second criterion that we consider is that of the minimization of the weighted standard of the frequency error. The weighting function, denoted W (ω), is defined by

"Ή = ⁽ S 0 s! I ^ω ω, i> ω ^ω . En

In the results that we present in this document, we always take ω _s = ω _p = -, but the proposed method applies for any positive weighting function. For a prototype filter p, whose Fourier transform is F (e ^jω ), the objective function to be minimized is written

Compared to the location criterion, which for a prototype of finite length is limited to. some elementary operations in the time domain, criterion (8) is much more costly in calculation. To avoid being penalized excessively in computation time, we use an original method of fast computation, calling upon only one FFT (Fast Fourier Transform) [14], which guarantees us nevertheless a great precision in computations. In addition, in appendix 7, we present, for any function W of a real variable with values in [0,1], an algorithm which further increases the speed of calculation of the objective function.

2 The conditions for perfect reconstruction

The conditions for perfect reconstruction of a modulated system, of the filter bank or transmultiplexer type, can generally be expressed as a function of the polyphase components of the prototype filter, p (n) of transform into z, P (z).

For example for a BFDM system, or OFDM, with 2M subcarriers, P (z) can be written in the form

where the Gι (z) are the polyphase components of order 2, according to which it is possible to write the conditions of perfect reconstruction for orthogonal or biorthogonal systems.

2.1 Biorthogonality and representation in scale

Let d and s be the integers defined by D - 2sM + d, s> 0 and 0 <d <2M - 1, we can then show that the biorthogonality conditions of BFDM / OQAM systems are given by [13]

- if 0 <d <M - 1 Gι (z) G _d -ι (z) + z ^~ G _M (z) G _{M +} - _t (z) = ^, if 0 <l <d

Gι (z) G _{2M + d} -ι (z) + G _{M +} ι (z) G _M - d-ι (z) =

2 'ά d + l ≤ l ≤ M - 1 (H) if M <d <2M - 1

Gι (z) G _d _; () + G _{M +} ι (z) G _d _ _M -.ι (z) = ^ -,

(12) if 0 <l <d - M

Gι (z) Ga-M + z ^ Gu + z) G _M , _d ^ z)

2 'if d + 1 - M <l <M - 1 (13) These conditions are exactly the same as those which have also been obtained for biorthogonal filter banks modulated in cosine, called DCT II and DCT IV, or modulated in exponential according to the technique called MDFT (Modified DFT) [5].

The conditions to be achieved according to the value of d can be directly satisfied if they are imposed on the coefficients p (n) of the prototype filter. For a given criterion, we then arrive at an optimization problem under constraints. It is also possible to take advantage of an embodiment structure, called a ladder, which guarantees by construction that the prototype filter will satisfy the conditions of biorthogonality. In this case we are reduced, for a given criterion, to an optimization problem without constraints. This is the approach we have developed.

The construction of a ladder implementation scheme is based on a procedure called "lifting scheme", a description of which can be found, for example, in reference [15]. As an example, let us choose the parameter d such that d = 2M - 1. Which for a BFDM / OQAM transmultiplexer will result in a reconstruction delay a = 2 (s + l) [13]. If we also assume that the prototype filter P has an initial length such that L = 2mM, with m positive integer, the reconstruction equation (12) is rewritten

G _l (z) G _2M ^ _l (z) + G _{M + l} (z) G _M -ι-ι (z) = ~, 1 = 0,. . . , - - 1,

ZM z ₍₎

with s an integer between 0 and m— 1 on which, as we have already said, the delay in reconstruction depends. We can show that thanks to the "lifting scheme", it is possible to obtain a prototype filter P ', of polyphase components G \, of greater length 2 (m +) M, but such that the reconstruction time is not modified and that the biorthogonality conditions are always checked. The procedure involves two matrices A, B and their inverses which are such that

and

> -ι -B (z)

(18) 1 where A (z) and B (z) are polynomials in z ^• '. The transition from prototype P to. P 'is then realized if, on the one hand, for each pair of polyphase components [G _I, G _M + I) ^and> on the other hand, each pair of component (Gd-ι, Gd-M-ι) -> ^are carried out transformations defined by the diagrams of Figure 1.

If in Figure 1, we take A (z) = CLQZ ^-1 and B (z) - 6 _o , with α ₀ and & o of the real quantities, we then increase the length of each polyphase component by 1, without modifying the delay .

In the same way one can preserve the property of biorthogonality by producing an increase in the delay, by introducing matrices C and D such as _z -ι

C = (19) c ₀ ⁱ j

-'c ¹ = 1 0 (20)

-co _z -i and

1 1

D (21)

0 z) _z -i

- —d ₀

D ^'1 = (22) 0 1 with c ₀ and do real numbers. The length of P '(z) is then 2 (m + 1) and the reconstruction delay a' - 2 (s' + 1) = 2 (s + 3) = a + 4. The corresponding scale diagrams are shown in figure 2.

These steps can be iterated to obtain, from an initial prototype filter, a final prototype with the desired length and reconstruction time. It is also possible to directly write the expression of the polyphase components as a function of the parameters linked to this representation on a so-called scale (AB, CD). To simplify the writing we will use the one proposed in [18] with the following notation

1 1

B (a) = -a

B- a) (24) 0 1 0

at

D (a) -

'^ W = 1 (26) To each quadruplet (Gι, G _M -ι-ι, G _M + G _2M -ι-ι) when l varies between 0 and Y - 1 then correspond 277? + 1 parameters noted at ^l _k , k = 1,. . . , 2m + 1 which allow to obtain the polyphase components by the following construction. We introduce the integers ι _x and j _\ such that if s is even, we have = 2jι and m = H + li + 1 and if s is odd s - 2 j + 1 and m =% + ji + 2. We set

1

Fn = o ^{' '} 1 ai, ^" 1 0 ^"

1 0 1 i (27)

We then have, for s even

[G z), G _{kI + l} (z)] =

-L = [1 1] F ₀ f [A (α) ^β " ₃ ) Il C ( _{+ 2j + 2} ) D ( _{+ 2j + 3} ), (28) ι = l and

(29)

For s odd, after defining FQ, we ask

Go = F ₀ C (a ₄ ^l ) B (a ₅ ^l ), (30) so that

and we have

C (Ω - (2 ₇ +4 lD (û 2ÎH 2J + 5 ₅ ) (32)

and

In all of the articles of which we are aware, the authors assume, or assert [5], that this structure is complete, that is to say that it covers all the biorthogonal solutions of RIF type. If we strictly limit ourselves to the matrices which we have just presented, this is not completely exact because, as we show it with the reference [18], there are defective cases, which nevertheless can be circumvented without degrading significantly. - the quality of the results. We will therefore take parametrization on a scale as a reference base. 2.2 Orthogonality and trellis representation

In the case where the prototype is required to be orthogonal, which then requires it to be also symmetrical, when it is assumed that the prototype of the analysis bench is identical to that of synthesis, the preceding conditions are simplified. The orthogonality conditions are then reduced to the following equation

G _t (z) Gι (z) + G _{l + M} (z) G _{M + l} (z) = ^ 0 <l <M - 1 (34) with 7 which is the paraconjugation operation, ie Gι (z ) = G (z ^{~ l} ).

This reconstruction condition, in the orthogonal case, which is that of OFDM / OQAM systems with 2M subcarriers [3] is also that of banks modulated in cosine with M sub-bands [7].

As in the biorthogonal case, there is also in orthogonal a representation which, by construction, guarantees that the relation (34) is always verified. In addition, this representation, called trellis, is complete [7]. If, to simplify the writing, we again take a length L - 2mM, the polyphase components then check for 0 </ <- 1, the relations which follow

Gι (z) z - ^ - ^ G _2M -ι-ι z), (35)

GM + I (Z) Z- ^ GM- ^ Z), (36) _y - (ml)

GI (Z) G ₂ M- _I - _I (Z) + G M + I (Z) G MII {Z) - (37)

2M To build the lattice representation we define the matrices

^' 1 0

Λ (z) = (38)

0 z- ^{1 '} z- ¹ 0 ^"

T (z) = (39) 0 1 •>

cos θ sin θ θ (θ) = (40) sin θ - cos θ

For each l = 0,. . . , M / 2 - 1, then there are m angles θ ₀ ^l , θ [,. . . , ^ - ι ^te ^ ^s that the components 2 -polyphases GI (Z), G _{M +} I (Z), GM- _I - _I (Z) and G ₂ M- _I - _I (Z) are obtained by the relations matrix:

[G _t (z) G _{M + l} (z)} = ( ₂ ) Θ (f [), (41)

G _2M -ι-ι {z) 1 m— 1 2)

GM-II (Z) 2M II θβ) r ( _* ) cos flή (4

A- = l sin f? Ή

The lattice representation therefore also makes it possible to reduce the dimension of the parameter space, while ensuring the properties of orthogonality and completeness, we will therefore take it as a reference in the orthogonal case. APPENDIX 2 The BFDM / OQAM application context

There are therefore several systems based on modulated filter banks and whose property of perfect reconstruction can be reduced to conditions given depending on the case by one or other of relations (10) to (13), in the case biorthogonal, or even to relation (34), in the orthogonal case.

By way of illustration, we will briefly summarize the way of arriving at equations (10) to (13) in the case of BFDM / OQAM multicarrier modulation. For more details on the calculations one can refer to [2, 11, 16].

1 Description of the continuous case

If 2τo and ₀ respectively designate the duration of a complex symbol time and the difference between carriers, we know that the complex envelope of a BFDM / OQAM signal can be written in the form [16]

+ oo K-l

^S W ⁼ Σ Σ ^a m., NJm, n (t). (43) n = - oom = 0 with

where ψ _m , n ^e is a phase term such as φ _{m> n} + ι - Ψ _m ; in ≡ § modulo π.

The maximum spectral efficiency is then obtained for ι ₀ ro =. At demodulation, the real biorthogonality conditions are realized if one demodulates on a base { _m , _n } such that {7 _m , _n> 7 _m , n} constitutes a pair of biorthogonal bases.

2 Description of the discrete case

If we take a number of carriers K = 2M, the discrete version of equations (43) and (44) results in the relations

+ ∞ 2M-1

^S W = Σ Σ Û! W, n7m, n [fc], (45) n = - oo m = 0 with

where p (n) is the causal prototype of length L and D is the integer parameter defined in paragraph 2.1. At reception to find the transmitted symbols we must project the BFDM / OQAM signal modulated on a base η _{m> n} which must be biorthogonal to the base of functions _m, „, we therefore carry out the following operation

where * denotes the conjugate complex. We can show that 7 _m , [fc] is written

ï ' _{m, n} [k] = 7 [fc - nM} e ^j & r ^{m (k} - ^nM - ^{! l)} f (48)

3 Formulation in the form of a transmultiplexer

The modulation and demodulation equations can also be in the form of. filter banks. Thus we have shown [11], [2] that by posing

/ _m (fc) = v ^ p [*] e>^"m<* - ^ (49) the BFDM modulation system corresponds to a synthesis bench. Similarly, at reception, if we set D - aM - b , with α and b positive integers, we can define a set of 2M filters (0 <m <2M - 1), which correspond to the analysis bench that we can use for demodulation

with p [k] = -γ [D - k].

In what follows we will limit ourselves to the case where p = p. Under these conditions if we express the analysis and synthesis filters as a function of the polyphase components Gι (z) of the prototype,

2M-1

F _m (z) = 2 ∑ e ^ '- ^ .-' G ^), (51)

; = O

we can then determine the conditions for perfect reconstruction in the form given by equations (10) to (13) [11, 2].

The case of OFDM / OQAM can be deduced with the same principle of calculation, or even as a particular case of the biorthogonal case. APPENDIX 3 Codes for OFDM / OQAM

Given the fact that, at the optimum θf is bounded between 0 and π, it is possible to consider only the cosines of these angular parameters. The new parameters therefore vary in the interval [0,1]. As some of the optimization programs consider only unbounded variables, we reduce ourselves to real parameters x to which the cosines of angles correspond by the application defined by

Note that if

10 ∞sfl ≈ - _i , (54)

VI + then

for θ in [0, π]. ic We consider the map u of {0,. . . , - 1} with values in the interval

[0,0.5] defined by ¹

* ^" <' ⁾ = Ï - ^

The angular coefficients in the base B are then given by

₀ cos) = Φ (∑ * P * (*))) (57)

For the moment we have used 3 codes. It is of course possible to add new ones.

Code # 0 - The trivial code. We choose K - Y and for space lZχ the space generated by the generating functions of the points Φ _M (1), 1 = 0, - - -, γ - 1, i.e. i _\ i, f 1 if x = φ _M (k - 1), _ccΛ *) = -uto = _{O otherwise} (58)

1. This app is of course not the only choice. In particular, we were able to observe with a code, 12, modification of code 2 where ΦM (1) ≈ ^ ti + ri

2M -, that we found results identical to those obtained with code 2.

0 In this case cosf? - = Φ (x _i ^l ₊₁ ), l - 0,. . . , γ - \, i = 0,. . . , m - 1.

Code n ° 1.- For K> 0, 71 _% is the subspace of polynomials of degree less than or equal to K - 1. We then obtain

cosθl = φ. (59)

Code n ° 2 - For K> 0, lZκ is still the subspace of polynomials of degree less than or equal to. K - 1, but the base chosen is that of the Chebyshev polynomials. We then obtain

cos. (60)

APPENDIX 4 Codes for the BFDM / OQAM

Code # 0 - The trivial code. It corresponds to K - - and the coefficients x ^l l = 0,. . . , γ - 1, j = 0,. . . , 2m are the coefficients α ^ ₊₁ , l = 0,. . . , γ - 1, j = 0,. . . 2m

Code n ° 1.- The space IZjς consists of polynomials of degree less than or equal to K - 1 represented by their coefficients. We obtain

(61)

Code n ° 2.- The space! Zχ is always made up of polynomials of degree less than or equal to K - 1 but the polynomials are represented by their Chebyshev development. We then obtain

X _jk (4φ _M (l) - 1), j = 0,. . . , 2m. (62)

where the T _k are the Chebyshev polynomials. Code n ° 3.- The space lZχ is the subspace of the polynomials p of degree less than or equal to 2K - 1 satisfying p (x +) + p (x -) = 2p (), that is to say say whose graph is symmetrical with respect to the point (, p ()). We then obtain

° i ₊ ι ≈ * J +, = 0, ..., 2m. (63)

Code n ° 4.- The space 1Zχ is the same as for code 3 but we choose _a representation of Chebyshev. We then obtain κ-ι ^α j + ι = ^χ ₃ ⁰ + @ ΦM (l) ^~ 1), j = 0, • • •, 2m. (64)

APPENDIX 5

The following theorem shows that one can obtain a BFDM filter of Doroslovacki localization as close as one wants to 1, whatever the values of the parameters M, m and s.

Theorem 1 - Let M be even, m> 1 integer, s integer in the interval 0 .. m— 1 and ε> 0, there is a prototype BFDM filter P (z) with parameters M, m and s satisfying ξ _mod [ P)> 1- ε.

Proof.- Let H (z) be any filter of length M defined by

M-1

H (z) = ∑ h, ₂ -, (65) ι = 0 with h _τ 0, ι = 0, ..., M - 1. We consider the filter P (z) defined by

2JW-1

P ( _z ) = z ^{~ sM} H (z) + ∑ z- ^l K, (z ^2M ). (66) τ = M

It is easy to see that the 2-polyphase components of P (z) are given by

J hz ^{~ s} if 0 <1 <M - 1, ^{tW ~} iv, (z) sïM <ι <2M-l. ^{0,)

The biorthogonality conditions (14) are then written

1 M hιI _2M -ι-ι (z) + h _M -ι-ιI <M + ι (z) = -, 1-0, .., -—- 1. (68)

We can then choose, for all l of 0..M / 2 - 1, K ₂ M- _I ( ^Z ) arbitrarily, then determine KMΛ _I (Z) such that

since

are not zero. For all a> 0, the filter

2M-1 P (z) = az- ^2sM H (z) + - ∑ z ^{~ l} K _t (z ^2M ) (70)

then also verifies the relations of biorthogonality. Thanks to the homogeneity of the function ξ _mo , we have

ξm d {Pa) ^~ Çmod (-Pa), (71) α When a tends to 0, the filter (- ^ P _a ) tends to the filter H (z) and by continuity of the function ξ _m0d , the second member of (71) tends to ξ _mw z (H). It then suffices to choose for H (z) the filter of length M which satisfies ξ _mod (H) = 1 and has small enough that ξ _mod (P _a )> 1 - ε. D

In other words, it suffices to refer to paragraph 1.1 and to choose for H (z) coefficients given by expression (6). The other coefficients of the prototype are then deduced directly thanks to expression (70).

It is also obviously possible to iterate this construction to obtain a smaller support for the significant part of the filter. We can also ensure that this significant part (H (z)) is carried by other adjacent polyphase components.

APPENDIX 6

A first paragraph gives a detailed study of the OFDM case for m - 1 when the cost function is localization or out-of-band energy.

Then the existence of a limit is studied experimentally in the case OFDM and BFDM for m> 2.

1 Asymtotic behavior when M -> • oo in the OFDM case for m = 1

1.1 Technical Lemmas

To fully exploit the behavior at the limit of prototype filters of length 2M when M becomes large (see section 1.2), we use the following 2 lemmas.

Lemma 1 - Let p (t) be a polynomial of degree 2k + l, k> 0 satisfying the property of symmetry: p (i) + p (l - i) = l, (72) and such that p (0) = 1 (and therefore p (l) = 0). Then there are constants α,;, i = 1,. . . , fc such as:

p (i) = i - t + (2t - 1) ∑ aj (l - if (73)

Demonstration.- See [18].

Lemma 2 - Let θ (t) be a continuous function on [0,] and 2n + 1 time continuously differentiable on [0,] with 1 <n <co and satisfying

Let h (t) be the function defined on [0,1] by

For k satisfying 1 <A: <n, h (i) is 2k + 1 differentiable on the left at t = if and only if

0 < ²ⁱ > = O _J i = l,. . . , fc. (76)

Demonstration - See [18]. Corollary - With an additional hypothesis, the analysis of θ (t) in a disc of the complex plane centered in t - of radius greater than, and also assuming that the degree of θ (t) is 2k + 1, we obtain the expression

θ (t) = (1 - t + (2t - 1) ∑ α, (1 - tγ), (77) where the a _t , i = 1,..., k are coefficients.

5

1.2 Convergence properties

For a given compact representation, and whatever the fixed value of m, the number of coefficients of this representation is constant and equal to mK where K is the degree of the code. In the particular case where m - 1, we note that the coefficients of the compact representation of optimal OFDM filters of length 2M, for a given cost function, converge towards a limit. We can therefore consider that this limit is almost reached with the coefficients of an optimal filter of length 2M with M large enough and consider the filters of parameter M 'different from M and of length 2M' with the same coefficients.

It is interesting to compare the value of the cost function on this result with the optimal value for the length 2M '. The ordinate of the curves in FIG. 13 represents the base 10 logarithm of the absolute value of the relative error committed for the location by replacing the optimal filter, for a value of M 'given on the abscissa, by the filter obtained by the process described above. 5 The opposite of this ordinate therefore represents, continuously, the number of exact decimal places in obtaining the optimal cost.

We consider the five values M = 128,256,1024,4096 = 2 ¹² and 32,768 = 2 ¹⁵ .

The curves in Figure 13 show

- that there is indeed a limit when M '→ oo since the curves have a horizontal asymptote,

- that the 8 parameters of the compact representation (code 2, degree 8) of the optimal Q filter for M = 1024, = 2048 allow to obtain filters of parameters M 'JJ ≈ 2M', M 'even, with the optimal localization to 10 ^{~ 6} of relative error near if M '> 2 ⁷ .

Figure 14 shows similar calculations for the out-of-band energy cost function and allows identical conclusions to be drawn. This property can then be used to calculate, as in section 1.1, analytical expressions for the transverse coefficients. 5

1.3 Regularity of the limit filter and simple analytical formulas

The analysis of the properties of the coefficients of the impulse response to the optimum reveals behaviors close to those described by these mathematical models, which allows us to deduce then simple formulas of the transverse coefficients which provide quasi-optimal results for large even values of M.

We present here the case where L - 2M, but we can specify that the method is generalized for values of m greater than 1 [18].

A prototype OFDM filter P (z) of parameters M even and of length 2M whose transverse coefficients are hi, i = 0,. . . , 2 - 1, has a lattice representation (trivial) with M / 2 angles θι, l = 0,. . . , M / 2 - 1 with, for 1 = 0,. . . , M / 2 - 1, hι = t -ii = sin 0j, h _M -ι-ι = h _M + ι = cos θi. (78)

As the θι are the values of a function on the interval [0,0.5] defined in the points -j ^ O-, 1 = 0,. . . , M / 2 - 1, then by composition with the sine and cosine functions, and with translation and interval symmetry functions

[0,0.5], the coefficients h _u l - 0,. . . , 2M - 1 can be represented as the values of a function defined on the interval [—1, + 1].

Location criterion

In the case of the location criterion we obtain the result represented in figure 15 in the case M = 32 and M = 1024. The coefficients of the filter of length 64 (M = 32) are represented by points while the plot is continuous for the filter of length 2048. The coefficients of the filters are normalized so that the total energy is equal to M. It can be seen that the limit curve is very regular.

The. Function θ (t), i € [0,0.5] depends on the 8 parameters x ₀ , Xι,. . . , x _& de la. compact representation for code 2, degree 8 by the formula

cos θ (t) = x _k T _k (4t - 1) (79)

where the T _f c are the Chebyshev polynomials. This can also be written θ (t) = arccot (a; (i)). (80)

With the coefficients x of the best optimal filter for M = 2 ¹⁵ , we obtain <3.57 x HT ⁹ , (81) | <1-72 x 10 ^{~ 8} . (82)

and we note that the function θ (t) is very close to a symmetrical function with respect to the point [^,]. These observations are in good coherence with the regularity of the limit curve carrying the transverse coefficients of the filter according to the results of lemmas 1 and 2.

We therefore seek a simpler expression of θ (i) of the form (77). The function 0 * (t) defined by

[t) = iLz ^ l ( _{π + t} + 2i ² - 12i ³ + δï ⁴ ) (83)

corresponds, in expression (77), to the choice k = 2 and to the values

1 4 α _x = - = -0.3183098861, α ₂ = - = -1.273239544. (84) 7T 7T

With k = 2, we then seek optimal values of the parameters ai and α so as to minimize the relative distance between the location of a filter calculated with the function θ (t) and the optimal location filter for values of M greater at a fixed M ₀ . Optimization is carried out with Cfsqp using a cost function of the minimax type.

With Mo - 1024 and minimizing the relative error on the location for 27 values of M whose logarithm to base 2 is equidistributed between 10 and 15, we find the values a ₀ = -0.306324700486135, α, = -1.38222431220756, (85 ) and denote by θ ₂ (t) the corresponding function.

With k = 3, and identical calculation conditions, we find ₀ = -0.33690242860365, d = -0.83833411092749, ₂ = -1.99179628625716,

(86) and denote by 0 ₃ (i) the corresponding function.

Finally, with k = 4, we find the values ₀ = -0.33144646457108, α, = -1.01784717458737, α ₂ = -0.43066697997064, α ₃ = -3.91628703543313, (87) and the corresponding function is called 0 ₄ (i).

FIG. 16 represents the relative errors made when these functions are used to calculate the optimal filters for localization. We denote by C ^* the curve relative to θ ^* (t), by C the curve relative to 0ι (t), etc.

We can also verify that for a filter PM (Z) = ∑ ² = o ^_1 hi ^{z ~ l} , OFDM / OQAM prototype with parameter M and length L = 2M, we have the following property

ÇMOD (M)> 0.90560. (88)

Energy criterion

When the cost function is out-of-band energy, the following results are obtained. Figure 17 is the analog of Figure 15. We note that the limit function is not differentiable at the origin but that it is regular in t =. The function θ (t) deduced from the compact representation of the best OFDM filter of length 2M for M = 2 ¹⁵ is well symmetrical with respect to the point (, f), but 0 (0) ≠ § and therefore 0 (1) ≠ 0 .

By a least squares adjustment, we find a function θ _e (t) of the form:

θ _e (τ) = ^ (l χ bo (t - \) + bι (t -) ³ )) (89) with b ₀ = -1.578661958412621, = 0.7990463095216372. (90)

However, as shown in Figure 18, the results are not as good as for localization. With more coefficients, we do not get better because the error committed is constant and linked to the lack of symmetry of

10 the limit curve θ (t). Introducing new code may be one way to improve this result.

2 The case m> 2

The results obtained for m = 1 which show, for the location and energy criteria, that the optimal filters have a compact representation ^ which becomes almost independent of M for the high values of this parameter, do not generalize for m > 2. It appears that the main distinction to be made then concerns the optimization criteria rather than the type, OFDM or BFDM, of the prototype.

2.1 Out-of-band energy criterion

For out-of-band energy and m> 2, it is possible to obtain directly, 0 without going through an interpolation process, a quasi-optimal prototype filter for large values of M. Thus we see in Figure 19 that the coefficients of the optimal filter for M = 128 (or for M = 1024) make it possible to construct a prototype filter for M <2 ¹⁵ whose cost is very close to the optimal cost.

In the case of BFDM Figures 20 and 21 show that, without being as convincing for the moment, the results obtained make it possible to guarantee, according to the value of initial M, the accuracy of the 3rd or 4th decimal place if m = 2 and of the 2nd or 3rd decimal if m = 3.

2.2 Location criteria

In the OFDM case, we have seen [18] that for the location and m, = 2 or m - 3, the coefficients of the compact representation (code 12, degree 8) of the

0 optimal filter for a fixed value Mo of M provide prototypes with a higher parameter M whose cost moves away from the optimal cost We deduce that we cannot identify the optimal prototype filter for a large value of M with a filter "limit".

The coefficients of the compact representation of the optimal filters therefore show a variation as a function of M, for M large. The examination of these coefficients in the case m = 2 therefore leads to seek dependent representations of M [18].

The examination of the coefficients obtained in the OFDM case for m = 2 leads to the search for a compact representation depending on the value of M. More precisely, the angular parameters θ ₀ and θ _\ are sought in the form

θ ₀ (t) = θ ₀₀ (t) - λPθn (t), ι (i) = + ^ „(t), (91)

Z where is a negative real and where

10

0oo 00 = (l - D + (2i - l). (92)

¹ ι = 0 θ _u (t) = - (l - 2t) ∑ β _l tX (93)

⁴ t = l

An optimization with CFSQP of the error committed on the location by comparison with all the optimal locations obtained for diffé¬

15 rents values of M gives, for example for k = 2, the following set of optimal coefficients

7 = -0.289654695596932 α-o = -0.300851581378120 α, = -0.723945110106017 a ₂ = -1.79709674616421

²⁰ _β ϋ = 0.713206997358430 βι = β ₂ = -2.02470362855665 2.25362085753772

According to the value of M, we obtain with this set of coefficients a precision which guarantees at least the accuracy of the second decimal place and which can> even reach the 5th decimal place. _i - To deal with the case of BFDM prototype filters, we are currently encountering the problem of degeneration. The introduction of a gain factor making it possible to retain only the only degenerate solutions should make it possible to solve this problem. In the current state it is therefore preferable in this case to simply apply the technique of reduction of parameters and, if necessary, that of rapid interpolation.

0 APPENDIX 7

1 Definitions and first properties

1.1 Definitions

For an integer m> 1, denote by E _m the set of functions h on - * the real interval] - m, + m [such that on each subinterval of] - m, + m [of the form Jf, ¹ ^, j = —2m, - 2m + 1,. . . , 2m - 1 the restriction of h is twice continuously differentiable and h and its derivatives admit limits at the limits of the sub-interval. In the rest of this chapter, we will say that a function of E _m is a prototype function with parameter m.

Example - For m = 1, E _x is the set of continuous functions h on [—1, -1- 1] twice continuously differentiable, except possibly in -, 0, and 0 h, h ', h "have left and right limits in -, 0,, right in -1, left in 1.

A prototype function will be called continuous if it is continuous at the points \, j = -2m + 1,. . . , 2m - 1 and if h (-m) = 0 and h (m) = 0.

Definition.- Let h in E _m and M be an even integer> 2. We designate by PM (Z) the filter of length 2m M whose transverse coefficients u, _% are _ given by

PM, _X = Λ [^ - j, ι = 0,. . . , 2mM - 1, (94) and we will say that PM (Z) is generated by h.

The points at which the values of the function h are thus calculated are never half-integers and form a regular subdivision of the interval 1 - m, -t- m [. 0 If the function h is even (resp. Odd), the PM filters (Z) are symmetrical

(resp. asymmetric).

For any filter P (~), denote by ξ (Jjw) its localization parameter in the sense of Doroslovacki and by JW (PM) its out-of-band energy.

1.2 Infinite limit for energy c We assume, without loss of generality for usual problems, that the function W (ω) admits an inverse Fourier transform and that ω _s <f. Let us also assume that W _M (ω) = M ^r (Mω). Then, when M tends to infinity, we can express the out-of-band energy of the prototype function using the following theorem, where, in the usual way, the letters in capital letters correspond to the Fourier transforms of those in lowercase, for example w denotes the inverse Fourier transform of W. Theorem 2.- We have \ H (σ) \ ² W (a) da _ J + ™ fh (t) h (u) w (i - u) du di

_Λ J ™ _TC ^J "^'" ^{PM) - _π / + h ² t) dt 7V fh? (T) dt

(95)

Demonstration .- ^ 8].

2 OFDM prototype functions

Definition.- Let h be a prototype function of E _m . We say that h is of OFDM class if, for all M, the PM filter (Z) is a symmetrical OFDM prototype filter.

The following theorem shows how the trellis representation of the OFDM prototype filters gives a characterization of the structure of the OFDM prototype functions of E _m using m angular functions.

Theorem 3 - Let h GE _m of class OFDM. U exist m functions θi (t) defined on the interval] 0, [, twice continuously differentiable and admitting limits in 0 and \ and verifying the following property. For all t, 0 <t <, we define the points Xi (t) = —m + i + t, i = 0,. . . , 2m— 1 of the interval] - m, + m [and the functions g _Q (t) and g ₂ (t) by m— 1 go (t) = ∑ h (x _2i (t)) X 92 ( t) = ∑ h (x _2i , ₁ (t)). Y ⁱ (96) i = 0 i = 0 where X is a variable. So, we have matrix equality

(9o (t) Mt)) ≈ α ^< (cos _o (t), sin o (t)) J [Λθ (0 * ()), (97)

where Θ and Θ (0) denote the m, atrices

1 0 cos 0 sin θ

Λ = θ (θ) = (98) 0 sin 0 - cos 0 and has a constant independent of t.

Demonstration.- The demonstration is trivial since the angles 0; (t) defined by the equality (97) depend continuously on values varying continuously from the function h.

3 Energy optimization

3.1 Other subdivisions

To set the P _M filter from the prototype function h, we used in (94) the points of abscissa ^{s 2 M} _2M that ^its ^ regularly spaced in interval (—m, 4- m) and are never half integers. Other choices are possible. For example, in all the previous sections, we made the choice with the function φ _M defined in (56) points ^{% + 1 ~} M + _I • ^ ^es ^ hey ⁰ - remes of the previous paragraph are also checked for this subdivision .

Another subdivision is considered below. This corresponds to the abscissa in the half-integer intervals which correspond to. the method of

5 integration of Gauss-Legendre. For this subdivision, the theorems are no longer verified. However, the construction of the corresponding Pu filter is very useful for calculating the values of Jw (h) -

3.2 Calculation of energy

It is assumed in this section that the function h is calculated from the data of m functions θi (t), i = 0,. . . , m - 1 as described in the theorem

10 3.

In this form, || / i || = 1 and therefore

h (u) h (t) w (t - u) dudt. (99)

To calculate the integral numerically (99), we cut the integration square into side squares whose sides have half-whole ends. On each of these small squares (there are 4m ² ), the integral is calculated by the scheme * - ^• > of Gauss-Legendre integration of order n, with n fixed on each of the two directions u and t .

Recall that the Gauss-Legendre scheme of order n makes it possible to evaluate the integral of a function / over the interval [—1, + 1] by the formula:

0 where the X _k , k = 1,. . . , n are the n roots of the Legendre polynomial of degree n, denoted P „(x), and p _k , k = 1,. . . , n of the weights calculated by the formula

2 (1 - 4),,

" ⁼ M ^'(101)

With such a choice of abscissas and weights, the evaluation of the integral of / is _< - exact if / is a polynomial of degree less than or equal to 2n - 1.

After having slightly transformed the. formula (100) to adapt it to a length interval, the integral takes the form

Jw {) = h uΛh uj) aij (102)

0 or

^{ai =} V5 ^W ^ ^{Uj ~ u} ^ ^{Pk (ui Pk} M '( ¹⁰³ )

° ù P _{k (u} i ₎ and P _{h (u)} are Gauss-Legendre weights depending on the place of Ui and U _j in the subdivision of one of the length intervals. The ai _j are the coefficients of a symmetric matrix A of order 2mn which does not depend on h but only on n and on the weight function W (a). We therefore evaluate once and for all the value of its coefficients before optimizing the cost function. The vector V = (h (uo), h (uι),..., H (u _mn -ι) of length 2mn is evaluated for given values of the parameters of the optimization problem and the function Jw {h) is given by

J _w (h) ~ V ^τ AV. (104)

Note - When the parameters of a compact representation code are used to calculate a PM filter (Z), the number of evaluations of the function h is mM (since the PM Z filter coefficients are completed) by symmetry ), then the energy is calculated with two real DFTs of length> 4mM, plus arithmetic operations (Appendix A). For the calculation at the limit with a function h depending on a similar meaning code, the number of evaluations of h is only 2 min, and the calculation of the quadratic form (104) is carried out with 0 (m ² n ² ) arithmetic operations In practice, we choose n = 6 or n = 10.

In this boundary calculation, we ensure orthogonality, or biorthogonality, by reducing ourselves to representations in lattice, or in scale, which can be written in the form of a function a (t) whose expression includes the one already seen for compact representation codes.

Code 1: the parameter a (t) is obtained by the formula dl ^α W = Σ <**, (105) i = 0 where d is the degree considered for this code and α, -, i = 0,. . . , d - 1 the corresponding d coefficients,

Code 2: the parameter a (t) is obtained by the formula

a (t) = ∑ αi7K4Z - 1), (106)

? = o where _T: is the Chebyshev polynomials, Code 3: the parameter (t) is obtained by the formula dl (t) = a ₀ + a ₁ t + t (l - 2t) Σ AFI- ² . (107) i = 2 3.3 First results in the OFDM case

We have optimized with CFSQP the coefficients of a prototype function for the energy cost function. The CFSQP program proves to be extremely sensitive to the limits given for the coefficients, to the degrees of the codes and to the initial values.

The best results are with the code 1 for all coeffi- cient and _c are given in Table 9 with the list of degrees. This table provides the value of E _∞ which designates the optimal out-of-band energy for the prototype function.

3.4 First results in the BFDM case

Optimization with CFSQP was also used to optimize the out-of-band energy of the BFDM type prototype functions. i n Table 10 gives the best results obtained for m = 1,. . . , 4, s =

0,. . . , m— 1. The limit prototypes are then used to construct filters of parameter M = 128 and the results are compared with the best result obtained for code 2 of degree 5 (E ° ₂₈ ) • - ^ _o o denotes the optimal out-of-band energy for the prototype function, Ei ₈ the out-of-band energy of the filter of length 2 x 120M which is deduced therefrom, and E ₂ g the optimum found by direct optimization on the compact representation of the filter .

APPENDIX 8 Tables

TAB. 1 - Better localization ξ _mod obtained for code 2, degree 5, by the program cfsqp kofdm.

TAB. 2 - Better localization ξ „obtained for code 0 by the program cfsqp kofdm.

TAB. 3 - Verification with cfsqp of the performance of rapid interpolation for localization ζ _MOD .

TAB. 4 - Verification, with the energy criterion, of the performance of rapid interpolation.

TAB. 5 - Better localization ξ, obtained for s = 1 and code 2, degree 3, by the program cfsqp kbfdm

TAB. 6 - Better localization ξ _mod obtained for s = 1 and code 2, degree 3, by the program dega_ kbfdm.

TAB. 7 - Best out-of-band energy (π / M, π) obtained for s = 1 and code 2, degree 3, by the program cfsqp_kbfdm.

TAB. 8 - Best out-of-band energy (π / M, π) obtained for s ≈ 1 and code 2, degree 3, by the dega_kbfdm program.

TAB. 9 - Better optimization of out-of-band energy for OFDM prototype functions with parameter m.

TAB. 10 - Best out-of-band energy (π / M, π) obtained for the BFDM prototype functions by the cfsqp ener program. APPENDIX 9

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Claims

1. Method for determining the filtering coefficients of a filter bank modulated in sub-bands, based on a prototype filter of length L intended to produce, by modulation, said filter bank modulated in sub-bands, characterized in that '' it includes the following stages:

(a) determination, for at least two values M less than the number of sub-bands of said modulated filter bank, as a function of a first space of parameters P, of a set of first filtering coefficients, corresponding to a global optimum the corresponding problem, noted {P, M, L}, for at least one predetermined criterion to be optimized;

(b) analysis of said set or sets of first filter coefficients, so as to determine a subset of filters corresponding to said optimums, to which a second space of parameters R is associated, with card R ≤ card P, allowing another representation, said compact, said filters; (c) optimization of a second set of filter coefficients, for at least two values of M and at least one value of L, using said second space of parameters R, corresponding to a global optimum of the corresponding problem, noted {R, M, L};

(e) storing at least one of said second sets of filter coefficients.

2. Method for determining filter coefficients according to claim

1, characterized in that it comprises the steps of: (d) applying for at least two values of M from step (c);

(f) rapid interpolation of a set of filter coefficients for a value of M corresponding to the number of sub-bands of said filter bank, from the second set or sets of stored coefficients obtained in step (d).

3. Method for determining filter coefficients according to any one of claims 1 and 2, characterized in that, in said step (c), said optimization is calculated for at least two values of M which are powers of 2.

4. Method for determining filter coefficients according to any one of claims 1 to 3, characterized in that, in said step (c), said optimization is calculated by taking account of at least one fixed value for at least one annex parameter A.

5. Method for determining filter coefficients according to claim 4, characterized in that the said additional parameter (s) A comprise a fixed time.

6. Method for determining filter coefficients according to any one of claims 1 to 5, characterized in that, in said step (b), said analysis takes account of at least one of the properties belonging to the group comprising: - linearity of the phase; the continuity of the parameters from one polyphase component to another; belonging to a predetermined range of angular parameters; properties of continuity and / or differentiability of limit curves, when the length tends towards infinity.

7. Method for determining filter coefficients according to any one of claims 1 to 6, characterized in that it comprises, after said step (c), a step of:

(d) validation of the results of the resolution of said second problem, for at least a low value of M.

8. Method for determining filter coefficients according to any one of claims 2 to 7, characterized in that it comprises, after said step (f), a step of:

(g) verification and / or optimization of the interpolated values for the corresponding problem {P, M, L} or {R, M, L}.

9. Method for determining filter coefficients according to any one of claims 1 to 8, characterized in that the said predetermined criterion or criteria to be optimized belong to the group comprising: localization in time and / or in frequency; - minimization of out-of-band energy.

10. Method for determining filter coefficients according to any one of claims 1 to 9, characterized in that, in said step (b), at least one function, called code, is implemented, associating with an optimal prototype filter P (z) an approximate representation P _x (z) of said optimal prototype filter P (z), expressed using said second set of parameters.

11. Method for determining filter coefficients according to any one of claims 1 to 10, characterized in that said modulated filter bank is used for the implementation of a multicarrier modulation system.

12. Method for determining filter coefficients according to claim

11, characterized in that said multicarrier modulation is an OFDM / OQAM modulation.

13. Method for determining filter coefficients according to claim

12, characterized in that said first space P is the space of the angular coefficients:

with card (P) = m * MI2 where: 2M is the number of subcarriers of said modulation; L = 2mM is the initial length of said prototype filter.

14. Method for determining filter coefficients according to claim

13, characterized in that said second space belongs to the group comprising: a space R _κ0 generated by the generating functions of the points φ _M (l), 1 = 0, ..., (M / 2) -l, with φ _M (l) a bijection from {0, (M / 2) -l} to {0, 1/2}; - a space R _KI corresponding to the subspace of polynomials of degree less than or equal to K- 1, K>0; a space R ^ corresponding to the subspace of the polynomials of degree less than or equal to Kl, K> 0, in the base of the polynomials of

Chebyshev.

15. Method for determining filter coefficients according to claim 14 and claim 10, characterized in that said codes are respectively: for said space R _κo : code n ° 0:

for said space R _KI : code n ° 1:

for

16. Method for determining filter coefficients according to claim 15, characterized in that said code No. 2 is used with degree 8.

17. Method for determining filter coefficients according to any one of claims 11 to 16, characterized in that said step (f) of interpolation implements a sub-step of searching for a polynomial of degree in log ₂ M as close as possible, in the least squares sense, to all the values already stored.

18. Method for determining filter coefficients according to claim

11, characterized in that said modulation is a BFDM / OQAM modulation.

19. Method for determining filter coefficients according to claim

18, characterized in that said first space P is the space of the coefficients in

with card (P) = (2 + l) M / 2 where: 2M is the number of subcarriers of said modulation; L = 2 M is the initial length of said prototype filter.

20. Method for determining filter coefficients according to claim

19, characterized in that said second space belongs to the group comprising: - a space R _κl consisting of polynomials of degree less than or equal to

K1, K> 0, represented by their coefficients; a space R _K2 made up of polynomials of degree less than or equal to K ~ l, K> 0, represented by their Chebyshev development; a space R _K3 made up of polynomials of degree less than or equal to K-1, K> 0, checking p (x + § + p (x- = 2pQ); a space R _K4 consisting of polynomials of degree less than or equal to Kl, K> 0, checking p (x + fy + p ( x-fy≈2p (§), represented by their development of Chebyshev.

21. Method for determining filter coefficients according to claim

20 and claim 10, characterized in that said codes are at least one of the following codes: code n ° 0: the coefficients

are the coefficients ^l _{J + 1} , l = 0, ..., - f- ^' = 0, ... 2m -

for said space R _K2 : code n ° 2: _] ^l ₊₁ = β < _k '(4φ _M (Dl), j = 0, ... m fc-0 for said space R _κ3 : code n ° 3:

for said space R _K4 : code n ° 4:

22. Method for determining filter coefficients according to any one of claims 18 to 21, characterized in that said optimization step (c) implements a local optimization algorithm and / or a genetic global minimization algorithm .

23. A method of determining filter coefficients according to any one of claims 15 to 22, characterized in that said prototype function is described using lattice representations, or in scale, in the form of a function a (t) such that: for code 1:

where d is the degree considered and a _t , i = 0, ..., dl the d coefficients correspondents; for code 2:

where T _{ are Chebyshev polynomials; - the code # 3: a (t) = a _Q + a ₁ t + t (l-2tyyot; t ^μ2.

24. Method for determining filter coefficients according to any one of claims 15 to 23, characterized in that said filter coefficients are obtained, for any α> 0, using the equation: P _a (z ) = ar ^2sM H (z) + i ^ (z ^2M )

where H (z) is a filter of length M verifying _mod (H) = 1, ξ _m0 being a function delivering a measure of discrete location of a filter; and is such that ξ _mod (PJ> 1 -ε.

25. Prototype filter for a modulated filter bank, characterized in that its filter coefficients are obtained using the method for determining filter coefficients according to any one of claims 1 to 24.

26. Modulated filter bank, characterized in that it is formed from a prototype filter whose filter coefficients are obtained using the method of determining filter coefficients according to any one of claims 1 to 24.

27. Terminal for receiving a multicarrier signal, characterized in that it implements a bank of modulated filters formed from a prototype filter whose filtering coefficients are obtained using the method of determining filter coefficients according to any one of claims 1 to 24.

28. Application of a modulated filter bank formed from a prototype filter whose filter coefficients are obtained using the method of determining filter coefficients according to any one of claims 1 to 24 to at least one of the fields belonging to the group comprising: multicarrier modulation of the BFDM / OQAM type; OFDM / OQAM type multicarrier modulation; coding in sub-bands.