FR2828600A1 - Method for determining filter coefficients of bank of modulated filters, receiver terminal and corresponding application - Google Patents

Abstract

A prototype filter of length L is designed to produce a modulated system with M sub-bands by the reduction of parameters. The method comprises the following steps: (1) Filter coefficient optimization (1) to obtain a global optimum for a chosen criterion and analysis of optimal solutions; (2) Interpretation (2) of a sub-set of filters solutions, and determining a compact representation; (3) Optimization (30) and verification (31) of a set of second filter coefficients; (4) Storage (40) of results for the set of second filter coefficients, fast interpolation (41) and verification (42) of optimized results. The filter coefficients are determined so that the bank of modulated filters is used for implementing a system of multi-carrier modulation, which is a modulation of type Orthogonal Frequency Division Multiplex/Offset Quadrature Amplitude Modulation (OFDM/OQAM), or of type Biorthogonal Frequency Division Multiplex/OQAM (BFDM/OQAM). Optimization implements a local optimization algorithm and/or a genetic algorithm for global minimization. A prototype filter for a bank of modulated filters has the filter coefficients determined by the method. A terminal for receiving a multi-carrier signal implements the bank of modulated filters. An application of the bank of modulated filters belongs to the group comprising the multi-carrier modulation of type BFDM/OQAM, the multi-carrier modulation of type OFDM/OQAM, and coding in sub-bands.

Description

- Method for determining the performance coefficients of a bench of

  modulated filters, prototype filter, modulated filter bank, terminal and

corresponding application.

  FIELD OF THE DISCLOSURE The field of the invention is that of digital filtering, and more specifically of modulated filter banks. In particular, the invention relates to

  determination of the prototype function of such modulated filterbanks.

  Modulated filter banks are a particularly important family in the family of filter banks. Their two main advantages 1 0 are: the possibility of implementation using fast algorithms using polyphase decompositions and fast transforms of the DCT (Discrete Cosine Transform) type for this purpose. which concerns modulated cosine banks, or DFT (Discrete Fourier Transform) type for banks modulated by complex exponentials; - a design method that for a given type of transform

  (DCT or DFT) involves only one or two prototype filters.

  Their applications are potentially very numerous.

  They are indeed adapted to multicarrier reception, to transmultiplexers, to subband coding. Thus, in the field of digital communications, the applicants of the present patent application have highlighted the formalism of modulated filter banks for systems. multicarrier modulation module using offset quadrature amplitude modulation (OQAM) for each carrier. It is 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 the 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 the audio coding with reduction of flow with possibly possibilities of improvement, in orthogonal [4], or in biorthogonal [5], of a standard of the 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]. In particular, it is shown that the design of the prototype filter leads to a nonlinear problem whose complexity increases with the length of the prototype. In the examples presented in this reference, the maximum length processed 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 prototype length is. in general, a multiple ranging from 1 to 4 of the number of carriers. It therefore appears necessary to have

  quick and efficient calculation methods.

  We now recall various categories of solutions that can be considered to solve this problem. We consider here the most frequent case which is the one where the shaping of the signal is carried out by a prototype filter of the 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

  orthogonality in continuous time on a medium of infinite duration.

  A defect of this technique is that, in discrete, the orthogonality is verified only approximately, and a prototype considered optimal in continuous time for a given criterion is not necessarily in the discrete case. This technique has, for example, been used in the context of the use of the Isotropic Orthogonal Transform Algorithm (IOTA) function [9] for the digital realization of an OFDM / OQAM modem. In this case, one relies on functions that form an orthogonal basis in continuous time and for which one conjectures that they constitute an optimum for the

time-frequency localization.

  In the case of digital processing, the orthogonality is no longer

  checked and the location of the discrete prototype is not directly optimized.

  2.2 Category 2: Optimization of prototypes with perfect qzasi reconstruction The "back-to-back" bank of filters, for example the modem taken in isolation for a transmission system, is then checked.

  approximately the condition of perfect reconstruction.

  According to this approach, the reconstruction is not perfect and the cost

  optimization of the coefficients can become prohibitive for long lengths.

  For example, in audio coding, the MPEG1 filter bank, layers 1 and 2, was calculated according to this principle [6]. It does not make it possible to obtain a perfect reconstruction. In more recent standards, the recommended filter banks should be

perfectly orthogonal.

  2.3 Category 3: Orthagonalization of RIF The discrete continuation of continuous orthogonalization as carried out by the IOTA method 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 orthogonality conditions.

  However, in this case, the calculation only takes into account the constraints

  orthogonality and does not really make it possible to optimize a precise criterion.

  This method has been used for example by ENST to synthesize prototypes of great length for OFDM / OQAM systems [10]. 2.4 Category 4: Optimization of orthogonal and biorhagonal RIF prototvpes The aim is both to optimize a given criterion and to verify 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 problem of nonlinear optimization, with constraints if one considers the transverse representation of the prototype and without constraints for representations of scale or lattice type. The difficulty is related to the large size of some problems, for example, systems

  OFDM / OQAM or BFDM / OQAM with hundreds or thousands of carriers.

  This problem is therefore faced when designing the biorthogonal and orthogonal prototypes of the BFDM / OQAM [11, 2] and OFDM / OQAM [3] systems. A parameterization more general than the lattice, since it also applies to oversampled orthogonal systems, is proposed in reference [10], but it does not simplify the problem of synthesis in

  the case of so-called critical sampling.

  A similar problem also arises in the context of sub-coding

band [5], [1].

  3. OBJECTIVES OF THE INVENTION The invention relates more particularly to the last category of solution. An object of the invention is to make it possible to synthesize prototypes

  to several hundred or thousands of coefficients.

  Compared to the categories of solutions 1 and 3 which already realize this objective, it should be remembered that: - Category 1 solutions lead to orthogonality losses which are all the more important as one tries to reduce the length of the prototype for a number of carriers, or sub

bands, given.

  - Category 3 solutions do not make it possible to optimize

precisely a given criterion.

  The object of the invention is in particular to overcome these disadvantages of

state of the art.

  More precisely, an object 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 criteria.

predetermined optimization.

  An object of the invention is thus to provide such prototype filters, whose lengths are significantly greater than the lengths that can be processed.

  so far by the known optimization methods.

  For example, in OFDM / OQAM, it is necessary to calculate prototypes for systems with more than 2048 carriers. Likewise, in BFDM / OQAM (where the heading of length 2048 for a system with 512 carriers was hardly achieved according to the prior art), the object of the invention is notably to enable the processing of length systems 8192 with 2048.

  carriers, for example in a reasonable calculation time.

  Another object of the invention is to provide, for a given prototype length, computationally incomparably shorter computing times than with

known optimization methods.

  It is another object of the invention to provide such prototype filters,

  which can be implemented easily and efficiently in terminals.

  These objectives, as well as others which will appear later, are achieved according to the invention by means of a method of determining filter coefficients of a modulated filter bank, based on a prototype filter of length L for producing, by modulation, a modulated system with M subbands, comprising the steps of: (a) determining, for at least two low values of M, as a function of a first parameter space P. of a set of first filter coefficients, corresponding to an overall optimum of the corresponding problem, noted {P. M, L}, for at least one predetermined criterion to be optimized; (b) analyzing said one or more sets of first filter coefficients, so as to determine a subset of filters corresponding to said optimums, which is associated with a second parameter space R. with card R <card P. allowing another representation, said compact, said filters; (c) optimizing a second set of filter coefficients, for at least two values of M and at least one value of L, using said second parameter space R. corresponding to an overall optimum of the corresponding problem, noted {R. M, L;

  (e) storing at least one of said second j 's with filter coefficients.

  In this way, as will be seen later, prototypes filters of very great length (for example some

  thousands) in a very short time, compared to known techniques.

  Advantageously, the method for determining filter coefficients of the invention further comprises a step of: (f) rapidly interpolating a set of filter coefficients for any value of M, from said second set 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 one characteristic of the invention, said optimization of said step (c) is computed taking into account at least one value set for at least one appendix parameter A. The at least one additional parameter A may in particular comprise a

fixed time.

  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 of a polyphase component to another; - belonging to a predetermined range of angular parameters; properties of continuity and / or derivability of limit curves,

  when the length goes to infinity.

  Preferably, the method of the invention comprises, after said step (c), a step of: (d) validating the results of the resolution of said second problem, for at least one low value of M. The method for determining filtering of the invention also advantageously comprises, after said step (f), a step of: (g) checking and / or optimizing the interpolated values for the problem

  corresponding {P. M, L} or {R M, L}.

  The one or more predetermined criteria to be optimized may notably belong to the group comprising: time and / or frequency localization;

  - the minimization of out-of-band energy.

  Advantageously, in said step (b), at least one function, called code, associates with an optimal prototype filter P (z) an approximate representation Px (z) of said optimal prototype filter P (z), expressed help

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

BFDM / OQAM modulation.

  In the case of OFDM / OQAM modulation, said first space P is advantageously the space of the angular coefficients: {, Oi <m-1.0 <1 2M 1} with card (P) = m * M / 2 o : 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 preferentially belongs to the group comprising: a space RKO generated by the generating functions of the points qJM (I), I = 0,..., (M / 2) -1, with M (I) a bijection of {0, (M / 2) 1} to {0,1 / 2}; - a RKI p ar e sp owing to the s or s-pace p o ly nmes of degree less than or equal to K-1, K> 0; a space RK2 corresponding to the subspace of polynomials of degree less than or equal to K-1, K> 0, in the base of the Chebyshev polynomials. According to various advantageous embodiments, said codes are 1 0 respectively: for said RKO space: code n 0: pk (X) = 1 (Mk-iiX: oi) M (k 1) - for said space RK7: code n 1 : cos (2Xik) M (I) k) k = 0 for said space RK2: code n 2:

COS (iXjkTk (4M (1) -1)).

  k = 0 In a particular embodiment, it is possible to use said code n 2

with degree 8.

  In a pretreatential manner, said interpolation step (f) implements a substep of searching for a polynomial of degree in 1og2 M approaching at best,

  in the least squares sense, the set of 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 {<x ,, j 1, ..., 2m + 1,1 = 0 ,,, ., 2M 1}, with card (P) = (2m + 1) M / 2 OR: 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 RK consisting of polynomials of degree less than or equal to K-1, K> 0, represented by their coefficients; a space RK2 consisting of polynomials of degree less than or equal to K-1, K> 0, represented by their Chebyshev development; a space RK3 consisting of polynomials of dome less than or equal to K-1, K> 0, satisfying p (x + 2) + p (x 2) = 2P (2); a space RK4 consisting of polynomials of degree less than or equal to K-1, K> 0, satisfying P (X + 2) + P (X 2) = 2P (2), represented by their

development of Chebyshev.

  According to various embodiments, said codes can be one of the following codes: code n 0: the coefficients x ', l = 0 ,, M2, j = 0, 2m are the coefficients OC! + h 1 = 0, ..., M2, j = 0, .., 2m -for said space RK]: code n 1: oc +, = i, x jk) M (1) k, j = O, ..., 2m k = 0 -for said space RK2: code n 2: kx + l = Ex, Tk (4M (1) 1), j, At 2m k = 0 -for said space RK3: code n 3: O (+ = XjO + iXjk () M (1) 2k 1, j = 0, ..., 2m k O for said space RK4: code n 4:

  oc'j +, = x j + i, xjkT2k, (2 4) M (1) -1), j = 0, ..., 2m.

  Advantageously, said optimization step (c) implements a local optimization algorithm and / or a global minimization genetic 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: a (t, ti FO od is the dogré considered and oci, i = 0,, d-1 the corresponding coefficients, - for the code n 2: a (t) = iTi (4t-1) where Tj are the polynomials of Chebyshev; code n 3:

a (t) = ao + c (, t + t (1-2t) iCciti-2.

  i = 2 According to a particular aspect of the invention, said filter coefficients are advantageously obtained, for any ot> O. using the equation: 2iL, -1 Po! (Z) = OCZ 2sMH (z) + x 2, Z iKi (Z2M) i M o H (z) is a filter of length M satisfying (mod (H) = 1, where n is a function delivering a measure of discrete location of a filter, and oc is such as Emod (Po :)> 1-E The invention also relates to: - prototype filters for modulated filter banks, the filtering coefficients of which are obtained using the method of determining filter coefficients described above modulated filter banks formed from a prototype filter whose filtering coefficients are obtained by means of the method for determining filtering coefficients described above; the reception terminals of a multicarrier signal; implementing a modulated filter bank formed from a prototype filter whose filtering coefficients are obtained using the method of determining filter coefficients described above. The invention also relates to the applications of a modulated filterbank formed from a prototype filter whose filtering coefficients are

  obtained by means of the filter coefficient determination method described above.

  on at least one of the domains belonging to the group comprising: BFDM / OQAM type multicarrier modulation; - OFDM / OQAM type multicarrier modulation; - subband coding. 4. Preferred Embodiments Other features and advantages of the invention will become more apparent

  clearly on reading the following description of embodiments

  preferred embodiments of the invention, given by way of simple illustrative and non-limiting examples, and the appended figures, among which: FIG. 1, commented on in appendix 1, illustrates the stages of "lifting" and "lifting dual" combined, without increased delay, corresponding to a scale representation for a BFDM / OQAM transmultiplexer; - Figure 2, also commented in Appendix 1, shows the stages of "lifting" and "lifting dual" combined, with increased delay, also retaining the property of biorthogonality; FIGS. 3A to 3D show the coefficients (as a function of 1 + 1), for 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; FIGS. 4A to 4D show the coefficients b1 (in function of 1 + 1), for mod, for M = 128 and respectively m = 1, ..., 4, and 0 <i <m-1 for the optimization of angular coefficients in the case of

1'OFDM / OQAM;

  FIGS. 5A to 5D show the coefficients (as a function of 1 + 1), optimal for the 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 an OFDM proton filter of length L = 256 for M = 64 optimized in localization: FIG. 6A: filter coefficients; FIG. 6B: frequency response; FIGS. 7A and 7B illustrate the responses of an OFDM prototype S filter of length L = 256 for M = 64 optimized for out-of-band energy: FIG. 7A: filter coefficients; FIG. 7B: frequency response; FIGS. 8A to 8D respectively show an interpolation of the first 4 coefficients of the optimal OFDM filters for the location for m = 1 and the code 2, degree 8; FIGS. 9A to 9D respectively show an interpolation of the 4 first coefficients of the optimal OFDM filters for the out-of-band energy for n = 1 and the code 2, degree S; 1S - Figure 10 illustrates the first four coefficients a! I (as a function of), (1) and for j = 1, ..., 4) of the scale representation M = 32, L = 192, s = O with localization optimization, in the case

BFDM / OQAM;

  FIG. 11 illustrates the first four coefficients c (as a function of l ,, (l) and for j = 1 .. 4) of the representation on a scale M = 32, L = 192, s = 0, with optirnisation of the out-of-band energy, in the case of BFDM / OQAM; FIGS. 12A and 12B illustrate the responses of a BFDM prototype optimized for localization for a length L = 256 for M = 64 and = 0: FIG. 12A: coefficients of the filter; FIG. 12B: frequency response; FIG. 13, commented on in appendix 6, shows the relative error corrected on the best location with the prototropic filters. OFDM pes with length L = tAI constructed from parameters e ^; optimal filters for code 2 of degree 8; - Figure 14, also commented on in Appendix 6, shows the relative error 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 the code 2 of step 8; FIG. 15, also commented on in appendix 6, is a support representation in [-1, +1] of the 2M transverse coefficients of the optimal filters for the location for M = 32 (cross) and M = 1024 (solid line); - Figure 16, also commented in Appendix 6, shows the relative error committed on the best out-of-band energy with OFDM prototype filters of length L = 2M built from 0º (t); FIG. 17, also commented on in appendix 6, is the analog of FIG. 15, when the cost function is the out-of-band energy; - Figure 18, also commented function in Appendix 6, illustrates the relative error committed on the best out-of-band energy with OFDM prototype filters of length L = 2M built from 61e (t); - Figure 19, also commented on 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 the code 2 of degree 8, for M being 128 and 1024; - Figure 20, also commented in Annex 6, shows the relative error

  comr.nise on the best out-banded energy with BFDM prototype filters -

  of length L = 4M (m = 2) constructed from optimal filter parameters for code 2 of degree 8, for M being 128 and 512; - Figure 21, also commented on 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 optimal filter parameters for the code 2 of degree 8, for M being 128 and 512; FIG. 22 is a simplified flow chart showing the general principle of the

method of the invention.

  4.1 General presentation The invention thus relates in particular to a method of rapid calculation of a prototype filter with real coefficients, denoted p (n), o for a filter of length L, n is an integer between 0 and L-1. To be eligible in the category of solutions sought, the coefficients p (n) must optimize a given criterion and

  verify biorthogonality or orthogonality conditions.

  Appendix 1 presents in detail two criteria (time frequency localization and minimization of out-of-band energy) that can be retained, as well as the biorthogonality conditions that apply to different families of banks.

modulated filters.

  To recall the application context of these preferential embodiments, the principle of obtaining a

  BFDM / OQAM transmultiplexer based on the modulation of a prototype p (n).

  4.2 The parameter reduction method The method of the invention, called parameter reduction, applies to a very wide set of modulated filter banks. It may be suitable, in particular, for any system for which equations (10) to (13) (annex 1) constitute sufficient conditions for the perfect reconstruction property: modulated DCT II and IV and MDFT I type banks 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 different analysis and synthesis benches [5, 16]. In a first step, the method used in a very general context is described. The way in which it applies to BFDM / OQAM and the OFDM / OQAM is then explained, and restrictions that may eventually exist in contexts are finally deduced.

different from those treated.

  4.2.1. General principle It is assumed from the outset that the function to be optimized, for a given criterion, is a prototype protector of length L which by modulation will produce

  a modulated system with M subbands.

- 15

  For such a prototype, the parameter reduction method is validated by

  the sequence of steps that follow, illustrated by the flowchart in Figure 22.

  1. Optimization (1) on a set of parameters, noted P. in principle in sufficient number, that is to say card P 2 number of degrees of freedom, to obtain the global optimum of the problem, noted {P. M, L}, for the chosen criterion, and analysis of the optimal solutions for low values of M. 2. Interpretation (2) of the properties of the solutions obtained as belonging to a subset of filters. In these subsets, we choose finite-dimensional sub-spaces where the filters

  can be represented with a reduced number of parameters.

  This representation is called "compact" in this patent application. The considered properties of the optimal filters obtained during the first step can be for example: the linearity of the phase; - the continuity of the parameters of a polyphase component to another; belonging to a restricted range of angular parameters (for example [0, 7]); - properties of continuity, differentiability of the limit curves

  when the length goes to infinity.

  This results in a representation with a new set of parameters, denoted R.

  deduced from P and which is said to be reduced when cardR <cardP.

  3. Optimization (30) 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, possibly, also fixed values for a set of parameters. other related parameters, noted A (including for example the delay), allowing to cover all the possible solutions of the problem. Verification 31, for small values of M, that the resolution of this problem, noted {R. M, L1, produces a non-optimum

  degraded, compared to that obtained for the problem {P. M, L}.

  4. Storage (40) of the best results obtained for a set of values of M, for example the powers of 2, and for a set of values of

  Fixed, when solving different problems {R. M, L}.

  Fast interpolation 41, without optimization, results for all values of M and verification 42 results with optimization of the problem

  corresponding type {P. M, L} or {P. M, L}.

  A set R of parameters which makes it possible to validate the steps 1 to 4 constitutes a compact representation which verifies the conditions required for

  our method of reducing the parameter space.

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

Annex 1.

  Moreover, it will be noted that, the problems to be solved being non-linear, several local and global optimization software were used to validate these results. 4.2.2 Case of the OFDM / OQAM We first present the case of the OFDM / OQAM for which we have already validated the whole procedure in 4 steps of figure 22, for

prototypes of length L = 2mM.

  4.2.2.1 Optimization of angular coefficients in P space We have chosen an initial parameter space which is the most usual one for this type of problem, and which is introduced in appendix 1 to paragraph 2.2, that is to say that of the angular coefficients {, 0i <m-1.0 <1 <M 1}. In this case we have card (P) = mxMI2 = number of degrees of freedom. The optimization problems to be solved are therefore without constraints and guarantee perfectly

  the orthogonality as well as the phase linearity of the prototype.

  The observation of the angular parameters obtained during direct optimization shows, as illustrated by FIGS. 3 to 5, that these, after

  reduction by period 2, can always be chosen in the interval [0,].

  On the other hand the corresponding parameters, that is to say those which are associated with the same index i for a given set of polyphase components I,

evolve continuously.

  4.2.2.2 Representation in compact form The objective is to obtain an approximate representation Px (z) of the optimal prototype filter P (z) by means of a reduced set of parameters {xk, lk <K, Oi <ml }. The prototype filter P (z) is said, in the present patent application, coded by (Xik) k = o,., K_t, i = o. ", And the application (Xik) ik1-Px (Z) is called a

  code. K is called the degree of the code.

  OFDM prototypes thus encoded are using mKparameters, and are said to be represented in compact form. It is usually 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 likely to 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 representable filters in compact form for this code. The number of parameters for the optimization problem is considerably reduced. For a given subspace Rk, the choice of a base B = (p ", PK) determines the choice of the 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 base to try to improve the conditioning: this leads to a

new code.

  Annex 3 presents three examples of codes that can advantageously

  to 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 K <M2 necessarily leads to a reduction of dimension of the space of the parameters and thus to a significant reduction of the computation time . Numerous optimizations of this type are described in documents [14, 18]. Table 1 (all the tables of the present patent application are grouped in Appendix 8) is an illustration of a set of results obtained, for the location criterion, with the code 2 and a local optimization software entitled CFSQP

  (Feasible Sequential Quadratic Programming in C) [19].

  In this table T is the execution time in seconds on a RISC6000 MHz, Nf is the number of calls to the objective function. It is clear that the gains in speed only exist for M 2 16, that is to say from the moment

start to need it.

  In comparison, the results in Table 2 obtained with the trivial code clearly show that, except for the low values of M 0, the smaller number of calls favors this code, the difference in time increases almost exponentially with M, whereas the reduced space does not penalize

  that very little quality of results.

  The examples of FIGS. 6 and 7 illustrate typical response forms of optimal solutions respectively for the location and out-of-band energy criteria. 4.2.2.4 Rapid interpolation versus M Comparing the relative efficiency of two available codes for location or energy optimization shows that the best results are obtained overall for code number 2 with degree 8. We have therefore retained these parameter values to generate a set {RK, M, L}

quasi-optimal 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 numbers

integers between 3 and 12.

  Examination of the angular coefficients and the optimum shows 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 log2M which best approaches, in the sense of the least squares, the set of points already obtained, for each parameter of the compact representation. The least-squares optimization has for

  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 localization and 1 ' 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 ú, is very small between the interpolated solutions, which are obtained almost instantaneously, and the

optimal solutions.

  Interpolation on a 1 / M scale could significantly improve

  interpolation results that are of lower quality.

  4.2.3 Case of BFDM / OQAM We now consider the situation detailed in paragraph

  2.1. of Annex 1, that is where L = 2mM and d = 2M - 1.

  4.2.3.1 Optimization of the coefficients oc The initial parameter space is then that of the coefficients in scale such that {oc "j = 1,, 2m + 1,1 = O ,, M l} .We thus have card (P) = (2m + 1) 2 which is the number of degrees of freedom.

  oc coefficients are without constraints and perfectly guarantee the biorthogonality.

  The observation of the parameters o: 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 exploitable

  to obtain compact representations.

  4.2.3.2 Representation in compact form Considering again the application defined by (54) (annex 3), we introduce new K-dimensional subspaces RK of the set of continuous functions defined on the interval [0]. , 0 5] RK is provided with a base B = (PI 'PK) and we associate with K-tuples of coordinates (Xj, ..., XjK), where oj varies between 1 and 2m + 1, coefficients ocj by the application: crj = (xj (ó (l)) (59) with cc j = X j + À À + X jK pK (60) Unlike the case of OFDM, the coefficients of the representation in scale are not angles, they are real parameters, without periodic behavior, so it is not necessary to use a function like the function).

  Codes that can be used are presented in Appendix 4.

  4.2.3.3 Optimization of the BFDM compact form As for the OFDM, the computing time gains of the compact form BFDM appear as soon as K <M12. Moreover, the results obtained in BFDM

  have a particularity when considering the criterion of localization.

  By way of example, the results obtained with "Cfsqp" can be examined.

  "Cfsqp" is a local optimization software. Thus with an initialization provided by a prototype with coefficients all zero, and almost whatever the limits of variation allowed for oc, we obtain the results reported at

table 5.

  Other tests were carried out with the software "Dega" (Differential Evolution Genetic Algorithm). "Dega" is a genetic algorithm of global minimization of a function defined on a set of continuous variables. With this program available on the web (http: //http.icsi.berkeley.edu/sorn/code.html !, the overall optimization capability allows for the optimal results obtained in Table 6. It should be noted that in this table and in the one presented next (see Table 8) the number of calls to the

  objective function (Nf) 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 the results of the OFDM / OQAM, delay and / or location value can be improved without significantly modifying the frequency behavior. Thus, in Figure 12, it can be seen that biorthogonality makes it possible to reduce the delay, here s = 0, with respect to what we obtain in orthogonal (see Figure 6). On the other hand, 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 an upcoming section (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 solutions that are not perfectly optimal. As the reference [14] shows, a single parameter makes it possible to make a distinction of type by its values, ie close to 0 for a non solution.

  defective or close to 1 for a detective solution.

  This particular behavior related 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 the OFDM / OQAM. 4.3 Case of Degenerate BFDM / OQAM Solutions with Optimal Location The case of the phenomenon of degeneration, underlined in paragraph 1.1 of Annex 1, leads to a more precise examination of the relevance of the criterion of

  localization in the biorthogonal case. This aspect is detailed in Annex 5.

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

  compact of prototype filters for the intermediate M values.

  According to another approach, detailed in appendix 6, one can try to understand the behavior when M tends towards the 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. One can therefore ask a first question: do these coefficients provide good prototype filters for a higher value of M and which can be arbitrarily large? The expected answer is that this is likely the case if the optimal coefficients tend towards a limit

when M goes to infinity.

  45. Prototyped functions The results established for the energy criterion assume in Expression (8) (Annex 1) that 9sP =, f. Naturally, the methods of parameter reduction and interpolation can be implemented with other values of these two parameters, but this nevertheless results in calculation times which become very high for the high values of M This is then naturally

  all the more critical as m is so high.

  Annex 7 proposes an approach called "prototype functions", which

  allows rapid optimization for any variant of the energy criterion.

  Moreover, in the case of the location criterion, the determination of prototype functions has also made it possible to establish theoretical limits,

  validating the previous results, but without improving them [18].

  4.6. Conclusion We have therefore described a method of reducing parameters that allows the

  rapid calculation of optimal Orthogonal and Biorthogonal modulated benches.

  This method makes it possible to deal with dimension problems, in number of sub-bands and length of the prototype, never reached. It is characterized in particular by: the method of reducing parameters; the rapid interpolation method; - 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 method of rapid interpolation can provide results for a large set of values of the number of 2M channels. In some cases, the stability of the compact representation even makes it possible to obtain the result even more directly, by avoiding the interpolation method, and by relying on simple analytical expressions. These methods have been illustrated with numerous examples that correspond to BFDM / OQAM and OFDM / OQAM type multicarrier modulation systems, which are also equivalent to the filter banks of the systems.

  used in subband coding.

l 24. ,,,

ANNEX 1

  1 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 time and frequency dimension are equally important, Mr. Alard

  [8] showed that the criterion of time-frequency localization was essential.

  1.1 Time-frequency localization For continuous-time signals, it is well known that the Gaussian function is the reference in time-frequency localization where it reaches

  the limit fixed by the principle of uncertainty.

  However, it has recently been shown that in order to measure the timing of discrete time signals, it is preferable to use modified definitions.

  temporal and frequency moments of order 2 l121.

  Thus for a discrete signal x whose norm is defined by XII = Ix (k) l2 (1) k = - the definitions proposed by Doroslovacki [12] for a real signal are the following: - temporal center of gravity: (k - 1/2) | (k) + x (k - 1) 12 T (X) = k oo + oo; (2) Ix (k) + x (k-1) 12 k = -oo - moment of order 2 in frequency: 1 + X 2 (X) = 11 112 | x (k): - x (k-1 ) l2; (3) k = -oo - moment of order 2 in time: I + 1 2 x (k) + x (k - 1) 2 m2 (X) = llxll2 (k - 2 - T (x)) | 2 | (4) We can then derive from these magnitudes a measure of discrete localization that we note (mod to differentiate it from the measure deduced in the references [131, [31 of the continuous case [8] (mo6 (X) = 1 (5 ) \ | 4m2 (X) 2 (X) For any discrete signal x of finite energy, ie such that x 12 (Z), we have 0 <(mOd (x) <1. The maximum localization, (mOd (x) = 1, is reached for a function xopt whose Fourier transform is expressed by Xopt (v) = Cl cos1rlK with CC and K> -2. The function xOpe can therefore be considered

  as a discrete homologous function of the continuous Gaussian function.

  Its expression in time is such that xopt (k + T (xopt)) = 2x r (K / 2 + 1-k) (K / Z + 1 + k) (6) where P (.) Is the Gamma function, with [F (k)] - = 0 if k <O. k Z; Xopt (0) is a non-zero constant and T (xopt) is the temporal center of gravity deduced from (2). To obtain a causal function in the interval [O, L-1], one can put K = L-1 and T (xopt) = 2 The criterion of localization has the particularity of being invariant by trans lation in time and frequency . This property has important consequences, which allows us in particular to identify a class of optimal solutions that can be obtained directly from the xopt function. This particular case

this is dealt with in Annex 5.

  1.2 The minimization of the out-of-band energy The second criterion which we consider is that of the minimization of the weighted norm of the error in frequency. The weighting function, denoted by W (w), is defined by W (w) = {0 if w> w (7)

  In the results that we present in this document, we always take

  days ws = wp = M 'but the proposed method applies for any positive weighting function. For a prototype filter p, whose Fourier transform is P (ei), the objective function to be minimized is written 1 2w Jw (P) = llpll2 W (w) IP (ei) 12dw (8) Compared to the criterion of localization, which for a prototype of finite length is limited to a few elementary operations in the time domain, criterion (8) is much more expensive in calculation. To avoid being penalized excessively in computation time, we use an original fast calculation method, using a single FFT (Fast Fourier Transform) 14], which nevertheless guarantees a high precision in the calculations. In addition, in Appendix 7, we present, for any function W of a real variable with values in [O, 1], an algorithm which increases even more the speed of

calculation of the objective function.

  2 Conditions of perfect reconstruction Conditions for perfect reconstruction of a modulated system, of type

  filter bank or transmultiplexer, can generally be expressed as

  polyphase components of the prototype filter, p (n) transform into

- Z. P (Z)

  For example for a BFDM system, or OFDM, with 2M subcarriers, P (z) can be written as 2M-1 P (Z) = Z-IGt (Z2M) (9) = o

  o G (z) are polyphase components of order 2M, depending on

  which it is possible to write the conditions of perfect reconstruction for

  orthogonal or biorthogonal systems.

  2.1 Biorthogonality and scale representation

  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 the BFDM / OQAM systems are given by l13] - if 0 <d <M-1 Gl (z) Gd-1 (Z) + Z-1GM + I (Z) GM + dl (Z) = 2M, (10) siO <1 <d z- (s-1) Gl (Z) G2M + d1 (Z) + GM + I (Z) GM + dI (Z) = 2M 'sid + 1 <I <M -1 (11) - if M <d <2M-1 Gl (Z) Gd_1 (z) + GM + I (Z) Gd-M1 (Z) = 2M (12) if 0 <I <dM zs Gl (Z) Gd-I (Z) + Z GM + I (Z) GM + d_I (Z) = 211, [if d + 1- M <I <M -1 (13) 3 (:) These conditions are exactly the same as those that were also obtained for cosmetically modulated biorthogonal filter banks, called DCT II and DCT IV, or modulated exponentially according to the so-called MDFT technique

(Modified DFT) [5.

  The conditions to be realized according to the value of d can be directly satisfied if they are imposed on the coefficients p (r) 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 a structure of realization, known as scale, which guarantees by construction that the prototype filter will satisfy the conditions of biorthogonality. In this case we reduce, for a given criterion, a problem of optimization without constraints. This is the approach we have developed. The construction of a scale realization scheme is based on a so-called "lifting scheme" procedure, which can be found, for example, one of the

in the reference [15].

* For 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 + 1) [13]. If we also suppose that the prototype filter P has an initial length such that L = 2 mM, with m positive integer, the reconstruction equation (12) is rewritten: G (z) G2M_I - / (z) + GM + 1 (z) GM __ / (Z) = 2.1, l = 0, 2 - 1, with s an integer between 0 and m-1 of which, as we have already said, depends the reconstruction delay. It can be shown that thanks to the "lift dry me", it is possible to obtain a prototype filter P ', polyphase components G, of greater length 2 (m + 1) M, but such that the reconstruction time is not modified and that the biorthogonality conditions are always verified. The procedure involves two matrices A, B and their inverses which are such that (A (z) 1) (15) (A (z) 1) (16) and B (1 B (z)) (17)

(1) (18)

  o A (z) and B (z) are z-i polynomials. The transition from the prototype P to P 'is then carried out if, on the one hand, for each pair of polyphase components (Gl, GM + 1) and, on the other hand, each pair of components (Gd-1, Gd-Ml ), we

  performs the transformations defined by the diagrams of Figure 1.

  If in figure 1, we take A (z) = aOz-i and B (z) = bo, with aO and bo

  realities, then increase the length of each polyphase component

of 1, without changing the delay.

  In the same way one can preserve the property of biorthogonality by producing an increase of the delay, by introducing matrices C and D such that (Zi 0) (19) Z-iC-i = 1/1 O1 (20) \ -cO z J and D (O z) (21) z-1D = (0 1) (22) with o and do real numbers. The length of P '(z) then equals 2 (m + 1) 11I and the reconstruction delay a' = 2 (s' + 1) = 2 (s + 3) = a + 4. The diagrams in

  corresponding scale are shown in Figure 2.

  These steps can be iterated to get from a prototype filter

  initial, a final prototype with the desired length and reconstruction time.

  It is also possible to directly write the expression of the polyphase components according to the parameters related to this so-called ladder representation (AB, CD). To simplify the writing we will use the one proposed in [18] with the following notation [Z 1] '() = [-oz-1 1]' (23) B (or3 = [È 1] B (o) [È 1] (24) C () = [Z 1 0] z 10 -6 (O) = [1 0] (25)

() [È Z-] () [0 1 (26)

  3u For each quadruplet (GI7GM-1-17GM + I, G2M-1-I) when I varies between 0 and M _ 1 then correspond to 2m + 1 parameters denoted o ', k = 1, ..., 2m + 1 which

  allow to obtain the polyphase components by the following construction.

  We introduce the integers i and jl such that if s is even, we have s = 2jl and m = i + jl + 1etsisestimpairs = 2jl + 1etm = il + jl + 2.0n pose

F [1 0] [1 '] [1 0] (27)

We then have for peers

[G (Z), GM + [(Z)] =

  ## STR2 ## ZD (l2il + 2; +2) C (2il + 2j + 1) x II B (82i + 3) A (02i + 2) Fo [1] (2 i = il, -l For odd s, after having defined Fo, we set Go = FoC (or1) B (ols) (30) 1S so that Z-1Go = zB1 (O5) C-1 (4) FO-1 (31) and we have [Gl (Z ), GM + I (Z) = he il 1 11 1] Go II A (or2i + 4) B (o2i + 5) I | C (oe2il + 2j + 4) D (2il + 2j + 5) (32) ) and [GM-1-1] 1 ZD (2il + 2j + 5) δ (2il + 2j + 4) X rIL B (2i + s) A (2i + 4) Z GO [1] () Overall articles which we know the authors sup- pose, or assert 151, that this structure is complete, that is to say that it covers all of the RIF-type biorthogonal solutions. we have just presented, this is not entirely accurate because, as we show at [18], there are

  defective cases, which can nevertheless be circumvented without significantly degrading

  the quality of the results. So we will take the parameterization in

scale as a baseline.

  2.2 Orthogonality and lattice representation In the case where the prototype is required to be orthogonal, which then forces it to be symmetrical, when it is assumed that the prototype of the analysis bench is identical to that of synthesis, the previous conditions are simplified. The orthogonality conditions are then reduced to the following equation G [(Z) G1 (Z) + G6 + M (Z) GM + (Z) = 2M O <I <M-1 (34) with. which is the paraconjugal operation, i.e. Gl (Z) = G! (Z-1) This condition of reconstruction, in the orthogonal case, which is that of the OFDM / OQAM systems with 2M subcarriers [3] is also that of the benches

  modulated in cosine with M sub-bands [7].

  As in the biorthogonal case, there is also an orthogonal representation of

  which, by construction, guarantees that the relation (34) is always verified.

  In addition, this representation, called lattice, is. If, to simplify the writing, we again take a length L = 2 mM, the polyphase components then check for 0 <I <M _ 1, the relations that follow G1 (z) = z (m 1) G2M_l_ (z), (35) G + 1 (z) = z- (ml) GM-1 (z), (36) z- (m-) Gl (Z) G2M-1-1 (Z) + GM + (Z) GM-1-l (Z) = 2M (37) To construct the lattice representation we define the matrices (Z) [0 zl] (38)

[Z-I O]

  2043 () = [Cine -cos] For each I = 0, ..., M / 2 - 1, then there exist m angles that are equal to the components 2M-polyphases G (z ), GM + (z), GM_ (z) and G2M - I (Z) are obtained by the matrix relations: [G6 (Z) GM + (Z)] = 2 [cosO0 sino] II A (Z) 43 ( k) (41) G2M-1 - (Z) = 1 (k) (Z) cos [GM_1_ (Z)] 2M ó k = 1) [sin e0] (42) The lattice representation therefore also makes it possible to reduce the dimension parameter space, while ensuring the orthogonality and

  completeness, we will take it as a reference in the orthogonal case.

ANNEX 2

  The BFDM / OQAM application context There are therefore several systems based on modulated filter banks and whose perfect reconstruction property can be reduced to the conditions given by one or the other of the relationships (10) at (13), in the biorthogonal case, or at the relation (34), in the orthogonal case. As an illustration, we will briefly summarize how to achieve equations (10) to (13) in the case of BFDM / OQAM multicarrier modulation. For more details on the calculations we can refer to

[2, 11, 16].

1 Description of the continuous case

  If 2rO and v0 respectively denote the duration of a complex symbol time and the difference between carriers, it is known that the complex envelope of a BFDM / OQAM signal can be written in the form l16.

+ oo K-

  s (t) = am, nm, n (t). (43) n = -oo m = 0 with m, n (t) = X (t-nTO) m (21) m (44) OR (Pm, n is a phase term such that iGm'2n + l- (Pm, 2n - 12r modulo T.

  The maximum spectral efficiency is then obtained for voTo = -. At the demo-

  the conditions of real blorthogonality are realized if one demodulates on a basis {) m, n} such that {) m, n'7m, n} constitutes a pair of bases

biortho-

gonal.

2 Discrete Case Description

  If one takes a number of carriers K = 2M, the version: discrete equations (43? And (44) results in the relations + oo 2M-1 s [k] = amnm, n [k;] '(45) n = -oo m = 0 with Ym, n [k] = V /] p [k-nM] ej2Mm (k nM-2) jn (46) op (n) is the causal prototype of length L and D is the integer parameter defined in section 2.1 At the reception to retrieve the transmitted symbols, we must project the modulated BFDM / OQAM signal on a base m, r which must be biorthogonal to the function base ^ / m, n, so we realize the following operation mm, = ('ym, s) = R {E, nr [k] s [k]} (47) where o * denotes the complex conjugate We can show that ym, r & [k] is written fm , n [k] = y [knM] ei2Mm (knM 2) jr & (48) 3 Formulation in the form of transmultiplexcur The modulation and demodulation equations can also be in the form of filter banks. ], [2] that by setting fm (k) = p [k] 2M (2) (49), the BFDM modulation system corresponds to a synthesis bench. eption, if we put D = aM-b, with a 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 the demodulation hm (k) = p [k] ei2Mm (k 2) (50) with p [k] = [Dk] In what follows we will limit ourselves to the case op = p. Under these conditions, if we express the analysis and synthesis filters as a function of the polyphase components G (z) of the prototype, Fm (Z) =, e2Mm (! - 2) z- [G [(Z2M) '(51) = o 2M-1Hm (Z) =, e 2M (2) Z G '(z). (52) = o we can then determine the perfect reconstruction conditions under

  the form given by equations (10) to (13) 111, 2].

  The case of the OFDM / OQAM can be deduced with the same principle of calculation,

  or as a special case of the biorthogonal case.

ANNEX 3

  Codes for OFDM / OQAM Taking into account that at the optimum is bounded between 0 and r, it is possible to consider only the cosines of these angular parameters. The new parameters therefore vary in the interval [0,1]. Since some of the optimization programs consider only unbounded variables, we get to real parameters x which correspond to the cosine of angles by the application defined by 4 (x) =. (53) Note that if cos =, (54) then sin = V '(55)

for in t0, rj.

  1S We consider the mapping M of {0,., M2 -I3 to values in the interval t70,0.5] defined bet

M (1) = M + (56)

  The angular coefficients in the base are then given by K-1 k \ COS () = XiPk ((M (l))) (57) For the moment we have used 3 codes. It is of course possible to

add new ones.

  Code n 0.- The trivial code. We choose K = 2-and for space K space

  generated by the generating functions of the points M (I), l = o, .., M _ I, that is,

  ie Pk (X) = 1 (k _) (X) = I if x = M (k 1) (58) {0 otherwise 1. This application is of course not the only choice possible. In particular it was found with a code, the 12, modification of the code 2 o éM (I) = 2, that one found

  identical results to those obtained with code 2.

  In this case cosel = (: ct + l), l = 0, ..., M-1, i = 0, ..., m-1.

  Code n 1.- For K> 0, IRK is the subspace of polynomials of degree less than or equal to K-1. We then obtain K-1 \ costs = d): ri) M (1)) (59) Code n 2.- For K> 0, 'RK is still the subspace of the polynomials of lower degree on equal to K - 1, but the chosen base is that of the Chebyshev polynomials. We then obtain

/ K-1 \

  COST = (E iTk (4 / M (1) - 1)). (60) k = 0

ANNEX 4

  Codes for the BFDM / OQAM Code n 0.- The trivial code. It corresponds to K = M and the coefficients xi, I = o '.., M _ 1, j = 0, ..., 2m are the coefficients ai ±' I = 0, ..., 2M _ 1, j = o, ..., 2m Code n 1.- The 17K space consists of polynomials of degree less than or equal to K-1 represented by their coefficients. We obtain K-1 a: + l = xj <M (I), j = O ...., 2m. (61) k = 0 Code n 2.- The space 1: K 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 K - i + i = xjTk (44M (I) - 1), j = 0, ..., 2m. (62) k = 0

  o The Tk are the Chebyshev polynomials.

  Code n 3.- The space 1? K is the subspace of polynomials p of degree less than or equal to 2K-1 satisfying p (x + 2) + p (X-2) = 2P (2), c ' ie whose graph is symmetrical with respect to the point (2, P (2)) We then obtain K-1 oj +; = X7 + Xj () M (1) - 2), j = 0, .., 2m. (63) Code n 4.The 17K space is the same as for code 3 but we choose a representation of Chebyshev. We then obtain K-1 <jL = XO + XJT2ki (2M (I) 1), j 0 ,, (64) k = 1

ANNEX 5

  The following theorem shows that one can obtain a localization BFDM filter.

  of Doroslovacki as close as 170n wants to 1, whatever the

values of parameters M, m and s.

  Theorem 1.- let M be even, m 1 integer, s integer in the interval O..m-1 and> 0, there exists a prototype filter BFDM P (z) of parameters M, rn and s satisfying (mod (P )> 1- Demonstration.- Let H (z) be any filter of length M defined by M-1 H (z) = hiZ, (65) i = o

with hi 7t 0, i = 0, ..., M-1.

  Consider the filter P (z) defined by 2M-P (Z) = ZH (Z) + z Ki (Z) (66) i = M It is easy to see that the 2M-polyphase components of P (z) are given by hiz-s if 0 <2 <M-1, 67) Gi () l Ki (Z) if M <i <2M-1. (The conditions of biorthogonality (14) are then written hK2M - (Z) + hM__KM + (Z) = 2M 'l = 0' 2 - 1. (68) We can then choose, for any I of 0..M / 2 - 1, K2M- (Z) arbitrarily, then determine KM + (Z) such that KM + (Z) = h (2M-h, K2M (z)), (69)

since hM__ 'is not zero.

  For all> O. the filter 1 2M- P. () z-2sMH (z) ± z Ki (z) also satisfies the biorthogonality relations. Thanks to the homogeneity of the function (, mod, we have (mod (Éa) = (mod (-Ea), (71) o When gold tends to 0, the filter ("P") tends towards the filter H ( z) and by continuity of the function (mod, the second member of (71) tends to (mod (H) It is then enough to choose for H (z) the filter of length M which satisfies (mod (H) = 1 and c small enough so that (mod (P)> 1 - ú. O In other words it is enough to refer to paragraph 1.1 and to choose for H (z) coefficients given by the expression (63. The other coefficients of

  prototype are then deduced directly thanks to the expression (70).

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

adjacent polyphases.

ANNEX 6

  A first paragraph gives a detailed study of the OFDM case for m = 1

  when the cost function is the location or the out-of-band energy.

  Then the existence of a limit is studied experimentally in the case

OFDM and BFDM for m> 2.

  s 1 Asymptotic behavior when M oo in the OFDM case for m = 1 1.1 Technical Lemmas To fully exploit the behavior at the limit of 2M prototype filters when III becomes large (see section 1.2), we use the 2

lemmas that follow.

  Lemma 1.- Let p (t) be a polynomial of degree 2k + 1, k> 0 satisfying the property of symmetry: p (t) + p (1 - t) = 1, (72) and such that p (0) = 1 (and thus p (1) = 0). Then there are constants ai, i = 1, ..., k such that: k p (t) = 1-t + (2t-1) aiti (1-t) i. (73) i = 1

Demonstration.- See [18,1.

  Lemma 2.- Let (t) be a continuous function on [0'2] and 2n + 1 times continuous

  nuement differentiable on [072] with 1 <n <x and verifying

 (0) = 2 '((2) = -. (74)

  Let h (t) be the function defined on [O, 1] by h (t) = {cos y (() - t) if <t <1 '(75) For k satisfying 1 <k <n, h (t ) is 2k + 1 differentiable on the left in t = 2 If and only if (2i) (1) = 0, i = 1, .. To, k. (76)

Demonstration.- See [181.

  Corollary.- With an additional hypothesis, the analycity of e (t) in a disk of the complex plane centered in = 2 of radius greater than 2 'and also assuming that the degree of 0 () is 2k + 1, we gets the expression 0 (t) = 2 (1 -t + (2t-1), has (1 -t)), (77) i = 1

  o the ai7i = 1, ..., k are coefficients.

  s 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 find 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 length 2M' with the same coefficients.

  It is interesting to compare the value of the cost function on this result to the optimal value for the length 2M '. The ordinate of the curves of FIG. 13 represents the logarithm at base 10 of the absolute value of the relative error made for the location by replacing the optimal filter, for a value

  of M 'given as abscissa, by the filter obtained by the method described above.

  1S The opposite of this order represents, therefore, continuously? the number of decimals

  exact in obtaining the optimal cost.

  We consider the five values M = 128,256,1024,4096 = 22 and 32768 = 2i3.

  The curves in Figure 13 show - that there is 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 filter for M = 1024 , L = 2048 allow to obtain parameters filters M ', L' = 2M ', M' by, with the optimal localization to

  -6 relative error if M,> 27.

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

  analytics for transverse coefficients.

  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 op timum reveals behaviors similar to those described by these thematic ma models, which allows us to deduce then simple formulas of the coefficients which provide quasi-optimal results for large even values of M. We present here the case where L = 2M, but it can be specified that the method

  generalizes for values of m greater than 1 [18].

  An OFDM prototype filter P (z) of parameters M even and of length 2M whose transverse coefficients are hi, i = O ...., 2M-1, has a representation trellis (trivial) with M / 2 angles é, l = 0, ..., M / 2 - 1 with, for I = 0, ..., M / 2 - 1, h = h2M_I_ '= sin, hM__ = hM + = cos. (78) As are the values of a function on the interval [0,0,5] defined in the points' ++, l = 0, ..., M / 2 - 1, then by composition with the sinus functions and cosine, and with translation and symmetry functions of the interval [0,0,5], the coefficients h'jl = 0, ..., 2M-1 can be represented as the

  values of a function defined on the interval [- 1, + 1].

  Case of localization criterion In the case of the localization 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 2048 length filter. 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.

  1S The function d (t), t [0,0.5] depends on the 8 parameters x0, x, ..., x8 of the compact representation for code 2, degree 8 by the formula cos0 (t) =, x (t ) = xkTb (4t 1) (79) where Tk are the Chebyshev polynomials. This can still be written H (t) = -arccot (x (t)). (80) With the coefficients xk of the best optimal filter for M = 25, we obtain e (0) = I. 570796330362582, 1 () - 72r1 <3.57 x 10-9, (81)

  e (2) = 0.7853981805419978, 1 (23 - 4 1 <1.72 X 10-8. (82)

  and we find that the function é (t) is very close to a symmetric function with respect to the point [2,74r]. 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 look for a simpler expression of (t) of the form (77).

  The function 0 * (t) defined by e * (t) = 2 (7r + t + 2t2 _ 12t3 + 8t4) (S3) corresponds, in the expression (77), to the choice k = 2 and the values a1 = - = - 0.3183098861, a2 = - - = - 1.273239544. (84): With k = 2, we then look for optimal values of the parameters a and a2 so as to minimize the relative distance between the location of a calculated filter with the function 0 (t) and the optimal location filter for values of M greater than one MB fixed. Optimization is conducted with Cfsqp in

  using a cost function of minimax type.

  With Mo = 1024 and minimizing the relative location error for 27 values of M whose base-2 log is equidistributed between 10 and 15, we find the values a0 = -0.3063247004S6135, al = -1. 38222431220756, ( 85)

  and e2 (t) denotes the corresponding function.

  With k = 3, and identical calculation conditions, we find a0 = - 0. 33690242860365, a = - 0.83833411092749, a2 = - 1.99179628625716, (86)

  and e3 (t) denotes the corresponding function.

  Finally, with k = 4, we find the values a0 = - 0.33144646457108, a = - 1. 01784717458737, a2 = - 0.43066697997064, a3 = - 3.91628703543313, (87)

  and the corresponding function is called (t).

  Figure 16 shows the relative errors made when using these functions to calculate optimal filters for localization. We denote by C * the curve relating to * (t), by C the curve relating to (t), and so on. We can also check that for a PM (Z) = z2Mo- hiZ-i filter, prototype

  OFDM / OQAM parameter M and length 1: = 2M, we have the

  following property (Moo (PM)> 0.90560. (88) Case of the energy criterion When the cost function is the out-of-band energy, the following results are obtained: Figure 17 is the analogue of Figure 15. note that the function

  limit is not derivable at the origin but that it is regular in t =.

  The function e (t) deduced from the compact representation of the best OFDM filter of length 2It1 for M = 215 is well symmetrical with respect to

  point (2, '4T), but 0 (0) 76' 2 and therefore (1) 0.

  By a least squares fit, we find a function 0e (t) of the form: e (t) = 4 (1 + bo (t-2) + b, (t-2))) (89) with bo = - 1. 78661L958412621, b1 = 0.79DO463095216372. (gO) However, as shown in Figure 18, the results are not as good as for the location. With more coefficients, one does not obtain better because the error committed is constant and related to the lack of symmetry of the curve é (t) limit. Introducing a new code can be a way

to improve this result.

  Lecasm> 2 The results obtained for m = 1, which show, for the loca lization and energy criteria, that the optimal filters have a compact representation which becomes quasi-independent of M for the strong values of this parameter, do not generalize. not for m> 2. It appears that the main distinction to make is then 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, without going through an interpolation process, a quasi-optimal prototype filter for large values of M Thus, we see in FIG. 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 <25 whose cost is very close to

optimal cost.

  In the case of the BFDM, figures 20 and 21 show that, without being so conclusive for the moment, the results obtained make it possible to guarantee, according to the value of initial M, the precision of the 3rd or 4th decimal place if m = 2 and

  2nd or 3rd decimal place if m = 3.

  2.2 Location criterion In the OFDM case, it has been found [18] that for the location and m = 2 or m = 3 the coefficients of the compact representation (code 12, degree 8) of the optimal filter for a fixed value Mo of M provide prototypes of higher parameter M whose cost moves away from the optimal cost It follows that we can not identify the optimal prototype filter for a large

  value of M with a 'limit' filter.

  The coefficients of the compact representation of the optimal filters show

  therefore a variation as a function of M, for M large. Examination of these coefficients

  in this case, m = 2 leads us to look for representations of M l18]. The examination of the coefficients obtained in the OFDM case for m = 2 leads to looking for a compact representation depending on the valenr of M. More precisely, the angular parameters' 0 and 0 are sought in the form o (t) = Hoo (t ) -M1l (t), l (t) = 2 + M7el1 (t), (91) o is a negative real and ok oo (t) = 2 (1 - t + (2t - 1) citi), (92) i = ok é1 (t) = 4 (1 - 2t), Biti (93) z = i An optimization with CFSQP of the error made on the localization by comparison with the set of optimal locations obtained for different values of M gives, for example for k = 2, the set of coefficients; optimal that follows

= -0.289654695596932

  crO = -0.3) 0851581378120 or = -0.723945110106017

02 = - 1.79709674616421

I = 0.713206997358436 -

31 = -2.02470362855665

p2 = 2.25362o85753772

  According to the value of M, we obtain with this set of coefficients a pre-

  which guarantees at least the accuracy of the second decimal place and which may

  even reach the 5th decimal place.

  To deal with the case of BFDM prototype filters, we are currently facing the problem of degeneration. The introduction of a gain factor to retain only the non-degenerate solutions should solve this problem. In the current state it is therefore preferable in this case to simply apply the technique of reducing parameters and, if necessary, that of

rapid interpolation.

ANNEX 7

  1 Definitions and first properties 1.1 Definitions For an integer m> 1, we denote by Em the set of functions h on c _ the real interval] -m, + m [as on each subinterval of] -m, + m [

  of the form] i, [, J = - 2m, - 2m + 1, ..., 2m-1 the restriction of h is.

  twice continuously differentiable and h and its derivatives admit limits across the sub-range. In the rest of this chapter, we will say

  that a function of Em is a prototype function of parameter m.

  Example: For m = 1, E is the set of continuous functions h on [1, + 1] twice continuously differentiable, except possibly in-2'0'- and h, h ', h "have limits to left and right in-2'0'- 'on the right in -1, on the left in 1. A prototype function will be called continuous if it is continuous in the points

  , j = - 2m + 1, ..., 2m-1 and if h (-m) = 0 and h (m) = 0.

  Definition.- Let h be in Em and M an even integer> 2. We denote by PM (Z) the filter of length 2mM whose coe / cients transversal PM.i are given by PMi = h (2i + 1 2mM), i = 0, ..., 2mM-1, (94) and we will say that PM (Z) is generated by h. The points in question are thus calculated the values of the function h are never half-integers and form a regular subdivision of the interval

-m, + m [.

  If the function h is even (or odd), the PM (Z) filters are symmetrical

(or antisymmetric).

  For any PM (Z) filter, we denote by (, (PM) its parameter of

  localization in the sense of Doroslovacki and by JW (PM) its out-of-band energy.

  1.2 Limit to infinity for energy We assume, without loss of generality for the usual problems, that the function W (w) admits a inverse Fourier transform and that ws <M. Let us also set that WM (W) = W (MW). Then, when M goes to infinity, we can express the out-of-band energy of the prototype function with the help of the following theorem, where, in the usual way, the letters in capital letters correspond to the Fourier transforms of those in lowercase, for example, v denotes the inverse Fourier transform of W. Theorem 2.- We have 1H () I2W (or) do _ i + mm mm h (t) h (l) w (tu) of the dt lm JW, .4 (PM) = 7E! + M h2 (tl dt 7r [+ mm h2 (t) dt (95) Demonstration - [18] 2 OFDM Prototype Functions Definition.- Let h be a prototype function of Em. that h is of OFDM class if, for any M, the PM (Z) filter is an OFDM symmetric prototype filter The following theorem shows how the lattice representation of the OFDM prototype filters gives a characterization of the structure of the pro functions

  OFDM totype of Em using m angular functions.

  Theorem 3.- Let h Em of OFDM class. n eist m functions di (t): defined over the interval] 0.2 [, deu; times continuously differentiable and admitting limits in 0 and 2 and verifying the following property. For all t, 0 <t <2, we define the points xi (t) = -m + i + t, i = 0, ..., 2m-1 of the interval] -m, + m [and the functions OJ (t) and g2 (t) per ml m-1 g0 (t) =, h (x2i (t)) Xi, 92 (t) = h (X2i + 1 (t)) Xi, (96) i = oi = oo X is a variable. Then we have the matrix equality m-1 (90 (t) ', g2 (t)) = or (cos o (t), sin o (t)) II AC) (ci (t)), (97 ) k = lo and 63 (f3) denote the matrices [0 X] 'e () [sin-cosH]' - (98)

  and o a constant independent of t.

  Demonstration.- The proof is trivial since the angles di (t)

  defined by equality (97) are continuously dependent on continuously

  the function h. 3 Energy optimization 3.1 Other subdivisions

  To define the PM filter from the prototype function h, we used

  in (94) the points of abscissa 2i + 2-MmM which are regularly spaced in the interval (-m, + m) and are never half-integers. Other choices are possible. For example, in all the previous sections, we did

  the choice with the function M defined in (56) of the points i + i vmr (; +.

  of the previous paragraph are also checked for this subdivision.

  Another subdivision is considered in the following. This corresponds to

  abscissae in the half-integer intervals corresponding to the method of in-

  tegration of Gauss-Legendre. For this subdivision, the theorems are no longer checked. However the construction of the corresponding PM filter is very

  useful for calculating the values of Jw (h).

  3.2 Computation of the energy One supposes in this section that the function h is computed from the data of m functions Bi (t), i = 0, ..., m-1 as described in Theorem 3. Under this form, llhll2 = 1 and therefore mk Jw (h) = Jh h (u) h (t) w (t-u) dadt. (99) -m -m; To numerically compute the integral (99), the square of integration is cut into side squares 2 whose sides have half-integer ends. Sure; each of these small squares (there are 4m2), we calculate the integral by the Gauss-Legendre integration scheme of order n, with n fixed on each of the two

u and t directions.

  Recall that the Gauss-Legendre scheme of order n makes it possible to evaluate the integral of a function f over the interval [-1, + 1] by the formula: f (t) dt Pkf (Xk), I , k =! (100) where xk, k = 1, ..., n are the n roots of the Legendre polynomial of degree n, denoted Pn (X), and Pk. k = 1,., n weights calculated by the formula rt Pn 1 (cA (101) With such a choice of abscissas and weights, the evaluation of the integral of f is

  exact if f is a polynomial of degree less than or equal to 2n-1.

  After having slightly transformed the formula (100) to adapt it to an interval of length 2 'the integral is put in the form

2mn - 2mn-

  Jw (h) = h (ui) h (nj) aij (102) i = o 3 = 0 o ai, j = 16W (7bj-bi) Pk (ni) Pk (uj), (103) o Pk (ui) ) and Pk (uj) are Gauss-Legendre weights depending on the place of oi and bj in the subdivision of one of the intervals of length 2. The ai, j are the coefficients of a symmetric matrix A d order 2mn which does not depend on h but only on n and the weight function W (ct). So we evaluate a

  times for the full value of its coefficients before optimizing the cost function.

  The vector = (h (no), h (u), ..., h (u2mn_) of length 2mn is evaluated for given values of the parameters of the optimization problem and the function Jw (h) is given by Jw ( h) VT A V. (104) Note.- When the parameters of a com pact representation code are used to compute a PM (Z) filter, the number of evaluations of the h function is mM (since the the coefficients of the PM (Z) filter are adjusted by symmetry), then the energy is calculated with two real DFTs> 4 mM, plus arithmetic operations (Appendix A) For the calculation at the limit with a function h depending on a code of similar meaning, the number of evaluations of h is only 2 minutes, and the calculation of the quadratic form (104) is done with O (m2n2) arithmetic operations

  1S In practice, we choose n = 6 or n = 10.

  In this computation at the limit, orthogonality, or biorthogonality, is ensured, by being reduced to representations in trellis, or in scale, which can be written in the form of a function a (t) whose expression resumes the one already seen

  for compact representation codes.

  Code 1: the parameter a (t) is obtained by the formula d- a (t3 = otiti, (105) i = ood is the degree considered for this code and ri, i = O ...., d - 1 the d corresponding coefficients, Code 2: the parameter a (t) is obtained by the formula d-1 a (t) = iTi (4t-1) (106) i = where the Ti are the polynomials of Chebyshev, Code 3: the parameter a (t) is obtained by the formula d-1 a (t) = o0 + ort + t (1 - 2t) oriti-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 bounds given for the coefficients, to the degrees of the codes and to the

initial values.

  The best results obtained are with the code 1 for all coefficients and are given in table 9 with the list of degrees. This table provides the value of Eoo which designates the optimum out-of-band energy for the pro iiotype function. 3.4 First results in the BFDM case An optimization with CFSQP was also used to optimize the energy

  out-of-band prototype functions of type BFDM.

  Table 10 gives the best results obtained for m = 1,., 4, s = 0, .., m-1. The limit prototypes are then used to construct M = 128 parameter filters and the results are compared with the best result obtained for code 2 of degree 5 (EO2l8d) Eoo denotes the optimum out-of-band energy for the prototype function, E128 the out-of-band energy of the 2 x 120M length filter deduced from it, and EO2ld the optimum found by a

  direct optimization on the compact representation of the filter.

ANNEX 8

TalbleauxL = 2M L = 4M = 6M = 8M

M = 4 Nf 480 2445 1290 4566

T 0.01 0.08 0.06 0.29

  (mod 0.95604756 0.98134765 0.98512417 0.98516308 M = 8 Nf 421 2549 3193 7613

T 0.01 0.14 0.26 0.83

  (mod 0.93097304 0.97416613 0.98110911 0.98135807 M = 16 Nb 409 571 1658 3212

T 0.03 0.06 0.25 0.62

  (mod O.9 1 807895 O.96978054 O.97976118 O.98030807 M = 32 Nf 630 625 1105 2486

T 0.06 0.11 0.29 0.9

  (mod 0.91176140 0.96554457 0.97929344 0.98000702 M = 64 Elf 300 972 1515 3369

T 0.05 0.32 0.76 2.33

  (mod 0.90866781 0.96070293 0.97909043 0.97988190 M = 128 Nf 590 898 1845 4373

T 0.18 0.57 1.79 5.99

  (mod 0.90714177 0.95525650 0.97892141 0.97978463 M = 256 Nf 322 640 3474 5652

T 0.2 0.8 6.72 15.4

  (mod 0.90638451 0.94947153 0.97867580 0.97968326 M = 512 Nf 416 985 3701 4243

T 0.47 2.43 14.38 26.31

  (mod 0.90600740 0.94366501 O.97825119 0.97818195 M = 1024 Nf 27 & 1051 4096 5740

T 0.64 5.39 40.29 86.28

  (mod D.90581923 0.93810774 0.97752077 0.97947747 TAB 1 - Best location l,; mo, obtained for code 2, degree 5, by the

cfsqp_kofdm program.

s

L = 2M L = 4M L = 6M L = 8M

M = 4 Nfrs 45 119 405 895

T 0.01 0.01 0.03

  (MOD O.95604756) 0.98134765 O.98512417 O.98516308 i M = 8 NJ 262 418 1342 2114

T O 0.02 0.06 0.11

  (mod 0.93097304 0.97416613 0.98110911 0.98135807 M = 16 Nf 415 1186 3747 6872

T 0.02 0.06 0.27 0.76

  (mod 0.91808152 0.96978110 0.97982131 0.98030906 M = 32 Nf 1168 4249 10594 30570

T 0.06 O.45 1.85 7.73

  (mod O.91177048 0.96554672 0.97935831 0.98000940 1 M = 64 Nb 3777 9509 26087 59254

T 0.4 2.41 11.14 55.45

  (mod 0.90868212 0.96070686 0.97913373 0.97988565 M = 128 Nf 11509 34523 91891 253010

T 2.82 31.21 176.1 943.35

  (mod 0.90715924 0.95526221 0.97894676 0.97978944 M = 256 Nf 33964 123705 330580 528428 T 30.78 467.01 2929.41> 3 hrs 1 Prod 0. 90640369 0.9494? 898 0.97868978 0.97968606

.. .. ...

  TAB. 2 - Better localization (mo <obtained for the code O by the program ctsqp_kofdm.

L = 2M L = 4M L = 6M L = 8M

  M = 48 (MOD 0.90970558 0.96279769 0.97908736 0.97984632

  -0.109 X 10-7 -0.872 X 10-6 -0.948 X 10-4 -0.847 X 10-4

  M = 96 (MOD 0.90766516 0.95758099 0.97898587 0.97982508

* C O -0.522 X 10-7 -0.143 X 10-4 -0.454 X 10-5

  M = 192 (MOD 0.90665509 0.95189868 0.97873054 0.97945047

  O -0.102 X 10-5 -0.659 X 10-4 -0.287 X 10-3

  M = 384 (MOD 0. 90615277 0.94606686 0.97845660 0.97951285

  O -0.147 X 10-6 -0.14 X 10-5 -0.12 X 10-3

  M = 768 (MOD 0.90590231 0.94037685 0.97761748 0.97520089

  C O -0.333 X 10-5 -0.259 X 10-3 -0.441 X 10-2

  M = 1536 (MOD 0.90577725 0.93506301 0.97686969 0.97680666

  O -0.663 X 10-6 -0.256 X 10-4 -0.268 X 10-2

  M - 3072 (MOD 0.90571477 0.93024393 0.97509817 0.79610305

  C O -0.157 X 10-4 -0.309 X 10-3 -0.187 X 10

  TAB. 3 - Checking with cfsqp performances of the rapid interpolation for the localization (MOD

L = 2M L = 4M L = 6M L = 8M

  M = 48 J 1.8968990 X 10-2 2.0369473 X 10-3 3.1732626 X 10-4 8.6231606 X 10-5

  0.158 X 10-6 0.358 X 10-5 0.207 X 10-4 0.72 X 10-4

  M = 96 J 1.8979757 X 10-2 2.0372893 X 10-3 3.1802581 X 10-4 8.6314095 X 10-5

  C 0.526 X 10-7 0.147 X 10-6 0.66 X 10-6 0.217 X 10-5

  M = 192 J 1.8982450 X 10-2 2.0373785 X 10-3 3.1820437 X 10-4 8.6337971 X 10-5

  & 0.526 X 10-7 0.112 X 10-5 0.641 X 10-5 0.224 X 10-4

  M = 384 T 1.8983123 X 10-2 2.0373982 X 10-3 3 1824668-X 10-4 8.6341719 X 10 5

  C 0.526 X 10-7 0.981 X 10-7 0.502 X 10-6 0.174 X 10-5

_.

  M = 768 J 1.8983292 X 10-2 2.0374055 X 10-3 3.1825934 X 10-4 8.6344623 X 10-5

  C 0.526 X 10-7 0.981 X 10-6 0.556 X 10-5 0.193 X 10-4

  M = 1536 J 1.8983333 X 10 = 2 2.0374051 X 10-3 3.1826051 X 10-4 8.6343466 X 10-5

  C O 0.147 X 10-6 0.565 X 10-6 0.195 X 10-5

  M = 3072 J 1 8983345 X 10-2 2 0374077 X 10-3 3.1826322 X 10-4 8.6345459 X 10-5

  C 0.105 X 10-6 0.122 X 10-5 0.691 X 10-5 0.24 X 10-4

  TAB.4 - Verification7 with the criterion of energy, performance of the interpo-

quick release.

L = 4M L = 6M L = 8M

M = 4 N 997 3518 94145

T 0.04 0.16 6.24

  (mod 0.98134765 0.98311206 0.98329853 M = 8 Nf 723 2236 13382

T 0.04 0.16 1.26

  (mod 0.97408002 0.97721012 0.97741087 M - 16 Nb 674 1773 8799

T 0.05 0.2 1.32

  (mod 0.96942291 0.97387771 0.97408999 M = 32 Nf 760 2617 13028

T 0.1 0.5I 3.39

  (mOd 0.96502414 0.97099214 0.97120973 TAB 5 - Best localsaton (mol obtained for s = 1 and code 2, degree 3,

by the program cfsqp_kbfdm.

| L = 4M L = 6M L = 8M

M = 4 1 Nf 3000061 3000061 3000061

T 71.46 91.32 112.69

  (mod 1.OOOOOOOO 1.OOOOOOOO 1.OOOOOOOO M = 8 Nf 3000061 3000061 3000061

T 114.39 153.61 195.12

  (mod 1.OOOOOOOO 1.O0OOOO () O 1.OOOOOOOO .. M = 16 Nb 3000061 3000061 3000061

T 200.59 278.28 358.65

  (mod 0.99999606 1.00000000 1.00000000 M = 32 N / A 3000061 3000061 3000061

T 372.44 525.51 1 685.65

  1 (mOd 0.99995021 0 9999809O | 0 99999982 TAB 6 - Best localsatzon (mo obtained for s = 1 and code 2, dogr 3,

by the dega_kbfdm program.

L = 4M L = 6M L = 8M

M = 4 Nf 671 1982 10643

T 0.06 0.22 1.36

  E 1.9624535 X 10-3 1.4848357 X 10-3 3.2321321 X 10-4

M = 8 Nf 1342 2584 11954

1 T 0.15 0.48 2.66

  E 2.0206358 X 10-3 1.5377920 X 10-3 4.6730159 X 10-4

M = 16 Nf 920 2533 10723

T 0.19 0.91 4.44

  E 2.0343188 X 10-3 1.5713923 X 10-3 7.328422 & X 10-4

M = 32 Nf 942 2643 5987

T 0.38 1.86 4.81

  _ E 2.0377072 X 10-3 1.5787295 X 10-3 7.8751671 X 10-4

  TAB. 7 - Best out-of-band energy (r / M, rr) obtained for s = 1 and the code

  27 degree 3, by the program cfsqp_kfdm.

I L = 4M L = 6M I L = 8M

1 M = 4Nf 3000061 3000061 3000061

113.85 182.7 205.7

  E 3.5291260 X 10-2 1.7559622 X 10-2 1.1731641 X 10-2

M = 8Nf 3000061 3000061 3000061

T 204.13 332.42 382.27

  E 3.8384568 X 10-2 2.1841209 X 10-2 2.I785398 X 10 = 12

M = 16 Nf 3000061 3000061 3000061

T 380.34 674.61 753.68

  E 3.7891304 X 10-2 2.0787967 X 10-2 1.8569596 X 10-2

M = 32 Nf 3000061 3000061 3000061

T 763.93 1343.73 1501.16

  E 3.7885664 X 10-2 2.1318771 X 10-2 1.7818542 X 10-2

  TAB.8 - Best out-of-band energy (lr / M, r) obtained for s = 1 and code

  2, degree 3, by the dega_kbfdm program.

  | m | Degrees of the codes | E 1 1.89834 x 10-2 2 8t8 2.037405 x 10-3 3 6.3, 6 3.189792 x 10-4 4 6,3,3,4 8.69177 x 10-5 5,4,3,2,3 1.838493 x 10 -5 6 4, 4,2,2,3,5 5.72311 x 10-6 TAB. 9 - Best optimization of out-of-band energy for functions

OFDM prototypes of parameter m.

  m = 1 m = 2 m = 3 m = 4 s = 0 Eo 218c. 1.8981327 x 10-2 2.9223829 x 10-3 1. 4497043 x 10-3 1.3165505 x 10-3 E 1.8983347 x 10-2 2.9230531 x 10-3 1. 3694251 x 10-3 1.0332733 x 10-3 E128 1.8981327 x 10- 2 2.9223829 x 10-3 1. 3692782 x 10-3 1.0330880 x 10-3 s = 1 Elld 2.0373402 x 10-3 1.5282759 x 10-3 3.1559744 x 10-4 E / W 2.0374056 x 10-3 4.8348134 x 10-4 _ E, z 2. 0373402 x 10-3 5.5106813 x 10-4 _ s = 2 E28 1.1072554 x 10-3 4.9753911 x 10-4 E / I: 3.1843398 x 10-4 1.0671547 x 10-4 E 3.1829879 x 10 -4 1. 0667812 x 10-4 s = 3 E28 5.0282284 x 10-4 E / \ li l: _

E128 _

  TAB. 10 - Best out-of-band erergy (r / M, 1r) obtained for functions

  BFDM prototypes by the program cfsqp_ener.

3Q

ANNEX 9

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57 2828600

Claims (28)

  A method for determining filter coefficients of a modulated filterbank, based on a prototype filter of length L intended to produce, by modulation, a modulated system with M subbands, characterized in that it comprises the steps following: (a) determination (1), for at least two weak values of M, as a function of a first parameter space P. of a set of first filter coefficients, corresponding to an overall optimum of the corresponding problem, noted {P. M, L}, for at least one predetermined criterion to be optimized; (b) analyzing (2) said one or more sets of first filter coefficients, so as to determine a subset of filters corresponding to said optimums, which is associated with a second parameter space R. with card R <P. another representation, said compact, of said filters; (c) optimizing (30) a second set of filter coefficients, for at least two values of M and at least one value of L, using said second parameter space R. corresponding to a global optimum of the problem correspondent, noted {R. M, L); (e) storing (40) at least one of said second joules of filter coefficients. 2. Method for determining filter coefficients according to claim 1, characterized in that it comprises a step (f) of rapid interpolation (41) of a set of filter coefficients for any value of M, to from said second or more sets of stored coefficients. 3. Method for determining filter coefficients according to one or other   Claims 1 and 2, characterized in that, in said step (c), said   optimization (30) 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   s8 2828600 optimis ation (30) is calculated taking into account at least one fixed value for at least one Annex parameter A. 5. A method of determining filter coefficients according to claim 4, characterized in that the or said additional parameters A include a fixed time. A method of determining filter coefficients according to any one of   claims 5, characterized in that, in said step (b), said analysis   (2) takes into account at least one of the properties belonging to the group comprising - linearity of the phase; - the continuity of the parameters of a polyphase component to another; - belonging to a predetermined range of angular parameters; properties of continuity and / or derivability of limit curves,   when the length goes to infinity.   7. A method of 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) validating (31) the results of the resolution of said second problem, for at least one low value of M. 8. Method for determining filter coefficients according to one or other   Claims 2 to 7, characterized in that it comprises, after said step (f),   a step of: (g) checking (42) and / or optimizing the interpolated values for the   corresponding problem {P. M, L} or {R M, L}.   9. A method of determining filter coefficients according to any one of   Claims 1 to 8, characterized in that the one or more predetermined criteria   to optimize belong to the group comprising: - location in time and / or in frequency;   - the minimization of out-of-band energy.   sg 2828600 10. A method of 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, associating with an optimal prototype filter P (z) an approximate representation P, (z) of said optimal prototype filter P (z),   expressed using said second set of parameters.   11. Method for determining filter coefficients according to one   any of claims 1 to 10, characterized in that said filter bank   modulated is used for the implementation of a multicarrier modulation system. A method of determining filter coefficients according to claim 11, characterized in that said multicarrier modulation is a modulation OFDM / OQAM.   13. A method of determining filter coefficients according to claim 12, characterized in that said first space P is the space of the angular coefficients: {'0i <m-1 O <l <M 1} with card (P) = m * M / 2 o: 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 RKO generated by the generating functions of the points M (I), I = 0, ..., (M / 2) -1, with (I) a bijection of {0, (M / 2) -1} to {0,1 / 2}; a space RKI corresponding to the subspace of polynomials of degree less than or equal to K-1, K> 0; a space RK2 corresponding to the subspace of polynomials of degree less than or equal to K-1, K> 0, in the base of the Chebyshev polynomials. 15. Method for determining filter coefficients according to the claim 2828600   14 and claim 10, characterized in that said codes are respectively: for said RKO space: code n 0: J lsix =) M (kl) pk (X) = 1 (Mk- (X1 Osinon - for said RKI space : code n 1: cos (iXik) M (I) k) k = 0 - for said space RK2: code n 2: CoS (ExikTk (4) M (1) -1)).   k = 0 16. Method for determining filter coefficients according to the claim   characterized in that said code n 2 is used with degree 8.   17. A method of determining filter coefficients according to any one of   Claims 11 to 16, characterized in that said interpolation step (f)   (41) implements a sub-step of finding a polynomial of degree in 1og2 M approaching at best, in the sense of least squares, the set of values already stored. 18. Method for determining filter coefficients according to the claim   11, characterized in that said modulation is a BFDM / OQAM modulation.   19. A method of determining filter coefficients according to claim 18, characterized in that said first space P is the space of the coefficients in scale such that {oc ', j = 1, ..., 2m + 1,1 = 0, ..., 2M 1}, with card (P) = (2m + 1) M / 2 o: 2M is the number of subcarriers of said modulation;   L = 2mM is the initial length of said prototype filter.   20. A method for determining filter coefficients according to claim 19, characterized in that said second space belongs to the group comprising: a space RK consisting of polynomials of degree less than or equal to K-1, K> 0, represented by their coefficients; a space RK2 consisting of polynomials of degree less than or equal to K-1, K> 0, represented by their Chebyshev development; a space RK3 consisting of polynomials of less than or equal to 61 2828600   K-1, K> 0, satisfying p (x + 2) + p (x 2) = 2P (2); a space RK4 consisting of polynomials of degree less than or equal to K-1, K> 0, satisfying P (X + 12) + P (X 2) = 2P (2), represented by their development of Chebyshev.   21. A method of determining filter coefficients according to claim 10, characterized in that said codes are at least one of the following codes: code n 0: the coefficients x; l = O ,, M2, j = 0.2m are the coefficients x +, l = 0, ..., 2, j = 0, ..., 2m - for said space RK1: code n 1: ori + I = iXk4M (I) k, j = 0, ..., 2m o: - for said space RK2: code n 2: + l => xkTk (4) M (l) -1), j = O ,, 2m k = 0 - for said space RK3: code n 3: K OCI + 1 = XJ + XJ ((M (1)) 2k 1, j = O, ..., 2m AO - for said space RK4: code n 4:   oel + = x J + ixkT2k l (2 M (l) - 1), j = 0, ..., 2m.   k = 0 22. A method of determining filter coefficients according to any one of   Claims 18 to 21, characterized in that said step (c) of optimization   (30) implements a local optimization algorithm and / or an algorithm   genetics of global minimization.   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 the code n 1: a (t) = i ti i = ood is the degree considered and oci, i = 0, ..., d-1 the coefficients 62 2828600   correspondents; for the code n 2: a (t) = iociT (4t-1) i = where the Ti are the Chebyshev polynomials; - for code 3: a (toc0 + cx, t + t (1-2t) ioci ti-2.   i = 2 24. A method of determining filter coefficients according to any one of   Claims 1 to 23, characterized in that said filter coefficients   are obtained, for any oc> 0, using the equation: Pa (Z) = az-2sMH (z) + iKi (72U) i = M o H (z) is a filter of length M satisfying ( mod (H) = 1, where Emo is a function delivering a discrete location measurement of a filter, and oc is such that (mod (Pa)> 1-E Prototype filter for modulated filter bank, characterized in that its filter coefficients are obtained using the method of 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 filtering coefficients are obtained using the method of   determination of filter coefficients according to any one of the 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 the   filter coefficients according to any one of claims 1 to 24.   28. Application of a modulated filterbank formed from a prototype filter whose filtering coefficients are obtained using the determination method   filter coefficients according to any one of claims 1 to 24 to   least one of the domains belonging to the group comprising: - BFDM / OQAM type multicarrier modulation; 63 2828600   OFDM / OQAM type multicarrier modulation;