METHOD AND DEVICE FOR MODULATING COMPLEX SYMBOLS, DEMODULATION METHOD AND DEVICE, AND CORRESPONDING COMPUTER PROGRAMS

Abstract

A method of modulating complex symbols is provided, which delivers a multiple carrier signal. The method performs the following acts for at least one base block of N×K complex symbols, where N and K are integers such that N>1 and K≧1: extending the base block to deliver a block of N×(2K−1) elements, referred to as an “extended” block; phase shifting the extended block, delivering a phase shifted extended block; filtering the phase shifted extended block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; mapping the N×(2K−1) filtered elements of said filtered block onto MK frequency samples, where M is the total number of carriers and M≧N; and transforming the MK frequency samples from the frequency domain to the time domain.

Claims

1. A modulation method comprising: modulating complex symbols with a modulation device, delivering a multiple carrier signal; wherein the modulating device performs the following steps, for at least one block of N×K complex symbols, referred to as a “base” block, where N and K are integers such that N>1 and K≧1: extending said base block to deliver a block of N×(2K−1) elements, referred to as an “extended” block, comprising: if K is odd: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; and 2K−2 columns comprising N(2K−2) elements, of which N(K−1) elements correspond to the remaining NK−N complex symbols of said base block and N(K−1) elements correspond to the conjugates of said remaining NK−N complex symbols of said base block; if K is even: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; two columns comprising 2N elements, of which N elements correspond to the real parts of said N second complex symbols of said base block, distinct from said N first complex symbols, and N elements correspond to the imaginary parts of said N second complex symbols; and 2K−4 columns comprising N(2K−4) elements, of which N(K−2) elements correspond to the remaining NK−2N complex symbols of said base block and N(K−2) elements correspond to the conjugates of said remaining NK−2N complex symbols of said base block; phase shifting said extended block, delivering a phase shifted extended block; filtering said phase shifted extended block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; mapping the N×(2K−1) filtered elements of said filtered block on MK frequency samples, where M is the total number of carriers and M≧N; and transforming said MK frequency samples from the frequency domain to the time domain, delivering said multiple carrier signal.

2. The modulation method according to claim 1, wherein said extending step performs the following sub-steps: randomly selecting said first N complex symbols of said base block, and allocating them to said reference column, corresponding to the central column of said extended block; if K is odd: determining the conjugates of the remaining N(K−1) complex symbols of said base block; allocating a first half of said remaining N(K−1) complex symbols and their respective conjugates to (K−1) columns to the left of said reference column, with a symmetrical relationship between said remaining complex symbols and their respective conjugates; and allocating a second half of said remaining N(K−1) complex symbols and their respective conjugates to (K−1) columns to the right of said reference column, with a symmetrical relationship between said remaining complex symbols and their respective conjugates; if K is even: randomly selecting said N second complex symbols of said base block; determining the real parts and the imaginary parts of said N second complex symbols; allocating one of said real or imaginary parts of each of said N second complex symbols to a central column from among the columns to the left of said reference column of said extended block, referred to as the “left central” column; allocating the other one of said real or imaginary parts of each of said N second complex symbols to a central column from among the columns to the right of said reference column of said extended block, referred to as the “right central” column; determining the conjugates of the remaining N(K−2) complex symbols of said base block; allocating a first half of said remaining N(K−2) complex symbols and their respective conjugates to columns to the left of said reference column, with a symmetrical relationship between said remaining complex symbols and their respective conjugates relative to said left central column; and allocating a second half of said remaining N(K−2) complex symbols and their respective conjugates to columns to the right of said reference column, with a symmetrical relationship between said remaining complex symbols and their respective conjugates relative to said right central column.

3. The modulation method according to claim 1, wherein said phase shifting step performs a phase shift row by row of said extended block, while multiplying elements of an (n+1).sup.th row of said extended block, with the exception of the element corresponding to said reference column, by a value equal to, where n lies in the range 0 to N−1.

4. A modulation method according to claim 1, wherein said phase shifted extended block C.sub.N×(2K−1).sup.E is obtained from the following equations:
C.sub.N×(2K−1).sup.E=J.sub.N×(2K−1) ⊙C.sub.N×(2K−1).sup.E with: $.Math.J_{N\times 2.Math.K-1}=\left[\begin{array}{c}{j}_{1\times 2.Math.K-1}^0\\ \mathrm{.Math.}\\ j_{1\times 2.Math.K-1}^{N-1}\end{array}\right]$ if K is even: ${\stackrel{\_}{C}}_{N\times 2.Math.K-1}^E=\{\begin{array}{c}{\stackrel{\_}{C}}_{n,k}^E={\left({\stackrel{\_}{C}}_{n,K-k}^E\right)}^{*}=C_{n,k},k\in \left[0,\left(K/2\right)-2\right]\\ C_{n,\left(K/2\right)-1}^E=\sqrt{2}.Math..Math.\left\{C_{n,\left(K/2\right)-1}\right\}\\ {\stackrel{\_}{C}}_{n,\left(K/2\right)-1+K}^E=\sqrt{2}.Math..Math.\left\{C_{n,\left(K/2\right)-1}\right\}\\ {\stackrel{\_}{C}}_{n,K-1}^E=C_{n,K/2}\\ {\stackrel{\_}{C}}_{n,k+K}^E={\left({\stackrel{\_}{C}}_{n,2.Math.K-1-k}^E\right)}^{*}=C_{n,k+K/2+1},k\in \left[0,\left(K/2\right)-2\right]\end{array}$ if K is odd: ${\stackrel{\_}{C}}_{N\times 2.Math.K-1}^E=\{\begin{array}{c}{\stackrel{\_}{C}}_{n,k}^E={\left({\stackrel{\_}{C}}_{n,K-k}^E\right)}^{*}=C_{n,k},k\in \left[0,\left(K-1\right)/2-1\right]\\ C_{n,K-1}^E=C_{n,\left(K-1\right)/2}\\ \begin{array}{c}{\stackrel{\_}{C}}_{n,k+K}^E={\left({\stackrel{\_}{C}}_{n,2.Math.K-1-k}^E\right)}^{*}=C_{n,k+\left(K-1\right)/2+1}=C_{n,k+\left(K-1\right)/2+1},\\ k\in \left[0,\left(K-1\right)/2-1\right]\end{array}\end{array}$ C.sub.N×K[C.sub.n,k].sub.n−0, . . . , N−1 et k−0, . . . , K−1 is said base block; n is an integer lying in the range 0 to N−1; k is an integer lying in the range 0 to K−1; * is the conjugate operator; ⊙ is the Hadamard product; and j.sub.1×(2K−1).sup.n is a vector in which all of the elements are equal to (√{square root over (−1)}).sup.n, with the exception of the element having the same index as the reference column, which element is equal to 1.

5. The modulation method according to claim 1, wherein said filtering step makes use of a filter of length 2K−1, such that the value of the filter coefficient of the same index as the reference column, referred to as the “reference” coefficient, is equal to 1, and the values of the other coefficients of the filter are symmetrical relative to the reference coefficient.

6. The modulation method according to claim 1, wherein for each row of the filtered block, the mapping step performs both a cyclic shift modulo MK, enabling the element belonging to the reference column to be brought into the ([(n+m) K mod MK]−1).sup.th position, where m is the index of the first carrier allocated to a given user, for m lying in the range 0 to M−N−1, and n is the index of the row lying in the range 0 to (N−1), and also sums the elements obtained after cyclic shifting column by column.

7. The modulation method according to claim 1, wherein, when N<M, a zero value is given to the first element of each column of the extended block preceding said reference column, or indeed a zero value is given to the last element of each column of said extended block following the reference column.

8. The modulation method according to claim 1, wherein K is strictly greater than 1.

9. The modulation method according to claim 1, wherein the modulation device also interleaves said frequency samples obtained at the output from said mapping step, prior to said step of transforming from the frequency domain to the time domain.

10. A modulation device comprising: a processor; a non-transitory computer-readable medium comprising instructions stored thereon, which when executed by the processor configure the processor to modulate complex symbols, delivering a multiple carrier signal, wherein the instructions comprise the following modules, that are activated for at least one block of N×K complex symbols, referred to as a “base” block, where N and K are integers such that N>1 and K≧1: an extension module for extending said base block to deliver a block of N×(2K−1) elements, referred to as an “extended” block, comprising: if K is odd: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; 2K−2 columns comprising N(2K−2) elements, of which N(K−1) elements correspond to the remaining NK−N complex symbols of said base block and N(K−1) elements correspond to the conjugates of said remaining NK−N complex symbols of said base block; if K is even: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; two columns comprising 2N elements, of which N elements correspond to the real parts of said N second complex symbols of said base block, distinct from said N first complex symbols, and N elements correspond to the imaginary parts of said N second complex symbols; 2K−4 columns comprising N(2K−4) elements, of which N(K−2) elements correspond to the remaining NK−2N complex symbols of said base block and N(K−2) elements correspond to the conjugates of said remaining NK−2N complex symbols of said base block; a phase shifter module for phase shifting said extended block, delivering a phase shifted extended block; a filter module for filtering said phase shifted extended block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; a mapping module for mapping the N×(2K−1) filtered elements of said filtered block on MK frequency samples, where M is the total number of carriers and M≧N; and a transformation module for transforming said MK frequency samples from the frequency domain to the time domain, delivering said multiple carrier signal.

11. A demodulation method comprising: demodulating a multiple carrier signal with a demodulation device, delivering at least one block of reconstructed complex symbols; wherein the demodulation device performs the following steps: transforming a said multiple carrier signal from the time domain to the frequency domain, delivering MK frequency samples, where M and K are integers such that M>1 and K≧1; mapping said MK frequency samples on a block of N×(2K−1) elements, where N is an integer such that M≧N>1, referred to as a “demapped” block; filtering said demapped block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; dephase-shifting a said filtered block, delivering a block referred to as a “dephase-shifted filtered” block; and reconstructing a base block from said dephase-shifted filtered block, delivering a block of reconstructed complex symbols, referred to as a “reconstructed” block, by: if K is odd: identifying a reference column of the N first elements in said dephase-shifted filtered block, delivering N first reconstructed complex symbols, and for the remaining N×(2K−1)−N elements of said dephase-shifted filtered block, summing elements of a row of said dephase-shifted filtered block in pairs with the conjugates of respective other elements of said row, delivering $\frac{N\times \left(2.Math.K-1\right)-N}{2}$ reconstructed complex symbols; if K is even: identifying a reference column of N first elements in said dephase-shifted filtered block, delivering N first reconstructed complex symbols, for 2N second elements of said dephase-shifted filtered block, summing real parts of said 2N second elements in pairs delivering N second reconstructed complex symbols, and for the remaining N×(2K−1)−3N elements of said dephase-shifted filtered block, summing elements of a row of said dephase-shifted filtered block in pairs with the conjugates of respective other elements of said row, delivering $\frac{N\times \left(2.Math.K-1\right)-3.Math.N}{2}$ reconstructed complex symbols.

12. The demodulation method according to claim 11, wherein said reconstructed block Ĉ.sub.N×K is obtained from the following equations: if K is odd: ${\hat{C}}_{n,k}=\{\begin{array}{c}{\hat{C}}_{n,k}={\tilde{C}}_{n,k}+{\tilde{C}}_{n,K-k}^{*},k\in \left[0,\left(K/2\right)-2\right]\\ {\hat{C}}_{n,\left(K/2\right)-1}=\sqrt{2}.Math.\left(.Math.\left\{{\tilde{C}}_{n,\left(K/2\right)-1}\right\}+j.Math..Math.\left\{{\tilde{C}}_{m\left(K/2\right)-1+K}\right\}\right)\\ {\hat{C}}_{n,K/2}={\tilde{C}}_{n,K-1}\\ {\hat{C}}_{n,k+K/2+1}={\tilde{C}}_{n,k+K}+{\left({\tilde{C}}_{n,2.Math.K-1-k}\right)}^{*}.Math.k\in \left[0,\left(K/2\right)-2\right]\end{array}$ if K is even: ${\hat{C}}_{n,k}=\{\begin{array}{c}{\hat{C}}_{n,k}={\tilde{C}}_{n,k}+{\tilde{C}}_{n,K-k}^{*},k\in \left[0,\left(K-1\right)/2-1\right]\\ {\hat{C}}_{n,\left(K-1\right)/2}={\tilde{C}}_{n,K-1}\\ {\hat{C}}_{n,k+\left(K-1\right)/2+1}={\tilde{C}}_{n,k+K}+{\left({\tilde{C}}_{n,2.Math.K-1-k}\right)}^{*},k\in \left[0,\left(K-1\right)/2-1\right]\end{array}$ with: $.Math.{\tilde{C}}_{N\times \left(2.Math.K-1\right)}=J_{N\times \left(2.Math.K-1\right)}^{*}\odot {\hat{C}}_{N\times \left(2.Math.K-1\right)}^E.Math..Math.J_{N\times 2.Math.K-1}=\left[\begin{array}{c}{j}_{1\times 2.Math.K-1}^0\\ \mathrm{.Math.}\\ j_{1\times 2.Math.K-1}^{N-1}\end{array}\right]$ Ĉ.sub.N×(2K−1).sup.E is said filtered block; {tilde over (C)}.sub.N×(2K−1)=[{tilde over (C)}.sub.n,l].sub.n=0, . . . , N−1 and l=0, . . . , 2K−1 Ĉ.sub.N×K=[Ĉ.sub.n,k].sub.n=0, . . . , N−1 and k=0, . . . , K−1 n is an integer lying in the range 0 to N−1; k is an integer lying in the range 0 to K−1; * is the conjugate operator; ⊙ is the Hadamard product; and j.sub.1×(2K−1).sup.n is a vector in which all of the elements are equal to (√{square root over (−1)}).sup.n, with the exception of the element having the same index as the reference column, which element is equal to 1.

13. The demodulation method according to claim 11, wherein, in order to construct each (n+1).sup.th row of said demapped block, said mapping step (23) extracts (2K−1) frequency samples from said MK frequency samples, from the ([(m+n)K−(K−1)] mod MK)+1) .sup.th frequency sample, where m is the index of the first carrier allocated to a given user.

14. A demodulation device comprising: a processor; a non-transitory computer-readable medium comprising instructions stored thereon, which when executed by the processor configure the processor to demodulate a multiple carrier signal, delivering at least one block of reconstructed complex symbols, wherein the instructions comprise the following modules: a transformation module for transforming said multiple carrier signal from the time domain to the frequency domain, delivering MK frequency samples, where M and K are integers such that M>1 and K≧1; a mapping module for mapping said MK frequency samples on a block of N×(2K−1) elements, where N is an integer such that M≧N>1, referred to as a “demapped” block; a filter module for filtering said demapped block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; a dephase shifting module for dephase-shifting said filtered block, delivering a block referred to as a “dephase-shifted filtered” block; and a reconstruction module for reconstructing a base block from said dephase-shifted filtered block, delivering a block of N×K reconstructed complex symbols, referred to as a “reconstructed” block, by: if K is odd: identifying a reference column of the N first elements in said dephase-shifted filtered block, delivering N first reconstructed complex symbols, and for the remaining N×(2K−1)−N elements of said dephase-shifted filtered block, summing elements of a row of said dephase-shifted filtered block in pairs with the conjugates of respective other elements of said row, delivering $\frac{N\times \left(2.Math.K-1\right)-N}{2}$ reconstructed complex symbols; if K is even: identifying a reference column of N first elements in said dephase-shifted filtered block, delivering N first reconstructed complex symbols, for 2N second elements of said dephase-shifted filtered block, summing real parts of said 2N second elements in pairs delivering N second reconstructed complex symbols, and for the remaining N×(2K−1)−3N elements of said dephase-shifted filtered block, summing elements of a row of said dephase-shifted filtered block in pairs with the conjugates of respective other elements of said row, delivering $\frac{N\times \left(2.Math.K-1\right)-3.Math.N}{2}$ reconstructed complex symbols.

15. A non-transitory computer-readable medium comprising a computer program stored thereon including instructions for performing a modulation method when the program is executed by a processor of a modulation device, wherein the method comprises: modulating complex symbols with the modulation device, delivering a multiple carrier signal; wherein the modulating device performs the following steps, for at least one block of N×K complex symbols, referred to as a “base” block, where N and K are integers such that N>1 and K≧1: extending said base block to deliver a block of N×(2K−1) elements, referred to as an “extended” block, comprising: if K is odd: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; and 2K−2 columns comprising N(2K−2) elements, of which N(K−1) elements correspond to the remaining NK−N complex symbols of said base block and N(K−1) elements correspond to the conjugates of said remaining NK−N complex symbols of said base block; if K is even: a column comprising N elements corresponding to N first complex symbols of said base block, referred to as a “reference” column; two columns comprising 2N elements, of which N elements correspond to the real parts of said N second complex symbols of said base block, distinct from said N first complex symbols, and N elements correspond to the imaginary parts of said N second complex symbols; and 2K−4 columns comprising N(2K−4) elements, of which N(K−2) elements correspond to the remaining NK−2N complex symbols of said base block and N(K−2) elements correspond to the conjugates of said remaining NK−2N complex symbols of said base block; phase shifting said extended block, delivering a phase shifted extended block; filtering said phase shifted extended block, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block; mapping the N×(2K−1) filtered elements of said filtered block on MK frequency samples, where M is the total number of carriers and M≧N; and transforming said MK frequency samples from the frequency domain to the time domain, delivering said multiple carrier signal.

Description

LIST OF FIGURES

[0141] Other characteristics and advantages of the invention appear more clearly on reading the following description of a particular implementation given merely by way of illustrative and nonlimiting example, and from the accompanying drawings, in which:

[0142] FIG. 1 shows the main steps performed by the modulation technique in a particular implementation of the invention;

[0143] FIG. 2 shows the main steps performed by the demodulation technique in a particular implementation of the invention;

[0144] FIGS. 3 and 4 show respectively the simplified structure of a modulator performing a modulation technique, and of a demodulator performing a demodulation technique in particular embodiments of the invention.

DETAILED DESCRIPTION OF AN IMPLEMENTATION OF THE INVENTION

[0145] The general principle of the invention relates both to a new technique for modulation on a multiplex of carriers, involving extending at least one block of complex symbols that are to be modulated, and delivering an extended block defining a specific pattern of symbols, and also to a corresponding new demodulation technique.

[0146] Specifically, using such a specific symbol pattern makes it possible to use symbols of complex-value at the input to the modulator and to satisfy the following condition for perfect reconstruction of the symbols: (C.sub.N×K=Ĉ.sub.N×K).

[0147] With reference to FIG. 1, there follows a description of the main steps performed by a modulation method in an implementation of the invention.

[0148] Such a method receives as input at least one base block of N×K complex symbols, written C.sub.N×K, with N>1 and K≧1, that it is desired to modulate.

[0149] The following notation is used

[00016]$C_{N\times K}=\left[\begin{array}{ccc}{C}_{0,0}& \mathrm{.Math.}& C_{0,K-1}\\ M& O& M\\ C_{N-1,0}& \mathrm{.Math.}& C_{N-1,K-1}\end{array}\right].$

Each complex symbol is written C.sub.n,k, with k being the index of the complex symbol in the symbol duration (i.e. of the core of the block), for 0≦k≦K−1, and n being the index of the subcarrier (i.e. of the row of the block), for 0≦n≦N−1. It should be observed that a complex symbol may be a data symbol, possibly having a zero value, or it may be a pilot.

[0150] The total number of available carriers is written M, with M≧N and with M being an even integer.

[0151] Consideration is given to a cellular communication system involving a plurality of users, N<M, with N being the number of carriers allocated to a user. For example, N is a multiple of 12.

[0152] During a first step 11, the base block C.sub.N×K is extended so as to obtain an extended block C.sub.N×(K−1).sup.E comprising N×(2K−1) elements. The number of columns is thus increased compared with the number of columns in the base block.

[0153] The elements forming the extended block are obtained from the complex symbols of the base block. Each element of the extended block corresponds either to a complex symbol of the base block, or to the conjugate of a complex symbol of the base block, or to the real or the imaginary part of a complex symbol of the base block (possibly multiplied by a factor of √{square root over (2)}).

[0154] Thus, if K is odd, the extended block C.sub.N×(K−1).sup.E comprises a column having N elements corresponding to N first complex symbols selected randomly from the base block, referred to as the “reference” column, and 2K−2 columns comprising N(2K−2) elements, of which N(K−1) elements correspond to the NK−N complex symbols remaining from the base block, and N(K−1) elements correspond to the conjugates of the NK−N complex symbols remaining of the base block.

[0155] If K is even, the extended block C.sub.N×(K−1).sup.E comprises a column comprising N elements corresponding to N first complex symbols selected randomly from the base block, referred to as the “reference” column, two columns comprising 2N elements, of which N elements correspond to the real portions of N second complex symbols of said base block that are distinct from the N first complex symbols, and N elements correspond to the imaginary portions of the N second complex symbols, and 2K−4 columns comprise N(2K−4) elements, of which N(K−2) elements correspond to the NK−2N remaining complex symbols of the base block and N(K−2) elements correspond to the conjugates of the NK−2N remaining complex symbols of the base block.

[0156] In a particular implementation of the invention, the reference column is the central column of the extended block. It is also possible to apply a permutation to the columns (and/or to the rows), so that the reference column does not correspond to the central column of the extended block.

[0157] As examples, and with the reference column as the central column: [0158] if N is equal to 3 and K is equal to 2, the base block:

TABLE-US-00001 a1 b1 a2 b2 a3 b3
may be extended in the following form:

TABLE-US-00002 Im(a2) a1 Im(b1) Im(a3) b3 Re(a2) Re(a3) b2 Re(b1) [0159] if N is equal to 3 and K is equal to 3, the base block:

TABLE-US-00003 a1 b1 c1 a2 b2 c2 a3 b3 c3
may be extended in the following form:

TABLE-US-00004 a1 a1* a2 a3 a3* b1 b1* b2 c2* c2 b3* b3 c1 c3 c3* [0160] if N is equal to 3 and K is equal to 4, the base block:

TABLE-US-00005 a1 b1 c1 d1 a2 b2 c2 d2 a3 b3 c3 d3
may be extended in the following form:

TABLE-US-00006 b1 Re(b2) b1* a1 a3* Re(c2) a3 a2 Im(b2) a2* b3 d2 Re(c3) d2* c1* Im(c2) c1 d1 d3 Im(c3) d3* [0161] if N is equal to 3 and K is equal to 5, the base block:

TABLE-US-00007 a1 b1 c1 d1 e1 a2 b2 c2 d2 e2 a3 b3 c3 d3 e3
may be extended in the following form:

TABLE-US-00008 b1 a2* a2 b1* b2 d2 b3 b3* d2* a1* d3* d3 a1 a3 c1 c2* c2 c1* c3* d1 d1* c3 e2 e1* e3* e3 e1 [0162] etc.

[0163] In more general manner, the extended block C.sub.N×(K−1).sup.E can be obtained from the following equations: [0164] if K is even:

[00017]${\stackrel{\_}{C}}_{N\times 2.Math.K-1}^E=\{\begin{array}{c}{\stackrel{\_}{C}}_{n,k}^E=\left({\stackrel{\_}{C}}_{n,K-k}^E\right)*=C_{n,k},k\in \left[0,\left(K/2\right)-2\right]\\ {\stackrel{\_}{C}}_{n,\left(K/2\right)-1}^E=\sqrt{2}.Math..Math.\left\{C_{n,\left(K/2\right)-1}\right\}\\ {\stackrel{\_}{C}}_{n,\left(K/2\right)-1+K}^E=\sqrt{2}.Math..Math.\left\{C_{n,\left(K/2\right)-1}\right\}\\ {\stackrel{\_}{C}}_{n,K-1}^E=C_{n,K/2}\\ {\stackrel{\_}{C}}_{n,k+K}^E=\left({\stackrel{\_}{C}}_{n,2.Math.K-1-k}^E\right)*=C_{n,k+K/2+1},k\in \left[0,\left(K/2\right)-2\right]\end{array}$ [0165] if K is odd:

[00018]${\stackrel{\_}{C}}_{N\times 2.Math.K-1}^E=\{\begin{array}{c}{\stackrel{\_}{C}}_{n,k}^E=\left({\stackrel{\_}{C}}_{n,K-k}^E\right)*=C_{n,k},k\in \left[0,\left(K-1\right)/2-1\right]\\ {\stackrel{\_}{C}}_{n,K-1}^E=C_{n,\left(K-1\right)/2}\\ {\stackrel{\_}{C}}_{n,k+K}^E=\left({\stackrel{\_}{C}}_{n,2.Math.K-1-k}^E\right)*=C_{n,k+\left(K-1\right)/2+1},k\in \left[0,\left(K-1\right)/2-1\right]\end{array}$

[0166] It should be observed that multiplying by the term in √{square root over (2)} for the real and imaginary parts serves to normalize the amplitudes of symbols in the various columns.

[0167] During a second step 12, the extended block C.sub.N×(K−1).sup.E is phase shifted, applying a different phase shift row by row. By way of example, such phase shifting implements multiplying all of the elements of a row of the extended block by a value equal to (√{square root over (−1)}).sup.n, where n is the index of the row lying in the range 0 to N−1, with the exception of the element corresponding to the reference column. The resulting block is a phase shifted extended block, written C.sub.N×(2K−1).sup.E.

[0168] For example, if J.sub.N×(2K−1) is used to designate the phase permutation matrix that is defined as follows:

[00019]$J_{N\times 2.Math.K-1}=\left[\begin{array}{c}{j}_{1\times 2.Math.K-1}^0\\ \mathrm{.Math.}\\ j_{1\times 2.Math.K-1}^{N-1}\end{array}\right]$

where j.sub.1×(2K−1).sup.n is a vector in which all of the elements are equal to (√{square root over (1)}).sup.n, with the exception of the element having the same index as the reference column, which element is equal to 1.

[0169] Thus, if the reference column is the first column of the extended block (e.g. following a permutation of columns), j.sub.1×(2K−1).sup.n is a vector such that the first element is equal to 1 and all of the other elements are equal to (√{square root over (1)}).sup.n. If the reference column is the central column of the extended block, then j.sub.1×(2K−1).sup.n is a vector such that the central element is equal to 1 and all of the other elements are equal to (√{square root over (1)}).sup.n.

[0170] The phase shifted extended block can then be obtained using the following equations:

C.sub.N×(2K−1).sup.E=J.sub.N×(2K−1) ⊙C.sub.N×(2K−1).sup.E

where ⊙ is the operator corresponding to the Hadamard product.

[0171] Returning to the above example proposed for K=4, the phase shifted extended block may be written in the following form:

TABLE-US-00009 b1 Re(b2) b1* a1 a3* Re(c2) a3 {square root over (−1)}a2 {square root over (−1)}Im(b2) {square root over (−1)}a2* b3 {square root over (−1)}d2 {square root over (−1)}Re(c3) {square root over (−1)}d2* −c1* −Im(c2) −c1 d1 −d3 −Im(c3) −d3*

[0172] During a third step at 13, the phase shifted extended block C.sub.N×(2K−1).sup.E is filtered, delivering a filter block X.sub.N×(2K−1) having N×(2K−1) filtered elements. By way of example, such filtering makes use of a filter of length 2K−1, such that the value of the filter coefficient of the same index as the reference column, referred to as the “reference” coefficient, is equal to 1, and the values of the other coefficients of the filter are symmetrical relative to the reference coefficient.

[0173] By way of example, the notation H.sub.N×(2K−1).sup.f is used for the filter matrix defined by:

H.sub.N×(2K−1).sup.f=1.sub.N×1h.sub.1×(2K−1).sup.f

where 1.sub.N×1 is a column vector made up of elements equal to 1, and h.sub.1×(2K−1).sup.f is a filter vector made up of the coefficients of the filter, having real values.

[0174] Thus, if the reference column is the first column of the extended block (e.g. following a permutation of columns), h.sub.1×(2K−1).sup.f is a vector such that h.sub.1×(2K−1).sup.f=[h.sub.0.sup.f,h.sub.1.sup.f, . . . ,h.sub.2K−2.sup.f,h.sub.2K−1.sup.f] in which the first element h.sub.0.sup.f is equal to 1 and all of the other elements are less than 1 (h.sub.0.sup.f>h.sub.1.sup.f> . . . >h.sub.2K−2.sup.f>h.sub.2K−1.sup.f). Thus, if the reference column is the second column of the extended block (e.g. following a permutation of columns), h.sub.1×(2K−1).sup.f is a vector such that h.sub.1×(2K−1).sup.f=[h.sub.1.sup.f,h.sub.0.sup.f ,h.sub.1.sup.f, . . . ,h.sub.2K−2.sup.f] in which the second element h.sub.0.sup.f is equal to 1 and all of the other elements are less than 1 and symmetrical relative to the second element h.sub.0.sup.f. If the reference column is the central column of the extended block, h.sub.1×(2K−1).sup.f is a vector such that h.sub.1×(2K−1).sup.f=[h.sub.K−1.sup.f,h.sub.K−2.sup.f, . . . ,h.sub.1.sup.f,h.sub.0.sup.f,h.sub.1.sup.f, . . . ,h.sub.K'12.sup.f,h.sub.K−1.sup.f] in which the central element h.sub.0.sup.f is equal to 1 and all of the other elements are symmetrical relative to the central element (and h.sub.0.sup.f>h.sub.1.sup.f> . . . >h.sub.K−2.sup.f>h.sub.K−1.sup.f).

[0175] In particular, the coefficients h.sub.k.sup.f of the filter may be calculated so as to comply with Nyquist's criterion, such that:

[00020]$h_k^f=\{\begin{array}{c}{\left(h_0^f\right)}^2=1\\ {\left(h_k^f\right)}^2+{\left(h_{K-k}^f\right)}^2=1,\mathrm{for}.Math..Math.k\in \left[1,K-1\right]\end{array}$

[0176] The filter block can then be obtained using the following equations:

X.sub.N×(2K−1)=H.sub.N×(2K−1).sup.f ⊙C.sub.N×(2K−1).sup.E

[0177] Repeating the preceding example proposed for K=4, and considering the filter vector h.sub.1×7.sup.f=[h.sub.3.sup.f, h.sub.2.sup.f, h.sub.1.sup.f, h.sub.0.sup.f , h.sub.2.sup.f, h.sub.3.sup.f], the filtered block may be written in the following form:

TABLE-US-00010 b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f Re(c2) .Math. h.sub.2.sup.f a3 .Math. h.sub.3.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f b3 .Math. h.sub.0.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f {square root over (−1)}d2* .Math. h.sub.3.sup.f −c1* .Math. h.sub.3.sup.f −Im(c2) .Math. h.sub.2.sup.f −c1 .Math. h.sub.1.sup.f d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f
During a fourth step 14, the N×(2K−1) filtered elements of the filtered block X.sub.N×(2K−1) are mapped onto MK frequency samples.

[0178] By way of example, for each row of the filtered block, such mapping performs a cyclic shift of nK positions, modulo MK, where n is the index of the row in the range 0 to (N−1), and a column by column sum of the elements obtained after cyclic shifting.

[0179] Repeating the above example for N=3 and K=4, and assuming that M=4, the following block is obtained after cyclic shifting by nK positions, modulo MK:

TABLE-US-00011 Col0 Col1 Col2 Col3 Col4 Col5 Col6 Col7 b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f Re(c2) .Math. h.sub.2.sup.f a3 .Math. h.sub.3.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. {square root over (−1)}a2* .Math. h.sub.1.sup.f b3 .Math. h.sub.0.sup.f h.sub.2.sup.f Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2 .Math. h.sub.1.sup.f {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f {square root over (−1)}d2* .Math. h.sub.3.sup.f −c1* .Math. h.sub.3.sup.f −Im(c2) .Math. h.sub.2.sup.f −c1 .Math. h.sub.1.sup.f d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f

[0180] Thereafter the following row vector is obtained after summing column by column:

TABLE-US-00012 Col0 Col1 Col2 Col3 Col4 Col5 Col6 Col7 b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f + Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2 .Math. h.sub.1.sup.f − {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}d2* .Math. h.sub.3.sup.f − d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f c1* .Math. h.sub.3.sup.f Im(c2) .Math. h.sub.2.sup.f c1 .Math. h.sub.1.sup.f
More generally, for each row of the filtered block, the mapping performs both a cyclic shift modulo MK, enabling the element belonging to the reference column to be brought into the ([(n+m) K mod MK]+1).sup.th position, where m is the index of the first carrier allocated to a given user, for m lying in the range 0 to M−N−1, and n is the index of the row lying in the range 0 to (N−1), and also sums the elements obtained column by column after cyclic shifting.

[0181] Repeating the above example with N=3 and M=K=4, and assuming m=0, the cyclic shifting modulo MK making it possible to bring the element belonging to the reference column to the ([(n+m)K]+1).sup.th position delivers the following block:

TABLE-US-00013 Col0 Col1 Col2 Col3 Col4 Col5 Col6 Col7 a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f Re(c2) .Math. h.sub.2.sup.f a3 .Math. h.sub.3.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f b3 .Math. h.sub.0.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f {square root over (−1)}d2* .Math. h.sub.3.sup.f −c1* .Math. h.sub.3.sup.f −Im(c2) .Math. h.sub.2.sup.f −c1 .Math. h.sub.1.sup.f Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f
Thereafter the following row vector is obtained after summing column by column:

TABLE-US-00014 Col0 Col1 Col2 Col3 Col4 Col5 Col6 a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f + Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f − {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f c1* .Math. h.sub.3.sup.f Im(c2) .Math. h.sub.2.sup.f Col7 Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2* .Math. h.sub.3.sup.f − d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f c1 .Math. h.sub.1.sup.f

[0182] This vector of MK frequency samples can be presented in the form of a column vector, in which each element corresponds to an entry of a module for transforming from the frequency domain to the time domain.

[0183] Returning to the generic expression for the filtered block, X.sub.N×(2K−1), the mapping step seeks to map each row x.sub.1×(2K−1).sup.n of the filtered block to the MK entries of a module for transforming from the frequency domain to the time domain. At the output from the mapping step, a column vector is obtained of size MK, written y.sub.MK×1, and that can be defined by the following equations:

[00021]$y_{\mathrm{MK}\times 1}=\underset{n=0}{\overset{N-1}{.Math.}}.Math.y_{\mathrm{MK}\times 1}^n$$y_{\mathrm{MK}\times 1}^n={\left(x_{l\times 2.Math.K-1}^n.Math.G_{2.Math.K-1\times \mathrm{MK}}^n\right)}^T$$G_{2.Math.K-1\times \mathrm{MK}}^n={\left[\begin{array}{c}{O}_{\left(m+n\right).Math.K-K+1\times 2.Math.K-1}\\ I_{2.Math.K-1}\\ O_{\left(M-m-n-1\right).Math.K\times 2.Math.K-1}\end{array}\right]}^T,\mathrm{for}.Math..Math.n\ne 0$$G_{2.Math.K-1\times \mathrm{MK}}^0={\left[\begin{array}{c}\left(O_{K\times K-1}.Math..Math.I_K\right)\\ O_{\mathrm{MK}-2.Math.K+1\times 2.Math.K-1}\\ \left(I_{K-1}.Math..Math.O_{k-1,K}\right)\end{array}\right]}^T,\mathrm{for}.Math..Math.m=n=0$ [0184] where I.sub.2K−1 is the unity matrix of size (2K−1)×(2K−1) and (.).sup.T is the transpose operator.

[0185] During an optional fifth step 15, it is possible to interleave the nonzero frequency samples that were not obtained from pilots. In the example described, consideration is given only to complex symbols of the data symbol type. It is thus possible to change the order of the N×K+K−1=15 nonzero frequency samples. Example, at the end of the optional interleaving step 15, the following vector ({tilde over (y)}.sub.MK×1).sub.T is obtained:

TABLE-US-00015 Col0 Col1 Col2 Col3 Col4 Col5 Col6 a1 .Math. h.sub.0.sup.f −d3* .Math. h.sub.3.sup.f Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f b1* .Math. h.sub.1.sup.f {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f Im(c2) .Math. h.sub.2.sup.f Col7 Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2* .Math. h.sub.3.sup.f − a3* .Math. h.sub.1.sup.f + −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f d1 .Math. h.sub.0.sup.f b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f c1 .Math. h.sub.1.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f c1* .Math. h.sub.3.sup.f

[0186] Finally, during a sixth step 16, the MK frequency samples, possibly after interleaving, are transformed from the frequency domain to the time domain, using a conventional transform, e.g. an inverse Fourier transform. This produces a column vector of size MK, written s.sub.MK×1, comprising the time samples of the multiple carrier signal.

[0187] For example, such a signal is obtained from the following equation:

s.sub.MK×1=F.sub.MK×MK.sup.H.{tilde over (y)}.sub.MK×1

(or s.sub.MK×1=F.sub.MK×MK.sup.H.y.sub.MK×1 if interleaving is not used) where F.sub.MK×MK.sup.H is a matrix representative of an inverse Fourier transform, with (.).sup.H being the conjugate transpose operator.

[0188] As mentioned above, the number of available carriers (M) may be greater than or equal to the number of carriers allocated to a user (N).

[0189] Thus, in the LTE system for example, only 300 carriers are used for transmitting payload data, out of the 512 carriers that are available. These 300 carriers are also grouped together into 25 groups of 12 carriers each, also known as “chunks”. Different chunks may be allocated to different users.

[0190] In the proposed modulation technique, if two adjacent chunks are allocated to different users when M>N, then the data in them may overlap, running the risk of causing interference.

[0191] In order to solve this problem, it is possible to give a zero value to the first element of each column of the extended block preceding the reference column, or indeed a zero value to the last element of each column of the extended block following the reference column.

[0192] In this way, zero value symbols are placed on the last or the first carrier of a chunk.

[0193] For example, for a base block C.sub.N×K, with N being the number of carriers allocated to a user (e.g. a value that is a multiple of 12), and using the above defined equations for the extended block C.sub.N×(K−1).sup.E, it is necessary to define zero value symbols on the last (or the first) carrier such that:

[00022]$\mathrm{For}.Math..Math.K.Math..Math.\mathrm{even}.Math.\text{:}.Math..Math.\{\begin{array}{c}{C}_{N-1,K/2-1}=.Math.\left\{C_{N-1,K/2-1}\right\}\\ C_{N-1,K/2+1.Math..Math.\mathrm{.Math.}.Math..Math.K-1}=0\end{array}.Math..Math.\mathrm{For}.Math..Math.K.Math..Math.\mathrm{odd}.Math.\text{:}.Math..Math.C_{N,\left(K-1\right)/2+1.Math..Math.\mathrm{.Math.}.Math..Math.K-1}=0$

[0194] In particular, if the columns of the base block are permutated randomly before the extension step, then the above equations need to be adapted to take this permutation into account.

[0195] In this way, the user is isolated from another user accessing the adjacent subband.

[0196] In particular, the energy saved by transmitting zero value symbols can be used advantageously to stimulate the remaining nonzero symbols, e.g. by increasing the order of a constellation or the coding rate.

[0197] With reference to FIG. 2, there follows a description of the main steps performed by a method of demodulating a multiple carrier signal in an implementation of the invention, enabling at least one block of complex symbols to be reconstructed. In particular, such a multiple carrier signal is generated using the method as described above. Demodulation thus performs operations that are duals of the operations performed for generating the multiple carrier signal.

[0198] During a first step 21, MK time samples s[k] are received, where k lies in the range 0 to MK−1, and M and K are integers such that M>1 and K≧1. The vector ŝ.sub.MK×1 made up of the MK time samples is transformed from the time domain into the frequency domain, e.g. by using a conventional transformation such as a Fourier transform.

[0199] This produces a column vector of size MK, e.g. written {tilde over (ŷ)}.sub.MK×1 if interleaving was performed at the modulation end, comprising MK interleaved frequency samples.

[0200] Returning to the example described above with reference to modulation, this gives by way of example ({tilde over (ŷ)}.sub.MK×1).sup.T defined as follows:

TABLE-US-00016 Col0 Col1 Col2 Col3 Col4 Col5 Col6 a1 .Math. h.sub.0.sup.f −d3* .Math. h.sub.3.sup.f Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f b1* .Math. h.sub.1.sup.f {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f Im(c2) .Math. h.sub.2.sup.f Col7 Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2* .Math. h.sub.3.sup.f − a3* .Math. h.sub.1.sup.f + −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f d1 .Math. h.sub.0.sup.f b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f − c1 .Math. h.sub.1.sup.f {square root over (−1)}a2 .Math. h.sub.3.sup.f c1* .Math. h.sub.3.sup.f

[0201] In general manner, such a signal is obtained from the following equation:

{tilde over (ŷ)}.sub.MK×1=F.sub.MK×MK.ŝ.sub.MK×1

(or ŷ.sub.MK×1=F.sub.MK×MK.ŝ.sub.MK×1 if interleaving was not performed at the modulation end) where F.sub.MK×MK is a matrix representative of a Fourier transform.

[0202] If interleaving is performed at the modulation end, deinterleaving is performed at the demodulation end during a second step 22, so as to recover the frequency samples in order. The vector (ŷ.sub.MK×1).sup.T obtained after deinterleaving may be written in the following form:

TABLE-US-00017 Col0 Col1 Col2 Col3 Col4 Col5 Col6 a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f + Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f − {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f c1* .Math. h.sub.3.sup.f Im(c2) .Math. h.sub.2.sup.f Col7 Col8 Col9 Col10 Col11 Col12 Col13 Col14 Col15 {square root over (−1)}d2* .Math. h.sub.3.sup.f − d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f c1 .Math. h.sub.1.sup.f

[0203] During a third step 23, the MK frequency samples are mapped onto a block of N×(2K−1) elements, with N being an integer such that M≧N>1, which block is referred to as the demapped block and is written {circumflex over (X)}.sub.N×(2K−1).

[0204] By way of example, in order to construct each (n+1).sup.th row of the demapped block, the mapping step extracts (2K−1) frequency samples from among of the MK frequency samples, from the ([(m+n) K−(K−1)] mod MK)+1).sup.th frequency sample.

[0205] Returning to the above-described example, the first row of the demapped block is constructed from the following seven frequency samples:

TABLE-US-00018 b1 .Math. h.sub.3.sup.f Re(b2) .Math. h.sub.2.sup.f b1* .Math. h.sub.1.sup.f a1 .Math. h.sub.0.sup.f a3* .Math. h.sub.1.sup.f + {square root over (−1)}a2 .Math. h.sub.3.sup.f Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + {square root over (−1)}a2* .Math. h.sub.1.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f
the second row of the demapped block is constructed from the following seven frequency samples:

TABLE-US-00019 a3* .Math. h.sub.1.sup.f + Re(c2) .Math. h.sub.2.sup.f + a3 .Math. h.sub.3.sup.f + b3 .Math. h.sub.0.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.f − {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}d2* .Math. h.sub.3.sup.f − {square root over (−1)}a2 .Math. h.sub.3.sup.f {square root over (−1)}Im(b2) .Math. h.sub.2.sup.f {square root over (−1)}a2* .Math. h.sub.1.sup.f c1* .Math. h.sub.3.sup.f Im(c2) .Math. h.sub.2.sup.f c1 .Math. h.sub.1.sup.f
the third row of the demapped block is constructed from the following seven frequency samples:

TABLE-US-00020 {square root over (−1)}d2 .Math. h.sub.1.sup.f − {square root over (−1)}Re(c3) .Math. h.sub.2.sup.f − {square root over (−1)}d2* .Math. h.sub.3.sup.f − d1 .Math. h.sub.0.sup.f −d3 .Math. h.sub.1.sup.f −Im(c3) .Math. h.sub.2.sup.f −d3* .Math. h.sub.3.sup.f c1* .Math. h.sub.3.sup.f Im(c2) .Math. h.sub.2.sup.f c1 .Math. h.sub.1.sup.f

[0206] By using a matrix representation, each row {circumflex over (x)}.sub.1×(2K−1).sup.n of the demapped block {circumflex over (X)}.sub.N×(2K−1) can be obtained from the following equations:

{circumflex over (x)}.sub.1×2K−1.sup.n={tilde over (y)}.sub.MK×1.sup.T(G.sub.2K−1×MK.sup.n).sup.T

with the matrix G being as defined for the modulation end and with

[00023]${\hat{X}}_{N\times \left(2.Math.K-1\right)}=\left[\begin{array}{c}{\hat{x}}_{1\times \left(2.Math.K-1\right)}^0\\ M\\ {\hat{x}}_{1\times \left(2.Math.K-1\right)}^{N-1}\end{array}\right].$

[0207] During a fourth step 24, the demapped block {circumflex over (X)}.sub.N×(2K−1) as obtained in this way is filtered, delivering a filtered block Ĉ.sub.N×(2K−1).sup.E of N×(2K−1) filtered elements. By way of example, such filtering makes use of a filter of length 2K−1, such that the value of the filter coefficient of the same index as the reference column, referred to as the “reference” coefficient, is equal to 1, and the values of the other coefficients of the filter are symmetrical relative to the reference coefficient. Such a filter is similar to the filter used at the modulation end.

[0208] Thus, returning to the above example, consideration is given to the demapped block defined by the three rows described above, which is multiplied by the filter matrix h.sub.1×7.sup.f=[h.sub.3.sup.f,h.sub.2.sup.f,h.sub.1.sup.f,h.sub.0.sup.f,h.sub.1.sup.f,h.sub.2.sup.f,h.sub.3.sup.f], in order to obtain the following filtered block:

TABLE-US-00021 b1 .Math. (h.sub.3.sup.f).sup.2 Re(b2) .Math. (h.sub.2.sup.f).sup.2 b1* .Math. (h.sub.1.sup.f).sup.2 a1 .Math. 1 a3* .Math. (h.sub.1.sup.f).sup.2 + Re(c2) .Math. (h.sub.2.sup.f).sup.2 + a3 .Math. (h.sub.3.sup.f).sup.2 + {square root over (−1)}a2 .Math. h.sub.3.sup.fh.sub.1.sup.f {square root over (−1)}Im(b2) .Math. (h.sub.2.sup.f).sup.2 {square root over (−1)}a2* .Math. h.sub.1.sup.fh.sub.3.sup.f a3* .Math. h.sub.1.sup.fh.sub.3.sup.f + Re(c2) .Math. (h.sub.2.sup.f).sup.2 + a3 .Math. h.sub.3.sup.fh.sub.1.sup.f + b3 .Math. 1 {square root over (−1)}d2 .Math. (h.sub.1.sup.f).sup.2 − {square root over (−1)}Re(c3) .Math. (h.sub.2.sup.f).sup.2 − {square root over (−1)}d2* .Math. (h.sub.3.sup.f).sup.2 − {square root over (−1)}a2 .Math. (h.sub.3.sup.f).sup.2 {square root over (−1)}Im(b2) .Math. (h.sub.2.sup.f).sup.2 {square root over (−1)}a2* .Math. (h.sub.1.sup.f).sup.2 c1* .Math. h.sub.3.sup.fh.sub.1.sup.f Im(c2) .Math. (h.sub.2.sup.f).sup.2 c1 .Math. h.sub.1.sup.fh.sub.3.sup.f {square root over (−1)}d2 .Math. h.sub.1.sup.fh.sub.3.sup.f − {square root over (−1)}Re(c3) .Math. (h.sub.2.sup.f).sup.2 − {square root over (−1)}d2* .Math. h.sub.3.sup.fh.sub.1.sup.f − d1 .Math. 1 −d3 .Math. (h.sub.1.sup.f).sup.2 −Im(c3) .Math. (h.sub.2.sup.f).sup.2 −d3* .Math. (h.sub.3.sup.f).sup.2 c1* .Math. (h.sub.3.sup.f).sup.2 Im(c2) .Math. (h.sub.2.sup.f).sup.2 c1 .Math. (h.sub.1.sup.f).sup.2

[0209] In more general manner, the extended block Ĉ.sub.N×(2K−1).sup.E can be obtained from the following equations:

Ĉ.sub.N×(2K−1).sup.E=H.sub.N×(2K−1).sup.f ⊙{circumflex over (X)}.sub.N×(2K−1)

[0210] During a fifth step 25, the filtered block is phase shifted. The resulting phase shifted filtered block is written Ĉ.sub.N×(2K+1).sup.E. By way of example, such phase shifting makes use of row by row phase shifting of the filtered block, multiplying elements of an (n+1).sup.th row of the filtered block by a value equal to (√{square root over (−1)}).sup.n, with n lying in the range 0 to N−1, with the exception of the element corresponding to the reference column, which is multiplied by 1. The phase shifting performed at the demodulation end is similar to the phase shifting performed at the modulation end.

[0211] Thus, returning to the above example, the phase shifted filtered block obtained after the phase shifting operation is:

TABLE-US-00022 b1 .Math. (h.sub.3.sup.f).sup.2 Re(b2) .Math. (h.sub.2.sup.f).sup.2 b1* .Math. (h.sub.1.sup.f).sup.2 a1 a3* .Math. (h.sub.1.sup.f).sup.2 + Re(c2) .Math. (h.sub.2.sup.f).sup.2 + a3 .Math. (h.sub.3.sup.f).sup.2 + {square root over (−1)}a2 .Math. h.sub.3.sup.fh.sub.1.sup.f {square root over (−1)}Im(b2) .Math. (h.sub.2.sup.f).sup.2 {square root over (−1)}a2* .Math. h.sub.1.sup.fh.sub.3.sup.f −ja3* .Math. h.sub.1.sup.fh.sub.3.sup.f + −jRe(c2) .Math. (h.sub.2.sup.f).sup.2 + −ja3 .Math. h.sub.3.sup.fh.sub.1.sup.f + b3 d2 .Math. (h.sub.1.sup.f).sup.2 + Re(c3) .Math. (h.sub.2.sup.f).sup.2 + d2* .Math. (h.sub.3.sup.f).sup.2 + a2 .Math. (h.sub.3.sup.f).sup.2 Im(b2) .Math. (h.sub.2.sup.f).sup.2 a2* .Math. (h.sub.1.sup.f).sup.2 jc1* .Math. h.sub.3.sup.fh.sub.1.sup.f jIm(c2) .Math. (h.sub.2.sup.f).sup.2 jc1 .Math. h.sub.1.sup.fh.sub.3.sup.f −{square root over (−1)}d2 .Math. h.sub.1.sup.fh.sub.3.sup.f + −{square root over (−1)}Re(c3) .Math. (h.sub.2.sup.f).sup.2 + −{square root over (−1)}d2* .Math. h.sub.3.sup.fh.sub.1.sup.f + d1 d3 .Math. (h.sub.1.sup.f).sup.2 Im(c3) .Math. (h.sub.2.sup.f).sup.2 d3* .Math. (h.sub.3.sup.f).sup.2 c1* .Math. (h.sub.3.sup.f).sup.2 Im(c2) .Math. (h.sub.2.sup.f).sup.2 c1 .Math. (h.sub.1.sup.f).sup.2

[0212] In more general manner, the phase shifted filtered block Ĉ.sub.N×(2K+1).sup.E can be obtained from the following equations:

Ĉ.sub.N×(2K+1).sup.E=J*.sub.N×(2K−1) ⊙Ĉ.sub.N×(2K−1).sup.E

where J.sub.N×(2K−1) is the phase permutation matrix defined at the modulation end.

[0213] During a sixth step 26, a block Ĉ.sub.N×K of N×K complex symbols is reconstructed row by row from the phase shifted block N×K.

[0214] Two situations are distinguished, depending on the value of K.

[0215] Thus, if K is odd, a reference column is identified having N first elements in the phase shifted filtered block, giving N first reconstructed complex symbols. For the N×(2K−1)−N remaining elements of the phase shifted filtered block, elements of a row of the phase shifted filtered block are summed in pairs with the conjugates of respective other elements of the same row, in such a manner as to obtain

[00024]$\frac{N\times \left(2.Math.K-1\right)-N}{2}$

reconstructed complex symbols.

[0216] If K is even, a reference column is identified having N first elements in the phase shifted filtered block, giving N first reconstructed complex symbols. For 2N second elements of the phase shifted filtered block, distinct from the N first elements, the real portions of the 2N second elements are summed in pairs so as to obtain N second reconstructed complex symbols. Finally, for the N×(2K−1)−3N remaining elements of the phase shifted filtered block, elements of a row of the phase shifted filtered block are summed in pairs with the conjugates of respective other elements of the same row, in such a manner as to obtain

[00025]$\frac{N\times \left(2.Math.K-1\right)-3.Math.N}{2}$

reconstructed complex symbols.

[0217] It should be observed that the manner of reconstructing the base block is not limited to the particular implementation described above. For example, if K is even, it is possible to choose to identify a reference column to obtain the N first reconstructed complex symbols, and then to reconstruct

[00026]$\frac{N\times \left(2.Math.K-1\right)-3.Math.N}{2}$

by summing in pairs elements and conjugate elements of the same line, followed by reconstructing N second complex symbols by summing in pairs the real portions of the 2N second elements.

[0218] Returning to the above example, the fourth column is identified in the phase shifted filtered block as being a reference column. This serves to obtain directly the first three reconstructed complex symbols: a1, b3 and d1.

[0219] Thereafter, 2N second elements are identified in the phase shifted filtered block, e.g. the following elements:

Re(b2).(h.sub.2.sup.f).sup.2, −jRe(c2)(h.sub.2.sup.f).sup.2+Im(b2).(h.sub.2.sup.f).sup.2, Re(c.sub.2).(h.sub.2.sup.f).sup.2+√{square root over (−1)}Im(b2).(h.sub.2.sup.f).sup.2, −√{square root over (−1)}Re(c3).(h.sub.2.sup.f).sup.2+Im(c2).(h.sub.2.sup.f).sup.2, Re(c3).(h.sub.2.sup.f).sup.2+jIm(c2).(h.sub.2.sup.f).sup.2, Im(c3).(h.sub.2.sup.f).sup.2,

and the real portions of these elements are summed in pairs.

[0220] This serves to obtain three new reconstructed symbols:

√{square root over (2)}(Re(Re(b2).(h.sub.2.sup.f).sup.2)+jRe(−jRe(c2)(h.sub.2.sup.f).sup.2+Im(b2).(h.sub.2.sup.f).sup.2))

√{square root over (2)}(Re(Re(c2).(h.sub.2.sup.f).sup.2+√{square root over (−1)}Im(b2).(h.sub.2.sup.f).sup.2)+jRe(−√{square root over (−1)}Re(c3).(h.sub.2.sup.f).sup.2+Im(c2).(h.sub.2.sup.f).sup.2))

√{square root over (2)}(Re(Re(c3).(h.sub.2.sup.f).sup.2+jIm(c2).(h.sub.2.sup.f).sup.2)+jRe(Im(c3).(h.sub.2.sup.f).sup.2))

[0221] It should be observed that multiplying by the term in √{square root over (2)} serves to normalize the amplitude of the reconstructed complex symbols.

[0222] For the remaining elements of the phase shifted filtered block, elements of a row of the phase shifted filtered block are summed in pairs with the conjugates of respective other elements of the same row, in such a manner as to obtain the following six reconstructed complex symbols:

(−ja3*.h.sub.1.sup.fh.sub.3.sup.f+a2.(h.sub.3.sup.f).sup.2)+(−ja3.h.sub.1.sup.fh.sub.3.sup.f+a2*.(h.sub.1.sup.f).sup.2)*

(a3*.(h.sub.1.sup.f).sup.2+√{square root over (−1)}a2.h.sub.1.sup.fh.sub.3.sup.f)*+(a3.(h.sub.3.sup.f).sup.2+√{square root over (−1)}a2*.h.sub.1.sup.fh.sub.3.sup.f)

(b1.(h.sub.3.sup.f).sup.2)+(b1*.(h.sub.1.sup.f).sup.2)*

(−√{square root over (−1)}d2.h.sub.1.sup.fh.sub.3.sup.f+c1*.(h.sub.3.sup.f).sup.2)*+(−√{square root over (−1)}d2*.h.sub.1.sup.fh.sub.3.sup.f+c1.(h.sub.1.sup.f).sup.2)

(d2.(h.sub.1.sup.f).sup.2+jc1*.h.sub.1.sup.fh.sub.3.sup.f)+(d2*.(h.sub.3.sup.f).sup.2+jc1.h.sub.1.sup.fh.sub.3.sup.f)*

(d3.(h.sub.1.sup.f).sup.2)+(d3*.(h.sub.3.sup.f).sup.2)*

[0223] The 12 complex symbols as reconstructed in this way can be placed in the reconstructed block Ĉ.sub.N×K, by taking account of the way in which the extended block is constructed from the base block at the modulation end.

[0224] In more general manner, the extended block Ĉ.sub.N×K can be obtained from the following equations: [0225] if K is even:

[00027]${\hat{C}}_{n,k}=\{\begin{array}{c}{\hat{C}}_{n,k}={\hat{C}}_{n,k}+{\tilde{C}}_{n,K-k}^{*},k\in \left[0,\left(K-1\right)/2-1\right]\\ {\hat{C}}_{n,\left(K-1\right)/2}={\tilde{C}}_{n,K-1}\\ {\hat{C}}_{n,k+\left(K-1\right)/2+1}={\tilde{C}}_{n,k+K}+{\left({\tilde{C}}_{n,2.Math.K-1-k}\right)}^{*},k\in \left[0,\left(K-1\right)/2-1\right]\end{array}$ [0226] if K is odd:

[00028]${\hat{C}}_{n,k}=\{\begin{array}{c}{\hat{C}}_{n,k}={\tilde{C}}_{n,k}+{\tilde{C}}_{n,K-k}^{*},k\in \left[0,\left(K/2\right)-2\right]\\ {\hat{C}}_{n,\left(K/2\right)-1}=\sqrt{2}.Math.\left(.Math.\left\{{\tilde{C}}_{n,\left(K/2\right)-1}\right\}+j.Math..Math.\left\{{\tilde{C}}_{m\left(K/2\right)-1+K}\right\}\right)\\ {\hat{C}}_{n,K/2}={\tilde{C}}_{n,K-1}\\ {\hat{C}}_{n,k+K/2+1}={\tilde{C}}_{n,k+K}+{\left({\tilde{C}}_{n,2.Math.K-1-k}\right)}^{*}.Math.k\in \left[0,\left(K/2\right)-2\right]\end{array}$

[0227] Furthermore, with reference to FIGS. 3 and 4 respectively, there follows a description of the simplified structure of a modulation device serving in particular to perform FB-OFDM type modulation and the structure of a demodulation device serving in particular to perform FB-OFDM type demodulation in a particular embodiment of the invention.

[0228] As shown in FIG. 3, a modulator in a particular embodiment of the invention comprises a memory 31 including a buffer memory, a processor unit 32, e.g. having a microprocessor μP and controlled by the computer program 33 performing the modulation method in an implementation of the invention.

[0229] On initialization, the code instructions of the computer program 33 are loaded by way of example into a random access memory (RAM) prior to being executed by the processor of the processor unit 32. The processor unit 32 receives as input at least one base block of complex symbols written C.sub.N×K. The microprocessor of the processor unit 32 performs the steps of the above-described modulation method in compliance with the computer program instructions 33 in order to generate a multiple carrier signal made up of MK time samples s.sub.MK×1. To do this, in addition to the buffer memory 31, the modulator comprises: [0230] an extender module for extending the base block C.sub.N×K, delivering a block of N×(2K−1) elements, referred to as an “extended” block C.sub.N×(2K+1).sup.E, and constructed as described above; [0231] a phase shifter module for phase shifting the extended block C.sub.N×(2K+1).sup.E, delivering a phase shifted extended block C.sub.N×(2K+1).sup.E; [0232] a filter module for filtering the phase shifted extended block C.sub.N×(2K+1).sup.E, delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block X.sub.N×(2K+1); [0233] a mapping module for mapping the N×(2K−1) filtered elements of the filtered block X.sub.N×(2K+1) onto MK frequency samples, delivering a vector y.sub.MK×1, with M being the total number of carriers and with M≧N; [0234] optionally an interleaving module for interleaving the frequency samples, delivering a vector {tilde over (y)}.sub.MK×1 of MK interleaved frequency samples; and [0235] a transformation module for transforming the possibly interleaved MK frequency samples from the frequency domain to the time domain, delivering the MK time samples s.sub.MK×1 forming the multiple carrier signal.

[0236] These modules are controlled by the microprocessor of the processor unit 32.

[0237] As shown in FIG. 4, a demodulator in a particular embodiment of the invention comprises a memory 41 including a buffer memory, a processor unit 42, e.g. having a microprocessor μP and controlled by the computer program 43 performing the demodulation method in an implementation of the invention.

[0238] On initialization, the code instructions of the computer program 43 are loaded by way of example into a RAM prior to being executed by the processor of the processor unit 42. The processor unit 42 receives as input MK time samples ŝ.sub.MK×1 forming the multiple carrier signal, in order to reconstruct at least one block of complex symbols Ĉ.sub.N×K. The microprocessor of the processor unit 42 performs the steps of the above-described de modulation method in application of the instructions of the computer program 43 in order to reconstruct at least one symbol block. To do this, in addition to the buffer memory 41, the demodulator comprises: [0239] a transformation module for transforming the MK time samples ŝ.sub.MK×1 of the multiple carrier signal from the time domain to the frequency domain, delivering MK frequency samples ŷ.sub.MK×1 (or possibly {tilde over (ŷ)}.sub.MK×1 if interleaving is performed at the modulation end), with M and K being integers such that M>1 and K>1; [0240] optionally a deinterleaving module for deinterleaving the frequency samples, delivering a vector ŷ.sub.MK×1 of MK deinterleaved frequency samples; [0241] a mapping module for mapping the possibly deinterleaved MK frequency samples ŷ.sub.MK×1 onto a block of N×(2K−1) elements, where N is an integer such that M≧N>1, referred to as a “demapped” block {circumflex over (X)}.sub.N×(2K+1); [0242] a filter module for filtering the demapped block {circumflex over (X)}.sub.N×(2K+1), delivering a block of N×(2K−1) filtered elements, referred to as a “filtered” block Ĉ.sub.N×(2K+1).sup.E; [0243] a dephase-shifting module for diphase-shifting the filtered block Ĉ.sub.N×(2K+1).sup.E, delivering a dephase-shifted block Ĉ.sub.N×(2K+1).sup.E; and [0244] a reconstruction module for reconstructing a base block from the dephase-shifted filtered block Ĉ.sub.N×(2K+1).sup.E, delivering a block of N×K reconstructed complex symbols, referred to as a “reconstructed” block Ĉ.sub.N×K, as described above.

[0245] These modules are controlled by the microprocessor of the processor unit 42.