Modulation method and device delivering a multicarrier signal, and corresponding demodulation method and device and computer program

Abstract

A method is provided for modulating data symbols, outputting a multi-carrier signal, implementing: a mathematical transform, which transforms data symbols from the frequency domain to a time domain, outputting transformed symbols; and a polyphase filtering, which filters the transformed symbols, outputting the multi-carrier signal. The polyphase filtering uses an expansion factor taking account of a compression factor , the compression factor being a number between 0 and 1 such that the multi-carrier signal can be transmitted at a Faster-Than-Nyquist rate.

Claims

1. A method comprising: modulating data symbols with a modulator device, outputting a multi-carrier signal, implementing: a mathematical transform step, which transforms the data symbols from a frequency domain to a time domain, outputting transformed symbols, a polyphase filtering step, which filters said transformed symbols, outputting said multi-carrier signal, wherein the polyphase filtering step applies an expansion to a prototype filter by an expansion factor that is a function of a compression factor , said compression factor being a number between 0 and 1 and strictly less than 1, and wherein said expansion factor is equal to the integer rounding of $\left[.Math.\frac{M}{2}\right]$ if said data symbols are real values, and said expansion factor is equal to the integer rounding of [.M] if said data symbols are complex values, where M is an integer equal to the number of carriers or to a size of said mathematical transform; and transmitting the multi-carrier signal at a faster-than-Nyquist rate.

2. The method according to claim 1, wherein modulating comprises a pre-processing step of said data symbols, implemented before said mathematical transform step, and said pre-processing step implements multiplication of said data symbols by a term taking account of said compression factor .

3. The method according to claim 1, wherein modulating comprises a step of post-processing said transformed symbols, implemented before said polyphase filtering step, said mathematical transform step outputting M transformed symbols, where M is an integer equal to the number of carriers, and said post-processing step implements repetition by block of said M transformed symbols, outputting b.sub.1 blocks of M transformed symbols, where b.sub.1 is an integer such that b.sub.1 >0, and a sub-block of b.sub.2 transformed symbols, where b.sub.2 is an integer such that 0 <b.sub.2 <M, and wherein said polyphase filtering step uses the prototype filter with a size L =b.sub.1M +b.sub.2 that uses b.sub.1 blocks of M transformed symbols and said sub-block of b.sub.2 transformed symbols as inputs.

4. The method according to claim 3, wherein b.sub.1 is equal to 4 and b.sub.2 is equal to 0.

5. The method according to claim 1, wherein, for an OFDM/OQAM type modulation and for data symbols with real values, said mathematical transform step implements a transformation from the frequency domain to the time domain comprising the following sub-steps: application of a partial inverse Fourier transform to said data symbols, outputting a first sub-set of C transformed symbols; from the first sub-set, obtain a second sub-set of (M-C) transformed symbols, said second sub-set of transformed symbols being complementary to said first sub-set of transformed symbols, to form the set of M transformed symbols; repetition and permutation of said M transformed symbols, outputting L transformed symbols, where L, M and C are integers such that L >M >C.

6. A modulator device for modulating data symbols, outputting a multi-carrier signal, comprising: a mathematical transform module configured to transform data symbols, from a frequency domain to a time domain, outputting transformed symbols; a polyphase filtering module configured to filter said transformed symbols, outputting said multi-carrier signal, wherein said polyphase filtering module applies an expansion to a prototype filter by an expansion factor that is a function of a compression factor , said compression factor being a number between 0 and 1 and strictly less than 1, and wherein said expansion factor is equal to the integer rounding of $\left[.Math.\frac{M}{2}\right]$ if said data symbols are real values, and said expansion factor is equal to the integer rounding of [. M] if said data symbols are complex values, where M is an integer equal to the number of carriers or to a size of said mathematical transform; and an output through which the multi-carrier signal is transmitted.

7. A method comprising: receiving a multi-carrier signal by a demodulator device; and demodulating the multi-carrier signal with the demodulator device, outputting estimated data symbols, implementing: a polyphase filtering step, which filters said multi-carrier signal, outputting data symbols in the time domain, and a mathematical transform step, which transforms said data symbols in the time domain, from a time domain to the frequency domain, outputting data symbols in the frequency domain, wherein said polyphase filtering step applies a decimation to a prototype filter by a decimation factor that is a function of a compression factor , said compression factor being a number between 0 and 1 and strictly less than 1, and wherein said decimation factor is equal to the integer rounding of $\left[.Math.\frac{M}{2}\right]$ if said data symbols before modulation are real values, and said decimation factor is equal to the integer rounding of [. M] if said data symbols before modulation are complex values, where M is an integer equal to the number of carriers or to a size of said mathematical transform.

8. The method according to claim 7, wherein demodulating comprises a post-processing step of data symbols in the frequency domain, outputting said estimated data symbols, said post-processing step implementing multiplication of said data symbols in the frequency domain by a term taking account of said compression factor .

9. A demodulating device for demodulating a multi-carrier signal, outputting estimated data symbols, comprising: an input, which receives the multi-carrier signal; a polyphase filtering module configured to filter said multi-carrier signal, outputting data symbols in the time domain, a mathematical transform module configured to transform said data symbols from a time domain to a frequency domain, outputting data symbols in the frequency domain, wherein said polyphase filtering module applies a decimation to a prototype filter by a decimation factor that is a function of a compression factor , said compression factor being a number between 0 and 1 and strictly less than 1, and wherein said decimation factor is equal to the integer rounding of $\left[.Math.\frac{M}{2}\right]$ if said data symbols before modulation are real values, and said decimation factor is equal to the integer rounding of [.M] if said data symbols before modulation are complex values, where M is an integer equal to the number of carriers or to a size of said mathematical transform.

10. A non-transitory computer-readable storage medium comprising a computer program stored thereon, which includes instructions for performing a method when this program is run using a processor of a modulator device, wherein the method comprises: modulating data symbols with a modulator device, outputting a multi-carrier signal, implementing: a mathematical transform step, which transforms the data symbols from a frequency domain to a time domain, outputting transformed symbols, a polyphase filtering step, which filters said transformed symbols, outputting said multi-carrier signal, wherein the polyphase filtering step applies an expansion to a prototype filter by an expansion factor that is a function of a compression factor , said compression factor being a number between 0 and 1 and strictly less than 1, and wherein said expansion factor is equal to the integer rounding of $\left[.Math.\frac{M}{2}\right]$ if said data symbols are real values, and said expansion factor is equal to the integer rounding of [.M] if said data symbols are complex values, where M is an integer equal to the number of carriers or to a size of said mathematical transform; and transmitting the multi-carrier signal at a faster-than-Nyquist rate.

Description

4. LIST OF FIGURES

(1) Other characteristics and advantages of the invention will become clearer after reading the following description of a particular embodiment, given as a simple illustrative example that is not in any way imitative and the appended drawings, on which:

(2) FIG. 1 shows the main steps used by the modulation method according to a particular embodiment of the invention;

(3) FIG. 2 shows the main steps used by the demodulation method according to a particular embodiment of the invention;

(4) FIGS. 3A to 3D show examples of OFDM/OQAM modulators capable of transmitting data at a Faster-Than-Nyquist rate;

(5) FIG. 4 shows an example of an OFDM/OQAM demodulator capable of receiving data at a Faster-Than-Nyquist rate;

(6) FIG. 5 shows an example of an OFDM modulator capable of transmitting data at a Faster-Than-Nyquist rate;

(7) FIG. 6 shows an example of an OFDM demodulator capable of receiving data at a Faster-Than-Nyquist rate;

(8) FIGS. 7 and 8 show the simplified structure of a modulator implementing a modulation technique and a demodulator implementing a demodulation technique according to a particular embodiment of the invention, respectively.

5. DESCRIPTION OF ONE EMBODIMENT OF THE INVENTION

(9) 5.1 General Principle

(10) The general principle of the invention is based on the use of a compression factor at an expander/decimator of a polyphase filter of a multi-carrier modulator/demodulator for Faster than Nyquist data transmission. Remember that for a theoretical compression factor equal to between 0 and 1, it is expected that the throughput will be multiplied by a factor of 1/ from the so-called Nyquist rate.

(11) More precisely, FIG. 1 shows the main steps involved in a modulation method according to one embodiment of the invention.

(12) Such a method receives data symbols as input, that may have real values denoted a.sub.m,n, or complex values, denoted c.sub.m,n.

(13) These data symbols undergo a mathematical transform 11 from the frequency domain to a transformed domain outputting transformed symbols. This step can use a conventional transform for example of the inverse Fast Fourier Transform type, or if data symbols have real values, a partial Fourier Transform outputting a first sub-set of transformed symbols, followed by construction of a second sub-set of transformed symbols from the first sub-set, from the technique disclosed in the French patent application FR 2 972 091 filed on 28 Feb. 2011 in the name of the Applicant.

(14) Polyphase filtering 12 is then done on the transformed symbols to shape the carriers. In particular, this polyphase filtering step uses an expansion factor taking account of a compression factor , the compression factor being a number between 0 and 1 capable of transmitting the multi-carrier signal at a Faster-Than-Nyquist rate.

(15) The polyphase filtering step is used to shape the carriers.

(16) In particular, if the data symbols have real values, the expansion factor is equal to the integer rounding of

(17) $\left[.Math.\frac{M}{2}\right]$
(i.e. to the integer closest to

(18) $\left[.Math.\frac{M}{2}\right]),$
where M is an integer equal to the size of the mathematical transform. If the data symbols have complex values, the expansion factor is equal to the integer rounding of [. M] (i.e. the integer closest to [. M]), where M is an integer equal to the size of the mathematical transform.

(19) The signal s obtained after the filtering operation is a multi-carrier signal.

(20) In particular, a pre-processing step 10 can be carried out before the data symbols transformation step. Such an optional step applies a phase shift to data symbols, and multiplies data symbols by a term taking account of the compression factor r. It is used particularly when the length of the prototype filter is even.

(21) The main steps used by a demodulation method according to the invention will now be presented with reference to FIG. 2.

(22) Such a method receives a multi-carrier signal as input.

(23) Polyphase filtering is then done on the multi-carrier signal during a first step 21, outputting data symbols in the transformed domain. Such polyphase filtering uses a decimation factor taking account of the compression factor .

(24) In particular, if data symbols before modulation have real values (for example for an OFDM/OQAM modulation), the decimation factor is equal to the integer rounding of

(25) $\left[.Math.\frac{M}{2}\right].$
If data symbols before modulation have complex values (for example for an OFDM modulation or oversampled OFDM modulation), the decimation factor is equal to the integer rounding of [. M].

(26) During a second step 22, a mathematical transform is applied on data symbols in the transformed domain, to transform them from the transformed domain to the frequency domain. This step uses a conventional transform, for example of the fast Fourier transform type.

(27) If the length of the prototype filter used is even, post-processing will be carried out on the data symbols in the frequency domain during a third step 23, outputting estimated data symbols .sub.m,n. In particular, this post-processing step multiplies data symbols in the frequency domain by a term taking account of the compression factor r. On the other hand, this step is optional if the length of the prototype filter is odd.

(28) 5.2 Example Embodiments

(29) The following describes various example embodiments of the invention, for modulation and demodulation of data symbols with complex or real values for transmission at a Faster than Nyquist (FTN) rate.

(30) One essential point is related to the definition of the FTN factor N.sub.f, a parameter that will be found in all the disclosed implementation schemes.

(31) 5.2.1 Notation

(32) The following describes the notation used in the remainder of the document: : FTN compression factor, 0<1; M: number of carriers, size of the mathematical transform, for example of the IFFT/FFT type; T.sub.e: sampling period;

(33) $F_0=\frac{1}{{\mathrm{MT}}_e}\text{:}$
inter-carrier spacing; T.sub.0=MT.sub.e; L: the length of the prototype filter such that L=b.sub.1M+b.sub.2, where b.sub.1 and b.sub.2 are integers such that b.sub.11 and 0b.sub.2M1; D=L1: delay parameter introduced to make the system causal; a.sub.m,n, c.sub.m,n: data symbols to be transmitted, that can be real or complex.

(34) 5.2.2 First Example Embodiment

(35) The following presents a first example embodiment according to which it is required to make an OFDM/OQAM modem for Faster-Than-Nyquist transmission.

(36) In addition to the previous notation, we define:

(37) $N=\frac{M}{2}$
the number of samples due to the offset of the OQAM;

(38) ${}_0=\frac{M}{2}{T}_e$ a FTN factor N.sub.f, such that N.sub.f is equal to the integer rounding of

(39) 0$\frac{M}{2};$ a phase term .sub.m,n, for example such as

(40) ${}_{m,n}=\frac{}{2}\left(m+n\right)+nm,$
where {1,0,1}, or

(41) ${}_{m,n}=\frac{}{2}n,$
or any other phase.

(42) It should be noted that if

(43) ${}_{m,n}=\frac{}{2}n,$
the phase term does not depend on m, and it is impossible to approach an orthogonal system when the compression factor tends to 1. If high compression factors are considered (tending towards 0), the choice of the phase term is not very important. It is sufficient that the receiver implements an equalization system that takes account of the selected phase law.

(44) A) Modulation

(45) FIGS. 3A to 3D show four OFDM/OQAM modulators for transmitting data at a Faster-Than-Nyquist rate.

(46) The first two modulators shown in FIGS. 3A and 3B use a pre-processing step, a mathematical transform step using a conventional inverse Fourier Transform module, and a polyphase filtering step.

(47) The last two modulators shown in FIGS. 3C and 3D use a mathematical transform step, adapting the technique disclosed in French patent application FR 2 972 091 mentioned above to take account of the FTN factor N.sub.f, and a polyphase filtering step.

(48) More precisely, the inventors proposed that the baseband equation of the OFDM/OQAM signal can be modified to obtain a modulator generating an FTN multi-carrier signal based on an OQAM modulation.

(49) Remember that the conventional OFDM/OQAM continuous signal can be written in the baseband in the following form:

(50) $s\left(t\right)=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{a}_{m,n}{f}_{m,n}\left(t\right)$
where f.sub.m,n(t)=g(tnT.sub.0/2)e.sup.j2mF.sup.0.sup.te.sup.jm,n, and g is the prototype filter.

(51) The data symbols a.sub.m,n have a real value and they can be obtained from a 2.sup.2K-QAM complex constellation, taking the real part and then the imaginary part successively.

(52) In sampling at rate

(53) $T_e=\frac{T_0}{M},$
and setting s[k]=s((kD/2)T.sub.e), the OFDM/OQAM discrete signal can then be written in the baseband in the following form:

(54) $s\left[k\right]=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n}{.Math.}{a}_{m,n}g\left[k-\mathrm{nN}\right]e^{j\frac{2}{M}m\left(k-\frac{D}{2}\right)}{e}^{j{}_{m,n}}$

(55) In the case of an orthogonal system (OFDM/OQAM), D is related to length L of the prototype filter by the relation D=L1. In the orthogonal case, the same functions base is used on transmission and reception and the constraint is to satisfy the following real orthogonality condition:
g.sub.m,n,g.sub.m,n.sub.R={g.sub.m,n,g.sub.m,n}={.sub..sup.g.sub.m,n(t)g*.sub.m,n(t)dt}=.sub.m,m.sub.n,n

(56) where .,..sub.R denotes the scalar real product.

(57) In the case of the conventional OFDM/OQAM, transmission is made at exactly the Nyquist rate. In other words, the real data symbols a.sub.m,n are transmitted at a rate such that

(58) $\frac{T_0}{2}{F}_0=\frac{1}{2},$
which corresponds to the condition T.sub.0F.sub.0=1 for the complex data symbols from which they are derived.

(59) According to the invention, the baseband equation of the OFDM/OQAM signal is modified as follows:

(60) $s\left(t\right)=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{a}_{m,n}g\left(t-n{}_0\right)e^{j2m{F}_{0}t}{e}^{j{}_{m,n}}$

(61) After sampling at rate T.sub.e, a normalized expression relative to T.sub.e, is obtained, i.e.:

(62) $s\left[k\right]=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{a}_{m,n}g\left[k-{\mathrm{nN}}_f\right]e^{j2\mathrm{mk}/M}{e}^{j{}_{m,n}}$

(63) This expression can be reformulated as follows:

(64) 0$s\left[k\right]=\underset{n=-}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j{}_{m,n}}{e}^{\frac{j2\mathrm{mk}}{M}}$

(65) It is required to implement the transform step in the form of an IFFT, to simplify the implementation of the modulator. The inventors have proposed to make modifications to the previous equation so that this IFFT can also simply generate the multi-carrier signal s[k].

(66) Two cases can then be distinguished, the non-causal case that will correspond to a fairly usual theoretical scheme in which the prototype filter g is assumed to be at the time origin, and the causal case in which the prototype filter is implemented starting from time 0.

(67) A.1) Non Causal Case

(68) In the first case, the above equation is considered and it is assumed that the prototype filter g[k], where

(69) $k\left[-\frac{L-1}{2},\frac{L-1}{2}\right],$
is not causal, in other words, it is centered on the origin k=0. An offset D/2 to the index of the Fourier transform is then introduced so that the causal index of the IFFT is also non-causal. Furthermore, we limit variations of the index of the Fourier Transform to the [0,L1] interval. One method of performing this process is to replace the index k of the Fourier transform by knN.sub.f+D/2. This gives the following formulated expression:

(70) $s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=-}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{\underset{\mathrm{transformation}\left(+\mathrm{post}\text{-}\mathrm{processing}\right)}{}}{\underset{m=0}{\overset{M-1}{.Math.}}\underset{\underset{\mathrm{pre}\text{-}\mathrm{processing}}{}}{a_{m,n}{e}^{j{}_{m,n}}{e}^{j2m\left({\mathrm{nN}}_f-\frac{D}{2}\right)/M}}{e}^{j2m\left(k-{\mathrm{nN}}_f+\frac{D}{2}\right)/M}}}$

(71) Such a modulator is shown in FIG. 3A. It comprises: a pre-processing module 30, implementing a multiplication of data symbols a.sub.m,n by a phase term e.sup.j.sup.m,n and a term taking account of the compression factor

(72) $e^{j2m\left({\mathrm{nN}}_f-\frac{D}{2}\right)/M},$ a mathematical transform module 31, implementing an inverse fast Fourier transform, outputting M transformed symbols, a post-processing module 33, implementing a cyclic repetition of transformed symbols, outputting L transformed symbols, and a polyphase filtering module 32, implementing, for the i-th transformed symbol multiplication of the prototype filter g by the factor

(73) $\left[i-\frac{L-1}{2}\right],$
denoted

(74) $g\left[i-\frac{L-1}{2}\right],$
an expansion by the FTN factor N.sub.f, and an offset of (i+1)z.sup.1, for i varying from 0 to L1.

(75) It should be noted that the pre-processing module 30 is optional if L is odd. Furthermore, the post-processing module 33 is also optional if L=M.

(76) This post-processing module 33 is used to adapt the number of transformed symbols at the output from the transform module 31 to the size of the prototype filter.

(77) More precisely, since the length L of the prototype filter is such that L=b.sub.1M+b.sub.2, where b.sub.1 and b.sub.2 are integers such that b.sub.11 and 0b.sub.2M1, the post-processing module 33 implements a block repetition of the M transformed symbols, outputting b.sub.1 blocks of M transformed symbols and a sub-block of b.sub.2 transformed symbols.

(78) The post-processing module 33, also called a cyclic extension block CYCEXD, can be written mathematically using an LM matrix in the following form:

(79) ${\mathrm{CYCEXD}}_{LM}={\left[\begin{array}{c}{I}_{MM}\\ \mathrm{.Math.}\\ I_{MM}\end{array}\right]}_{LM}$

(80) where I is the identity matrix.

(81) For example, we can choose b.sub.1=4 and b.sub.2=0 if it is required to use a conventional Iota type prototype filter. We can choose b.sub.1=1 and b.sub.2=0 if it is required to use a conventional TFL type prototype filter.

(82) Thus, due to the periodicity property of the Fourier transform, a filtering operation can be performed with a cyclic extension per block, followed by multiplication operation with a single factor, then followed by a parallel-to-series conversion.

(83) The polyphase filtering module 32 is conventional, and its operation is not described in more detail.

(84) A.2) Causal Case

(85) In the second case, the reformulated equation of the multi-carrier signal s[k] is modified slightly as follows:

(86) $s\left[k\right]=\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j{}_{m,n}}{e}^{j2\mathrm{mk}/M}$
in which it is assumed that the prototype filter g[k], where k [0,L1], is causal, in other words it starts at index k=0. Thus, it is no longer necessary to introduce an offset D/2 to the index of the Fourier transform, and the signal s[k] can be expressed as follows:

(87) $s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{\underset{\mathrm{transformation}\left(+\mathrm{post}-\mathrm{processing}\right)}{}}{\underset{m=0}{\overset{M-1}{.Math.}}\underset{\underset{\mathrm{pre}-\mathrm{processing}}{}}{a_{m,n}{e}^{j{}_{m,n}}{e}^{j2m\left({\mathrm{nN}}_f-D/2\right)/M}}{e}^{j2m\left(k-{\mathrm{nN}}_f\right)/M}}}$

(88) Such a modulator is shown in FIG. 3B. It comprises: a pre-processing module 30, implementing multiplication of data symbols a.sub.m,n by a phase term e.sup.j.sup.m,n and a term taking account of the compression factor

(89) $e^{j2m\left({\mathrm{nN}}_f-\frac{D}{2}\right)/M},$ a mathematical transform module 31, implementing an inverse fast Fourier transform, outputting M transformed symbols, a post-processing module 33, implementing a cyclic repetition of transformed symbols, outputting L transformed symbols, and a polyphase filtering module 32, implementing for the i-th transformed symbol multiplication by the i-th coefficient of the prototype filter g denoted g[i], an expansion by the FTN factor N.sub.f, and an offset of (i+1)z.sup.1, for i varying from 0 to L1.

(90) Once again, it should be noted that the pre-processing module 30 is optional if L is odd, and the post-processing module 33 is optional if L=M. Non-causal modulator structure (FIG. 3A) and causal modulator structure (FIG. 3B) are similar, all that is modified in the filtering module 32 are the indexes of the prototype filter g. Therefore, operation of the various modules is not discussed again in detail.

(91) It should be noted that the two modulators shown in FIGS. 3A and 3B can process complex data symbols c.sub.m,n, rather than real data symbols a.sub.m,n. Thus, a new structure of OFDM/OQAM modulators is disclosed that can operate with complex values either at twice the Nyquist rate if the compression factor is equal to 1, or faster than twice the Nyquist rate if the compression factor is less than 1.

(92) Other example embodiments of the modulator will now be disclosed with reference to FIGS. 3C and 3D, using mathematical transform step by adapting the technique described in the French patent application FR 2 972 091 mentioned above, and polyphase filtering step.

(93) The example shown lies in the context of a causal filter and real data symbols. However, the solution described below can be applied in the context of a non-causal filter.

(94) More precisely, the above equation for the multi-carrier signal is reconsidered:

(95) 0$s\left[k\right]=\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j{}_{m,n}}{e}^{j2m\left({\mathrm{nN}}_f-D/2\right)M}{e}^{j2m\left(k-{\mathrm{nN}}_f\right)/M}$

(96) Using a property of symmetry at the output from the IFFT, the OQAM multicarrier modulation at the FTN rate can be made from a partial IFFT, also called a clipped IFFT and denoted IFFTe, and therefore does not require a complete IFFT. This new scheme reduces the operational complexity by a factor of the order of 2.

(97) Two cases are considered below depending on the value of the phase term .sub.m,n.

(98) In a first case, we assume

(99) ${}_{m,n}=\frac{n}{2}.$

(100) Considering the above equation for the multi-carrier signal, and assuming D=L1=b.sub.1M+b.sub.21, we can write s[k] as follows:

(101) $s\left[k\right]=\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j\frac{n}{2}}{e}^{j2m\left({\mathrm{nN}}_f-D/2\right)M}{e}^{j2{\mathrm{mk}}_1/M}$$s\left[k\right]=\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]e^{j\frac{n}{2}}\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{-jm\left(b_2-1\right)/M}{e}^{j2{\mathrm{mk}}_2/M}$
where k.sub.1=knN.sub.f [0,L1] and k.sub.2=k.sub.1+nN.sub.fb.sub.1N.

(102) If b.sub.21 is odd, in other words b.sub.21=2q1 where q , the previous equation can be reformulated as follows:

(103) $s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{\underset{\mathrm{permut}+\mathrm{CYCEXD}}{}}{e^{j\frac{n}{2}}\underset{\underset{\mathrm{IFFTe}+\mathrm{HSExt}}{}}{\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{jm/M}{e}^{j2{\mathrm{mk}}_2/M}}}}$
where the indexes are such that k.sub.1=knN.sub.f [0,L1] and k.sub.2=k.sub.1+nN.sub.fb.sub.1Nq.

(104) A modulator capable of generating such a signal s[k] is shown in FIG. 3C. It comprises: a mathematical transform module 35, implementing a partial inverse fast Fourier transform, outputting C transformed symbols, reconstruction of (MC) transformed symbols, outputting all transformed symbols, and if ML, permutation and repetition of the transformed symbols, outputting L transformed symbols, a polyphase filtering module 32, implementing for the i-th transformed symbol multiplication by the i-th coefficient of the prototype filter g denoted g[i], an expansion by the FTN factor N.sub.f, and an offset of (i+1)z.sup.1, for i varying from 0 to L1.

(105) The following describes the principle of the mathematical transform module 35.

(106) We start by applying a partial inverse fast Fourier transform in an IFFTe module 351, to M data symbols multiplied by the term e.sup.jm/M, for m varying from 0 to M1. The result obtained is M/2 transformed symbols. It should be noted that the IFFTe module can use different known IFFT algorithms.

(107) The vector obtained at the output from this module 351 is denoted U.sub.n such that:

(108) $U_n\left[k_1\right]=\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{jm/M}{e}^{j2{\mathrm{mk}}_1/M}$$\mathrm{for}{k}_1=0,\mathrm{.Math.},\frac{M}{2}-1.$

(109) We then reconstruct the set of M transformed symbols in an HSExt-Permut-CYCEXD module 352, by applying a hermitian symmetry:

(110) $U_n\left[M-1-k_1\right]=U_n^{*}\left[k_1\right]\mathrm{for}{k}_1=0,\mathrm{.Math.},\frac{M}{2}-1.$

(111) If ML, a permutation and a repetition of the transformed symbols is applied, outputting L transformed symbols such that:
U.sub.n[k.sub.2]=U.sub.n[mod(k.sub.1+nN.sub.fb.sub.1Nq),M]
for k.sub.1=0, . . . ,M1,

(112) or also:

(113) U.sub.n[k.sub.2]=U.sub.n[mod(k.sub.2,M)] for k.sub.2=0, . . . ,L1 and n even,

(114) U.sub.n[k.sub.2]=U*.sub.n[mod(k.sub.2,M)] for k.sub.2=0, . . . ,L1 and n odd.

(115) Thus, L transformed symbols are obtained and input in the polyphase filtering module 32.

(116) Operation of the polyphase filtering module 32 is conventional, and is not described in more detail.

(117) If b.sub.21 is even, in other words b.sub.21=2q where q , then the previous equation for s[k] may be reformulated as follows:

(118) $s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}g\left[k-{\mathrm{nN}}_f\right]\underset{\underset{\mathrm{permut}+\mathrm{CYCEXD}}{}}{e^{j\frac{n}{2}}\underset{\underset{\mathrm{IFFTe}+\mathrm{HSExt}}{}}{\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j2{\mathrm{mk}}_2/M}}}}$
in which the indexes are such that k.sub.1=knN.sub.f [0,L1] and k.sub.2=k.sub.1+nN.sub.fb.sub.1Nq.

(119) FIG. 3D shows a modulator used to generate such a signal s[k]. It comprises: a mathematical transform module 35, implementing a partial inverse fast Fourier transform, outputting C transformed symbols, reconstruction of (M-C) transformed symbols, outputting all transformed symbols, and if ML, permutation and repetition of the transformed symbols, outputting L transformed symbols, a polyphase filtering module 32, implementing for the i-th transformed symbol multiplication by the i-th coefficient of the prototype filter g, denoted g[i], expansion by the FTN factor N.sub.f, and offset of (i+1)z.sup.1, for i varying from 0 to L1.

(120) The following describes the principle of the mathematical transform module 35 in this case.

(121) Firstly, a partial inverse fast Fourier transform is applied in an IFFTe module 353, to the M data symbols. The result thus obtained is

(122) $\frac{M}{2}+1$
transformed symbols. It should be noted again that the IFFTe module can make use of different known used IFFT algorithms.

(123) The vector obtained at the output from this module 353 is denoted U.sub.n, as follows:

(124) $U_n\left[k_1\right]=\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j2{\mathrm{mk}}_1/M}$$\mathrm{for}{k}_1=0,\mathrm{.Math.},\frac{M}{2}.$

(125) The set of M transformed symbols is then reconstructed in an HSExt-Permut-CYCEXD module 354 by applying an hermitian symmetry:

(126) $U_n\left[M-k_1\right]=U_n^{*}\left[k_1\right]\mathrm{for}{k}_1=1,\mathrm{.Math.},\frac{M}{2}-1.$

(127) If ML, a permutation and a repetition of transformed symbols are applied, outputting L transformed symbols, such that:
U.sub.n[k.sub.2]=U.sub.n[mod(k.sub.1+nN.sub.fb.sub.1Nq,M)]
for k.sub.1=0, . . . ,M1.

(128) The result thus obtained is L transformed symbols input into the polyphase filtering module 32. Once again, operation of the polyphase filtering module 32 is conventional and will not be described in more detail.

(129) We will now consider a second case, in which the phase term is equal to

(130) 0${}_{m,n}=\frac{\left(n+m\right)}{2}.$

(131) In this case, in which orthogonality with a low compression factor can be restored, the multi-carrier signal s[k] uses the same expressions as those defined above for cases with odd and even b.sub.21, with a difference in the indexes:

(132) $\mathrm{if}{b}_2-1\mathrm{is}\mathrm{odd}:s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}\left[k-n{N}_f\right]\underset{\underset{\mathrm{permut}+\mathrm{CYCEXD}}{}}{e^{j\frac{n}{2}}\underset{\underset{\mathrm{IFFTe}+\mathrm{HSExt}}{}}{\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{jm/M}{e}^{j2{\mathrm{mk}}_2/M}}}}$$\mathrm{if}{b}_2-1\mathrm{is}\mathrm{even}:s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}\left[k-n{N}_f\right]\underset{\underset{\mathrm{permut}+\mathrm{CYCEXD}}{}}{e^{j\frac{n}{2}}\underset{\underset{\mathrm{IFFTe}+\mathrm{HSExt}}{}}{\underset{m=0}{\overset{M-1}{.Math.}}{a}_{m,n}{e}^{j2{\mathrm{mk}}_2/M}}}}$
in which the indexes are such that k.sub.1=knN.sub.f[0,L1] and

(133) $k_2=k_1+\frac{M}{4}+{\mathrm{nN}}_f-b_{1}N-q.$

(134) Partial inverse fast Fourier transform operations and reconstruction operations of all transformed symbols are similar to the previous case and will not be described in more detail.

(135) On the other hand, permutation and repetition of transformed symbols outputting L transformed symbols, uses the following function:

(136) $U_n\left[k_2\right]=U_n\left[\mathrm{mod}\left(k_1+\frac{M}{4}+{\mathrm{nN}}_f-b_{1}N-q,M\right)\right]$$\mathrm{for}{k}_1=0,\mathrm{.Math.},M-1.$

(137) The rest of the processing is identical and will not be described further.

(138) B) Demodulation

(139) An OFDM/OQAM demodulator for receiving data at a Faster-Than-Nyquist rate is disclosed below with reference to FIG. 4.

(140) Such a demodulator uses processing inverse to that done by the modulator.

(141) In particular, assuming that data symbols a.sub.m,n are reals, the general expression for the demodulator is given by:

(142) $y_{m_0,n_0}=\left\{\underset{k}{.Math.}s\left[k\right]{}_{m_0,n_0}\left[k\right]\right\}$
in which the index (m.sub.0, n.sub.0) corresponds to the targeted demodulation position in the time-frequency plane. The variation interval of the index k depends on the fact that the system is assumed to be causal or non-causal. In the following, we will derive the demodulator structure for the causal case. The non-causal case is easily deduced from the causal case.

(143) By defining g.sub.m.sub.0.sub.,n.sub.0[k] in the above equation, we obtain:

(144) $y_{m_0,n_0}=\left\{\underset{k}{.Math.}s\left[k\right]\left[k-n_0{N}_f\right]e^{-j{}_{m_0,n_0}}{e}^{-\frac{j2m_0\left(k-D/2\right)}{M}}\right\}$

(145) All that remains is to determine the variation interval of k in the summation part. Assuming that the prototype filter g[k] is a causal filter with length L, the sum is limited to k [n.sub.0N.sub.f,n.sub.0N.sub.f+L1]. In the same way as for the modulator, the variation of the index of the Fourier transform has to be limited to [0,M1] so that a conventional FFT can be applied with its periodicity rule.

(146) Consequently, the inventors proposed the following modifications:

(147) $y_{m_0,n_0}=\left\{\underset{\underset{\mathrm{FFT}}{}}{\underset{k=n_0{N}_f}{\overset{n_0{N}_f+M-1}{.Math.}}\underset{\underset{\mathrm{filtering}+\mathrm{CYCCOMB}}{}}{\underset{l=0}{\overset{L/M}{.Math.}}s\left[k+lM\right]\left[k+lM-n_0{N}_f\right]}{e}^{-\frac{j2m_0\left(k-n_0{N}_f\right)}{M}}}\underset{\underset{\mathrm{post}\text{-}\mathrm{processing}}{}}{e^{-j{}_{m_0,n_0}}{e}^{-j2m_0\left(n_0{N}_f-D/2\right)/M}}\right\}$

(148) Such a demodulator is shown in FIG. 4, and comprises: a polyphase filtering module 41, implementing an offset of (i+1)z.sup.1, decimation by the FTN factor N.sub.f, multiplication by the coefficient [i] of the prototype filter g denoted g[i], for i varying from 0 to L1, outputting L data symbols in the transformed domain; a pre-processing module 44, implementing extraction of M data symbols in the transformed domain (for example time domain); a mathematical transform module 42 implementing a fast Fourier transform, outputting M data symbols in the frequency domain, un post-processing module 43 implementing multiplication of data symbols in the frequency domain by a phase term e.sup.j.sup.m,n and a term taking account of the compression factor

(149) $e^{-j2m\left({\mathrm{nN}}_f-\frac{D}{2}\right)/M},$
and an extraction module of the real part 45.

(150) It should be noted that the pre-processing module 44 is optional if L=M.

(151) Similarly, the post-processing module 43 is optional if L is odd.

(152) Finally, if the modulated data symbols are complex and not real, the module for extraction of the real part 45 is also optional.

(153) The pre-processing module 44 is used to adapt the number of data symbols in the transformed domain at the output from the filtering module 41 to the size of the mathematical transform module.

(154) More precisely, the pre-processing module 44, also called the cyclic combination block CYCCOMB, implements extraction of a block of M data symbols in the transformed domain.

(155) The pre-processing module 44 can be mathematically described by a ML matrix that is the transpose of the CYCEXD matrix:
CYCCOMB.sub.ML=CYCEXD.sub.Lm.sup.T

(156) Operation of the other modules is conventional. Therefore, they are not described in more detail.

(157) Such a demodulator is particularly adapted to demodulating a multi-carrier signal constructed from the modulator according to any one of FIGS. 3A to 3D.

(158) 5.2.3 Second Example Embodiment

(159) The following discloses a second example embodiment in which an attempt is made to make an OFDM modem for Faster-Than-Nyquist transmission.

(160) In this context, data symbols, denoted c.sub.m,n, can be complex.

(161) An FTN factor N.sub.f is defined in addition to the previous notation, such that N.sub.f is equal to the integer rounding of M.

(162) A) Modulation

(163) We will present an OFDM modulator for Faster-Than-Nyquist transmission of data, with reference to FIG. 5.

(164) More precisely, the inventors proposed to modify the baseband equation of the OFDM signal, to obtain a modulator generating an FTN multi-carrier signal based on an OFDM modulation (i.e. with a complex orthogonality if =1).

(165) It will be remembered that for an OFDM modulation on M carriers, it is required to transmit complex data symbols c.sub.m,n (m I={0, . . . ,M1} and n ). The usual representations for these symbols correspond to alphabets used for amplitude modulations such as QAM and PSK (Phase Shift Keying).

(166) The conventional OFDM signal is written as follows in baseband:

(167) $s\left(t\right)=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{c}_{m,n}{f}_{m,n}\left(t\right)$ where f.sub.m,n(t)=f(tnT.sub.0)e.sup.j2mF.sup.0.sup.t e.sup.j.sup.m,n and f(t) is an integrable square function also called a prototype function; T.sub.0 is the duration of a multicarrier symbol; F.sub.0 is the space between two successive carriers; .sub.m,n is a phase term that can be chosen arbitrarily; j.sup.2=1.

(168) In this context, the orthogonality condition, in other words the condition that minimizes the error in the presence of AWGN type disturbance, is made considering the adapted filter in reception. In other words, the scalar product of basic functions used for both modulation and demodulation must be such that:
(f.sub.m,n,f.sub.m,n=.sub..sup.f.sub.m,n(t)f*.sub.mn(t)dt=.sub.n,n.sub.m,m

(169) If it is required to transmit at the Nyquist rate in addition to an orthogonal system, it is necessary to impose F.sub.0T.sub.0=1. In other words, the Nyquist condition described above (WT=) must be satisfied for each carrier m. The difference in the multiplication factor () is due to the fact that this case considers the transmission of complex and non-real data symbols.

(170) According to the invention, the baseband equation of the OFDM signal is modified as follows in order to transmit data at a Faster-Than-Nyquist rate:

(171) $s\left(t\right)=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{c}_{m,n}\left(t-n{T}_0\right)e^{j2m{F}_{0}t}$

(172) Sampling at rate T.sub.e, the OFDM discrete signal normalized relative to the rate T.sub.e can be written in baseband in the following form:

(173) 0$s\left[k\right]=\underset{m=0}{\overset{M-1}{.Math.}}\underset{n=-}{\overset{+}{.Math.}}{c}_{m,n}\left[k-n{N}_f\right]e^{j2\mathrm{mk}/M}$$s\left[k\right]=\underset{\underset{\mathrm{filtering}}{}}{\underset{n=0}{\overset{+}{.Math.}}\left[k-n{N}_f\right]\underset{\underset{\mathrm{IFFT}+\mathrm{CYCEXD}}{}}{\underset{m=0}{\overset{M-1}{.Math.}}\underset{\underset{\mathrm{pre}\text{-}\mathrm{processing}}{}}{c_{m,n}{e}^{j2mn{N}_f/M}}{e}^{j2m\left(k-{\mathrm{nN}}_f\right)/M}}}$

(174) In the following, we will derive the structure of the modulator in the causal case. The non-causal case is easily derived from the causal case.

(175) Such a modulator is shown in FIG. 5. It comprises: a pre-processing module 50, implementing multiplication of data symbols c.sub.m,n by a term taking account of the compression factor e.sup.j2mnN.sup.f.sup./M, a mathematical transform module 51, implementing an inverse fast Fourier transform, outputting M transformed symbols, a post-processing module 53, implementing a cyclic repetition of transformed symbols, outputting L transformed symbols, and a polyphase filtering module 52, applying for the i-th transformed symbol, multiplication by the i-th coefficient of the prototype filter g denoted g[i], expansion by the FTN factor N.sub.f, and an offset of (i+1)z.sup.1, for i varying from 0 to L1.

(176) As in the above examples, it can be seen that if L is odd, the pre-processing module 50 is optional. Furthermore, if L=M, the post-processing module 53 is optional.

(177) Operation of the various modules has been described above with reference to FIGS. 3A to 3D. Therefore, it will not be repeated here.

(178) B) Demodulation

(179) An OFDM demodulator, capable of receiving data at a Faster-Than-Nyquist rate, is described in the following with reference to FIG. 6.

(180) Such a demodulator uses processing inverse to that done by the modulator.

(181) The general expression for the demodulator is given by:

(182) $y_{m_0,n_0}=\underset{k}{.Math.}s\left[k\right]{}_{m_0,n_0}\left[k\right]$
where the index (m.sub.0, n.sub.0) corresponds to the targeted demodulation position in the time-frequency plane. The variation interval of the index k depends on whether or not the system is assumed to be causal. We derive the structure of the demodulator in the causal case below. The non-causal case is easily derived from the causal case.

(183) In defining g.sub.m.sub.0.sub.,n.sub.0[k] in the above equation, we obtain:

(184) $y_{m_0,n_0}=\underset{\underset{\mathrm{FFT}}{}}{\underset{k=n_0{N}_f}{\overset{n_0{N}_f+M-1}{.Math.}}\underset{\underset{\mathrm{filtering}+\mathrm{CYCOMB}}{}}{\underset{l=0}{\overset{L/M}{.Math.}}s\left[k+\mathrm{lM}\right]\left[k+\mathrm{lM}-n_0{N}_f\right]}{e}^{-j2m_0\left(k-n_0{N}_f\right)/M}}\underset{\underset{\mathrm{post}\text{-}\mathrm{processing}}{}}{e^{-j2m_0{n}_0{N}_f/M}}$

(185) Such a demodulator is shown in FIG. 6. It comprises: a polyphase filtering module 61, applying an offset of (i+1)z.sup.1, decimation by the FTN factor N.sub.f, multiplication of the prototype filter g by the coefficient [i], denoted g[i], for i varying from 0 to L1, outputting L data symbols in the transformed domain; a pre-processing module 64, applying extraction of M data symbols in the transformed domain (for example the time domain); a mathematical transform module 62, applying a fast Fourier transform, outputting M data symbols in the frequency domain; and a post-processing module 63, applying multiplication of data symbols in the frequency domain by a term taking account of the compression factor e.sup.j2mnN.sup.f.sup./M.

(186) It should be noted that if L=M, the pre-processing module 64 is optional.

(187) Similarly, if L is odd, the post-processing module 63 is optional.

(188) Operation of all modules is similar to that described with reference to FIG. 4. Therefore, they will not be described in more detail.

(189) In particular, such a demodulator is suitable for demodulating a multi-carrier signal constructed using the modulator in FIG. 5.

(190) 5.3 Structure of the Modulator or Demodulator

(191) Finally, the simplified structure of a modulator modulating a multi-carrier signal and the structure of a demodulator demodulating a multi-carrier signal according to one particular embodiment of the invention, are described with reference to FIGS. 7 and 8 respectively.

(192) As shown in FIG. 7, such a modulator comprises a memory 71 comprising a buffer memory, a processing unit 72 for example including a microprocessor P and controlled by the computer program 73, implementing the modulation method according to one embodiment of the invention.

(193) On initialization, the code instructions of the computer program 73 may for example be loaded into a RAM memory before being run by the processor in the processing unit 72. The processing unit 72 receives the real data symbols a.sub.m,n or complex data symbols c.sub.m,n, as inputs. The microprocessor in the processing unit 72 implements the steps of the modulation method described above, in accordance with the instructions of the computer program 73, to generate a multi-carrier signal. To achieve this, in addition to the buffer memory 71, the modulator includes a mathematical transform module for transforming data symbols from the frequency domain to a transformed domain, and a polyphase filtering module for filtering transformed symbols, applying an expansion factor taking account of a compression factor .

(194) These modules are controlled by the microprocessor in the processing unit 72.

(195) As shown in FIG. 8, such a demodulator has a memory 81 comprising a buffer memory, a processing unit 82 implements, for example comprising a microprocessor P and controlled by the computer program 82, implementing the demodulation method according to one embodiment of the invention.

(196) On initialization, the code instructions of the computer program 83 may for example be loaded in a RAM memory before being run by the processor in the processing unit 82. The processing unit 82 receives a multi-carrier signal as input. The microprocessor in the processing unit 82 implements the steps of the demodulation method described above, according to the instructions of the computer program 83, to estimate transmitted data symbols. To achieve this, in addition to the buffer memory 81, the demodulation device comprises a polyphase filtering module for filtering the multi-carrier signal, outputting data symbols in the transformed domain, a mathematical transform module for transforming data symbols in the transformed domain, from a transformed domain to the frequency domain, outputting data symbols in the frequency domain and a module for post-processing of data symbols in the frequency domain, outputting estimated data symbols. The post-processing module implements multiplication of data symbols in the frequency domain by a term taking account of a compression factor , and the polyphase filtering module uses a decimation factor taking account of the compression factor .

(197) These modules are controlled by the microprocessor of the processing unit 82.

(198) Although the present disclosure has been described with reference to one or more examples, workers skilled in the art will recognize that changes may be made in form and detail without departing from the scope of the disclosure and/or the appended claims.