Estimation of a time, phase and frequency shift of an OQAM multicarrier signal

Abstract

A method is provided for receiving an OQAM multi-carrier signal, which implements a step of estimating, in the time domain, at least one time, phase, and/or frequency shift of the multi-carrier signal. The estimation step implements at least one estimator in order to estimate the time shift, referred to as a time estimator, and/or at least one estimator for estimating the phase shift, referred to as a phase estimator, and/or at least one estimator for estimating the frequency shift, referred to as a frequency estimator. The multi-carrier signal includes at least one preamble, and at least one of the estimators takes into account coefficients of a prototype filter used in transmission in order to shape at least one preamble inserted into the multi-carrier signal.

Claims

1. A method for receiving an OQAM multicarrier signal comprising: estimating, in the time domain, at least one shift, said at least one shift being a time shift or a phase shift of said OQAM multicarrier signal, said estimating being implemented by at least one estimator configured to estimate said at least one shift, wherein: said at least one estimator is a time estimator when said at least one shift is said time shift, said at least one estimator is a phase estimator when said at least one shift is said phase shift, and said at least one estimator estimates said at least one shift based on coefficients of a prototype filter used at transmission to shape at least one preamble inserted into said OQAM multicarrier signal.

2. The method for receiving according to claim 1, wherein said estimating implements, in the time domain, a first sub-step of estimating said time shift followed by a second sub-step of estimating said phase shift.

3. The method for receiving according to claim 2, wherein said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. \overset{M / 4 - 1}{\underset{k = 0}{.Math.}} h (k) h (M / 2 - 1 - k) r (k + \tilde{}) r (M / 2 - 1 - k + \tilde{}) .Math. and$ ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. \overset{M / 4 - 1}{\underset{k = 0}{.Math.}} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \tilde{}) r (M - 1 - k + \tilde{}) .Math.$ and said phase estimator implements said second sub-step of estimating said phase shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \frac{1}{2} {\overset{M / 4 - 1}{\underset{k = 0}{.Math.}} h (k) h (M / 2 - 1 - k) r (k + \hat{}) r (M / 2 - 1 - k + \hat{})} and$ ${\hat{}}_{{LS}_{2}} = \frac{1}{2} {\overset{M / 4 - 1}{\underset{k = 0}{.Math.}} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \hat{}) r (M - 1 - k + \hat{})}$ with: {circumflex over ()}.sub.LS.sub.1 or {circumflex over ()}.sub.LS.sub.2 being the estimation of said time shift at output of the time estimator, and {circumflex over ()} being the estimation of the time shift fixed at input of the phase shift estimator; {circumflex over ()}.sub.LS.sub.1 or {circumflex over ()}.sub.LS.sub.2 being the estimation of said phase shift at output of the phase estimator; {. } being the argument of a complex number; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; k being an integer such that 0kM/41; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; h(k) being coefficients of said prototype filter used at transmission; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being said time shift of the received OQAM multicarrier signal relative to said sent multicarrier signal; being said phase shift of the received OQAM multicarrier signal relative to said sent multicarrier signal.

4. The method for receiving according to claim 2, wherein said time shift is estimated using a parameter corresponding to a position of at least one pair of maximum values obtained at transmission for said OQAM multicarrier signal, said position being defined relative to the outputs of a step of transformation, from the frequency domain into the time domain implemented at transmission, of a set of data symbols forming said at least one preamble, called preamble symbols.

5. The method for receiving according to claim 4, wherein, for preamble symbols p.sub.m,n defined at input of said step of transformation from the frequency domain to the time domain implemented at transmission, by: ${\begin{matrix} p_{m, n} = \sqrt{2} / 2 for m = 4 p and m = 1 + 4 p \\ p_{m, n} = - \sqrt{2} / 2 for m = 2 + 4 p and m = 3 + 4 p \end{matrix} and$ with p.sub.m,n being a preamble symbol associated with a sub-carrier with index m at the instant n, p an integer, and 0mM1, said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M / 2 - 1 + \tilde{}) .Math., {\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 2 + \tilde{}) r (M - 1 + \tilde{}) .Math.,$ with: {circumflex over ()}.sub.LS.sub.1 or {circumflex over ()}.sub.LS.sub.2 being the estimation of said time shift at output of said time estimator; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); k being an integer, s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being a time shift of the received OQAM multicarrier signal relative to the sent multicarrier signal; being a phase shift of the received OQAM multicarrier signal relative to the sent multicarrier signal.

6. The method for receiving according to claim 4, wherein for preamble symbols p.sub.m,n defined at input of said step of transformation from the frequency domain to the time domain implemented at transmission by: ${\begin{matrix} p_{2 m, n} = \sqrt{2} / 2 \\ p_{2 m + 1, n} = j \sqrt{2} / 2 \end{matrix} for 0 m M / 2 - 1$ with p.sub.m,n being a preamble symbol associated with a sub-carrier with index m at the instant n, and j.sup.2=1, said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 4 - 1 + \tilde{}) r (M / 4 + \tilde{}) .Math., {\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.,$ with: {circumflex over ()}.sub.LS.sub.1 or {circumflex over ()}.sub.LS.sub.2 being the estimation of said time shift at output of said time estimator; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); k being an integer; s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being a time shift of the received OQAM multicarrier signal relative to the sent multicarrier signal; being a phase shift of the received OQAM multicarrier signal relative to the sent multicarrier signal.

7. The method for receiving according to claim 4, wherein, for preamble symbols p.sub.m,n defined at input of the step of transformation from the frequency domain to the time domain implemented at transmission by: * ${\begin{matrix} p_{2 m, n} = \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1$ with p.sub.m,n being a preamble symbol associated with a sub-carrier with index m at the instant n, said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 4 - 1 + \tilde{}) r (M / 4 + \tilde{}) .Math., {\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.,$ ${\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) + r (M / 4 + \tilde{}) r (3 M / 4 - 1 + \tilde{}) .Math.,$ with: {circumflex over ()}.sub.LS.sub.1, {circumflex over ()}.sub.LS.sub.2 or {circumflex over ()}.sub.LS.sub.3 being the estimation of said time shift at output of said time estimator; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); k being an integer; s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being a time shift of the received OQAM multicarrier signal relative to the sent multicarrier signal; being a phase shift of the received OQAM multicarrier signal relative to the sent multicarrier signal.

8. The method for receiving according to claim 4, wherein, for preamble symbols p.sub.m,n defined at the input of the step of transformation from the frequency domain to the time domain implemented at transmission by: ${\begin{matrix} p_{2 m, n} = {(- 1)}^{m} \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1$ with p.sub.m,n being a preamble symbol associated with a sub-carrier with index m at the instant n said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M / 2 - 1 + \tilde{}) .Math., {\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 2 + \tilde{}) r (M - 1 + \tilde{}) .Math., {\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M - 1 + \tilde{}) + r (M / 2 - 1 + \tilde{}) r (M / 2 + \tilde{}) .Math.,$ with: {circumflex over ()}.sub.LS.sub.1, {circumflex over ()}.sub.LS.sub.2 or {circumflex over ()}.sub.LS.sub.3 being the estimation of said time shift at output of said time estimator; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; r(k) being the received OQAM multicarrier signal, such that r(k)=s(k)e.sup.j+b(k); k being an integer; s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being a time shift of the received OQAM multicarrier signal relative to the sent multicarrier signal; being a phase shift of the received OQAM multicarrier signal relative to the sent multicarrier signal.

9. The method for receiving according to claim 4, wherein for preamble symbols p.sub.m,n defined at input of the step of transformation from the frequency domain to the time domain implemented at transmission, by: ${\begin{matrix} p_{m, n} = 1 & if \mod (m, 4) = 0 \\ p_{m, n} = 0 & else \end{matrix} for 0 m m - 1$ with p.sub.m,n being a preamble symbol associated with a sub-carrier with index m at the instant n, said time estimator implements said first sub-step of estimating said time shift according to one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M / 2 - 1 + \tilde{}) + r (M / 4 - 1 + \tilde{}) r (M / 4 + \tilde{}) .Math.$ ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 2 + \tilde{}) r (M - 1 + \tilde{}) + r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.$ ${\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M - 1 + \tilde{}) + r (M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) + r (M / 4 + \tilde{}) r (3 M / 4 - 1 + \tilde{}) + r (M / 2 - 1 + \tilde{}) r (M / 2 + \tilde{}) .Math.$ with: {circumflex over ()}.sub.LS.sub.1, {circumflex over ()}.sub.LS.sub.2 or {circumflex over ()}.sub.LS.sub.3 being the estimation of said time shift at output of said time estimator; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max being a predetermined maximum value of {tilde over ()}; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); k being an integer; s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being a time shift of the received OQAM multicarrier signal relative to the sent multicarrier signal; being a phase shift of the received OQAM multicarrier signal relative to the sent multicarrier signal.

10. A device for receiving an OQAM multicarrier signal comprising: at least one estimator which estimates, in the time domain, at least one shift, said at least one shift being a time shift or a phase shift of said OQAM multicarrier signal, wherein said at least one estimator is a time estimator when said at least one shift is said time shift, said at least one estimator is a phase estimator when said at least one shift is said phase shift, and wherein said at least one estimator estimates said at least one shift based on coefficients of a prototype filter used at transmission to shape at least one preamble inserted into said OQAM multicarrier signal.

11. A non-transitory computer readable medium comprising a computer program stored thereon and comprising instructions for implementing a method for receiving an OQAM multicarrier signal when the program is executed by a processor of a receiver device, wherein the method comprises: estimating, in the time domain, at least one shift, said at least one shift being a time shift or a phase shift of said OQAM multicarrier signal, wherein wherein said estimating is based on coefficients of a prototype filter used at transmission to shape at least one preamble inserted into said OQAM multicarrier signal.

12. A method for receiving an OQAM multicarrier signal comprising: estimating, in the time domain, at least one shift, said at least one shift being a time shift or a phase shift of said OQAM multicarrier signal, said estimating being implemented by at least one estimator configured to estimate said at least one shift, wherein: said at least one estimator is a time estimator when said at least one shift is said time shift, said at least one estimator is a phase estimator when said at least one shift is said phase shift, wherein said estimating is based on coefficients of a prototype filter used at transmission to shape at least one preamble inserted into said OQAM multicarrier signal to estimate, in the time domain, said at least one shift, and wherein said time estimator implements one of the following equations: ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. {.Math.}_{k = 0}^{M / 4 - 1} h (k) h (M / 2 - 1 - k) r (k + \tilde{}) r (M / 2 - 1 - k + \tilde{}) .Math. and$ ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. {.Math.}_{k = 0}^{M / 4 - 1} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \tilde{}) r (M - 1 - k + \tilde{}) .Math.$ and said phase estimator implements one of the following equations: ${\hat{}}_{{LS}_{1}} = \frac{1}{2} {{.Math.}_{k = 0}^{M / 4 - 1} h (k) h (M / 2 - 1 - k) r (k + \tilde{}) r (M / 2 - 1 - k + \tilde{})} and$ ${\hat{}}_{{LS}_{2}} = \frac{1}{2} {{.Math.}_{k = 0}^{M / 4 - 1} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \tilde{}) r (M - 1 - k + \tilde{})}$ with: {acute over ()}.sub.LS.sub.1 or {acute over ()}.sub.LS.sub.2 being the estimation of said time shift at output of said time estimator, and {circumflex over ()} being the estimation of the time shift fixed at input of the phase shift estimator; {acute over ()}.sub.LS.sub.1 or {acute over ()}.sub.LS.sub.2 being the estimation of said phase shift at output of said phase estimator; {. } being the argument of a complex number; M being the number of sub-carriers of an OQAM symbol of said OQAM multicarrier signal; k being an integer such that 0kM/41; {tilde over ()} being an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ( )}.sub.max being a predetermined maximum value of {tilde over ()}; h(k) being coefficients of said prototype filter used at transmission; r(k) being the received OQAM multicarrier signal such that r(k)=s(k)e.sup.j+b(k); s(k) being a sent multicarrier signal; b(k) being a Gaussian white noise; being said time shift of the received OQAM multicarrier signal relative to said sent multicarrier signal; being said phase shift of the received OQAM multicarrier signal relative to said sent multicarrier signal.

Description

5. LIST OF FIGURES

(1) Other characteristics and advantages of the invention shall appear more clearly from the following description of a particular embodiment given by way of a simple, illustratory and non-exhaustive example and from the appended drawings, of which:

(2) FIG. 1, described with reference to the prior art, presents a classic OFDM/OQAM modulation scheme;

(3) FIGS. 2A and 2B respectively illustrate a general system of transmission implementing a technique of reception according to one embodiment of the invention, and the implemented estimation step;

(4) FIG. 3 represents the amplitude of the outputs of the frequency/time transformation module for a given preamble;

(5) FIGS. 4 and 5 illustrate the effect of the prototype filters on the peak values obtained as outputs of the frequency/time transformation module for specific preambles;

(6) FIGS. 6 and 7 illustrate the cost functions of two examples of phase and/or time shift estimators according to the first embodiment of the invention;

(7) FIGS. 8 and 9 show the performance values obtained by using examples of phase and/or time shift estimators according to the first embodiment of the invention;

(8) FIGS. 10 and 11 represent the amplitude of the outputs of the frequency/time transformation module for a preamble required for estimating a time and/or frequency shift;

(9) FIG. 12 illustrates the cost function of an example of time and/or frequency shift estimators according to the invention;

(10) FIG. 13 shows the performance values obtained by using examples of time and/or frequency shift estimators according to the invention;

(11) FIG. 14 presents the simplified structure of a receiver implementing a technique of reception according to one embodiment of the invention.

6. DESCRIPTION OF ONE EMBODIMENT OF THE INVENTION

(12) 6.1 General Principle

(13) The invention is situated in the context of transmission systems implementing an OFDM/OQAM or BFDM/OQAM type modulation and implementing preambles, and proposes a technique for simplifying the operation of synchronization of the receivers.

(14) It can be noted that, since these systems of transmission can be implemented in the form of banks of filters, they are also called FBMC/OQAM (Filter Bank Multicarrier/OQAM >>), or more generally OQAM systems.

(15) The general principle of the invention relies on the use on the reception side of an estimator for at least one shift in time, phase and/or frequency of the multicarrier signal, each of these estimators taking account of the prototype filter used at transmission to shape the preamble. Such a prototype filter can be very short and, at transmission, can generate a preamble that is localized on a single OQAM multicarrier symbol.

(16) In particular, if a multipath channel is considered, the estimators proposed according to the invention can be used to estimate the maximum delay introduced by the channel and therefore compensate accurately for the channel effects.

(17) 6.2 Example of Implementation

(18) 6.2.1 Transmission System

(19) Referring to FIG. 2A, a general system of transmission implementing a technique of reception according to the invention is presented.

(20) More specifically, the operations implemented on the sending side, namely the operations of pre-processing, conversion from the frequency domain to the time domain and filtering of the modulator 10, are known and have already been described with reference to FIG. 1.

(21) It can be noted however, that in the case of a transmission system implementing a preamble, the incoming data symbols to be transmitted on a sub-carrier m at the instant n can be denoted as c.sub.m,n instead of a.sub.m,n, and can be of complex values for the preamble symbols instead of being of real values for the payload data symbols. For example, the data symbols at the instants n=0 and n=1 are preamble symbols denoted as p.sub.m,n, and the data symbols at the next instants are payload data symbols denoted as a.sub.m,n.

(22) The FBMC/OQAM multicarrier signal at output of the modulator 10 can therefore be expressed in the following form:

(23) $s [k] = {.Math.}_{m = 0}^{M - 1} {.Math.}_{n = -}^{n = +} c_{m, n} h ({kT}_{s} - n \frac{M}{2} T_{s})^{j \frac{2}{M} m (k - D / 2) T_{s}}^{j_{m, n}},$

(24) with c.sub.m,n=p.sub.m,n symbols of complex value for the shaping of the preamble (preferably of real or imaginary pure value) and c.sub.m,n=a.sub.m,n data symbols of real value for the shaping of the payload part, T.sub.s the sampling time chosen such that the symbol duration T=MT.sub.s with M=2N is the number of carriers. Again, it can be recalled that distinct prototype filters can be used to shape the preamble and the payload part of the multicarrier signal s[k].

(25) The passage into the transmission channel can deform the multicarrier signal s[k], by introducing a Gaussian white additive noise, a time shift (also called a delay) and/or a phase shift , and/or a frequency shift of the multicarrier signal.

(26) The multicarrier signal received can therefore be expressed in the following form: r[k]=s[k]e.sup.j+b[k] and/or r(k)=e.sup.j2[c(k) custom character s(k)]+b(k) if the frequency shift occurs for example at transmission. It is also possible to take account of a frequency shift introduced at reception,

(27) with b[k] a centered circular complex additive white Gaussian noise with a spectral density .sub.b.sup.2, custom character representing a product of convolution and c(k) the response of a multipath channel.

(28) On the reception side, it is sought to estimate these time shift parameters , a phase shift and a frequency shift during an estimation step 21 and then to compensate for the multicarrier signal received during a compensation step 22 in order to synchronize the receiver.

(29) At output of the compensation step 22, a synchronized signal is obtained that can be expressed in the following form:
r.sub.synchro[k]=r[k+{circumflex over ()}]e.sup.j{circumflex over ()} and/or r.sub.synchro[k]=r[k+{circumflex over ()}]e.sup.j2

(30) This synchronized signal can then be demodulated conventionally during a demodulation step 23 in order to obtain an estimation of the sent data symbols {tilde over (c)}.sub.m,n.

(31) 6.2.2 Estimation of the Time and Phase Shift According to the First Embodiment

(32) Here below, a more detailed description of the estimation step 21 is provided according to a first embodiment of the time shift parameters and phase shift parameters , especially represented by FIG. 2B.

(33) The inventors of the present patent application, who are also inventors of the French patent application FR 1151590 filed on 28.sup.th Feb. 2011 on behalf of the same Applicant, have, in the above-mentioned patent application, shown particular relationships between the different outputs of the frequency/time transformation module 12 and polyphase filtering module 13 implemented at transmission.

(34) More specifically, they have shown that the outputs of the frequency/time transformation modules are conjugate in sets of two, and that the polyphase components of the prototype filter are para-conjugate in sets of two.

(35) It is therefore possible to use this symmetry to re-use a part of the results of the multiplications occurring at different instants of filtering, and thus reduce the complexity of the filtering.

(36) It is also possible, according to the present invention, to use these relationships to reduce the complexity of the estimations implemented during the estimation step 21.

(37) Thus, by choosing a delay such that D=qM1, with q an integer, the inventors have shown that the following relationships are obtained at output of the frequency/time transformation module 12 whatever the incoming data symbols (payload data or preamble data):

(38) ${\begin{matrix} u_{k, n} = {(- 1)}^{n} u_{M / 2 - k - 1, n}^{*} & for 0 k M / 4 - 1 \\ u_{M / 2 + k, n} = {(- 1)}^{n} u_{M - k - 1, n}^{*} & for 0 k M / 4 - 1 \end{matrix}$
with: u.sub.m,n a transformed signal associated with the output with index m of the frequency/time transformation step 12 at an instant n; * the conjugate operator.

(39) Consequently, if the instant n=0 is considered and whatever the type of preamble (random or deterministic), the following will be obtained for 0kM/41 and a prototype filter with a length L=qM, with q an integer such that q1:
h[M/21k]s[k]=h[k]s*[M/21k] and
h[M1k]s[M/2+k]=h[M/2+k]s*[M1k]

(40) When there is no noise, i.e. considering r[k]=s[k]e.sup.j, these relationships can also be written as follows
h[M/21k]r[k+]=h[k]r*[M/21k+]e.sup.j2(1)
h[M1k]r[M/2+k+]=h[M/2+k]r*[M1k+]e.sup.j2(2)

(41) for 0kM/41.

(42) It is possible to define two estimators, on the basis of a least square type of measurement between the outputs of the frequency/time transformation module 12 verifying a property of conjugation.

(43) Thus, referring to FIG. 2B, according to this first embodiment, the estimation step (21) successively implements a first sub-step for estimating a time shift (2110) followed by a second sub- step for estimating (2111) of a phase shift.

(44) The first estimator implementing the first sub-step for estimating a time shift (2110) is based on the relationship (1) and consists in minimizing the following cost function:

(45) $_{1} (\hat{}, \hat{}) = {.Math.}_{k = 0}^{M / 4 - 1} {.Math. h [M / 2 - 1 - k] r [k + \hat{}] - h [k] r^{*} [M / 2 - 1 - k + \hat{}]^{j 2 \hat{}} .Math.}^{2}$

(46) By developing this equation, it is obtained:

(47) $_{1} (\hat{}, \hat{}) = {.Math.}_{k = 0}^{M / 4 - 1} h^{2} [M / 2 - 1 - k] {.Math. r [k + \hat{}] .Math.}^{2} + {.Math.}_{k = 0}^{M / 4 - 1} h^{2} [k] {.Math. r [M / 2 - 1 - k + \hat{}] .Math.}^{2} - 2 .Math. {^{- 2 j \hat{}} {.Math.}_{k = 0}^{M / 4 - 1} h [k] h [M / 2 - 1 - k] r [k + \hat{}] r [M / 2 - 1 - k + \hat{}]}$ $_{1} (\hat{}, \hat{}) = (\hat{}) - (\hat{}, \hat{})$ $with :$ $(\hat{}) = {.Math.}_{k = 0}^{M / 2 - 1} h^{2} [M / 2 - 1 - k] {.Math. r [k + \hat{}] .Math.}^{2} and$ $(\hat{}, \hat{}) = 2 .Math. {^{- 2 j \hat{}} {.Math.}_{k = 0}^{M / 4 - 1} h [k] h [M / 2 - 1 - k] r [k + \hat{}] r [M / 2 - 1 - k + \hat{}]}$

(48) It can be noted that the quantity ({circumflex over ()}) is independent of the estimation of the phase shift {circumflex over ()} and weakly linked to the time shift {circumflex over ()}. As a consequence, the first LS estimator amounts to jointly maximizing the function:

(49) $({\hat{}}_{LS}, {\hat{}}_{LS}) = \underset{\hat{}, \hat{}}{argmax} (\hat{}, \hat{})$

(50) To obtain the estimation of the phase shift {circumflex over ()}.sub.LS delivered by the second estimation sub-step (2111) of a phase shift, the estimation of the time shift {circumflex over ()} is fixed in the previous equation, and the estimation of the phase shift {circumflex over ()} is made to vary, thus leading to the following phase estimator:

(51) ${\hat{}}_{{LS}_{1}} = \frac{1}{2} {{.Math.}_{k = 0}^{M / 4 - 1} h (k) h (M / 2 - 1 - k) r (k + \hat{}) r (M / 2 - 1 - k + \hat{})}$

(52) where {. } designates the argument of a complex number.

(53) Consequently, on the basis of the above equations, the following estimator for the time shift is obtained:

(54) $\begin{matrix} {\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{argmax} .Math. {.Math.}_{k = 0}^{M / 4 - 1} h (k) h (M / 2 - 1 - k) r (k + \tilde{}) r (M / 2 - 1 - k + \tilde{}) .Math. . & (3) \end{matrix}$

(55) It can be noted that, for this time estimator, the used preamble can be equal to an FBMC/OQAM multicarrier symbol (with the duration M/2) because the relationships on which this estimator is based (the relationships between the first M/2 samples) remain valid and are not affected by the following FBMC/OQAM multicarrier symbol, which is shifted by M/2 samples relative to the preamble. This observation can be applied even for a prototype filter length greater than M.

(56) The second estimator is based on the relationship (2) and minimizes the following cost function:

(57) 0 $_{2} (\hat{}, \hat{}) = {.Math.}_{k = 0}^{M / 4 - 1} {.Math. h [M - 1 - k] r [M / 2 + k + \hat{}] - h [M / 2 + k] r^{*} [M - 1 - k + \hat{}]^{j 2 \hat{}} .Math.}^{2}$

(58) By following the same development as above, the following phase and time estimators are obtained:

(59) $\begin{matrix} {\hat{}}_{{LS}_{2}} = \frac{1}{2} {{.Math.}_{k = 0}^{M / 4 - 1} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \hat{}) r (M - 1 - k + \hat{})} {\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{argmax} .Math. {.Math.}_{k = 0}^{M / 4 - 1} h (M - 1 - k) h (M / 2 + k) r (M / 2 + k + \tilde{}) r (M - 1 - k + \tilde{}) .Math. & (4) \end{matrix}$

(60) It is noted that for this time estimator, the preamble preferably comprises a zero FBMC/OQAM multicarrier symbol before the payload data. The used preamble can therefore be equal to two FBMC/OQAM multicarrier symbols.

(61) It is noted that the two phase estimators and the two time estimators proposed require prior knowledge on the prototype filter used at transmission but over a shorter range of correlation (M/2 samples instead of M). The complexity of computation of the estimators is therefore reduced by half.

(62) 6.2.3 Examples of Preambles and Time Estimators Simplified According to the First Embodiment

(63) In order to improve the estimation of the time synchronization performed prior to the phase synchronization, two types of preambles and the associated estimators are proposed here below, enabling a further reduction of the complexity of implementation of the synchronization.

(64) Thus, a simplification of the computations can be obtained for the estimators proposed here above and for specific preambles, leading to better performing estimators. This simplification is due to the presence of peak values at the output of the frequency/time transformation module, also called maximum values, for the preamble. The position of these peak values depends on the used preamble. The performance of the estimators is related to the position of this peak values relative to the coefficients of the prototype filter used at transmission. An optimum choice of the position of these peak values is made in such a way that they are filtered by fairly big filter coefficients.

(65) To this end, different examples of preambles are proposed, enabling at least one of the following criteria to be fulfilled: the quantity of data contained in the preamble and therefore the length of the preamble must be reduced to the utmost in order to increase the spectral efficiency; the possibility of obtaining high performance, i.e. the capacity to make a precise estimation of the delay {circumflex over ()} which depends exactly on the design of the estimator; the possibility of being used for other functions, such as estimation of the transmission channel.

(66) In the examples here below, on the basis of these criteria, the inventors have proposed two main types of preamble, with several variants for each type, and the corresponding estimators. The choices of preambles and estimators are evaluated in considering a prototype filter optimized for a time-frequency localization with a length L=M, denoted as TFL.

(67) However, the invention is not limited to the use of such a prototype filter and the principle proposed can be applied equivalently to prototype filters having an arbitrary length, especially a multiple of M, with M as an even-parity value.

(68) A) First Type of Preamble

(69) First of all, two novel preambles are presented, which are particularly promising because they can be used for other functions, such as channel estimation.

(70) According to a first example, a preamble emitted at the instant n and having the following structure is considered:

(71) ${\begin{matrix} p_{m, n} = \sqrt{2} / 2 for m = 4 p and m = 1 + 4 p \\ p_{m, n} = - \sqrt{2} / 2 for m = 2 + 4 p and m = 3 + 4 p \end{matrix}$

(72) with p.sub.m,n a preamble symbol associated with a sub-carrier having an index m at the instant n, p an integer, and 0mM1.

(73) According to a second example, a preamble sent at the instant n having the following structure is considered:

(74) ${\begin{matrix} p_{2 m, n} = \sqrt{2} / 2 \\ p_{2 m + 1, n} = j \sqrt{2} / 2 \end{matrix} for 0 m M / 2 - 1.$

(75) In particular, it may be recalled that it can be necessary to add one or more columns of zero preamble symbols before the payload part, such as p.sub.m,n+i=0 for i=1, 2, . . . .

(76) The addition of such a column of zero preamble symbols makes it possible to separate the first OQAM multicarrier symbol of the preamble (corresponding to n=0) from the payload part of the multicarrier signal in order to preserve the properties of conjugation.

(77) By using the time estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) the preamble can be formed by a single column of preamble symbols (for n=0 for example), since the relationships of conjugation obtained between the first M/2 samples of the multicarrier signal are valid even if the next column (for n=1 for example) belongs to the payload part of the multicarrier signal.

(78) By contrast, if the time estimator {circumflex over ()}.sub.LS.sub.2 defined in equation (4) is used, the preamble must be formed by at least two columns of preamble symbols, of which a first one is defined by the previous equations with p.sub.2m,n={square root over (2/2)} and p.sub.2m+1,n=j{square root over (2/2)}, and at least a second one is defined by p.sub.m,n+1=0.

(79) A preamble according to the second example makes it possible to obtain peak values at the outputs with indices M/41 and M/4 of the frequency/time transformation step rather than at the outputs with indices 0 and M/21 in the first example. This is worthwhile because, if a preamble according to the first example is considered, after filtering by the coefficient h[0] of the prototype filter, the peak value of the output with index 0 will be attenuated, and therefore the multiplication of the two peak values at the outputs with indices 0 and M/21 in the equation of the time estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) will be disturbed by the noise and the multipath channel. It is therefore desirable to propose this second example of preamble, which enables the position of the peak values to be shifted.

(80) The preambles according to these two first examples therefore have pairs of peak values at output of the step of frequency/time transformation, at transmission, at different positions.

(81) Thus, the preamble according to the first example has maximum values at the outputs of the frequency/time transformation module 12 with indices 0, M/21, M/2 and M1. The preamble according to the second example has maximum values at the outputs of the frequency time/transformation module 12 with indices M/41, M/4,3M/41 and 3M/4.

(82) Using the fact that pairs of peak values can be generated by the FBMC/OQAM modulator at particular positions in choosing an appropriate preamble and in taking account of the intrinsic properties of symmetry/conjugation intrinsic to the modulator, it is possible to further simplify the operation of estimation of the phase and time shifts of the multicarrier signal in reducing the complexity of the computations associated with the estimators.

(83) More specifically, it can be deduced that the peak values of the signal received contribute preponderantly to the computation of the delay in using the previous estimators defined in the equations (3) and (4). These peak values are the least affected in the presence of noise or of a multipath channel. It is therefore possible to make these estimators more robust and less complex by overlooking the computation of the products of the samples that do not correspond to peak values. Simplified or reduced LS estimators are then obtained.

(84) Therefore, having prior knowledge of the peak values of the signal sent, it is possible, for each preamble, to obtain the reduced LS estimators corresponding to the previous estimators.

(85) Thus, the preamble according to the first example is considered, the first reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) can be expressed in the following form:

(86) ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{argmax} .Math. r (\tilde{}) r (M / 2 - 1 + \tilde{}) .Math.,$
since the first pair of peak values is obtained at the outputs with indices 0 and M/21,
and the second reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) can be expressed in the following form:

(87) ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{argmax} .Math. r (M / 2 + \tilde{}) r (M - 1 + \tilde{}) .Math.,$
since the second pair of peak values is obtained at the outputs having indices M/2 and M1.

(88) If the preamble according to the second example is considered, the first reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) can be expressed in the following form:

(89) ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{argmax} .Math. r (M / 4 - 1 + \tilde{}) r (M / 4 + \tilde{}) .Math.,$
since the first pair of peak values is obtained at the outputs having indices M/41 and M/4;
and the second reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) can be expressed in the following form:

(90) ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{argmax} .Math. r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.,$
since the second pair of peak values is obtained at the outputs having indices 3M/41 and 3M/4.

(91) B) Second Type of Preamble

(92) Here below, three other examples of preambles formed by a column of preamble symbols are presented, wherein half the carriers have a zero value and possibly one or more columns of preamble symbols, of which all the carriers have a zero value, called a zero symbols.

(93) Such preambles are particularly valuable in that they can be used to obtain an additional property of conjugation.

(94) Thus, if an FBMC/OQAM transmission system implementing a number M of carriers that is a multiple of 4 and a time lag D=qM1 (with D=L1 in the orthogonal case and DL1 in the bi-orthogonal case) is taken, and if the reasoning described in the French patent application FR 1151590 mentioned here above is followed, it can be shown that the outputs of the frequency/time transformation step applied to the symbols of the preamble are related as follows:

(95) ${\begin{matrix} u_{k, n} = {(- 1)}^{n} u_{M / 2 - k - 1, n}^{*} & for 0 k M / 4 - 1 \\ u_{M / 2 + k, n} = {(- 1)}^{n} u_{M - k - 1, n}^{*} & for 0 k M / 4 - 1 \\ u_{k, n} = {(- 1)}^{n} u_{M - k - 1, n}^{*} & for 0 k M / 2 - 1 \end{matrix}$
with: u.sub.m,n a transformed symbol associated with the output with index m of the frequency/time transformation step 12 at an instant n; * the conjugate operator.

(96) It is considered for example that the first column of preamble symbols is sent at an even-parity instant n.

(97) Taking account of this novel property of conjugation, a novel estimator can be defined. Indeed, for 0kM/41 the following relationship is obtained if the prototype filter is symmetrical and has a length L=M:
s[k]=s*[M1k] for 0kM/21.

(98) It can be noted that if a prototype filter with a length L=qM is considered, it is desirable to add 2q1 columns of zero preamble symbols at the end of the preamble to separate the preamble (after modulation) from the payload data. The above relationship therefore remains valid for 0kqM/21.

(99) When there is no noise, the above relationship leads to the following expression of the received signal:
r[k+]=r*[M1k+]e.sup.j2 for 0kM/21:

(100) From this relationship, it is possible to define a novel estimator that does not require the zero-setting of the first coefficient of the prototype filter. Indeed, the cost function to be minimized is given by:

(101) $_{3} (\hat{}, \hat{}) = {.Math.}_{k = 0}^{M / 2 - 1} {.Math. r [k + \hat{}] - r^{*} [M - 1 - k + \hat{}]^{j 2 \hat{}} .Math.}^{2}$
which leads to the following phase estimator and time estimator:

(102) 0 $\begin{matrix} {\hat{}}_{{LS}_{3}} = \frac{1}{2} {{.Math.}_{k = 0}^{M / 2 - 1} r (k + \hat{}) r (M - 1 - k + \hat{})} and {\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. {.Math.}_{k = 0}^{M / 2 - 1} r (k + \tilde{}) r (M - 1 - k + \tilde{}) .Math. . & (5) \end{matrix}$

(103) Such estimators especially have high performance if they are combined with the use of one of the following preambles.

(104) Thus, according to a third example, a preamble sent at the instant n having the following structure is considered:

(105) ${\begin{matrix} p_{2 m, n} = \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1$
with p.sub.m,n a preamble symbol associated with a sub-carrier having an index m at the instant n.

(106) According to a fourth example, a preamble sent at the instant n having the following structure is considered:

(107) ${\begin{matrix} p_{2 m, n} = {(- 1)}^{m} \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1.$

(108) The preambles according to the third and fourth examples make it possible to obtain pairs of peak values at output of the frequency/time transformation step at different positions.

(109) Thus, the preamble according to the third example makes it possible to obtain maximum values at the outputs of the frequency/time transformation module 12 with indices M/41, M/4, 3M/41 and 3M/4. The preamble according to the fourth example makes it possible to obtain maximum values at the outputs of the frequency/time transformation module 12 having indices 0, M/21, M/2 and M1. As already indicated, the position and the number of these peaks can have an impact on the complexity and the performance of the estimator.

(110) Here below, a fifth example of a preamble is proposed, permitting to limit the maximum value of the pairs of peaks obtained at output of the frequency/time performance operation, in order to limit the PAPR (peak to average power ratio) of the multicarrier signal sent.

(111) The preamble according to this fifth example at the instant n has for example the following structure:

(112) ${\begin{matrix} p_{m, n} = 1 & if \mod (m, 4) = 0 \\ p_{m, n} = 0 & else \end{matrix} 0 m M - 1.$

(113) In this case, the maximum values are obtained at the outputs with indices 0, M/41, M/4, M/21, M/2 and M1, as illustrated in FIG. 3 for a number of carriers M=128, where the X axis represents the indices of the outputs of the frequency/time transformation module, and the Y axis represents the amplitude of the preamble at output of the frequency/time transformation module.

(114) Such a preamble thus makes it possible to distribute the maximum values on a greater number of outputs of the frequency/time transformation module, with lower amplitudes and therefore to reduce the PAPR of the sent signal.

(115) As already indicated with reference to the first two examples of preambles, it can be necessary, in these three last examples, to add one or more columns of zero preamble symbols in the preamble after modulation, before the payload part, in order to preserve the properties of conjugation after filtering.

(116) Thus, if we use the time estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) or the time estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5) the preamble must be formed by at least two columns of preamble symbols, a first one defined by the preceding equations, and at least a second one defined by p.sub.m,n+1=0 for 0mM1. More specifically, to use a prototype filter with a length L=qM, it is desirable to insert 2q1 columns of zero-preamble symbols into the preamble to separate the first column of preamble symbols defined by the preceding equations and the payload data.

(117) By contrast, to use the time estimator {circumflex over ()}.sub.LS.sub.1 as defined in the equation (3), it can happen that the preamble is formed by only one column of preamble symbols (for n=0 for example).

(118) As described here above with reference to the first two examples of preambles, it is also possible to reduce/simplify the estimators proposed and increase their performance in taking account of the position of the pairs of peak values generated by the FBMC/OQAM modulator.

(119) Thus, if we consider the preamble according to the third example, the first reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (3) can be expressed in the following form:

(120) ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 4 - 1 + \tilde{}) r (M / 4 + \tilde{}) .Math.,$
since the first pair of peak values is obtained at the outputs with indices M/41 and M/4,
the second reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) can be expressed in the following form:

(121) ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.,$
since the second pair of peak values is obtained at the outputs with indices 3M/41 and 3M/4, and
the third reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5) can be expressed in the following form:

(122) ${\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) + r (M / 4 + \tilde{}) r (3 M / 4 - 1 + \tilde{}) .Math.,$
since the two pairs of peaks are obtained at the outputs with indices M/41, M/4, 3M/41 and 3M/4.

(123) If we consider the preamble according to the fourth example, the first reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) can be expressed in the following form:

(124) ${\hat{}}_{{LS}_{1}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M / 2 - 1 + \tilde{}) .Math.,$
since the first pair of peak values is obtained at the outputs with indices 0 and M/21;
the second reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) can be expressed in the following form:

(125) ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 2 + \tilde{}) r (M - 1 + \tilde{}) .Math.,$
since the second pair of peak values is obtained at the outputs with indices M/2 and M1,
and the third reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5) can be expressed in the following form:

(126) ${\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M - 1 + \tilde{}) + r (M / 2 - 1 + \tilde{}) r (M / 2 + \tilde{}) .Math.,$
since the two pairs of peaks are obtained at the outputs with indices 0, M/21, M/2 and M1.

(127) If we consider the preamble according to the fifth example, the first reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) can be expressed in the following form:

(128) 0 ${\hat{}}_{{LS}_{1}} = \underset{\hat{}}{\arg \max} .Math. r (\hat{}) r (M / 2 - 1 + \hat{}) + r (M / 4 - 1 + \hat{}) r (M / 4 + \hat{}) .Math.$
since the peak values are obtained at the outputs with indices 0, M/21, M/41 and M/4;
the second reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) can be expressed in the following form:

(129) ${\hat{}}_{{LS}_{2}} = \underset{\tilde{}}{\arg \max} .Math. r (M / 2 + \hat{}) r (M - 1 + \tilde{}) + r (3 M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) .Math.$
since the peak values are obtained at the outputs with indices M/2, M1, 3M/41, 3M/4,
and the third reduced estimator corresponding to the estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5) can be expressed in the following form:

(130) ${\hat{}}_{{LS}_{3}} = \underset{\tilde{}}{\arg \max} .Math. r (\tilde{}) r (M - 1 + \tilde{}) + r (M / 4 - 1 + \tilde{}) r (3 M / 4 + \tilde{}) + r (M / 4 + \tilde{}) r (3 M / 4 - 1 + \tilde{}) + r (M / 2 - 1 + \tilde{}) r (M / 2 + \tilde{}) .Math.$

(131) It is seen therefore that the simplified/reduced estimators proposed according to the invention perform well as compared with the non-simplified estimators.

(132) Indeed, when there is noise and a multipath channel, the samples of the multicarrier signals with low amplitudes will be disturbed and therefore reduce the performance of the estimator. It is therefore proposed, in at least one embodiment of the invention, to eliminate these samples and to keep only the peak values of the multicarrier signal which for their part are less affected by the noise and the channel.

(133) This leads to obtaining reduced estimators which have better performance than the non-simplified estimators as defined in the equations (3), (4) and (5) because these simplified estimators reduce the number of peak values around the value to be estimated.

(134) 6.2.4 Estimation of Time and Frequency Shifts According to the Second Embodiment

(135) Here below, a more detailed description is given of the step 21 of estimation, according to a second embodiment, of the time shift parameter and frequency shift parameter especially represented in FIG. 2B.

(136) Just as in the case of the first embodiment, on the basis of the French patent application FR 1151590 filed on 28.sup.th Feb. 2011 on behalf of the present Applicant, the second embodiment of the invention delivering a time and frequency estimation derives profit from the particular relationships between the different outputs of the frequency/time transformation module 12 and polyphase filtering module 13 implemented at transmission, in relying on the demonstration of the inventors who have shown that the outputs of the frequency/time transformation module are conjugate in sets of two and the polyphase components of the prototype filter are para-conjugate in sets of two.

(137) It is therefore possible to use this symmetry to reutilize a part of the results of the multiplications that take place at different filtering times and thus to reduce the complexity of the filtering.

(138) It is also possible according to the present invention to use these relationships to reduce the complexity of the time and frequency estimators implemented during the successive sub-steps of timing estimation (2120) and frequency estimation (2121) of the estimation step 21 according to a second embodiment.

(139) Thus, in choosing a time lag such that D=qM1, with q an integer and for a particular preamble formed by two FBMC/OQAM symbols, for which half of the carriers (even-parity and odd-parity) are zero, such that:

(140) the first symbol sent at the instant n=0 such that:

(141) ${\begin{matrix} p_{2 m, 0} = d with d or d �� \\ p_{2 m + 1, 0} = 0 \end{matrix} or {\begin{matrix} p_{2 m, 0} = 0 \\ p_{2 m + 1, 0} = d with d or d �� \end{matrix}$
with p.sub.m,n a preamble symbol associated with a sub-carrier with an index m at the instant n=0, and 0mM/21, with M the number of sub-carriers of an OQAM symbol of said multicarrier signal, the value of d, which is a pure real value or a pure imaginary value being capable of being adjusted as a function of a compromise between the performance of the estimator and the PAPR (peak to average power ratio) value of the multicarrier signal sent;
while the following symbol sent at the instant n=1 corresponds to a zero symbol such that:
p.sub.m,1=0, for 0mM1
the inventors have shown that the following relationships are obtained at output of the frequency/time transformation module for this choice of preamble:
u.sub.k,n=(1).sup.nu.sub.M/2+k,n for 0kM/21
with:

(142) u.sub.m,n a transform symbol associated with the output with index m of the frequency/time transformation step 12 at an instant n.

(143) Consequently, if we re-write the relationship representing the FBMC/OQAM multicarrier signal at output of the modulator expressed in the following form:

(144) $s [k] = {.Math.}_{m = 0}^{M - 1} {.Math.}_{n = -}^{n = +} c_{m, n} h ({kT}_{s} - n \frac{M}{2} T_{s})^{j \frac{2}{M} m (k - D / 2) T_{s}}^{j_{m, n}}$

(145) with c.sub.m,n=p.sub.m,n data symbols of complex value for the shaping of the preamble (preferably with pure real or imaginary value), and c.sub.m,n=a.sub.m,n data symbols of real value for the shaping of the payload part, T.sub.s the sampling time chosen such that the symbol duration T=MT.sub.s with M=2N the number of carriers. Again, it may be recalled that distinct prototype filters can be used to shape the preamble and the payload part of the multicarrier signal s[k], in being restricted to M1 first samples and for 0kM/21 a modulated preamble signal is obtained, given by the following relationship:

(146) $s_{p} [k] = {.Math.}_{m = 0}^{M - 1} p_{m, 0} h [k]^{j \frac{2}{M} m (k - D / 2)}^{j_{m, 0}} = h [k] \underset{\underset{u_{k, 0}}{}}{{.Math.}_{m = 0}^{M - 1} p_{m, 0}^{j \frac{2}{M} m (k - D / 2)}^{j_{m, 0}}}$

(147) Given the previous relationship between each transformed symbol u.sub.m,n for the particular preambles used for the estimation of the time shift and the frequency shift according to the second embodiment of the invention, the preamble signal appropriately weighted by multiplying with the coefficients of the prototype filter, for example at the instant k by the coefficient h[M/2+k], and at the instant M/2+k by h[k], presentsa pseudo-periodicity expressed for 0kM/21 by the relationship:
h[M/2+k]s.sub.p[k]=h[k]s.sub.p[M/2+k](6)

(148) When there is no noise and for a flat channel (c(k)=1), i.e. if we consider r[k]=e.sup.j2s[k], minimizing the two terms of the relationship (6) amounts to minimizing the following quantity:

(149) $\begin{matrix} (\hat{f}, \hat{}) = \underset{\hat{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {.Math. h [k + M / 2] r [k + \tilde{}] - h [k] r [k + \tilde{} + M / 2]^{- j 2 \tilde{f} \frac{M}{2}} .Math.}^{2}} & (7) \end{matrix}$

(150) It is possible to define two estimators on the basis of a least squares (LS) type measurement between the outputs of the frequency/time transformation module 12 verifying a property of conjugation.

(151) Thus, with respect to FIG. 2B, according to this first embodiment, the estimation step (21) successively implements a first sub-step for estimating a time shift (2120) followed by a second sub-step for estimating (2121) of a frequency shift.

(152) The first estimator implementing the first sub-step for estimating a time shift (2120) is based on the minimizing of the relationship (7) and consists in minimizing the following cost function in writing A=h[k+M/2]r[k+{tilde over ()}], B=h[k]r[k+{tilde over ()}+M/2] and

(153) $= 2 \tilde{f} \frac{M}{2}$
and in recalling that |AB|.sup.2=(AB)(A.sup.*B.sup.*):

(154) $\begin{matrix} (\hat{f}, \hat{}) = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {(A - B) (A^{*} - B^{*}^{- j})}} \\ = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {{.Math. A .Math.}^{2} + {.Math. B .Math.}^{2} - 2 {{AB}^{*}^{- j}}}} \\ = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {{.Math. A .Math.}^{2} + {.Math. B .Math.}^{2}} - .Math. 2 {.Math.}_{k = 0}^{M / 2 - 1} {AB}^{*} .Math. \cos \\ (- + {.Math.}_{k = 0}^{M / 2 - 1} {AB}^{*})} \end{matrix}$

(155) We obtain a minimum relative to the frequency shift {tilde over ()} when the cosine is equal to 1, i.e. =AB.sup.* from these developments, we then obtain the time estimator implementing the first sub-step for estimating a time shift according to the following relationship:

(156) ${\hat{}}_{LS} = \underset{\tilde{}}{\arg \max} [2 .Math. R_{1} (\tilde{}) .Math. - Q_{1} (\tilde{})},$
and the frequency estimation implementing the second sub-step for estimating a frequency shift according to the following equation:

(157) 0 ${\hat{f}}_{LS} = \frac{1}{2 N} \arg {R_{1} ({\hat{}}_{LS})}, with :$ $N a discrete time shift, R_{1} (\tilde{}) = {.Math.}_{k = 0}^{M / 2 - 1} h (k) h (k + M / 2) r^{*} (k + \tilde{}) r (k + M / 2 + \tilde{}), Q_{1} (\tilde{}) = {.Math.}_{k = 0}^{M / 2 - 1} {h (k)}^{2} {.Math. r (k + \tilde{}) .Math.}^{2} + {h (k + M / 2)}^{2} {.Math. r (k + M / 2 + \tilde{}) .Math.}^{2},$
{circumflex over ()}.sub.LS the estimation of the time shift at output of said time estimator;
{circumflex over ()}.sub.LS the estimation of the frequency shift at output of said phase estimator;
M the number of sub-carriers of an OQAM symbol of said multicarrier signal;
k an integer such that 0kM/21;
{tilde over ()} an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max is a maximum predetermined value of {tilde over ()}
h(k) coefficients of said prototype filter used at transmission;
r(k) is a received multicarrier signal such that r(k)=e.sup.j2[c(k) custom character s(k)]+b(k) if the frequency shift occurs for example at transmission. It is also possible to take account of the frequency shift introduced at reception;
a convolution product;
c(k) a response of a multipath channel;
s(k) a sent multicarrier signal;
b(k) a Gaussian white noise;
the time shift of the multicarrier signal relative to the multicarrier signal sent;
the frequency shift of the received multicarrier signal relative to the sent multicarrier signal.

(158) 6.2.5 Examples of Preambles and Time and Frequency Estimators Amplified According to the Second Embodiment

(159) In order to improve the estimation of the time synchronization performed preliminarily to the synchronization in frequency, a reduction is proposed here below of the complexity of the computations that can be done for the estimators proposed here above and for specific preambles, leading to better performing estimators.

(160) This simplification is due to the presence of peak values at the output of the frequency/time transformation module, also called peak values or maximum values for the preamble. The position of these peak values depends on the preamble used. The performance of the estimators is related to the position of these peak values relative to the coefficients of the prototype filter used at transmission. An optimal choice of the position of these peak values is done in such a way that they are filtered by fairly large coefficients of the filter.

(161) Thus, using the particular preamble proposed here above for the time and frequency synchronization formed by two FBMC/OQAM symbols, of which half of the carriers (even-parity and odd-parity) are zero values such that:

(162) the first symbol sent at the instant n=0 such that:

(163) ${\begin{matrix} p_{2 m, 0} = d with d or d �� \\ p_{2 m + 1, 0} = 0 \end{matrix} or {\begin{matrix} p_{2 m, 0} = 0 \\ p_{2 m + 1, 0} = d with d or d �� \end{matrix}$
with p.sub.m,n a preamble symbol associated with a sub-carrier with index m at the instant n=0, and 0mM/21, with M the number of sub-carriers of an OQAM symbol of said multicarrier signal, the value d being capable of being adjusted as a function of a compromise between the performance of the estimator and the PAPR (peak to average power ratio) value of the multicarrier signal sent;
while the following symbol sent at the instant n=1 corresponds to a zero symbol such that:
p.sub.m,1=0, for 0mM1,
and in being limited solely to the high-amplitude outputs of the frequency/time transformation module namely in being limited to peak values, it is possible to obtain amplified time and frequency estimators.

(164) In other words, this example of an even-parity preamble will necessarily have the form p.sub.2m+1=0 and p.sub.2m=d , with d as a pure real or imaginary value, the sign of d depending on the value of m.

(165) Indeed, again on the basis of the minimization of the relationship (7), in assuming A=h[k+M/2]r[k+{tilde over ()}], B=h[k]r[k+{tilde over ()}+M/2] and

(166) $= 2 \tilde{f} \frac{M}{2}$
and in recalling that |ABe.sup.j|.sup.2=(ABe.sup.j)(A.sup.*B.sup.*e.sup.j) the following cost function is minimized:

(167) $\begin{matrix} \begin{matrix} (\hat{f}, \hat{}) = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {(A - B^{- j}) (A^{*} - B^{*}^{- j})}} \\ = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {{.Math. A .Math.}^{2} + {.Math. B .Math.}^{2} - 2 {{AB}^{*}^{- j}}}} \\ = \underset{\tilde{f}, \tilde{}}{\arg \min} {{.Math.}_{k = 0}^{M / 2 - 1} {{.Math. A .Math.}^{2} + {.Math. B .Math.}^{2}} - \\ .Math. 2 {.Math.}_{k = 0}^{M / 2 - 1} {AB}^{*} .Math. \cos (- + {.Math.}_{k = 0}^{M / 2 - 1} {AB}^{*})} \end{matrix} & (8) \end{matrix}$

(168) The expressions of A and B depend only on , using the particular preamble proposed here above for the time and frequency synchronization, and in being limited only to the high-amplitude outputs of the high-amplitude frequency/time transformation module, i.e. in being limited to the peak values, for example for the positions of the following maximum values k, M/21k, M/2+k, M1k, with k an integer, we obtain a minimum of the equation (8) relative to the frequency shift {tilde over ()} when the cosine is equal to 1, i.e. {tilde over ()}({tilde over ()})=R({tilde over ()}), with:
R({tilde over ()})=h(k)h(k+M/2)r*(k+{tilde over ()})r(k+M/2+{tilde over ()})+h(M/21k)h(M1k)r*(M/21k+{tilde over ()})r(M1k+{tilde over ()})

(169) Because the cosine is equal to 1, we then obtain the following time estimate:

(170) On the basis of these developments, we then obtain the time estimator implementing the first sub-step for estimating a time shift according to the following equation:

(171) $\begin{matrix} {\hat{}}_{LS} = \underset{\tilde{}}{\arg \max} {2 .Math. R_{1} (\tilde{}) .Math. - Q_{1} (\tilde{})} & (9) \end{matrix}$
where
Q({tilde over ()})=h(k+M/2).sup.2|r(k+{tilde over ()})|.sup.2h(M1k).sup.2|r(M/21k+{tilde over ()})|.sup.2+h(k)|r(M/2+k+{tilde over ()}).sup.|2h(M/21k).sup.2|r(M1k+{tilde over ()})|.sup.2

(172) Once the time shift has been estimated, it is possible to estimate the frequency shift:
{tilde over ()}({tilde over ()})=R({tilde over ()})

(173) In addition, since |R({tilde over ()})| and Q({tilde over ()}) have peak values for {tilde over ()}={tilde over ()} it is possible to replace the difference of the equation (9) by the product according to the following equation:

(174) $\begin{matrix} {\hat{}}_{LS} = \underset{\tilde{}}{\arg \max} {.Math. R_{1} (\tilde{}) .Math. Q_{1} (\tilde{})}, & (10) \end{matrix}$
thus reducing the complexity of the time estimation sub-step.

(175) Consequently, we obtain the simplified frequency estimator implementing the second sub-step for estimating a frequency shift according to the following equation:

(176) ${\hat{f}}_{LS} = \frac{1}{2 N} \arg {R_{1} ({\hat{}}_{LS})},$
with M the number of sub-carriers of an OQAM symbol of said multicarrier signal:
{circumflex over ()}.sub.LS the estimation of said time shift at output of said time estimator;
{circumflex over ()}.sub.LS the estimation of said frequency shift at output of said phase estimator;
M the number of sub-carriers of an OQAM symbol of said multicarrier signal;
{tilde over ()} an integer such that 0{tilde over ()}{tilde over ()}.sub.max with {tilde over ()}.sub.max is a predetermined maximum value of {tilde over ()}
h(k) the coefficients of said prototype filter used at transmission;
r(k) a received multicarrier signal such that r(k)=e.sup.j2[c(k) custom character s(k)]+b(k);
a convolution product;
c(k) the response of a multipath channel;
s(k) a sent multicarrier signal;
b(k) a Gaussian white noise;
the time shift of the received multicarrier signal relative to said sent multicarrier signal;
the frequency shift of the received multicarrier signal relative to said sent multicarrier signal.

(177) Thus, for a value of k equal to M/41 we obtain the following expressions of R({tilde over ()}) and Q({tilde over ()}):
R({tilde over ()})=h(M/41)h(3M/41)r*(M/41+{tilde over ()})r(3M/41+{tilde over ()})+h(M/4)h(3M/4)r*(M/4+{tilde over ()})r(3M4+{tilde over ()})
and
Q({tilde over ()})=h(3M/41).sup.2|r(M/41+{tilde over ()})|.sup.2+h(3M/4).sup.2|r(M/4+{tilde over ()})|.sup.2+h(M/41)|r(3M/41+{tilde over ()}).sup.|2h(M/4).sup.2|r(3M/4+{tilde over ()})|.sup.2

(178) FIG. 12 illustrates the cost function of this estimator in using a TFL type prototype filter and the length L=M, for a signal-to-noise ratio of 10 dB, a number of carriers M=128, a delay /Ts=13 and a frequency shift =0,002, which amounts to the standardized value v=0,256. This FIG. 12 clearly shows a highly significant peak value for the value of the time shift introduced into the system.

(179) 6.3 Generation of Preambles

(180) 6.3.1 Generation of Preambles for the Estimation of a Time Shift and a Phase Shift According to the First Embodiment

(181) Naturally, other preambles can be used, making it possible to obtain one or more pairs of peak values at output of the frequency/time transformation module at specific positions. Such preambles perform well when they are used for a receiver according to the invention and can also be used by any unspecified classic receiver.

(182) Here below, we propose two examples of algorithms to generate such preambles. It is assumed to this end that the number of carriers M is a power of 2 such that M8, the length of the prototype filter is L=M and similarly the number of non-zero preamble symbols P.sub.m,n is limited to M, giving only one column of preamble symbols at a given point in time. As already indicated, if a lengthier prototype filter is used, it is enough to add columns of zero preamble symbols in the preamble to separate it from the payload data.

(183) On these assumptions, only the preamble symbols p.sub.m,o corresponding to the column of preamble symbols at the instant n=0, also called pilots, must be taken into account and the signal at output of the frequency/time transformation module can be written, in considering at input and output only discrete sequences and in removing the need for the time index n, in the following form:

(184) $u_{k} = {.Math.}_{m = 0}^{M - 1} p_{m}^{j \frac{2}{M} m (k - (M - 1) / 2)}^{j \frac{}{2} m}$
which after simplification gives:

(185) $\begin{matrix} u_{k} = {.Math.}_{m = 0}^{M - 1} p_{m}^{- j \frac{}{2} m}^{j \frac{m}{M}^{j \frac{2 mk}{M}}} \\ = {.Math.}_{m = 0}^{M - 1} p_{m}^{j_{m, k}} \end{matrix}$ $with$ $_{m, k} = - \frac{}{2} m + \frac{m}{M} + 2 \frac{mk}{M} .$

(186) The determining of the preambles can be implemented by: searching for the values of the pilots producing one or more pairs of peak values u.sub.k with maximum values (at output of the frequency/time transformation module); assuming initially that the pilots are either pure real values or pure imaginary values.

(187) Owing to the relationships of symmetry at output of the frequency/time transformation module referred to here above, i.e. u.sub.M/21k=u.sub.k.sup.* and u.sub.M1k=u.sub.M/2+k.sup.* for 0kM/41, the peaks always occur in pairs.

(188) Besides, it is desirable that these pairs of peaks should appear in a limited number to prevent their energy from being dispersed over an excessively great number of outputs u.sub.k of the frequency/time transformation module.

(189) For example, if it is sought to obtain a pair of peak values of maximum amplitude in k.sub.0 and (M/21k.sub.0) , it is necessary to determine the values of the pilots p.sub.m giving the maximum of the expression max.sub.p.sub.m|u.sub.k.sub.0|.sup.2. Knowing that .sub.k|u.sub.k|.sup.2=|u.sub.k.sub.0|.sup.2, the choice of the objective function should at the same time ensure that, outside the position of the peak values, the output values u.sub.k of the frequency/time transformation module show low energy.

(190) The two algorithms proposed here below can be used to determine purely real preambles respectively producing one or two pairs of peak values. They can of course extend to the generating of purely imaginary preambles or mixed preambles, combining purely real preamble symbols and purely imaginary preamble symbols or even complex preamble symbols.

(191) 5.3.1.1 Determining a Single Pair of Peak Values

(192) As indicated here above, obtaining a peak value (peak) at the output with index k.sub.0 of the frequency/time transformation module amounts to maximizing the expression |u.sub.k.sub.0|.

(193) Now, according to the simplified equation of the value u.sub.k proposed here above, it is noted that all the phases with indices m and Mm , for 1mM/21, are symmetrical relative to the imaginary axis. Indeed, the quantities

(194) $^{j_{0, k_{0}}} and^{j_{M / 2, k_{0}}}$
are pure real and imaginary values respectively and are equal to 1 and j(1).sup.k.sup.0 in assuming that M=2.sup.q with q3 as an integer.

(195) Indeed, the term of the phase with index Mm is such that:

(196) 0 $_{M - m, k_{0}} = - \frac{}{2} (M - m) + \frac{(M - m)}{M} + 2 \frac{(M - m) k_{0}}{M}$ $\begin{matrix} _{M - m, k_{0}} = \frac{}{2} m + - \frac{m}{M} - 2 \frac{{mk}_{0}}{M} \\ = -_{m, k_{0}} \end{matrix}$

(197) It is therefore possible to write the simplified equation of the value u.sub.k as follows:

(198) $u_{k_{0}} = p_{0} + {j (- 1)}^{k} p_{M / 2} + {.Math.}_{m = 1}^{M / 2 - 1} p_{m}^{- j \frac{}{2} m}^{j \frac{m}{M}}^{j2 \frac{m k_{0}}{M}} - {.Math.}_{m = 1}^{M / 2 - 1} p_{M - m}^{j \frac{}{2} m}^{- j2 \frac{m k_{0}}{M}}$ $u_{k_{0}} = p_{0} + {j (- 1)}^{k} p_{M / 2} + {.Math.}_{m = 1}^{M / 2 - 1} {p_{m}^{j_{m, k_{0}}} - p_{M - m}^{- j_{m, k_{0}}}}$

(199) If it assumed that the symbols of the preambles p.sub.m have values coming from a constellation derived from a binary phase-shift keying modulation, the problem amounts to determining the signs of the preamble symbols p.sub.m.

(200) Now, the pairs of preamble symbols p.sub.m and p.sub.Mm that contribute to the sum in the above equation have arguments .sub.m,k and (.sub.m,k+) that position them symmetrically relative to the imaginary axis. Therefore, depending on their respective signs, the resulting vector associated with each of these preamble symbols is either a pure real value or a pure imaginary value.

(201) The maximizing procedure can therefore be implemented by the following algorithm:

(202) 1) specifying the desired value of the position k.sub.0 for the peak value;

(203) 2) choosing the signs of the preamble symbols p.sub.0 and p.sub.M/2;

(204) 3) computing, for each index m such that 1mM/21, the value of .sub.m,k.sub.0 and comparing its projection on the axis of the real values (cosine) and its projection on the axis of the imaginary values (sine) which amounts to testing the following condition: if |cos(.sub.m,k.sub.0)||sin(.sub.m,k.sub.0)| then bool(m)=0 else bool(m)=1, where the operator bool represents a two-state variable;

(205) 4) if bool(m)=0, assigning opposite signs to the preamble symbols p.sub.m and p.sub.Mm, i.e. p.sub.m=p.sub.Mm. This sign is chosen such that Sign(p.sub.m)*Sign(cos(.sub.m,k.sub.0))=Sign(p.sub.0). If bool(m)=1, assigning the same sign to the preamble symbols p.sub.m and p.sub.Mm, i.e. p.sub.m=p.sub.Mm. This sign is chosen such that Sign(p.sub.m)*Sign(sin(.sub.m,k.sub.0))=Sign((1).sup.k.sup.0 p.sub.M/2).

(206) For example, it is assumed that a search is made to obtain a peak value at the position with index k.sub.0=M/4. We consider preamble symbols such as p.sub.0=p.sub.M/2=1. The associated phase is such that .sub.m,k.sub.0=m/M, which implies that |cos(.sub.m,k.sub.0)||sin(.sub.m,k.sub.0) for 1mM/4 and |cos(.sub.m,k.sub.0)|<|sin(.sub.m,k.sub.0)| for M/4+1mM/21.

(207) In complying with the different steps of the algorithm proposed here above, we obtain the following preamble:

(208) $p_{m} = {\begin{matrix} 1 & for 0 m 3 M / 4 \\ - 1 & for 3 M / 4 + 1 m M - 1 \end{matrix}$

(209) If we consider that the symbols of the preamble can also take imaginary values, it is possible to obtain other even better performing preambles owing to the presence of additional degrees of freedom.

(210) 6.3.1.2 Determining Two Pairs of Peak Values

(211) Here below, we propose a second algorithm used to generate a preamble leading to the obtaining of two pairs of peak values at output of the frequency/time transformation module.

(212) It is assumed here that the preamble symbols have a value such that p.sub.m=1, it is sought to determine two pairs of peak values separated by M/2 samples. The two first peak values to be maximized are then localized at the positions k.sub.0 and M/2+k.sub.0, and the two other values are deduced from relationships of symmetry.

(213) The output of the frequency/time transformation module corresponding to the index M/2+k.sub.0 can be expressed as follows:

(214) $\begin{matrix} u_{M / 2 + k_{0}} = {.Math.}_{m = 0}^{M - 1} p_{m}^{- j \frac{}{2} m}^{j \frac{m}{M}}^{j \frac{2 m (M / 2 + k_{0})}{M}} \\ = {.Math.}_{m = 0}^{M - 1} {(- 1)}^{m} p_{m}^{- j \frac{}{2} m}^{- j \frac{m}{M}}^{j \frac{2 {mk}_{0}}{M}} \end{matrix}$

(215) By comparing this equation and the simplified equation of the value u.sub.k proposed in relation to the first algorithm, it can be noted that the optimizing (i.e. the maximizing) of the two outputs with indices k.sub.0 and M/2+k.sub.0 leads to a contradiction at the signs.

(216) Indeed, if identical optimizing choices are made for u.sub.k.sub.0 and u.sub.M/2+k.sub.0 (which amounts to fixing the signs of p.sub.0 and p.sub.M/2), it can be noticed that the optimizing of the even-parity indices of m leads to the same distribution of the signs for u.sub.k.sub.0 and u.sub.M/2+k.sub.0. However, the optimizing of the odd-parity indices leads to two choices which have opposite signs. Therefore, if a gain is made on the resultant vectors of u.sub.k.sub.0, there is a loss on the same resultant vector of u.sub.M/2+k.sub.0 and vice versa.

(217) To obtain peak values of a same amplitude for the values u.sub.k.sub.0 and u.sub.M/2+k.sub.0, it is therefore necessary to preserve the optimizing of the even-parity indices and accurately choose the resultant vectors of the odd-parity indices of u.sub.k.sub.0 and u.sub.M/2+k.sub.0, so as to maximize the final resultants for having the same amplitude is imposed.

(218) To this end, it is possible first of all to note that the phase with index m (even-parity or odd-parity) and the phase with index M/2+m are shift by /2. Indeed, we have:

(219) $_{M / 2 + m, k_{0}} = - \frac{}{2} (M / 2 + m) + \frac{(M / 2 + m)}{M} + 2 \frac{(M / 2 + m) k_{0}}{M}$ $_{M / 2 + m, k_{0}} = - \frac{}{2} m + \frac{}{2} + \frac{m}{M} + k_{0} + 2 \frac{m k_{0}}{M} = \frac{}{2} + k_{0} +_{m, k_{0}}$

(220) Hence, the phase difference between .sub.M/2+m,k.sub.0 and .sub.m,k.sub.0 is /2, with a sign that depends on the parity of k.sub.0. This means that if we maximize p.sub.m to optimize the outputs u.sub.k.sub.0 having even-parity indices, the resultant imaginary vector is equal, as a norm, to the real resultant.

(221) It is necessary henceforth to choose the optimal way for adding the resultant vectors of the odd-parity indices, considering that the resultant vectors of u.sub.k.sub.0 are opposite to those resulting from u.sub.M/2+k.sub.0, as indicated here above.

(222) In both cases, it can be said that if the optimization of the pair (m, Mm) has a maximum projection on the axis of the real values, then the optimization of the pair (M/2+m,M/2m) necessarily has a maximum projection on the axis of the imaginary values and vice versa. Now, it can be recalled that the resultant vectors of these two pairs have the same amplitude.

(223) The optimal solution is therefore to obtain a resultant vector of the even-parity indices that is perpendicular to that of the odd-parity indices. This can be done by choosing, for example, the pair (p.sub.m,p.sub.Mm) which maximizes the real resultant vector of u.sub.k.sub.0 and the pair (p.sub.M/2+m, p.sub.M/2m) qui which maximizes the imaginary resultant vector of u.sub.M/2+k.sub.0.

(224) The optimizing procedure can therefore be implemented by the following algorithm:

(225) 1) specifying the value of the position k.sub.0 for the peak value and deducing the value of the position M/2+k.sub.0;

(226) 2) choosing the signs of the preamble symbols p.sub.0 and p.sub.M/2;

(227) 3) computing, for each index m such that 1mM/21, the value of .sub.m,k.sub.0 and comparing its projection on the axis of real values (cosine) and its projection on the axis of the imaginary values (sine), which amounts to testing the following condition: if |cos(.sub.m,k.sub.0)||sin(.sub.m,k.sub.0) then bool(m)=0 else bool(m)=1, where the operator bool represents a two-state variable;

(228) 4) if bool(m)=0: if m is an odd-parity value: assigning opposite signs to the preamble symbols p.sub.m and p.sub.Mm. This sign is chosen in such a way that Sign (p.sub.m)*Sign(cos(.sub.m,k.sub.0))=Sign(p.sub.0) in a first case, or Sign (p.sub.m)*Sign(cos(.sub.m,k.sub.0))=Sign(p.sub.0) in a second case; if m is an even-parity value: assigning opposite signs to the preamble symbols p.sub.m and p.sub.Mm, such that Sign(p.sub.m)*Sign(cos(.sub.m,k.sub.0))=Sign(p.sub.0);

(229) if bool(m)=1: assigning opposite signs to the preamble symbols p.sub.m and p.sub.Mm. This sign is chosen such that Sign(p.sub.m)*Sign(sin(.sub.m,k.sub.0))=Sign((1).sup.k.sup.0 p.sub.M/2) in the first case, or Sign(p.sub.m)*Sign(sin(.sub.m,k.sub.0)=Sign((1).sup.k.sup.0 p.sub.M/2) in the second case.

(230) It can be noted that several choices are possible on the final directions of the odd-parity resultants.

(231) For example, it is assumed that it is sought to obtain a peak value at the position with index k.sub.0=M/4 and in following the different steps of the algorithm proposed here above, we obtain the same preamble as for the first algorithm by inversing the signs of the odd-parity indices for 1mM/41 and for 3M/4+1mM1.

(232) Again, as for the first algorithm, if we consider that the preamble symbols can also take imaginary values, it is possible to obtain other even better performing preambles, owing to the presence of additional degrees of freedom.

(233) 6.3.2 Generating Preambles for the Estimation of a Time Shift and a Frequency Shift According to the Second Embodiment

(234) As mentioned here above, the particular case of the joint estimation of time and frequency shifts requires particular preambles formed by two FBMC/OQAM symbols, of which half of the carriers (even-parity or odd-parity) are zero such that:

(235) the first symbol sent at the instant n=0 such that:

(236) ${\begin{matrix} p_{2 m, 0} = d with d or d �� \\ P_{2 m + 1, 0} = 0 \end{matrix} and$
with p.sub.m,n a preamble symbol associated with a sub-carrier with index m a the instant n=0, and 0mM/21, with M the number of sub-carriers of an OQAM symbol of said multi-carrier signal, the value of d being capable of being adjusted as a function of a compromise between the performance of the estimator and the PAPR (peak to average power ratio) value or a peak factor of the multi-carrier signal sent;
while the following symbol sent at the instant n=1 corresponds to a zero symbol such that:
p.sub.m,1=0, for 0mM1.

(237) The shape of the above preamble corresponds to the preambles known as even-parity preambles. By contrast, it must be noted that, when zeros are placed at the even-parity indices, the preambles obtained are called odd-parity values.

(238) The method for generating preambles for the estimation of a time shift and frequency shift uses the principle explained with reference to the generation of preambles for the estimation of a time shift and a phase shift and especially the relationships of symmetry at output of the frequency/time transformation module explained here above, i.e. u.sub.M/21k=u.sub.k.sup.* and u.sub.M1k=u.sub.M/2+k.sup.* for 0kM/41, implying that the peaks occur always in pairs.

(239) Besides, the type of particular preambles explained here above and required for the estimation of a time shift and a frequency shift implies an additional symmetry given by:
u.sub.k=u.sub.M/2+k,n for 0kM/21(11).

(240) As a consequence, such a preamble implies at least two pairs of peak values.

(241) When it is desired to obtain exactly two pairs of peak values, the invention uses the algorithm implementing the maximizing method used to determine a single pair of peak values, relating to the estimation of a time shift and a phase shift as explained here above (paragraph 5.3.1.1) according to two approaches: implementing steps 1 to 4 of the algorithm and adding a step 5) setting the pilots of odd-parity indices at zero, i.e. p.sub.2m+1,0=0, in order to prevent the performance of unnecessary computations, the step 3 is performed in considering only values of m that are even-parity or odd-parity values.

(242) It must be noted that the algorithm implementing the maximizing procedure used to determine two pairs of peak values pertaining to the estimation of a time shift and a phase shift as described in detail here above (paragraph 5.3.1.2), does not make it possible to produce more pairs of peak values for the particular preamble required for estimating a time shift and a frequency shift. Indeed, owing to the relationship of symmetry (11), the additional peak values obtained are found in the same positions but with lower amplitudes.

(243) Thus, a single algorithm is implemented for the estimation of a time shift and a frequency shift, this algorithm giving, at the even-parity positions, a value equal to d according to the values of m, with d being a pure real value or a pure imaginary value.

(244) Such preambles are illustrated by FIGS. 10 and 11 respectively for a number of carriers M=16 and for k=M/41, or for a number of carriers M=32 and for k=M/81, where the X axis represents the indices of the outputs of the frequency/time transformation module and the Y axis represents the amplitude of the preamble at output of the frequency/time transformation module.

(245) In particular, when a number of carriers M=32 and for k=M/81, the algorithm for generating preambles relating to the estimation of a time shift and a frequency shift delivers a preamble such that:
{p.sub.2m,0}={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}.

(246) 6.4 Effect of the Prototype Filter

(247) If we consider again the scheme of the FBMC/OQAM modulator of FIGS. 1 and 2, it is seen that the data symbols obtained at output of the pre-processing module 11 and entering the frequency/time transformation module 12 depend on the time lag D introduced by the prototype filter.

(248) Now, to have the three relationships of conjugation presented with reference to the second type of preamble, namely:

(249) ${\begin{matrix} u_{k, n} = {(- 1)}_{u_{M / 2 - k - 1, n}}^{n *} & for 0 k M / 4 - 1 \\ u_{M / 2 + k, n} = {(- 1)}_{u_{M - k - 1, n}}^{n *} & for 0 k M / 4 - 1 \\ u_{k, n} = {(- 1)}_{u_{M - k - 1, n}}^{n *} & for 0 k M / 2 - 1 \end{matrix}$
it is necessary that the time lag D should be such that D=qM1, with q an integer greater than or equal to 1 (which corresponds to L=qM for the OFDM/OQAM case).

(250) Furthermore, to be able to use the third estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5), the length of the preamble should be equal to the length of the prototype filter. Therefore, and by way of an example, for a length L=qM, the preamble should be formed by 2q columns of preamble symbols (corresponding to the points in time n, n+1, . . . , n+2q1). As described here above, the first column of preamble symbols comprises specific pilots and the other 2q1 columns of preamble symbols have zero values.

(251) The inventors of the present patent application have noted that the prototype filter used at transmission has a direct influence on the performance of the estimators. Indeed, to preserve a high amplitude of peak values obtained at output of the frequency/time transformation module, these values must be filtered (i.e. multiplied) by filtering coefficients which are not negligible relative to the other coefficients. Thus, if the peak obtained has to be filtered by a low-value coefficient, it is desirable to increase the size of the preamble to shift the position of this peak.

(252) It is of course possible to use the first estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) and the second estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) for prototype filters of a length L=qM in using the preambles defined here above, possibly by adding only one column of zero preamble symbols.

(253) However, it is well known that the most significant coefficients of most prototype filters localized in time and in frequency are distributed on the M/2 coefficients that are around the centre of these prototype filters.

(254) As a consequence, in using for example the preambles according to the second or third examples above and the corresponding reduced estimators, it is possible to derive benefit from one of the two pairs of peak values obtained at the positions {M/41,M/4} or {3M/41,3M/4} at output of the frequency/time transformation module in adding q1 columns of zero-preamble symbols to the preamble and by filtering the preamble by the coefficients h[qM/2M/41] and h[qM/2M/4].

(255) For example, FIG. 4 illustrates the peak values obtained at the positions {M/41,M/4} and {3M/41,3M/4} at output of the frequency/time transformation module before filtering, in using the preamble Pn sent at the instant n according to the third example described here above, namely:

(256) ${\begin{matrix} p_{2 m, n} = \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1$

(257) We consider a prototype filter 40 illustrated in dashes with a length L=M. The peak values will therefore be filtered by the coefficients {h[M/41],h[M/4],h[3M/41],h[3M/4]} of the prototype filter which have fairly large amplitudes to preserve the peak values once the signal is sent.

(258) It can be noted that the M/2 first samples of the multi-carrier signal sent (corresponding to the preamble for which the first pair of peak values is a part) preserve a relationship of conjugation (plus or minus one coefficient) even if the following OQAM multi-carrier symbol sent at the instant n+1, denoted as Sn+1, is not zero. As a consequence, the first estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) can be used. By contrast, the M/2 following samples lose the relationships of conjugation if the symbol Sn+1 is not zero and the second estimator {circumflex over ()}.sub.LS.sub.2 defined in the equation (4) cannot be applied.

(259) FIG. 5 illustrates another example of peak values obtained at the positions at output of the frequency/time transformation module before filtering, in using a preamble Pn sent at the instant n according to the third example described here above and in considering a prototype filter 50 illustrated in dashes having a length L=qM=2M.

(260) It can be noted that, for the M/2 first samples of the multi-carrier signal sent (corresponding to the preamble for which the first pair of peak values corresponding to the positions {M/41,M/4} is a part), the first pair of peak values will be weakened because of filtering by low-amplitude coefficients. It is therefore preferable to use the second pair of peak values (corresponding to the positions {3M/41,3M/4}), which is filtered by a pair of high-amplitude coefficients {h[3M/41],h[3M/4]}, to estimate the time shift.

(261) In particular, for the second pair of peak values not to be disturbed by the OQAM multi-carrier symbol sent at the following instants, the OQAM multi-carrier symbol sent at the instant n+1, denoted as Sn+1, is preferably zero. It can be noted however that, in the case illustrated in FIG. 5, even if the symbol Sn+1 is not zero, the disturbance caused by the samples of this symbol Sn+1 is not too inconvenient because these samples, which will get added to the peak values, are filtered by low-value.

(262) More generally, it is noted that the first pair of peak values is preferably used for computing the estimator reduced for q as an odd-parity value and that the second pair of peak values is preferably used for the computation of the estimator reduced for q as an even-parity value.

(263) As a consequence, it is possible to generalize the reduction of the first estimator defined in the equation (3) and the second estimator defined in the equation (4) whatever the value of q, as an integer greater than or equal to 1, by the following equation:

(264) $\begin{matrix} {\hat{}}_{{LS}_{4}} = \underset{\tilde{}}{\arg \max} .Math. r (q M / 2 - M / 4 - 1 + \tilde{}) r (q M / 2 - M / 4 + \tilde{}) .Math. \end{matrix}$

(265) FIG. 6 illustrates the cost function .sub.4 of this estimator by using a TFL type prototype filter with a length L=4M, for a signal-to-noise ratio of 15 dB, a number of carriers M=128, a delay /Ts=10 and a preamble according to the third example described here above.

(266) This FIG. 6 shows two peaks at a distance of M/2 samples. The first peak is due to the multiplication of the two maximum values obtained at output of the frequency/time transform module localized at the positions {3M/41,3M/4}, and filtered by the coefficients of the prototype filter {h[qM/2M/41], h[qM/2M/4]}. The second peak is due to the multiplication of the two maximum values obtained at output of the frequency/time transform module localized at the positions {M/41,M/4}, and filtered by the coefficients of the prototype filter {h[qM/2+M/41],h[qM/2+M/4]}, which is equivalent to +M/2 in the fourth estimator {circumflex over ()}.sub.LS.sub.4. It can be noted that the relationship h[qM/2+M/4]s[qM/2+M/41]=h[qM/2+M/41]s*[qM/2+M/4] remains valid even if we add q1 columns of zero-preamble symbols in the preamble because the q.sup.th column will have no effect on the peak values {M/41,M/4} of the preamble after filtering, these peak values being situated in the first half of the outputs of the frequency/time transformation module.

(267) It is also possible to derive benefit from the two pairs of peak values obtained at output of the frequency/time transformation module at the positions {M1,0} or {M/21,M/2}, by using for example the preamble according to the fourth example here above and the corresponding reduced estimators.

(268) In this case, the two pairs of peak values obtained at the above-mentioned positions must be filtered by filtering coefficients {h[qM/21],h[qM/2]}, and it is necessary to add q columns of zero M preamble symbols into the preamble.

(269) More generally, it is observed that the pair of peak values corresponding to the positions {M1,0} is preferably used in the computation of the estimator reduced for q as an even-parity value and that the pair of peak values corresponding to the positions {M/21,M/2} is preferably used for the computation of the estimator reduced to q as an odd-parity value.

(270) As a consequence, it is possible to generalize the reduction of the first estimator defined in the equation (3) and of the second estimator defined in the equation (4), whatever the value of q as an integer greater than or equal to 1, by the following equation:

(271) $\begin{matrix} {\hat{}}_{{LS}_{5}} = \underset{\tilde{}}{\arg \max} .Math. r (q M / 2 - 1 + \tilde{}) r (q M / 2 + \tilde{}) .Math. \end{matrix}$

(272) FIG. 7 illustrates the cost function .sub.5 of this estimator in using a TFL type prototype filter with a length L=4M, for a signal-to-noise ratio of 15 dB, a number of carriers M=128, a delay /Ts=10 and a preamble according to the fourth example described here above.

(273) If we add q1 columns of zero-preamble symbols to the preamble, we obtain practically the same performance in terms of cost because the payload OQAM multi-carrier symbol following the preamble, corresponding to the column of payload data symbols at the instant q, which will get added to the preamble, will be filtered by the M first coefficients of a prototype filter which generally have low amplitude as compared with the following M coefficients.

(274) The performance of the estimators according to the invention is presented with reference to FIGS. 8 and 9 for the estimators relating to the estimation of a time shift and a phase shift according to the first embodiment, and to FIG. 13 for the estimators of a time shift and a frequency shift estimation according to the second embodiment.

(275) More specifically, FIG. 8 proposes a comparison between the first estimator {circumflex over ()}.sub.LS.sub.1 defined in the equation (3) and the third estimator {circumflex over ()}.sub.LS.sub.3 defined in the equation (5), before and after simplification in using the preambles according to the second type of preamble presented. The first reduced estimator {circumflex over ()}.sub.LS.sub.1 by using the first preamble according to the fourth example is denoted as {circumflex over ()}.sub.LS.sub.1.sub.R. The third reduced estimator {circumflex over ()}.sub.LS.sub.3 by using the preamble according to the third example is denoted as {circumflex over ()}.sub.LS.sub.3.sub.R.

(276) The curves obtained illustrate the performance of the estimators after simplification in terms of root mean square error (RMSE) relative to the signal-to-noise ratio (SNR). The performance of the reduced estimators is due especially to the improvement of their robustness against the effect of a multipath channel and Gaussian noise, in eliminating the most disturbed samples.

(277) The root mean square error is expressed here in the following form:

(278) 0 $R M S E (/ T) = \frac{1}{M} \sqrt{\frac{1}{re} {.Math.}_{i = 1}^{re} {(- {\hat{}}_{i})}^{2}}$

(279) where re is the number of realizations, {circumflex over ()}.sub.i the estimated value of the delay at the i.sup.th iteration and T is the symbol time (with T=M in discrete values).

(280) According to FIG. 8, the first reduced estimator {circumflex over ()}.sub.LS.sub.1.sub.R by using the first preamble according to the fourth example is the best performing estimator.

(281) FIG. 9 illustrates the performance of the same estimators in terms of RMSE as a function of the number of carriers for a signal-to-noise ratio of 10 bB and in the presence of an Ma type transmission channel. The estimators always show a gain after reduction. The first reduced estimator {circumflex over ()}.sub.LS.sub.1.sub.R by using the preamble according to the fourth example is always the best performing preamble because the samples which come into the computation of the estimator have the greatest amplitudes after filtering and therefore are the least disturbed during transmission.

(282) In terms of complexity, the first and second estimators defined by the equations (3) and (4) which use the correlation between the first and the second half of the samples of the preambles after filtering are less complex in terms of operational computation than the third estimator defined by the equation (5) using the correlation between all the samples of the preamble. After simplification of these estimators, in using the presence of the peak values after the frequency/time transformation step, the complexity of these estimators becomes negligible.

(283) FIG. 13 illustrates the performance of a time shift estimator according to the first embodiment corresponding to the reduced estimator {circumflex over ()}.sub.LS.sub.1.sub.R by using the preamble according to the fourth example called {circumflex over ()}.sub.LS.sub.o, and a time shift estimator according to the second embodiment pertaining to the equation (10) called {circumflex over ()}.sub.LS.sub.n, in the presence of a Gaussian noise and in the presence of an Ma type transmission channel.

(284) For the estimator {circumflex over ()}.sub.LS.sub.o, the result is obtained for a preamble such that:

(285) ${\begin{matrix} p_{2 m, 0} = {(- 1)}^{m} \sqrt{2} / 2 \\ p_{2 m + 1, n} = 0 \end{matrix} for 0 m M / 2 - 1, et p_{m, n + 1} = 0, for 0 m M - 1$

(286) This preamble verifies especially the property imperatively required for the estimation of a time shift and a frequency shift according to the second embodiment, since one carrier in every two is zero. The two preambles of the two estimators compared according to the first and second embodiments have thus the same energy.

(287) In presence of a purely Gaussian noise, it appears that the estimator {circumflex over ()}.sub.LS.sub.n performs better while, in the presence of a Ma type transmission channel, the estimator {circumflex over ()}.sub.LS.sub.o gives a slight advantage.

(288) 6.5 Structure of the Receiver

(289) Finally, referring to FIG. 14, we present the simplified structure of a receiver comprising an OFDM/OQAM or BFDM/OQAM demodulator implementing a technique for receiving according to the example described here above.

(290) Such a receiver comprises a memory 101 comprising a buffer memory, a processing unit 102 equipped for example with a microprocessor P and driven by the computer program 103 implementing the method for receiving according to the invention.

(291) At initialization, the code instructions of the computer program 103 are for example loaded into a RAM and then executed by the processor of the processing unit 102. The processing unit 102 inputs a multi-carrier signal r[k]. The microprocessor of the processing unit 102 implements the steps of the method for receiving described here above according to the instructions of the computer program 103 to carry out an estimation of the delays in time and/or in phase and/or in frequency affecting the received multi-carrier signal r[k]. To this end, the receiver comprises, in addition to the buffer memory 101, means for estimating comprising at least one time shift estimator, and/or at least one phase shift estimator and/or at least one frequency shift estimator taking account of the coefficients of a prototype filter used at transmission to shape the preamble. These means are driven by the microprocessor of the processing unit 102. 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.

Estimation of a time, phase and frequency shift of an OQAM multicarrier signal

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H04L27/2684

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H04L27/2698

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H04L27/2631

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H04L27/2654

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