Method and module for estimating transmission chanels of a multi-antenna multi-carrier system

Abstract

A method and a module for estimating transmission channels in a multi-antenna system. A matrix A is calculated which is constructed in the form of blocks from training sequences and an appropriate Fourier matrix. For a receive antenna RXj concerned, the method and the module calculate Nt impulse responses in the time domain by multiplying Np pilot symbols extracted from a frequency-domain signal Rj(n) obtained after demodulation of a time-domain signal received by the receive antenna RXj concerned by a product of matrices comprising the pseudo-inverse matrix of the product of the Hermitian matrix of the A matrix with the A matrix enabling decorrelation of modulated carriers adjacent null carriers.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 a, 1b and 1c are illustrations of known frame constructions of types 1, 2 and 3, respectively, for a system with two send antennas.

FIG. 2 is a flowchart of an estimation method of the invention.

FIG. 3 is a diagram of one particular transmission system of the invention enabling implementation of a method of the invention.

FIG. 4 is a graphical representation of simulation results.

FIGS. 5 a, 5b, and 5c are plots of the real part of the coefficient of the channel as a function of the carrier index.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 2 is a flowchart of an estimation method of the invention.

An estimation method of the invention estimates transmission channels in a multi-antenna system. A transmission channel links a send antenna TX_i to a receive antenna RX_j. A multi-antenna system uses N_t≧2 send antennas, at least one receive antenna, and a time-frequency frame for each send antenna. A time-frequency frame determines the temporal location of payload data symbols and pilot symbols on the various carriers. The time-frequency frame can further include null symbols, but as these are not involved in the estimation method, they are not referred to in this description. Nor does the description refer to the guard carriers, as they are not involved in the estimation method either. The pilot symbols for the same send antenna form a training sequence. The N_t training sequences are temporally offset from each other within a time interval. They are known to the receiver and enable it to estimate N_t impulse responses that correspond to the N_t transmission channels. The data symbols and pilot symbols are frequency-modulated by an orthogonal multiplexer to form orthogonal symbols that are sent by the send antenna connected to the orthogonal multiplexer in the form of a multi-carrier signal with N_FFT carriers including N_p pilot carriers and null carriers. Depending on the time-frequency frame concerned, the pilot symbols can be distributed over one or more orthogonal symbols. In one particular embodiment the modulation is OFDM modulation.

An estimation method of the invention is executed in the receiver after a demodulation step which demodulates a time-domain signal received by the receive antenna RX^j concerned by means of an FFT of size N_FFT to obtain a frequency-domain signal R^j(n) and to extract therefrom N_p pilot symbols. In one particular embodiment, the demodulation is OFDM demodulation. The size N_FFT of the FFT is typically determined as a function of the size of the inverse FFT applied on sending. These sizes are typically made identical.

The estimation method 1 of the invention includes a step 2 of calculating a matrix A and, for a given receive antenna RX_j, a step 3 of calculating the N_t impulse responses in the time domain of the N_t channels linking a send antenna TX_i to the respective receive antenna RX_j. To obtain the N_r×N_t channels, it is necessary to repeat step 3 for the various receive antennas RX_j.

The calculation step 2 calculates a matrix A constructed in the form of blocks from training sequences and a Fourier matrix with dimensions N_FFT×N_FFT.

The step 3 of calculating the N_t impulse responses in the time domain multiplies the N_p pilot symbols with a product of matrices comprising the pseudo-inverse matrix of the product of the Hermitian matrix of the A×A matrix. This multiplication by the product of particular matrices decorrelates the modulated carriers adjacent null carriers.

FIG. 3 is a diagram of a particular transmission system implementing a method of the invention.

The transmission system SY includes a multi-carrier sender device EM and a receiver device RE. The sender device EM is connected to N_t send antennas TX₁, . . . , TX_Nt, where N_t≧2. The receiver device RE is connected to N_r receive antennas RX₁, . . . , RX_Nr, where Nr≧1.

In the example shown, the sender device EM includes a channel coder module CdC, a bit interleaver EB, a binary to symbol coding module CBS, a space-time coding module CET, and as many OFDM multiplexers MX as there are send antennas.

The channel coding module CdC codes input source data corresponding to one or more signals, typically telecommunication signals, to supply coded output data dc, for example using a convolutional code.

The bit interleaver interleaves the bits of the coded data in accordance with a particular interleaving law to supply interleaved coded data dce.

The symbol to binary coding module CBS transforms the interleaved coded data dce into complex data symbols sc, for example by BPSK (binary phase-shift keying), QPSK (quadraphase-shift keying) or 16-QAM (quadrature amplitude modulation) modulation.

The space-time coding module CET determines from the complex data symbols sc a two-dimensional matrix of data symbols sd, for example in accordance with an Alamouti-type code, and generates pilot symbols.

Each OFDM multiplexer modulates the data symbols sd and pilot symbols sp inserted at the input of the multiplexer by sinusoidal sub-carriers having orthogonal functions that are the conjugate Fourier components of an inverse Fourier transform of size N_FFT corresponding to the number of carriers of an OFDM multiplex. The N_FFT carriers comprise N_mod modulated data carriers and N_p pilot carriers. The set of N_FFT carriers is referred to as an OFDM symbol. The OFDM symbols can include null carriers and guard carriers that are not modulated. The output of an OFDM multiplexer constitutes the time-domain OFDM signal without guard intervals.

Prior to sending, the sender device inserts a guard interval into the OFDM signal.

The signals sent include reference sequences. For each send antenna concerned, a reference sequence is determined by the pilot symbols contained in one or more OFDM symbols. The sequence is determined by the number g of these OFDM symbols, the amplitude of the pilot symbols, and the pilot carriers. The reference sequences are identical to each other but time-shifted relative to each other by an interval Δt least equal to the maximum length L of the transmission channels, Δ_t≧L, so that the impulse responses of the various channels do not interfere with each other. The time to transmit of a channel or the transmission delay introduced by the channel is referred to as the length of the channel. The interval Δt must be understood as a number of time intervals between samples. The time shift between reference sequences is typically applied before the OFDM modulation by phase-shifting the pilot symbols in the various reference sequences in the frequency domain.

If c(p), p=0, . . . , N_p−1 denotes the pilot symbol sent on the carrier frequency of index p at the send antenna TX₁, the pilot symbol sent in the same OFDM symbol period at the antenna TX_i is equal to:

$\begin{array}{cc}{c\left(p\right)}^i=c\left(p\right){}^{-j2\pi \frac{p\left(i-l\right)\Delta t}{N_P}}& \left(1\right)\end{array}$

Moreover, the condition for determining all the impulse responses is N_t×Δt≦N_p.

In the example illustrated, the receiver device RE includes as many OFDM demultiplexers DX₁, . . . , DX_Nr as there are receive antennas RX₁, . . . , RX_Nr. It further includes a channel estimation module EsT, an equalization module EgL, a symbol to binary decoding module DbS, a bit de-interleaver Dbt, and a channel decoder module DEC.

Each signal received by a receive antenna is processed by an OFDM demultiplexer. An OFDM demultiplexer applies the function that is the inverse of the function of an OFDM multiplexer. Thus an OFDM demultiplexer applies a direct Fourier transform of size N_FFT to the received signal, after elimination of the guard time. An OFDM demultiplexer demodulates a time-domain signal received by the receive antenna RX_j concerned by means of an FFT of size N_FFT to obtain a frequency-domain signal R^j(n) and extract N_p pilot symbols from it.

The outputs of the OFDM demultiplexer supply frequency-domain OFDM signals R^j(n) that are processed by the estimation module and the equalization module.

The signal from the receive antenna RX_j can be expressed in the frequency domain at the time n, i.e. after elimination of the guard time and after OFDM demodulation, in the form of a column vector of dimension N_FFT:

$\begin{array}{cc}{R}^j\left(n\right)=\sum _{i=1}^{N_t}\mathrm{diag}\left\{X^i\left(n\right)\right\}Fh^{j,i}+{\Xi }^j\left(n\right)& \left(2\right)\end{array}$

where X′(n) is a vector of dimension N_FFT, the OFDM symbol sent at the time n at the antenna TX_j;

F is the Fourier matrix with dimensions N_FFT×N_FFT;

h^j,i(n) is the column vector representing the L samples of the frequency response of the sub-channel linking the send antenna TX_i to the receive antenna RX_j; and

Ξ^j(n) is the column vector of dimension N_FFT representing the Fourier transform of Gaussian additive white noise.

To simplify the calculations, which is advantageous, the estimation method processes the demodulated signal according to equation (2) only over the period Δt:

$\begin{array}{cc}{R}^j\left(n\right)=\sum _{i=1}^{N_t}\mathrm{diag}\left\{X^i\left(n\right)\right\}F^{\prime }h^{\mathrm{\prime j},i}+{\Xi }^j\left(n\right)& \left(3\right)\end{array}$

F′ is the matrix containing the first Δt columns of the Fourier matrix F with dimensions N_FFT×N_FFT and h′^i,j is a column vector of size Δt such that:

h^′j,i=[h^j,iT,0_Δt-L]^T  (4)

A diagonal matrix having the column vector x on its diagonal is denoted diag{x}:

$\begin{array}{cc}\mathrm{diag}\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]& \left(5\right)\end{array}$

The Fourier matrix F is a square matrix with dimensions N_FFT×N_FFT of the following form:

$\begin{array}{cc}\begin{array}{c}F=\frac{1}{\sqrt{N_{\mathrm{FFT}}}}\\ \left[\begin{array}{ccccc}1& 1& 1& \dots & 1\\ 1& w_{N_{\mathrm{FFT}}}& w_{N_{\mathrm{FFT}}}^2& \dots & w_{N_{\mathrm{FFT}}}^{N_{\mathrm{FFT}}-1}\\ ⋮& ⋮& ⋮& \dots & ⋮\\ 1& w_{N_{\mathrm{FFT}}}^{N_{\mathrm{FFT}}-1}& w_{N_{\mathrm{FFT}}}^{2\left(N_{\mathrm{FFT}}-1\right)}& \dots & w_{N_{\mathrm{FFT}}}^{\left(N_{\mathrm{FFT}}-1\right)\left(N_{\mathrm{FFT}}-1\right)}\end{array}\right]\\ \mathrm{in}\mathrm{which}w_{N_{\mathrm{FFT}}}={}^{-j\frac{2\pi }{N_{\mathrm{FFT}}}}.\end{array}& \left(6\right)\end{array}$

The matrix F′ is deduced directly from F:

$\begin{array}{cc}{F}^{\prime }=\frac{1}{\sqrt{N_{\mathrm{FFT}}}}\left[\begin{array}{ccccc}1& 1& 1& \dots & 1\\ 1& w_{N_{\mathrm{FFT}}}& w_{N_{\mathrm{FFT}}}^2& \dots & w_{N_{\mathrm{FFT}}}^{\Delta t-1}\\ ⋮& ⋮& ⋮& \dots & ⋮\\ 1& w_{N_{\mathrm{FFT}}}^{N_{\mathrm{FFT}}-1}& w_{N_{\mathrm{FFT}}}^{2\left(N_{\mathrm{FFT}}-1\right)}& \dots & w_{N_{\mathrm{FFT}}}^{\left(\Delta t-1\right)\left(\Delta t-1\right)}\end{array}\right]& \left(7\right)\end{array}$

Each OFDM symbol Xⁱ(n) sent at the time n by the send antenna TX_i can be broken down into a vector containing the data symbols and a vector containing the pilot symbols:

X ⁱ(n)=Sⁱ(n)+Bⁱ(n)  (8)

where Sⁱ(n) is the vector of the payload data symbols with dimension N_FFT and Bⁱ(n) is the vector of the pilot symbols with the same dimension.

Equation (3) can therefore be expressed in the form:

$\begin{array}{cc}{R}^j\left(n\right)=\sum _{i=1}^{N_t}\mathrm{diag}\left\{S^i\left(n\right)+B^i\left(n\right)\right\}F^{\prime }h^{\mathrm{\prime j},i}+{\Xi }^j\left(n\right)& \left(9\right)\end{array}$

Knowing that Δt≧L, we can impose Δt=L. With this condition, the accumulation of received vectors corresponding to the g orthogonal symbols sent containing pilot symbols constituting a training sequence can be expressed in the form:

$\begin{array}{cc}\begin{array}{c}{R}^j={\left[{R^j\left(0\right)}^T\dots {R^j\left(g-1\right)}^T\right]}^T\\ ={\mathrm{Th}}^{\mathrm{\prime j}}+{\mathrm{Ah}}^{\mathrm{\prime j}}+{\Xi }^j\end{array}& \left(10\right)\end{array}$

where R^j is a column vector of dimension N_FFT×g;

T is a matrix with dimensions (N_FFT×g)×(N_t×Δt) containing the payload data symbols during the training sequences (of the N_t send antennas);

$\begin{array}{cc}T=\left[\begin{array}{ccc}{S}_{\mathrm{diag}}^1\left(0\right)F^{\prime }& \dots & S_{\mathrm{diag}}^{N_1}\left(0\right)F^{\prime }\\ ⋮& & ⋮\\ S_{\mathrm{diag}}^1\left(g-1\right)F^{\prime }& \dots & S_{\mathrm{diag}}^{N_t}\left(g-1\right)F^{\prime }\end{array}\right]& \left(11\right)\end{array}$

A is a matrix with dimensions (N_FFT×g)×(N_t×Δt) containing the pilot symbols during the training sequences (of the N_t send antennas);

$\begin{array}{cc}A=\left[\begin{array}{ccc}{B}_{\mathrm{diag}}^1\left(0\right)F^{\prime }& \dots & B_{\mathrm{diag}}^{N_1}\left(0\right)F^{\prime }\\ ⋮& & ⋮\\ B_{\mathrm{diag}}^1\left(g-1\right)F^{\prime }& \dots & B_{\mathrm{diag}}^{N_1}\left(g-1\right)F^{\prime }\end{array}\right]& \left(12\right)\end{array}$

h^′j is the column vector of dimension N_t×Δt containing the N_t impulse responses;

h^′j=[h^j,lT, . . . , h^j,NiT]^T  (13)

Ξ^j is the column vector of dimension N_FFT×g containing g representations of the Fourier transform of Gaussian additive white noise.

The channel estimation module includes means for calculating the matrix A using equation (12), typically instructions of a program that in a preferred embodiment is incorporated in an electronic circuit, a calculation unit such as a processor or the like whose operation is then controlled by the execution of the program.

To estimate the impulse response vector from the received signal RJ, Barhumi uses the following equation:

ĥ^j=A⁺R^j  (14)

in which the symbol ⁺ signifies the pseudo-inverse. The pseudo-inverse operation is the following operation:

A ⁺=(A^HA)⁻¹A^H  (15)

To eliminate the interference terms caused by the data symbols present in the OFDM symbol, the product of A⁺ and T must be equal to a null matrix, which is made possible by imposing non-contiguous sets of pilot symbols and data symbols. Furthermore, calculating the impulse responses can be simplified by considering in the equations only the frequencies dedicated to channel estimation, i.e. by limiting the calculations to the pilot carriers:

ĥ^j=Ã⁺{tilde over (R)}^j  (16)

where {tilde over (R)}^j is the matrix with dimensions (N_p×g)×1 extracted from R^j considering only the pilot carriers; andà is the matrix with dimensions (N_p×g)×(N_t×Δt);

$\begin{array}{cc}\tilde{A}=\left[\begin{array}{ccc}{\tilde{B}}_{\mathrm{diag}}^1\left(0\right){\tilde{F}}^{\prime }& \dots & {\tilde{B}}_{\mathrm{diag}}^{N_t}\left(0\right){\tilde{F}}^{\prime }\\ ⋮& & ⋮\\ {\tilde{B}}_{\mathrm{diag}}^1\left(g-1\right){\tilde{F}}^{\prime }& \dots & {\tilde{B}}_{\mathrm{diag}}^{N_t}\left(g-1\right){\tilde{F}}^{\prime }\end{array}\right]& \left(17\right)\end{array}$

The estimated value ĥ^j is therefore a column vector containing the N_t impulse responses. To obtain separately the various frequency responses of the sub-channels necessary for the equalization phase, the method effects vector windowing of the estimated vector ĥ^j.

If there are null sub-carriers at the edges of the spectrum, the Barhumi solution, equations (14) and (16), leads to edge effects, i.e. to discontinuities at the edges of the spectrum, and this degrades performance.

A solution according to the invention effects an estimate of the impulse responses ĥ^j using the following equation instead of equation (16):

ĥ ^j=(Ã^HÃ)⁺Ã^H{tilde over (R)}^j  (18)

According to this equation, the estimate of the vector of the impulse responses h^j is equal to a product of matrices with the received signal {tilde over (R)}^j. The product of matrices is equal to the product of the pseudo-inverse matrix of the product of the Hermitian matrix of the à matrix with the à matrix with the Hermitian matrix of Ã.

The multiplication by the pseudo-inverse matrix of the matrices product Ã^H×à decorrelates the modulated carrier and null carrier regions.

The channel estimation module includes means for calculating the N_t impulse responses ĥ^j using equation (18), typically instructions of a program which in a preferred embodiment is incorporated in an electronic circuit, a calculation unit such as a processor or the like of operation that is then controlled by the execution of the program.

Knowing the parameters of the multi-antenna system (the size N_FFT of the FFT, the number N_mod of modulated carriers, the time shift Δt between reference sequences, the number g of pilot OFDM symbols forming a reference sequence, and the amplitude c(p) of the pilot symbols), an estimation method and module of the invention advantageously pre-calculate the product of matrices corresponding to the expression:

(Ã^HÃ)⁺Ã^H  (19)

In a preferred embodiment, the estimation method, respectively the estimation module, multiplies this pre-calculated product of matrices with the demodulated OFDM signal to obtain an estimate of the N_t channels accumulated in the expression for ĥ^j according to equation (18).

To obtain an estimate of the N_r×N_t channels, the calculation of ĥ^j must be repeated for j varying from 1 to N_r.

The vector expression for the channel ĥ^j,i linking the particular send antenna i, for i assuming a value from 1 to N_t, to the particular receive antenna j, for j assuming a value from 1 to Nr, is obtained by making a selection from the N_p successive samples of the column vector ĥ^j calculated for the receive antenna j, starting from the sample (i−1)×N_p.

In one particular embodiment of the method, the calculation of the N_t×N_r impulse responses previously described for the N_p pilot carriers is completed by an interpolation that can be linear in time or linear in frequency (one-dimensional (1D) interpolation) to obtain the coefficients of each channel for all of the carriers modulated with payload data. The interpolation can be of an order higher than one.

The following two examples illustrate the calculation of the expression for  and, for the first example, also the expression (19): (Ã^HÃ)⁺Ã^H.

FIRST EXAMPLE

The parameters of the system take the following values: N_FFT=8, N_mod=6, and Δt=2. The unmodulated sub-carriers are assumed to be situated at the two ends of the spectrum. It is assumed that only one OFDM symbol is necessary for estimating the channel (g=1), and the pattern of that OFDM symbol, which constitutes a reference sequence, is represented in Table 2. The first column represents the frequency-domain indices and the second the associated data type (N=null carrier, P=pilot symbol, D=payload data).

TABLE 2
Pattern of pilots distributed in the time-frequency plane
0 N
1 P
2 D
3 P
4 D
5 P
6 P
7 N

The transmission system comprises N₂=2 send antennas and N_r=1 receive antenna. Equation (2) gives the expression for the pilot symbol modulating the carrier p for the reference sequence sent by the send antenna TX_i, with i=1 or 2:

${c\left(p\right)}^1=c\left(p\right){}^{-j2\pi \frac{p\left(1-1\right)2}{4}}=c\left(p\right)$ ${c\left(p\right)}^2=c\left(p\right){}^{-j2\pi \frac{p2}{4}}=c\left(p\right){}^{-j\pi p}$

c(p)¹ is therefore the pilot symbol modulating the carrier frequency p for the reference sequence sent by the send antenna TX₁ and c(p)² is the pilot symbol modulating the carrier frequency p for the reference sequence sent by the send antenna TX₂.

In the example, c(p)=1 for all the pilot carriers. The vectors of the pilot symbols B¹ and B² and the extracted vectors limited to the pilot carriers {tilde over (B)}¹ and {tilde over (B)}² are then expressed in the form:

$B^1=\left[\begin{array}{c}0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 1\\ 0\end{array}\right]\Rightarrow {\tilde{B}}^1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]B^2=\left[\begin{array}{c}0\\ 1\\ 0\\ {}^{-j\pi }\\ 0\\ {}^{-j2\pi }\\ {}^{-j2\pi }\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 0\\ -1\\ 0\\ 1\\ -1\\ 0\end{array}\right]\Rightarrow {\tilde{B}}^2=\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right]$

The Fourier matrix F is as follows:

$\begin{array}{c}F=\\ \frac{1}{\sqrt{8}}\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& \begin{array}{c}0.707-\\ 0.707j\end{array}& -j& \begin{array}{c}-0.707-\\ 0.707j\end{array}& -1& \begin{array}{c}-0.707+\\ 0.707j\end{array}& j& \begin{array}{c}0.707+\\ 0.707j\end{array}\\ 1& -j& -1& j& 1& -j& -1& j\\ 1& \begin{array}{c}-0.707-\\ 0.707j\end{array}& j& \begin{array}{c}0.707-\\ 0.707j\end{array}& -1& \begin{array}{c}0.707+\\ 0.707j\end{array}& -j& \begin{array}{c}-0.707+\\ 0.707j\end{array}\\ 1& -1& 1& -1& 1& -1& 1& -1\\ 1& \begin{array}{c}-0.707+\\ 0.707j\end{array}& -j& \begin{array}{c}0.707+\\ 0.707j\end{array}& -1& \begin{array}{c}0.707-\\ 0.707j\end{array}& j& \begin{array}{c}-0.707-\\ 0.707j\end{array}\\ 1& j& -1& -j& 1& j& -1& -j\\ 1& \begin{array}{c}0.707+\\ 0.707j\end{array}& j& \begin{array}{c}-0.707+\\ 0.707j\end{array}& -1& \begin{array}{c}-0.707-\\ 0.707j\end{array}& -j& \begin{array}{c}0.707-\\ 0.707j\end{array}\end{array}\right]\end{array}$

in which the symbol j represents the square root of minus one, i.e. the solution of the equation j²=−1. The matrices F′ and {tilde over (F)}′ are expressed in the following form:

$\begin{array}{c}{F}^{\prime }=\frac{1}{\sqrt{8}}\left[\begin{array}{cc}1& 1\\ 1& 0.707-0.707j\\ 1& -j\\ 1& -0.707-0.707j\\ 1& -1\\ 1& -0.707+0.707j\\ 1& j\\ 1& 0.707+0.707j\end{array}\right]\Rightarrow \\ {\tilde{F}}^{\prime }=\frac{1}{\sqrt{8}}\left[\begin{array}{cc}1& 0.707-0.707j\\ 1& -0.707-0.707j\\ 1& -0.707+0.707j\\ 1& 0.707+0.707j\end{array}\right]\end{array}$

The matrix à is then expressed in the following form:

$\tilde{A}=\frac{1}{\sqrt{8}}\left[\begin{array}{cccc}1& 0.707-0.707j& 1& 0.707-0.707j\\ 1& -0.707-0.707j& -1& 0.707+0.707j\\ 1& -0.707+0.707j& 1& -0.707+0.707j\\ 1& 0.707+0.707j& -1& -0.707+0.707j\end{array}\right]$

whence the following expressions for Ã^H, (Ã^HÃ)⁺ and (Ã^HÃ)⁺Ã^H:

$\begin{array}{c}{\tilde{A}}^H=\frac{1}{\sqrt{8}}\left[\begin{array}{cccc}1& 1& 1& 1\\ \begin{array}{c}0.707+\\ 0.707j\end{array}& \begin{array}{c}-0.707+\\ 0.707j\end{array}& \begin{array}{c}-0.707-\\ 0.707j\end{array}& \begin{array}{c}0.707-\\ 0.707j\end{array}\\ 1& -1& 1& -1\\ \begin{array}{c}0.707+\\ 0.707j\end{array}& \begin{array}{c}0.707-\\ 0.707j\end{array}& \begin{array}{c}-0.707-\\ 0.707j\end{array}& \begin{array}{c}-0.707+\\ 0.707j\end{array}\end{array}\right]\\ {\left({\tilde{A}}^H\tilde{A}\right)}^{+}=\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right]\\ {\left({\tilde{A}}^H\tilde{A}\right)}^{+}{\tilde{A}}^H=\left[\begin{array}{cccc}0.707& 0.707& 0.707& 0.707\\ 0.5+0.5j& -0.5+0.5j& -0.5-0.5j& 0.5-0.5j\\ 0.707& -0.707& 0.707& -0.707\\ 0.5+0.5j& 0.5-0.5j& -0.5-0.5j& -0.5+0.5j\end{array}\right]\end{array}$

SECOND EXAMPLE

The parameters of the system take the following values: N_FFT=8, N_mod=6, and Δ_t=2. The transmission system comprises N_t=3 send antennas and N_r=1 receive antenna. The number of pilot carriers N_p=4 is now insufficient because the condition N_t×Δ_t≦N_p is not satisfied. It is therefore necessary for this transmission system to have at the minimum N_p=6, for example N_p=N_mod=6. The unmodulated sub-carriers are assumed to be situated at the two ends of the spectrum. It is assumed that only one OFDM symbol is needed to estimate the channel (g=1), and the pattern of that OFDM symbol, which constitutes a reference sequence, is represented in Table 3. The first column represents the frequency-domain indices and the second column the associated data type (N=null carrier, P=pilot symbol).

TABLE 3
Pattern of pilots distributed in the time-frequency plane
0 N
1 P
2 P
3 P
4 P
5 P
6 P
7 N

Equation (2) gives the expression for the pilot symbol modulating the carrier p for the reference sequence sent by the send antenna TX_i, with i=1, 2 or 3:

${c\left(p\right)}^1=c\left(p\right){}^{-j2\pi \frac{p\left(1-1\right)2}{4}}=c\left(p\right)$ ${c\left(p\right)}^2=c\left(p\right){}^{-j2\pi \frac{p2}{6}}=c\left(p\right){}^{-jp\frac{2\pi }{3}}$ ${c\left(p\right)}^3=c\left(p\right){}^{=j2\pi \frac{p\times 2\times 2}{6}}=c\left(p\right){}^{-jp\frac{4\pi }{3}}$

c(p)¹ is therefore the pilot symbol modulating the carrier frequency p for the reference sequence sent by the send antenna TX₁, c(p)² is the pilot symbol modulating the carrier frequency p for the reference sequence sent by the send antenna TX₂, and C(p)³ is the pilot symbol modulating the carrier frequency p for the reference sequence sent by the send antenna TX₃.

In the example, c(p)=1 for all the pilot carriers. The vectors of the pilot symbols B¹, B² and B³ and the extracted vectors limited to the pilot carriers, {tilde over (B)}¹, {tilde over (B)}² and {tilde over (B)}³ are then expressed in the form:

$B^1=\left[\begin{array}{c}0\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 0\end{array}\right]\Rightarrow {\tilde{B}}^1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\\ 1\end{array}\right]B^2=\left[\begin{array}{c}0\\ 1\\ {}^{-j\frac{2\pi }{3}}\\ {}^{-j\frac{4\pi }{3}}\\ {}^{-j\frac{6\pi }{3}}\\ {}^{-j\frac{8\pi }{3}}\\ {}^{-j\frac{10\pi }{3}}\\ 0\end{array}\right]\Rightarrow {\tilde{B}}^2=\left[\begin{array}{c}1\\ {}^{-j\frac{2\pi }{3}}\\ {}^{-j\frac{4\pi }{3}}\\ 1\\ {}^{-j\frac{8\pi }{3}}\\ {}^{-j\frac{10\pi }{3}}\end{array}\right]$ $B^3=\left[\begin{array}{c}0\\ 1\\ {}^{-j\frac{4\pi }{3}}\\ {}^{-j\frac{8\pi }{3}}\\ {}^{-j\frac{12\pi }{3}}\\ {}^{-j\frac{16\pi }{3}}\\ {}^{-j\frac{20\pi }{3}}\\ 0\end{array}\right]\Rightarrow {\tilde{B}}^3=\left[\begin{array}{c}1\\ {}^{-j\frac{4\pi }{3}}\\ {}^{-j\frac{8\pi }{3}}\\ 1\\ {}^{-j\frac{4\pi }{3}}\\ {}^{-j\frac{2\pi }{3}}\end{array}\right]$

The Fourier matrix F is as follows:

$\begin{array}{c}F=\\ \frac{1}{\sqrt{8}}\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& \begin{array}{c}0.707-\\ 0.707j\end{array}& -j& \begin{array}{c}-0.707-\\ 0.707j\end{array}& -1& \begin{array}{c}-0.707+\\ 0.707j\end{array}& j& \begin{array}{c}0.707+\\ 0.707j\end{array}\\ 1& -j& -1& j& 1& -j& -1& j\\ 1& \begin{array}{c}-0.707-\\ 0.707j\end{array}& j& \begin{array}{c}0.707-\\ 0.707j\end{array}& -1& \begin{array}{c}0.707+\\ 0.707j\end{array}& -j& \begin{array}{c}-0.707+\\ 0.707j\end{array}\\ 1& -1& 1& -1& 1& -1& 1& -1\\ 1& \begin{array}{c}-0.707+\\ 0.707j\end{array}& -j& \begin{array}{c}0.707+\\ 0.707j\end{array}& -1& \begin{array}{c}0.707-\\ 0.707j\end{array}& j& \begin{array}{c}-0.707-\\ 0.707j\end{array}\\ 1& j& -1& -j& 1& j& -1& -j\\ 1& \begin{array}{c}0.707+\\ 0.707j\end{array}& j& \begin{array}{c}-0.707+\\ 0.707j\end{array}& -1& \begin{array}{c}-0.707-\\ 0.707j\end{array}& -j& \begin{array}{c}0.707-\\ 0.707j\end{array}\end{array}\right]\end{array}$

in which the symbol j represents the square root of minus one, i.e. the solution of the equation j²=−1. The matrices F′ and {tilde over (F)}′ are expressed in the following form:

$\begin{array}{c}{F}^{\prime }=\frac{1}{\sqrt{8}}\\ \left[\begin{array}{cc}1& 1\\ 1& 0.707-0.707j\\ 1& -j\\ 1& -0.707-0.707j\\ 1& -1\\ 1& -0.707+0.707j\\ 1& j\\ 1& 0.707+0.707j\end{array}\right]\Rightarrow {\tilde{F}}^{\prime }=\frac{1}{\sqrt{8}}\left[\begin{array}{cc}1& 0.707-0.707j\\ 1& -j\\ 1& -0.707-0.707j\\ 1& -1\\ 1& -0.707+0.707j\\ 1& j\end{array}\right]\end{array}$

The matrix à is then expressed in the following form:

$\tilde{A}=\frac{1}{\sqrt{8}}\left[\begin{array}{ccc}{\tilde{B}}_{\mathrm{diag}}^1{\tilde{F}}^{\prime }& {\tilde{B}}_{\mathrm{diag}}^2{\tilde{F}}^{\prime }& {\tilde{B}}_{\mathrm{diag}}^3{\tilde{F}}^{\prime }\end{array}\right]$

FIGS. 4, 5a, 5b and 5c set out simulation results comparing the performance of the known Barhumi technique and the performance of a method of the invention. The simulations relate to a MIMO OFDM system considering a time-selective and frequency-selective BRAN channel E adapted to an external environment type MIMO context, having the characteristics as set out in Table 4 in Appendix B. The frame considered is that described in the Barhumi paper; the set of modulated carriers is divided into two non-contiguous sub-sets: a set of pilot carriers and a set of payload data.

FIG. 4 is a plot of the bit error rate (BER) for 16QAM modulation with a time shift Δt equal to 128. The channel estimation effected at the pilot frequencies is completed by linear interpolation in the time domain to estimate the set of coefficients for the various modulated frequencies. The curve 1, C1, corresponds to a perfect estimate, the curve 2, C2, corresponds to a Barhumi estimate, and the curve 3, C3, corresponds to an estimate in accordance with the invention. Comparing the curves shows that the method of the invention has the advantage of a lower bit error rate than the Barhumi method of the same signal-to-noise ratio and, furthermore, that this difference increases in proportion to the increasing signal-to-noise ratio.

FIG. 5 a is a plot of the real part of the coefficient of the channel as a function of the index of the carrier, which varies from 0 to 1023, knowing that the Fourier transform (FFT) has a size of 1024 samples and that the ratio Eb/N0 is equal to 20 dB. FIGS. 5b and 5c zoom in on the FIG. 5a trace around the null carrier indices, respectively at the lower edge of the spectrum (155 to 190) and at the upper edge of the spectrum (835 to 870), knowing that the null carriers correspond to the indices 0 to 159 and 864 to 1023. The curve 1, C4, corresponds to the true values of the coefficients of the channel, the curve 2, C5, corresponds to a Barhumi estimate, and the curve 3, C6, corresponds to an estimate in accordance with the invention. The curve C2 shows significant edge effects near the null carriers at the edges of the spectrum, as previously indicated in the description of the prior art, around the carrier indices 160 and 863. These effects are not observed with a method according to the invention.

APPENDIX A

TABLE 1
Type de frame used	[Teng]	[Moon], [Larsson], [Baek]	[Barhumi]
Size of FFT window	1024
Number of OFDM	32
symbols in a frame
Modulated carriers	704
Payload data	75%	84.4%	88.6%
Pilot symbols	18.75%	9.4%	5.1%
Other (guard, synch)	6.25%
Modulated data	22528

APPENDIX B

TABLE 4
Nt                               2
Nr                               2
Carrier frequency                5200 MHz
Sampling frequency               50 MHz
Space between carriers           48.828125 kHz
Cyclic prefix ratio              0.211
OFDM symbol time                 24.8 μs
Length of convolutional code     7
Efficiency of code               ½
Channel decoding algorithm       SOVA
FFT size                         1024
Number of OFDM symbols per frame 32
Modulated carriers               704
Null carriers                    320
Payload data                     19968
Pilot symbols                    1152
Other (Synch, Guard)             1408
Data                             22528

Claims

1. A method of estimating transmission channels in a multi-antenna system including Nt send antennas, where Nt is greater than or equal to 2, and at least one receive antenna, wherein a time-frequency frame for each send antenna comprises pilot symbols forming a training sequence and data symbols of a payload signal, the Nt training sequences, time-shifted relative to each other, being known to the receiver and enabling the receiver to estimate Nt impulse responses corresponding to the Nt transmission channels between a send antenna and the respective receive antenna RXj, the data symbols and the pilot symbols being frequency-modulated by an orthogonal multiplexer to form orthogonal symbols that are sent by the send antennas in the form of a multi-carrier signal with NFFT carriers including Np pilot carriers and null carriers, the method comprising: a step of calculating a matrix A constructed in the form of blocks from training sequences and from the Fourier matrix with dimensions NFFT×NFFT;wherein for a particular receive antenna RXj, the method comprises a step of calculating the Nt impulse responses in the time domain by multiplying Np pilot symbols extracted from a frequency-domain signal Rj(n) obtained after demodulation of a time-domain signal received by the particular receive antenna RXj by means of an FFT of size NFFT by a product of matrices comprising the pseudo-inverse matrix of the product of the Hermitian matrix of the A matrix with the A matrix enabling decorrelation of modulated carriers adjacent null carriers.

2. The method according to claim 1, wherein the calculation of the Nt impulse responses is limited to the Np pilot carriers.

3. The method according to claim 2, wherein the matrices product is expressed as follows: (AHA)+AH where AH is the Hermitian matrix of the matrix A and + designates the pseudo-inverse matrix.

4. The method according to claim 2, wherein the calculation of the Nt impulse responses is extended to carriers modulated by data by means of interpolation.

5. The method according to claim 1, wherein the step of calculating the Nt impulse responses is repeated for each receive antenna concerned of the multi-antenna system.

6. A module for estimating transmission channels in a multi-antenna system using Nt send antennas, where Nt is greater than or equal to 2, and at least one receive antenna, wherein a time-frequency frame for each send antenna comprises pilot symbols forming a training sequence and data symbols of a payload signal, the Nt training sequences, time-shifted relative to each other, being known to the receiver and enabling the receiver to estimate Nt impulse responses corresponding to the Nt transmission channels between a send antenna and a respective receive antenna RXj, the data symbols and the pilot symbols being frequency-modulated by an orthogonal multiplexer to form orthogonal symbols that are sent by the send antennas in the form of a multi-carrier signal with NFFT carriers including Np pilot carriers and null carriers, wherein the nodule comprises: means for calculating a matrix A constructed in the form of blocks from training sequences and from a Fourier matrix with dimensions NFFT×NFFT; andwherein for a receive antenna RXj concerned, the means for calculating Matrix A comprises means for calculating the Nt impulse responses in the time domain by multiplying Np pilot symbols extracted from a frequency-domain signal Rj(n) obtained after demodulation of a time-domain signal received by the receive antenna RXj concerned by means of an FFT of size NFFT by a product of matrices comprising the pseudo-inverse matrix of the product of the Hermitian matrix of the A matrix with the A matrix enabling decorrelation of modulated carriers adjacent null carriers.

7. A receiver for a multi-antenna system including a transmission channel estimation module according to claim 6.

8. A multi-antenna system including a receiver including a transmission channel estimation module according to claim 6.

9. A computer program on an information medium, including program instructions adapted to implement a channel estimate method according to claim 1, with said program being loaded into and executed in a receiver.

10. An information medium containing program instructions adapted to implement a channel estimation method according to claim 1, with said program being loaded into and executed in a receiver.