Several types of OFDM multi-carrier modulation are known at this time.
Of these, the nearest to a standard modulation technique includes a particularly simple equalization system based on inserting a guard time. This guard time, also called a cyclic prefix, provides good performance in the face of echoes at the price of a loss of spectral efficiency.
As a matter of fact, to guarantee that all information received comes from the same multi-carrier symbol, no payload information is transmitted during the guard time. This is an effective way to combat echo phenomena caused both by the Doppler effect and by multiple propagation paths resulting in intersymbol interference (ISI) in the receiver.
In a system with a plurality of transmit antennas, the interference observed in the receiver originates from the presence of a plurality of signals transmitted simultaneously by the various transmit antennas, which causes what is called interantenna interference or spatial interference.
Transmit/receive systems comprising multiple antennas are already well known. One such system is represented diagrammatically in FIG. 1. The system includes a transmitter EM, at least two transmit antennas TX1, TX2, one or more receive antennas RX1, RX2, and a receiver RE. The signals radiated by a transmit antenna are transmitted to the receive antennas via the propagation channel CT. By way of simplification, the propagation channel often includes the transmit and receive antennas. Some of these systems employ spatial coding, space-time coding or space-frequency coding associated with multi-carrier modulation, in particular OFDM modulation, making it possible to exploit the space-time diversity of these systems.
The first systems proposed all employed orthogonal space-time block codes. Alamouti, in “A Simple Transmit Diversity Technique for Wireless Communications”, IEEE J. Sel. Areas Comm., 1998, 16, (8), pp. 1451-1458, describes the first system employing, for two transmit antennas, an orthogonal space-time block code with an efficiency of 1 (where the efficiency is defined as the ratio between the number N of different data symbols transmitted and the number L of multi-carrier symbol times during which they are transmitted). The term data symbol commonly refers to the output of a converter module that formats the information to be transmitted according to a given constellation (QPSK, QAM, etc.). A multi-carrier symbol represents a multiplex produced by a multi-carrier multiplexer, commonly referred to as an OFDM multiplex, which entails distributing the data symbols between the sub-carriers of the multiplex and summing the data symbols weighted by Fourier coefficients. This weighted summation is effected by means of an inverse Fourier transform. It is routine for some sub-carriers to be reserved for particular symbols such as pilot symbols. In the remainder of this document the term symbol refers to various types of symbols and in particular to data symbols and pilot symbols.
With OFDM modulation, it is commonly assumed that the channel on each sub-carrier is flat. Ignoring the introduction of noise, this amounts to assuming that the effect of the propagation channel is reflected in each symbol transmitted on a sub-carrier, with a given OFDM symbol time, by multiplication by a single complex coefficient.
FIG. 2 illustrates space-time coding in accordance with an Alamouti scheme for two transmit antennas TX1, TX2 and one receive antenna RX1. In this situation, Q=2 independent symbols S1, S2 are coded during a time period equal to two OFDM symbol times.
The principle of what is proposed by S. Alamouti is to create a coding scheme ensuring decoupling of the transmitted symbols on reception and thus enabling linear maximum likelihood decoding. The coding orthogonal pattern is expressed by the following matrix C:
                    C        =                  (                                                                      s                  1                                                                              s                  2                                                                                                      -                                      s                    2                    *                                                                                                s                  1                  *                                                              )                                    (        1        )            
Referring to FIG. 2, and assuming that the channel does not vary over at least one OFDM symbol time, the equations at the input of the receiver are:at time t1: y1=h1s1+h2s2+n1  (2)at time t2: y2=−h1s*2+h2s*1+n2  (3)where t2=t1+T, T being the OFDM symbol time, n1, n2 is the noise introduced by the propagation channel CT at times t1 and t2, respectively, and h1, h2 are the respective coefficients of the propagation sub-channels from the transmit antennas TX1 and TX2 to the receive antenna RX1, assumed to be constant over at least the OFDM symbol time T.
These equations may be expressed in matrix form:
                                          (                                                                                y                    1                                                                                                                    y                    2                                                                        )                    =                                                                      (                                                                                                              s                          1                                                                                                                      s                          2                                                                                                                                                              -                                                      s                            2                            *                                                                                                                                                s                          1                          *                                                                                                      )                                ⁢                                  (                                                                                                              h                          1                                                                                                                                                              h                          2                                                                                                      )                                            +                              (                                                                                                    n                        1                                                                                                                                                n                        2                                                                                            )                                      =                                          C                ⁢                                  h                  →                                            +                              b                →                                                    ⁢                                  ⁢                  with          ⁢                      :                          ⁢                                  ⁢                                            h              →                        =                          (                                                                                          h                      1                                                                                                                                  h                      2                                                                                  )                                ,                                    and              ⁢                                                          ⁢                              b                →                                      =                          (                                                                                          n                      1                                                                                                                                  n                      2                                                                                  )                                                          (        4        )            
The matrix of the code C satisfies the following orthogonality condition, in which I2 is the identity matrix of dimension two:
                                          C            H                    ⁢          C                =                              ∑                          i              =              1                                      Q              =              2                                ⁢                                          ⁢                                                                  s                i                2                                                    ⁢                          I              2                                                          (        5        )            
On reception, the received signals y1 and y2 can be expressed in the following form, reflecting the effect of the channel on the transmission of the signals according to an Alamouti scheme, and assuming that the channel is constant over at least one OFDM symbol time:
                                          y            →                    =                                    (                                                                                          y                      1                                                                                                                                  y                      2                      *                                                                                  )                        =                                                                                (                                                                                                                        h                            1                                                                                                                                h                            2                                                                                                                                                                            -                                                          h                              2                              *                                                                                                                                                            h                            1                            *                                                                                                                )                                    ⁢                                      (                                                                                                                        s                            1                                                                                                                                                                            s                            2                                                                                                                )                                                  +                                  (                                                                                                              n                          1                                                                                                                                                              n                          2                          *                                                                                                      )                                            =                                                H                  ⁢                                      s                    →                                                  +                                  n                  →                                                                    ⁢                                  ⁢                  with          ⁢                      :                          ⁢                                  ⁢                              H            =                          (                                                                                          h                      1                                                                                                  h                      2                                                                                                                                  -                                              h                        2                        *                                                                                                                        h                      1                      *                                                                                  )                                ,                                    s              →                        =                          (                                                                                          s                      1                                                                                                                                  s                      2                                                                                  )                                ,                                    and              ⁢                                                          ⁢                              n                →                                      =                          (                                                                                          n                      1                                                                                                                                  n                      2                      *                                                                                  )                                                          (        6        )            where hi are the coefficients of the propagation channel and * is the “conjugate” operator.
The equations (6) satisfy the following equation:HHH=∥h∥2I2=(|h1|2+|h2|2)I2  (7)where H is the hermitian operator.
Decoding simply consists in multiplying the received equivalent vector {right arrow over (y)} by the matrix HH. The output vector is written:{tilde over ({right arrow over (y)}=HH{right arrow over (y)}=HH(H{right arrow over (s)}+{right arrow over (n)})=∥h∥2{right arrow over (s)}+{tilde over ({right arrow over (n)}  (8)
The symbols s1 and s2 can therefore be decoded using two threshold detectors that satisfy the optimum maximum likelihood (ML) decision criterion.
Orthogonal Frequency Division Multiplex/Offset
Quadrature Amplitude Modulation (OFDM/OQAM) is an alternative to standard OFDM modulation and was introduced to avoid the loss of spectral efficiency caused by the introduction of a guard time.
To be more precise, OFDM/OQAM modulation does not require a guard time (cyclic prefix) by means of a judicious choice of the prototype function for modulating each of the signal carriers, which makes it possible to locate each of the carriers accurately in the time-frequency space. One example of a prototype function is the Iota function, described in particular in patent application FR 2 733 869, which has the feature of being identical to its Fourier transform.
To combat interference, the standard approach is to apply the Alamouti scheme to OFDM/OQAM modulation. With this type of modulation, the symbols am,n transmitted on a carrier m at a time n are real symbols and the coding matrix may be expressed in the form:
                    C        =                  (                                                                      a                                      m                    ,                    n                                                                                                a                                      m                    ,                                          n                      +                      1                                                                                                                                            -                                      a                                          m                      ,                                              n                        +                        1                                                                                                                                          a                                      m                    ,                    n                                                                                )                                    (        9        )            
The above formula may be concisely expressed in the form:
                    C        =                  (                                                                      a                  1                                                                              a                  2                                                                                                      -                                      a                    2                                                                                                a                  1                                                              )                                    10        )            
With OFDM/OQAM modulation, the receiver receives the symbols ai phase-shifted by the channel coefficients, to which is added the intrinsic interference Ii caused by the real orthogonality. Even assuming a perfect channel estimate, it is therefore a priori impossible to recover the transmitted symbols using a single receive antenna. Ignoring noise, the symbols received after modulation on a carrier m at respective times t′1 and t′2 where t′2=t′1+T may be expressed in the following form:r1=h1a1+I1(a1)+h2a2+I1(a2)  (11)r2=h2a1+I2(a1)−h1a2+I2(a2)  (12)where I1(ai) is the intrinsic interference affecting the symbol ai at time t′1, which depends on the adjoining symbols on each side of the symbol ai at time t′1, and where I2(ai) is the intrinsic interference affecting the symbol ai at time t′2, which depends on the adjoining symbols on each side of the symbol ai at time t′2. The interference Ii includes interference caused by the simultaneous transmission of signals by a plurality of antennas and therefore includes spatial interference. Despite the orthogonality (or quasi-orthogonality) that exists between a symbol and its intrinsic interference, and despite a knowledge of the channel that is assumed to be perfect, it is not possible to solve this system of equations using the same approach as for standard OFDM. This system comprises two equations in six unknowns, the two transmitted symbols a1, a2 and the four interference terms I1(a1), I1(a2), I2(a1), I2(a2).
In contrast to standard OFDM transmission using an Alamouti scheme, OFDM/OQAM transmission using real space-time coding generates on reception, by construction, interference relative to a transmission time that makes it impossible to process interference between two successive transmission times. As a matter of fact, assuming that I1(a1) and I1(a2) are respectively equal to I2(a1) and I2(a2) cannot be envisaged because:                the intrinsic interference I1(ai)(i=1,2) depends on the adjacent symbols (in the time-frequency plane) on each side of the symbol ai at time t′1.        the intrinsic interference I2(ai)(i=1,2) depends on the adjacent symbols (in the time-frequency plane) on each side the symbol ai at time t′2; and        the neighbors of the symbol ai at time t′1 are different from the neighbors of the symbol ai at time t′2.        
Consequently, for i=1,2, I1(ai)≠I2(ai).
Furthermore, simulation has verified that the variance of the intrinsic interference is equal to the variance of the real symbols. Considering all this interference as noise, the overall noise level is increased enormously, and it is not possible to obtain an acceptable bit error rate Teb.
Consequently, with OFDM/OQAM modulation, it is not possible to use an Alamouti scheme in the transmitter to combat interference in the receiver.