With orthogonal MC-CDMA, each data symbol is simultaneously
transmitted at N binary phase shift-keying (BPSK) narrowband subcarriers,
each separated by F/Tb Hz where F is an integer.Due to its special signal
structure, MC-CDMA signals will not experience significant linear distortion
in fading channels where the symbol duration, 
, is much larger than the delay spread, 
. As shown in the transmitter model of Fig. 1, each of the N subcarrier
waveforms is modulated (multiplied) by a single (time constant) chip belonging
to a spreading code of length N. Different users transmit at the same set
of subcarriers but with a different spreading code in the frequency domain.
In contrast to Direct-Sequence CDMA.
(or other wideband) signals, MC-CDMA signals are not significantly affected
by delay spreads due to their narrowband composition. This modulation technique
should not be confused with transmitting multiple DS-CDMA signals at different
frequencies. Note that the signal structure of MC-CDMA is similar to that
of Orthogonal Frequency Division Multiplexing (OFDM). but the manner in which the signals are
used is very different.
While the general shape of each individual subcarrier will not experience significant linear distortion after transmission, the flat fading that each subcarrier with experience results in an amplitude scaling. Unequal scaling a different subcarriers distorts the orthogonality between users. While the conventional detection methods of equal gain combining (EGC) and maximal ratio combining (MRC) may be sufficient in an additive white Gaussian noise (AWGN) channel, these detection methods do not directly address the issues of orthogonality and of interference cancellation. Consequently, these detection methods do not perform as well in interference limited channels.
In this paper, we will apply Wiener filtering to the detection of MC-CDMA signals. Wiener filtering is optimal in a mean-squared error sense with respect to both the noise and the interference. In most fading channels, the determination of the Wiener coefficients is performed adaptively. In this paper, we are primarily concerned with the theoretical limits and not the implementation aspects of how to track the channel fluctuations. Thus, it is assumed that accurate estimates of the complete channel state information (i.e., the fading at the subcarriers) are available and that the Wiener coefficients are chosen accordingly.
corresponds to the data symbol of the mth user. If user m is inactive, then 
. Define the code matrix C to
be
, of the mth user.
Using the vector and matrix notation described above, an equivalent discrete vector
representation of the transmitted signal described in Fig. 1 is
is the carrier frequency and 
is a unit pulse that is non-zero for 
and zero otherwise. Note that in a distortionless
channel, that data symbol of the mth user can be recovered using the orthogonality of the
codes as indicated by the following operation
, over the symbol duration.
Using this assumption, the index k will be suppressed throughout the rest of this paper.
Applying the received signal to the receiver model shown in Fig. 2, the equivalent
discrete representation of the received signal is
represents the component of the received signal at the ith subcarrier, the
channel matrix H is defined to be
,
is a vector containing the corresponding AWGN terms with 
representing
the noise term at the ith subcarrier with power 
. Obviously, the discrete representation of the demodulated signal
is an approximation. However, under the condition 
, each subcarrier faces a relatively flat channel.
represents the optimal weighting coefficients. These coefficients
can also be viewed as an amplitude equalization to compensate for the fading at the
subcarriers. Arbitrarily choosing m = 0 as the desired signal, the optimal choice of the
equalization vector D in the mean-squared sense can be determined to be
represents the
cross-correlation vector between the desired symbol, a0, and the received signal vector,
Y. In the formulation of this method, it is assumed that estimates of the channel
amplitudes, 
, are
available and are consequently treated as deterministic constants.
.
, the equalization coefficient should be small to
avoid excessive amplification of the noise. For large 
, the correction factor should be proportional to
taking the inverse of the channel, 
, in order to restore the orthogonality between users.
and 
. 
radians in a complex gaussian fading channel is given in
.
and 
is the local-mean power of the mth user. Defining 
as a vector of dimension N whose elements, hi for i =
0, 1, ..., N-1, are the complex correlated gaussian fading variables at
the different subcarriers, the autocorrelation matrix of 
, 
where 
denotes the
conjugate transpose of 
, can be determined. The vector 
can be generated
by taking linear combinations of independent, idenctically distributed (i.i.d.)
unit normal gaussian random variables, using
transformation matrix A is 
matrix with the eigenvectors of 
as its column vectors and 
is a diagonal matrix with the
eigenvalues of 
as its
entries. The relationship between the fading variables of
is
.
.
.
appeared reasonably accurate for SNRs up to 12dB and for N = 64 subcarriers.
As the number of subcarriers is increased, the accuracy of the analytical
results improves.. It was also shown that most of the benefits of frequency
diversity can be achieved even in a channel with a significant degree of correlated
fading at the subcarriers.