Joint Transmitter and Receiver Optimization for Improper-Complex Second-Order Stationary Data Sequence

In this paper, the transmission of an improper-complex
second-order stationary data sequence is considered over a strictly
band-limited frequency-selective channel. It is assumed that the
transmitter employs linear modulation and that the channel output
is corrupted by additive proper-complex cyclostationary noise.
Under the average transmit power constraint, the problem of minimizing
the mean-squared error at the output of a widely linear
receiver is formulated in the time domain to find the optimal transmit
and receive waveforms. The optimization problem is converted
into a frequency-domain problem by using the vectorized Fourier
transform technique and put into the form of a double minimization.
First, the widely linear receiver is optimized that requires, unlike
the linear receiver design with only one waveform, the design
of two receive waveforms. Then, the optimal transmit waveform
for the linear modulator is derived by introducing the notion of the
impropriety frequency function of a discrete-time random process
and by performing a line search combined with an iterative algorithm.
The optimal solution shows that both the periodic spectral
correlation due to the cyclostationarity and the symmetric spectral
correlation about the origin due to the impropriety are well
exploited.