Introduction - Signal Processing and Optimization for Transceiver Systems - page 7

q ( n )
d

F ( z )
d

s ( n ) H ( z ) + G ( z )

d

s ( n )

d

precoder channel equalizer

Figure 1.6 . An all-discrete equivalent of the digital communication system.

Here H d ( z ) is the transfer function of an equivalent discrete-time channel. It
is the z -transform of an equivalent digital channel impulse response h d ( n ), that
is

∞

h d ( n ) z −n . H d ( z )= (1 . 13)

n = −∞

Similarly, F d ( z )and G d ( z ) are the transfer functions of the discrete-time pre-
coder and equalizer. The subscript d (for “discrete”), which is just for clarity,
is usually dropped. In practice H d ( z ) is causal and can be approximated by a
finite impulse response, or FIR ,systemsothat

L

h d ( n ) z −n . H d ( z )= (1 . 14)

n =0

The problem of optimizing the precoder F d ( z ) and equalizer G d ( z )forfixed
channel H d ( z ) and fixed noise statistics will be addressed in later chapters.

1.4 MIMO channels

The transceivers described so far have one input signal s ( n ) and a corresponding
output

s ( n ) . These are called single-input single-output ,or SISO , transceivers.
An important communication system that comes up frequently in this topic is
the multi-input multi-output ,or MIMO , channel. Figure 1.7 shows a MIMO
channel assumed to be linear and time-invariant with a transfer function matrix
H ( z ), usually an FIR system:

L

h ( n ) z −n .

H ( z )= (1 . 15)

n =0

The sequence h ( n ) , called the MIMO impulse response, is a sequence of matrices.
If the channel has P inputs and J outputs then H ( z ) has size J

P, and so
does each of the matrices h ( n ) . The MIMO communication channel is used to
transmit a vector signal s ( n )with M components:

×

s M− 1 ( n )] T .

s ( n )=[ s 0 ( n ) s 1 ( n ) ... (1 . 16)