Signal Processing for Wireless Transceivers - Signal Processing Systems - page 172

SM can apply different so called layering solutions. A layer refers to a coded
data stream which can be multiplexed to transmit antennas using different schemes.
In horizontal layering, each stream is transmitted from different antenna, which
makes the spatial separation of the streams somewhat more straightforward. Vertical
layering multiplexes each stream to all transmit antennas, which enables achieving
spatial diversity amongst encoded bits, but complicates the receiver processing.
In the forthcoming discussion on the receiver design in Sects. 2.2 - 2.4 , we assume
for the simplicity of notation that P N identity matrix), i.e.,

no precoding without loss of generality. If precoding is applied, we just need to
replace H by HP in the discussion below.

= I N (where I N is a N ×

2.2 Optimum Detector and Decoding

The ultimate target of the receiver processing is to reproduce the true transmitted
information bit sequence at the FEC decoder output. This is of course usually
not perfectly possible, because of the random noise, fading, interference and other
sources of distortion in the radio channel and in the communication equipment.
Therefore, a pragmatic optimum receiver would minimize the probability of decod-
ing errors given the received observation y in ( 1 ) . Such an approach would lead
to jointly optimum decoding, demodulation and equalization, which is practically
too complex to be realized [ 83 ] . This is the reason, why practical receivers are
partitioned as shown in Figs. 1 b and 2 . Therein the equalizer and demodulator
process the received signal y to provide an estimate of the coded bit sequence b
in a form applicable for the FEC decoder, which then provides the final estimate of
the information bit sequence.
If there were no FEC coding, the optimum detector would simply make a hard
decision by finding the most likely transmitted data symbol vector x given the
observed received signal y ,or x MAP = denotes

the conditional probability density (or mass) function (PDF) (depending on the
context). We also assume herein that the channel matrix H is perfectly known. In
the receiver context p

arg min x ∈ Ω N p ( x | y ) , where p ( x | y )

is usually called as the a posteriori probability (APP),
and the optimum detector is the maximum APP (MAP) receiver, which minimizes
the average probability of symbol sequence decision error; the same principle
has also been called maximum likelihood sequence estimation (MLSE) in the ISI
channel context [ 95 ] . By Bayes rule p

( x | y )

) .
Thus, if there is no a priori information or all the possible modulation symbols
are equally likely, the maximization in the MAP sequence detector reduces to the
maximum likelihood (ML) sequence detector x ML =

( x | y )= p ( x , y ) / p ( y )= p ( y , x ) p ( x ) / p ( y

arg min x ∈ Ω N p ( y | x ) . In the

Gaussian channel with known channel realization, p is the Gaussian PDF

the ML detection reduces to finding the constellation points with the minimum
Euclidean distance (ED) to the received signal vector y ,or

( y | x )

2

x ML = arg min

x

N ||

y − Hx || . (2)

∈ Ω