mohatta2015.pdf

54 MATCHED FILTERING
where the elements of H can be determined from the system model. Note that the
elements in u may be correlated. Similar to (2.52), the MLSD solution is given by
s = arg. max [z - HAq] H C" 1 [z - HAq], (2.101)
where S G denotes the set of all possible symbol vectors s.
2.5 AN EXAMPLE
Consider the Long Term Evolution (LTE) cellular system [Dah08j. This is an
evolution of a 3G system, providing higher data rates. On the downlink (base
station to mobile device), OFDM is used. Let's consider a single transmit antenna
and single receive antenna. We will assume a front-end filter (partially) matched to
p(i), which is approximately the same as matching to p(t)a(t). Let's also assume
that the delay spread is less than or equal to the length of the cyclic prefix.
These assumptions give us the classic OFDM receiver scenario. In [Wei71] it
is shown that for a particular symbol period, by discarding the portions of the
received signal corresponding to the cyclic prefix of the earliest arriving path, ISI
from symbols within the same symbol period as well as symbols form other symbol
periods is avoided. As a result, MF makes sense and the matched filter decision
variables can be generated using a Discrete Fourier Transform (DFT), which can
be implemented efficiently using a Fast Fourier Transform (FFT).
2.6 THE LITERATURE
An interesting history of MF can be found in [Kai98], which attributes the first
(classified) publication of the idea (applied to radar) to [Nor43]. An early tutorial
on MF is [Tur60a]. The development of the log-likelihood function for a continuous
time signal is based on the more rigorous development in [Wha71]. In [VTr68],
several rigorous approaches for obtaining a sufficient statistic for detection are provided,
giving rise to the matched filter. The expression for MF in colored noise is
based on [VTr68], though a development from maximizing SNR can be found in
[Wha71]. Details regarding the WMF can be found in [For72]. MF given discretetime
received signal samples is considered in [Mey94].
The use of the Schwartz inequality to derive the matched filter can be found in
[Tur60a]. The complex form of the Schwartz inequality can be found in [Sch05].
With multiple receive antennas, spatial matched filtering has a long history
[Bre59]. MF in a purely spatial dimension is referred to as maximal ratio combining
(MRC). To reduce complexity, a subset of antenna signals can be combined
[Mol03], referred to as generalized selection diversity.
Much work has been done to find closed-form expressions for the MFB for various
channel models. Here we give a few examples for cellular communications channels.
Analysis for fading channels usually employs the characteristic function approach
[Tur60b, Tur62]. MFB error rate averaged over fading medium coefficients is derived
for channels with two paths in [Maz91] and for those with more than two paths in
[Kaa94, Lin95]. The MFB for rapidly varying channels (variation within a symbol