Adaptive Channel Estimation in Wireless Communications - Adaptive Signal Processing in Wireless Communications - page 42

baud-rate sampled channel response are quite sensitive to symbol timing errors. Ini-
tially, in the trained case, fractional sampling was investigated to “robustify” the equal-
izer performance against timing errors. The model (2.21) does not apply to fractionally
spaced samples, i.e., when the sampling interval is a fraction of the symbol duration.
The fractionally sampled digital communications signal is a cyclostationary signal [9]
that may be represented as a vector stationary sequence using a time series representa-
tion (TSR) [9, section 12.6]. Suppose that we sample at N times the baud rate with signal
samples spaced T s / N seconds apart where T s is the symbol duration. Then a TSR for the
sampled signal is given by

L

∑ 0

yn () = hlsn l ()( − +; =,,, , ) vni () ( 12  N ) (2.22)

i i i

l =

where now we have N samples every symbol period, indexed by i . Notice, however, that
the information sequence s ( n ) is still one sample per symbol. It is assumed that the signal
incident at the receiver is first passed through a receive filter whose transfer function
equals the square root of a raised cosine pulse, and that the receive filter is matched to the
transmit filter. The noise sequence in (2.22) is the result of the fractional-rate sampling
of a continuous-time filtered white Gaussian noise process. Therefore, the sampled noise
sequence is white at the symbol rate, but correlated at the fractional rate. Stack N con-
secutive received samples in the n th symbol duration to form an N vector y ( n ) satisfying

L

∑ 0

y () n = h ()( l sn l − +, ) v () n (2.23)

l =

where h ( n ) is the vector impulse response of the SIMO-equivalent channel model given
by

T

=  

  ,

h () nhnhn () ()  h () n (2.24)

1 2 N

and y ( n ) and v ( n ) are defined similarly. A block diagram of model (2.22) is shown in
Figure 2.4 .

2.3 Channel Estimation

We first consider three types of channel estimators within the framework of maximizing
the likelihood function. (Unless otherwise noted, the underlying channel model is given
by the time-invariant model (2.23).) In general, one of the most effective and popular
parameter estimation algorithms is the maximum likelihood (ML) method. The class of
maximum likelihood estimators are optimal asymptotically.