Adaptive Channel Estimation in Wireless Communications - Adaptive Signal Processing in Wireless Communications - page 38

yn hnsn vn () = ()() + . () (2.10)

Finally, a time-non-selective and frequency-non-selective channel is modeled as

yn hs nvn () = + , () () (2.11)

where h is a random variable (or a constant).

2.2.1.2 Autoregressive (AR) Models
It is possible to accurately represent a WSSUS channel by a large-order AR model; see
[ 34 ] and references therein. However, it is far more common to use a first-order AR
model ( AR (1)) given by [34, 35]

( = − ( ) + () ,

hnl α h nl 1 wn (2.12)

c c

where α c is the AR coefficient, and the driving noise w c ( n ) is zero-mean complex Gauss-
ian with variance σ w 2 and statistically independent of h ( n - 1; l ). Assume that h ( n ; l ) is
also zero-mean complex Gaussian with variance σ h 2 . hen [35]

1

{ } ,

() − ( )

∗

α = Ehnlhn l 1 (2.13)

c

2

σ

h


 


 .

2

2 2

σσ α
wc

= −

(2.14)

h c

2.2.1.3 Basis Expansion Models
Recently, basis expansion models (BEMs) have been widely investigated to represent dou-
bly selective channels in wireless applications [3, 13, 36, 46, 61], where the time-varying
taps are expressed as superpositions of time-varying basis functions in modeling Dop-
pler effects, weighted by time-invariant coefficients. Candidate basis functions include
complex exponential (Fourier) functions [13, 36], polynomials [3], discrete prolate sphe-
roidal sequences [61], etc. In contrast to AR models that describe temporal variation
on a symbol-by-symbol update basis, a BEM depicts the evolution of the channel over
a period (block) of time. Intuitively, the coefficients of the BEM approximation should
evolve much more slowly in time than the channel, and hence are more convenient to
track in a fast-fading environment.
Suppose that we include the effects of transmit and receive filters in the time-variant
impulse response h ( t ;τ) in (2.1). Suppose that this channel has a delay spread τ d and a
Doppler spread f d . Consider the k t h block of data consisting of an observation window of
T B symbols where the baud-rate data samples in the block are indexed as n = n k , n k + 1,
… , n k + T B - 1, n k := ( k - 1) T B . If 2 f d τ d < 1 (underspread channel), the complex exponential
basis expansion model (CE-BEM) representation of h ( n ; l ) in (2.6) is given by