Adaptive Coded Modulation for Transmission over Fading Channels - Adaptive Signal Processing in Wireless Communications - page 79

is given by R x (τ N - m +1 - τ N - n +1 ), and the correlation vector of the in-phase component of ˆ
with the in-phase component of the fading of interest is given by ρ , where ρ i = R x (τ N - i +1 ).
he parameter σ 2 in equation (3.2) is the mean square error of a minimum mean square
error (MMSE) estimator [72, p. 54] of the in-phase (or quadrature) fading of interest,
and is given by

=− + ( )

− 1

2 T 2

σ ρ Σ σ

I

ρ . (3.4)

x ∈

Understanding the Rician density in equation (3.2) and the expression for the Rician
noncentrality parameter s in equation (3.3) is key to designing effective adaptive coded
modulation schemes. In particular, for s = 0, the Rician probability density function
in equation (3.2) is equivalent to a Rayleigh probability density function, indicating
that coded modulation structures designed for Rayleigh fading channels are pertinent
for application when s is small; likewise, as s → ∞, the (properly normalized) Rician
density function approaches a delta function, thus indicating that the effective channel
approaches an AWGN channel. Since coded modulation schemes for Rayleigh fading
channels differ greatly from AWGN schemes, the interpretation of equation (3.3) is used
extensively in the design of structures, as demonstrated in section 3.3.3.
There is one limitation to directly employing the result in equation (3.2). In particular,
it presumes that the autocorrelation function R X ( τ ) of the random process X ( t ) is known
at the transmitter; however, this autocorrelation function can vary greatly in wireless
systems [51, p. 88-89]. Thus, it must generally be estimated [18, 19], either implicitly or
explicitly, or uncertainties in it must be worked into system design [24, 27]. To address
both possibilities in one framework, the autocorrelation function will be assumed to lie
in some uncertainty class R, which matches the approach taken in [27] directly. If it can
be accurately estimated through techniques as described in [18, 19], this class can be
shrunk accordingly (in the limit to a single autocorrelation function).
Thus, given the model for the system measurements and the measurements of the
autocorrelation function, one should be able to ascertain (1) how much predictor error σ 2
will generally be in the system, and (2) what is the uncertainty class R over which some
sort of robustness will be maintained. Understanding both of these for a given system
configuration will be the key to understanding the design of the coded modulation. In
particular, the former will allow the choice of the coded modulation structure, while the
latter will allow one to design on that structure.

3.3 Adaptivity in Single-Input Single-Output Systems

3.3.1 Information Theoretic Bounds
Before considering the derivation of practical signaling schemes that adapt to the mul-
tipath fading, it is instructive to consider the improvement in Shannon capacity that is
available when CSI is made available to the transmitter. Assume that the sequence of
zero-mean complex Gaussian channel fading coefficients affecting the transmitted sym-
bols forms an IID sequence; in other words, an IID Rayleigh fading channel is assumed.