Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

Chapter 2
Spectral Representation of Wide-Sense Stationary Processes
The primary purpose of this chapter is to establish a rigorous mathematical development
for the spectral representation of a real wide-sense stationary stochastic noise process. Following
this development, the original definition of the one-sided power spectral density is
reviewed and related to the more modern two-sided definition. In Section 2.3 the spectral
representation of a real noise process is extended to a complex-valued noise process. Consequently,
this complex process is used in Section 2.4 to develop the fundamental results of
narrow-band stochastic processes. Finally, a specialization is made to the Gaussian noise
process where the probability density function for two jointly Gaussian random variables
is derived.
2.1 The Real Process
Let x(t) be a real wide-sense stationary (WSS) stochastic noise process with zero mean
and continuous two-sided power spectral density G(f). From the theories of shot and
thermal noise, it can be assumed that such a process, with a suitable choice for G(f),
can represent the background noise in a radio receiver. The correlation function of x(t) is
defined by
R(τ) E{x(t + τ)x(t)} =
∫ ∞
−∞
G(f)e i2πfτ df, (2.1)
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