A Signal Theoretic Introduction to Random Processes

10POWER SPECTRAL DENSITY THEORY

10.1 INTRODUCTION

The most widely used approach in the physical sciences for characterizing a random process is via a power spectral density, that is, the decomposition of the signals defined by a random process into their constituent sinusoidal components. The power spectral density for a random process has been introduced in Chapter 7. This chapter extends the discussion of a power spectral density with the definition of the cross power spectral density and the consideration of the power spectral density of a sum of random processes. The power spectral density of a periodic pulse train, signalling random processes, quantization error signals, shot noise random processes, 1/f noise, and a correlated jittered random process are established. Further theory and results for the power spectral density, consistent with the approach taken, can be found in Howard (2002).

10.2 POWER SPECTRAL DENSITY THEORY

Consider a random process defined on [0, T ], with a signal sample space

(10.1)

that is based on the sample space S of experimental outcomes underpinning the random process. The sample space S defines a random variable Ω with a probability mass function for the countable case and a probability density function for the uncountable case.

In terms ...