Random Signals

Abstract

Random signals arise in all fields of science and engineering. Measurements of all physical variables are never perfect, and the imperfections can be viewed as random noise. Thus, measurement of any variable over time results in a random signal. If the measurement is analog, then the signal is continuous, and if digital then the signal is discrete. Random signals are also known as stochastic processes or random processes. In the discrete-time case, they are called time series. This chapter provides an introduction to the mathematical tools needed to handle random signals.

Appendix 3.1 Brief Review of the Two-Sided Z-Transform

This section provides a brief review of the two-sided z-transform.

The two-sided z-Transform of sequence

$$\left \{ {G\left( k \right)} \right\} = \left\{ { \ldots ,G\left( { - 1} \right), G\left( 0 \right), \ldots , G\left( i \right), \ldots } \right\}$$

is defined as

$${\varvec{G}}\left( z \right) = \cdots + G\left( { - 1} \right)z^{1} + G\left( 0 \right) + \cdots + G\left( i \right)z^{ - i} + \cdots ,$$

where $z$ is a time advance operator. It is also possible to define the z-transform using the Laplace transform, the delay theorem for Laplace transforms, and with the definition of $z = {\text{e}}^{sT}$, where $T$ is the sampling period. With the second definition, the z-transform inherits some properties of the Laplace transform. In particular, the z-transform is a linear transform.

• Linear:

$${\mathcal{Z}}\left\{ {a{\varvec{f}}\left( k \right) + b{\varvec{g}}\left( k \right)} \right\} = a{\varvec{F}}\left( z \right) + b{\varvec{G}}\left( z \right).$$

Convolution Theorem

The z-transform of the convolution of two sequences is the product of their transforms

$${\mathcal{Z}}\left\{ {\mathop \sum \limits_{i = - \infty }^{\infty } G\left( {k - i} \right){\varvec{f}}\left( i \right)} \right\} = {\varvec{G}}\left( z \right){\varvec{F}}\left( z \right),$$

$$\left\{ {{\varvec{f}}\left( k \right)} \right\} = \left\{ { \ldots ,{\varvec{f}}\left( { - 1} \right),{\varvec{f}}\left( 0 \right), \ldots , {\varvec{f}}\left( i \right), \ldots } \right\}.$$

Response of DT System

The response of linear system to any input sequence $\left\{ {{\varvec{f}}\left( k \right)} \right\}$ is the convolution summation of the input and the impulse response sequence

$${\varvec{x}}\left( k \right) = \mathop \sum \limits_{i = - \infty }^{\infty } G\left( i \right){\varvec{f}}\left( {k - i} \right) = \mathop \sum \limits_{i = - \infty }^{\infty } G\left( {k - i} \right){\varvec{f}}\left( i \right).$$

By the convolution theorem, the z-transform of the response is the product

$${\varvec{X}}\left( {\varvec{z}} \right) = {\varvec{G}}\left( {\varvec{z}} \right){\varvec{F}}\left( {\varvec{z}} \right),\user2{ G}\left( {\varvec{z}} \right) = \mathop \sum \limits_{{{\varvec{i}} = - \infty }}^{\infty } {\varvec{G}}\left( {\varvec{i}} \right){\varvec{z}}^{{ - {\varvec{i}}}} .$$

The impulse response sequence and the z-transfer function are z-transform pairs

$$\begin{array}{*{20}c} {G\left( i \right)} \\ {{\text{impulse response}}} \\ \end{array} \mathop \leftrightarrow \limits^{{\quad \quad \rm{\mathcal{Z}}\quad \quad }} \begin{array}{*{20}c} {\user2{G}\left( z \right)} \\ {{\text{transfer function}}} \\ \end{array} .$$

Even Function

For an even function $G\left( n \right) = G\left( { - n} \right)$, the z-transform satisfies

$${\varvec{G}}\left( {\varvec{z}} \right) = \mathop \sum \limits_{{{\varvec{m}} = - \infty }}^{\infty } {\varvec{G}}\left( {\varvec{m}} \right){\varvec{z}}^{{ - {\varvec{m}}}} = \mathop \sum \limits_{{{\varvec{m}} = - \infty }}^{\infty } {\varvec{G}}\left( { - {\varvec{m}}} \right){\varvec{z}}^{{ - {\varvec{m}}}} .$$

Substituting $l = - m$ gives

$${\varvec{G}}\left( {\varvec{z}} \right) = \mathop \sum \limits_{{{\varvec{l}} = - \infty }}^{\infty } {\varvec{G}}\left( {\varvec{l}} \right){\varvec{z}}^{{\varvec{l}}} = {\varvec{G}}\left( {{\varvec{z}}^{{ - {\mathbf{1}}}} } \right),$$

that is,

$${\varvec{G}}\left( z \right) = {\varvec{G}}\left( {z^{ - 1} } \right).$$

Hence, poles are located symmetrically w.r.t. the unit circle.