APPENDIX C

Review of Signals and Systems

This appendix provides a summary of some important results in the area of signal representation and linear time invariant systems. Any engineering student embarking on a serious study of probability and random processes should be familiar with these concepts and hence a rigorous development is not attempted here. Rather, this review is intended as a brief refresher for those who need it and also as a quick reference for some important results that are used throughout the text. In this appendix, attention is focused on deterministic signals and systems in both continuous and discrete time.

Definition C.1: Consider a periodic signal x(t) whose period is T_o . That is, x(t) = x(t + T_o ) for all t. The inverse of the period f_o = 1/T_o is called the fundamental frequency of x(t) and any frequency, f_n = nf_o , which is a multiple of the fundamental frequency is called a harmonic.

Any periodic signal (subject to some mild constraints known as the Dirichlet conditions) can be represented as a linear combination of complex exponential signals, exp(j2πf_nt), whose frequencies are at the harmonics of the signal. That is, if x (t) is periodic with period T_o, then

(C.1)

This is known as the Fourier Series expansion and the series coefficients can be computed according to

(C.2)

Since the signal is periodic, the integral in the previous expression can be taken over any convenient interval of length T_o. In general, the series coefficients are complex numbers, and it is common to express them in terms of their magnitude and phase, x_n = |x _n | exp . The Fourier series coefficients display the frequency content of periodic signals.

For signals which are not periodic, the Fourier transform can be used to display the frequency content of a signal. The Fourier transform of a signal is given by

(C.3)

and the inverse Fourier Transform is

(C.4)

Sometimes we use the notation x(t) ↔ X(f) to indicate that x(t) and X(f) are a Fourier Transform pair. A table of some common Fourier Transform pairs is provided in Table E.1. Some of the more important properties of Fourier Transforms are listed below.

For signals in discrete time, x [n], a Discrete-time Fourier Transform (DTFT) is defined according to:

(C.5)

and the inverse DTFT is

(C.6)

Since X(f) is periodic with period of 1, the integral in the previous equation can be taken over any interval of length 1. It is common to view the DTFT using discrete frequency samples as well. In that case, the definition of the DTFT and its inverse is modified to give the N -point DTFT:

(C.7)

(C.8)

Alternatively, by replacing exp (j2πf) with z in the definition of the DFT, we get the z-transform:

(C.9)

The inverse z-transform is given by a complex contour integral,

(C.10)

where the contour of integration is any closed contour which encircles the origin in the counterclockwise direction and is within the region of convergence of X(z). Because of the complicated nature of the inverse transform, it is common to compute these inverse transforms via tables. A table of some common z-transform pairs is provided in Table E.2.

These various transform representations of signals are particularly useful when studying the passage of signals through linear time-invariant (LTI) systems.

Definition C.2: Suppose when x(t) is input to a system, the output is y(t). The system is said to be time-invariant if the input x(t − t_o ) produces an output of y(t − t_o ). That is, a time delay in the input produces the same time delay on the output but no other changes to the output. Furthermore, suppose the two inputs x₁(t) and x₂(t) produce the two outputs y₁(t) and y₂(t), respectively. Then, the system is linear if the input ax₁ (t) + bx₂(t) produces the output ay₁ (t) + by2(t) for any constants a and b. Identical definitions apply to discrete time systems as well.

A direct consequence of the linearity of a system is the concept of superposition which states that if the input can be written as a linear combination of several terms x(t)= a₁ x₁(t) + a₂ x₂ (t) + … + a_nx_n (t), then the corresponding output can be written as the same linear combination of the corresponding outputs y(t) = a₁ y₁(t) + a₂ y₂(t) + … + a_ny_n (t). Any LTI system can be described in terms of its impulse response, h (t). If the input is a delta (impulse) function, δ(t), the output is then the impulse response y(t) = h(t). For any LTI system, the input/output relationship is given in terms of the impulse response according to the convolution integral

(C.11)

If the input is a complex exponential at some frequency f, i.e., x(t) = exp(j 2 πft), then the corresponding output is then

(C.12)

That is, the output will also be a complex exponential whose magnitude and phase have been adjusted according to the complex number H(f). This function of frequency is called the transfer function of the system and is the Fourier Transform of the impulse response. Since complex exponentials form eigenfunctions of any LTI system, when studying LTI systems, it is convenient to decompose signals into linear combinations of complex exponentials. If x(t) is a periodic signal it can be written as a linear combination of complex exponentials through its Fourier Series representation,

(C.13)

Then using the concept of superposition together with the previous result, we find that the output of an LTI system, when x(t) is input is

(C.14)

Hence the Fourier series coefficients of the input and output of an LTI system are related by the simple form

(C.15)

A similar relationship holds for the Fourier transforms of non-periodic signals. Taking Fourier Transforms of both sides of (C.11) and using the convolution property of Fourier Transforms results in

(C.16)

Identical relationships hold for the DFT and z-transforms as well.