25-08-2017, 09:32 PM

Representation of Signals

INTRODUCTION

Signals may be classified as predictable or as unpredictable, as analog or discrete and

of finite or infinite duration. Deterministic signals are defined exactly as a function of time.

They can be periodic (such as a sinewave or squarewave) or they can be an aperiodic “one

shot” signal. Deterministic signals contain no information because their future is completely

predictable by the receiver. They are easy to model and are useful since they can provide a

reasonably accurate evaluation of communication system performance. Stochastic signals, on

the other hand, are unpredictable and thus can communicate information. Although the time

waveform of a stochastic signal is random, the signal power may be predictable. Examples of

stochastic signals are thermal noise in electronic circuits, (i.e. “background hiss”) and

information signals such as voice or music.

An analog signal, denoted x(t), is a continuous function of time and is uniquely

determined for all t. When a physical signal such as speech is converted to an electrical signal

by a microphone, we have an electrical analog of the physical waveform. An equivalent

discrete-time signal, denoted as x(kT), exists only at discrete instants. It is characterized by a

sequence of values that exist at specific times, kT, where k is an integer and T is normally a

fixed time interval. On the other hand, a continuous time signal may be restricted to a set of

discrete amplitudes. A signal that is discrete in both time and amplitude is referred to as a

digital signal. Furthermore, these discrete digital signal amplitudes can be represented by a

set of numbers (codes) and, as such, can be stored in a computer memory. Pulse code

modulation (PCM) is an example of a digital signal. These categorizations are illustrated in

Figure 2-1.

SIGNAL POWER

Instantaneous power is the voltage-current product at a specific time, p(t) = v(t) i(t),

while average power is P = 〈p(t)〉 = 〈v(t) i(t)〉 where 〈•〉 indicates time average. With

normalized power, the load resistance is assumed to be 1 Ω, current is numerically equal to

voltage and average normalized power is simply the time average of voltage squared, PN =

〈v(t)2〉. Root-mean-square (rms) voltage is the square root of normalized power, Vrms = PN

For example, the 12 volt supply in an automobile has normalized power of 144 ‘watts’.

Voltage Gain

The decibel unit of power gain has been generalized to

describe the voltage ratio in systems where the input and output

impedances are not defined. For example, an ideal transformer

with turns ratio 1:2 will have no power gain (0 dB) but will

have a voltage gain of 2 (6 dB). Voltage gain is useful to

describe electronic amplifiers which have high input impedance

and low output impedance. In this case, voltage gain is constant

but the power gain varies with the load resistance.

FREQUENCY SPECTRUM REPRESENTATION OF PERIODIC WAVEFORMS

A periodic waveform is composed of repeated copies of a waveform segment where

the segment length is equal to the repetition period. These waveforms are classed as power

signals since the duration is assumed to be infinite and there is no decay. Examples of

periodic waveforms are a sinusoid, a square wave and a triangular wave. A lengthy example

might be a continuously repeating musical recording. We are particularly interested in pulse

waveforms since they are used in sampling, an essential foundation of digital communication.