1.3 Continuous-Time Signals 671.3 CONTINUOUS-TIME SIGNALS
That signals are functions of time-carrying information is easily illustrated with a recorded voice
signal. Such a signal can be thought of as a continuously varying voltage, generated by a microphone,
that can be transformed into an audible acoustic signal—providing the voice information—by means
of an amplifier and speakers. Thus, the speech signal is represented by a function of time
v(t), tb≤t≤tf (1.1)wheretbis the time at which this signal starts, andtfthe time at which it ends. The functionv(t)
varies continuously with time, and its amplitude can take any possible value (as long as the speakers
are not too loud!). This signal obviously carries the information provided by the voice message.
Not all signals are functions of time alone. A digital image stored in a computer provides visual
information. The intensity of the illumination of the image depends on its location within the image.
Thus, a digital image can be represented as a function of two space variables(m,n)that vary discretely,
creating an array of values calledpicture elementsorpixels. The visual information in the image is thus
provided by the signalp(m,n)where 0≤m≤M−1 and 0≤n≤N−1 for an image of sizeM×N
pixels. Each of the pixel values can be represented, for instance, by 256 gray scale values or 8 bits/pixel.
Thus, the signalp(m,n)varies discretely in space and in amplitude. A video, as a sequence of images
in time, is accordingly a function of time and of two space variables. How their time or space variables
and their amplitudes vary characterizes signals.
For a time-dependent signal, time and amplitude vary continuously or discretely. Thus, according
to the independent variable, signals arecontinuous-timeordiscrete-timesignals—that is,ttakes an
innumerable or a finite set of values. Likewise, the amplitude of either a continuous-time or a discrete-
time signal can vary continuously or discretely. Thus, continuous-time signals can be continuous-
amplitude as well as discrete-amplitude signals. Continuous-amplitude, continuous-time signals are
calledanalog signalsgiven that they resemble the pressure variations caused by an acoustic signal. A
continuous-amplitude, discrete-time signal is called adiscrete-time signal. Adigital signalhas discrete
time and discrete amplitude. If the samples of a digital signal are given as binary codes the signal is
called abinary signal.
A good way to illustrate the signal classification is to consider the steps needed to process the voice
signalv(t)in Equation (1.1) with a computer. As indicated above, inv(t)time varies continuously
betweentbandtf, and the amplitude also varies continuously, and we assume it could take any
possible real value (i.e.,v(t)is an analog signal). As such,v(t)cannot be processed with a computer.
It would require to store an innumerable number of signal values (even whentbis very close totf)
and for an accurate representation of the amplitude valuesv(t), we might need a large number of
bits. Thus, it is necessary to reduce the amount of data without losing the information provided by
the signal. To accomplish that, we sample the signal by taking signal values at equally spaced times
nTs, wherenis an integer andTsis thesampling period, which is appropriately chosen for this signal
(in Chapter 7 we will learn how to choseTs).