Signals and Systems - Electrical Engineering

10 C H A P T E R 0: From the Ground Up!

with minimal information loss. Chapters 1, 7, and 8 will provide the necessary information about continuous-

time and discrete-time signals, and show how to convert one into the other and back. The sampling theory

presented in Chapter 7 is the backbone of digital signal processing.

0.3.1 Continuous-Time and Discrete-Time Representations

There are significant differences between continuous-time and discrete-time signals as well as in their

processing. A discrete-time signal is a sequence of measurements typically made at uniform times,

while the analog signal depends continuously on time. Thus, a discrete-time signalx[n] and the

corresponding analog signalx(t)are related by a sampling process:

x[n]=x(nTs)=x(t)|t=nTs (0.1)

That is, the signalx[n] is obtained by samplingx(t)at timest=nTs, wherenis an integer andTsis

thesampling periodor the time between samples. This results in a sequence,

{···x(−Ts)x( 0 )x(Ts)x( 2 Ts)···}

according to the sampling times, or equivalently

{···x[−1]x[0]x[1]x[2]···}

according to the ordering of the samples (as referenced to time 0). This process is calledsamplingor

discretizationof an analog signal.

Clearly, by choosing a small value forTswe could make the analog and the discrete-time signals look

very similar—almost indistinguishable—which is good, but this is at the expense of memory space

required to keep the numerous samples. If we make the value ofTslarge, we improve the memory

requirements, but at the risk of losing information contained in the original signal. For instance,

consider a sinusoid obtained from a signal generator:

x(t)=2 cos( 2 πt)

for 0≤t≤10 sec. If we sample it everyTs 1 =0.1 sec, the analog signal becomes the following

sequence:

x 1 [n]=x(t)|t=0.1n=2 cos( 2 πn/ 10 ) 0 ≤n≤ 100

providing a very good approximation to the original signal. If, on the other hand, we letTs 2 =1 sec,

then the discrete-time signal becomes

x 2 [n]=x(t)|t=n=2 cos( 2 πn)=2 0≤n≤ 10

See Figure 0.5. Although forTs 2 the number of samples is considerably reduced, the representation

of the original signal is very poor—it appears as if we had sampled a constant signal, and we have

thus lost information! This indicates that it is necessary to come up with a way to chooseTsso that

sampling provides not only a reasonable number of samples, but, more importantly, guarantees that

the information in the analog and the discrete-time signals remains the same.