Signals and Systems - Electrical Engineering

452 C H A P T E R 8: Discrete-Time Signals and Systems

just like those for continuous-time signals. However, some of these basic signals do not display

the mathematical complications of their continuous-time counterparts. For instance, the discrete-

impulse signal is defined at every integer value in contrast with the continuous-impulse response,

which is not defined at zero.

n The discrete approximation of derivatives and integrals provides an approximation of diffe-

rential equations, representing dynamic continuous-time systems by difference equations.

Extending the concept of linear time invariance to discrete-time systems, we obtain a convo-

lution sum to represent LTI systems. Thus, dynamic discrete-time systems can be represented

by difference equations and convolution sums. A computationally significant difference with

continuous-time systems is that the solution of difference equations can be recursively obtained,

and that the convolution sum provides a class of systems that do not have a counterpart in the

analog domain.

8.2 Discrete-Time Signals

Adiscrete-time signalx[n]can be thought of as a real- or complex-valued function of the integer sample

indexn:

x[.] :I→R(C)

n x[n] (8.1)

The above means that for discrete-time signals the independent variable is an integern, the sample

index, and that the value of the signal atn,x[n], is either a real- or a complex-value function. Thus,

the signal is only defined at integer valuesn—no definition exists for values between the integers.

Remarks

n It should be understood that a sampled signal x(nTs)=x(t)|t=nTsis a discrete-time signal x[n]that is a

function of n only. Once the value of Tsis known, the sampled signal only depends on n, the sample index.

However, this should not prevent us in some situations from considering a discrete-time signal obtained

through sampling as a function of time t where the signal values only exist at discrete times{nTs}.

n Although in many situations discrete-time signals are obtained from continuous-time signals by sampling,

that is not always the case. There are many signals that are inherently discrete—think, for instance, of a

signal consisting of the final values attained daily by the shares of a company in the stock market. Such

a signal would consist of the values reached by the share in the days when the stock market opens. This

signal is naturally discrete. A signal generated by a random number generator in a computer would be

a sequence of real values and can be considered a discrete-time signal. Telemetry signals, consisting of

measurements—for example, voltages, temperatures, pressures—from a certain process, taken at certain

times, are also naturally discrete.

nExample 8.1

Consider a sinusoidal signal

x(t)=3 cos( 2 πt+π/ 4 ) −∞<t<∞