Signals and Systems - Electrical Engineering

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

8.4 What Have We Accomplished? Where Do We Go from Here?....................

As you saw in this chapter the theory of discrete-time signals and systems is very similar to the the-

ory of continuous-time signals and systems. Many of the results in the continuous-time theory are

changed by swapping integrals for sums and differential equations for difference equations. However,

there are significant differences imposed by the way the discrete-time signals and systems are gener-

ated. For instance, the discrete frequency can be considered finite but circular, and it depends on the

sampling time. Discrete sinusoids, as another example, are not necessarily periodic. Thus, despite the

similarities there are also significant differences between the continuous-time and the discrete-time

signals and systems.

Now that we have a basic structure for discrete-time signals and systems, we will continue developing

the theory of linear time-invariant discrete-time systems using transforms. Again, you will find a great

deal of similarity but also some very significant differences. In the next chapters, carefully notice the

relation that exists between the Z-transform and the Fourier representations of discrete-time signals

and systems, not only with each other but with the Laplace and Fourier transforms. There is a great

deal of connection among all of these transforms, and a clear understanding of this would help you

with the analysis and synthesis of discrete-time signals and systems.

Problems............................................................................................

8.1. Discrete sequence—MATLAB

Consider the following formula

x[n]=x[n−1]+x[n−3] n≥ 3

x[0]= 0

x[1]= 1

x[2]= 2

Find the rest of the sequence for 0 ≤n≤ 50 and plot it using the MATLAB functionstem.

8.2. Finite-energy signals—MATLAB

Given the discrete signalx[n]=0.5nu[n]:

(a) Use MATLAB to plot the signalx[n]forn=− 5 to 200.

(b) Is this a finite-energy discrete-time signal? That is compute the infinite sum

∑∞

n=−∞

|x[n]|^2

Hint: Show that

∑∞

n= 0

αn=

1

1 −α