Signals and Systems - Electrical Engineering

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

8.6. Generation of periodic discrete-time signals—MATLAB

Periodic signals can be generated by obtaining a period and adding shifted versions of this period. Suppose

we wish to generate a train of triangular pulses. A period of the signal is

x[n]=r[n]− 2 r[n−1]+r[n−2]

wherer[n]=nu[n]is the discrete-time ramp signal.

(a) Carefully plotx[n].

(b) Let

y[n]=

∑∞

k=−∞

x[n− 2 k]

and carefully plot it. Indicate the periodNofy[n].

(c) Write a MATLAB script to generate and plot the periodic signaly[n].

8.7. Expansion and compression of discrete-time signals—MATLAB

Consider the discrete-time signalx[n]=cos( 2 πn/ 7 ).

(a) The discrete-time signal can be compressed by getting rid of some of its samples (downsampling).

Consider the downsampling by 2. Write a MATLAB script to obtain and plotz[n]=x[2n]. Plot also

x[n]and compare it withz[n]. What happended? Explain.

(b) The expansion for discrete-time signals requires interpolation, and we will see it later. However, a first

step of this process is the so-calledupsampling. Upsampling by 2 consists in defining a new signal

y[n]such thaty[n]=x[n/2]forneven, andy[n]= 0 otherwise. Write a MATLAB script to perform

upsampling onx[n]. Plot the resulting signaly[n]and explain its relation withx[n].

(c) Ifx[n]resulted from sampling a continuous-time signalx(t)=cos( 2 πt)using a sampling periodTs

and with no frequency aliasing, determineTs. How would you sample the analog signalx(t)to get the

downsampled signalsz[n]? That is, choose a value for the sampling periodTsto getz[n]directly from

x(t). Can you chooseTsto gety[n]fromx(t)directly? Explain.

8.8. Absolutely summable and finite-energy discrete-time signals—MATLAB

Suppose we sample the analog signalx(t)=e−^2 tu(t)using a sample periodTs= 1.

(a) Expressing the sampled signal asx(nTs)=x[n]=αnu[n], what is the corresponding value ofα? Use

MATLAB to plotx[n].

(b) Show thatx[n]is absolutely summable—that is, show the following sum is finite:

∑∞

n=−∞

|x[n]|

(c) If you know thatx[n]is absolutely summable, could you say thatx[n]is a finite-energy signal? Use

MATLAB to plot|x[n]|andx^2 [n]in the same plot to help you decide.

(d) In general, for what values ofαare signalsy[n]=αnu[n]finite energy? Explain.

8.9. Discrete-time periodic signals

Determine whether the following discrete-time sinusoids are periodic or not. If periodic, determine its

periodN 0.

x[n]=2 cos(πn−π/ 2 )

y[n]=sin(n−π/ 2 )

z[n]=x[n]+y[n]

v[n]=sin( 3 πn/ 2 )