Signals and Systems - Electrical Engineering

Problems 357

5.20. Filter for half-wave rectifier
Suppose you want to design a dc source using a half-wave rectified signalx(t)and an ideal filter. Letx(t)
be periodic,T 0 = 2 , and with a period

x 1 (t)=

{

sin(πt) 0 ≤t≤ 1

0 1 <t≤2,

(a) Find the Fourier transformX()ofx(t), and plot the magnitude spectrum including the dc and the first

three harmonics.

(b)Determine the magnitude and cut-off frequency of an ideal low-pass filterH(j)such that when we

havex(t)as its input, the output isy(t)= 1. Plot the magnitude response of the ideal low-pass filter.

(For simplicity assume the phase is zero.)

5.21. Passive RLC filters—MATLAB
Consider an RLC series circuit with a voltage sourcevs(t). Let the values of the resistor, capacitor, and
inductor be unity. Plot the poles and zeros and the corresponding frequency responses of the filters with
the output the voltage across the
(a) Capacitor
(b)Inductor
(c)Resistor
Indicate the type of filter obtained in each case. Use MATLAB to plot the poles and zeros, the magnitude,
and the phase response of each of the filters obtained above.

5.22. AM modulation and demodulation
A pure tonex(t)=4 cos( 1000 t)is transmitted using an AM communication system with a carrier
cos(10,000t). The output of the AM system is

y(t)=x(t)cos(10,000t)

At the receiver, to recoverx(t)the sent signaly(t)needs first to be separated from the thousands of other

signals. This is done with a band-pass filter with a center frequency equal to the carrier frequency, and the

output of this filter then needs to be demodulated.

(a) Consider an ideal band-pass filterH(j). Let its phase be zero. Determine its bandwidth, center

frequency, and amplitude so we get as its output 10 y(t). Plot the spectrum ofx(t), 10y(t), and the

magnitude frequency response ofH(j).

(b)To demodulate 10 y(t), we multiply it bycos(10,000t). You need then to pass the resulting signal

through an ideal low-pass filter to recover the original signalx(t). Plot the spectrum of

z(t)= 10 y(t)cos(10,000t)

and from it determine the frequency response of the low-pass filterG(j)needed to recoverx(t). Plot

the magnitude response ofG(j).

5.23. Ideal low-pass filter—MATLAB
Consider an ideal low-pass filterH(s)with zero phase and magnitude response

|H(j)|=

{

1 −π≤≤π

0 otherwise

(a) Find the impulse responseh(t)of the low-pass filter. Plot it and indicate whether this filter is a causal

system or not.