Signals and Systems - Electrical Engineering

Problems 415

(c) Design a band-pass filter to get rid of the hum. Plot the output of the band-pass filter.

(d) Is any of these alternatives better than the others? Explain.

6.20. Demodulation of AM—MATLAB
The signal at the input of an AM receiver is

u(t)=m 1 (t)cos( 20 t)+m 2 (t)cos( 100 t)

where the messagesmi(t),i=1, 2are the outputs of a low-pass Butterworth filter with inputs

x 1 (t)=r(t)− 2 r(t− 1 )+r(t− 2 )

x 2 (t)=u(t)−u(t− 2 )

respectively. Suppose we are interested in recovering the messagem 1 (t).

(a) Design a 10 th-order low-pass Butterworth filter with half-power10 rad/sec. Implement this filter using

MATLAB and find the two messagesmi(t),i=1, 2using the indicated inputsxi(t),i=1, 2, and

plot them.

(b) To recover the desired messagem 1 (t), first use a band-pass filter to obtain the desired signal con-

tainingm 1 (t)and to suppress the other. Design a band-pass Butterworth filter with a bandwidth of

10 rad/sec, centered at 20 rad/sec and order 10 that will pass the signalm 1 (t)cos( 20 t)and reject the

other signal.

(c) Multiply the output of the band-pass filter by a sinusoidcos( 20 t)(exactly the carrier in the transmitter),

and low-pass filter the output of the mixer (the system that multiplies by the carrier frequency cosine).

Design a low-pass Butterworth filter of bandwidth10 rad/sec, and order 10 to filter the output of the

mixer.

(d) Use MATLAB to display the different spectra. Compute and plot the spectrum ofm 1 (t),u(t),the output

of the band-pass filter, the output of the mixer, and the output of the low-pass filter. Write numeric

functions to compute the analog Fourier transform and its inverse.

6.21. Quadrature AM—MATLAB
Suppose we would like to send the two messagesmi(t),i=1, 2, created in Problem 6.20 using the same
bandwidth and to recover them separately. To implement this, consider the QAM approach where the
transmitted signal is
s(t)=m 1 (t)cos( 50 t)+m 2 (t)sin( 50 t)
Suppose that at the receiver we receives(t)and that we only need to demodulate it to obtainmi(t),i=1, 2.
Design a low-pass Butterworth filter of order 10 and a half-power frequency10 rad/sec(the bandwidth of
the messages).
(a) Use MATLAB to plots(t)and its magnitude spectrum|S()|. Write numeric functions to compute the
analog Fourier transform and its inverse.
(b) Multiplys(t)bycos( 50 t), and filter the result using the low-pass filter designed before. Use MATLAB
to plot the result and to find and plot its magnitude spectrum.
(c) Multiplys(t)bysin( 50 t), and filter the result using the low-pass filter designed before. Use MATLAB
to plot the result and to find and plot its magnitude spectrum.
(d) Comment on your results.