Description
1. This problem reviews basic parameters for FM. Consider a tone signal with amplitude 2V at frequency 5kHz that is frequency modulated using a VCO with frequency
sensitivity 10kHz=V . Compute: (1)the frequency deviation; (2)the ; (3)the bandwidth estimate given by Carsonís rule. Also specify whether this should be considered
narrowband or wideband FM.
2. In this problem, you will be synthesizing and studying various analog modulations of
a speciÖc signal:
m (t) = 1
1 + (t 10)2
which you will represent over the time span 0 t 20 sec. The spectrum is:
M (f) = ej40f e
2jfj
The 90% energy containment bandwidth B (i.e., 90% of the energy is contained in the
range jfj B) is given by:
B =
ln 10
4
= 0:1832Hz
Some other information you will Önd useful for this problem: the Hilbert transform is:
m^ (t) = t 10
1 + (t 10)2
and the integral is:
v (t) = Z t
=1
m ( ) d =
2
+ tan1
(t 10)
where here tan1
is the principal branch, returning values in the range =2 to =2.
For example, v (1) = 0, v (10) = =2, v (1) = .
We are going to use a carrier frequency of fc = 2Hz, and a sampling rate of fs = 10Hz.
(a) In MATLAB, set up a time vector t with values 0 t 20 at sample times (i.e.,
integer multiples of 1=fs). All your time domain signals should use this t vector.
You will note t has length approximately (but not exactly) 2000.
(b) The way you are going to compute spectra is to use § t of length N = 4096, via the
usual zero padding. After applying § tshift, the proper frequency vector should
span roughly from fs=2 to fs=2. Generate the correct frequency vector f. All
you frequency domain plots should use this f vector (unless speciÖed otherwise
below). Also, for any of the frequency plots in decibels (not all will be in dB),
make sure the vertical bounds are set so interesting features are visible (i.e., donít
let is go down to 1million dB or s
(c) First, plot jM (f)j on a decibel scale, but only over the frequency range jfj 1Hz.
Show the full values (i.e., donít limit the vertical axis) so you can observe, in
particular, how small the magnitude spectrum becomes at 1Hz.
(d) Next, generate two AM signals, at 10% and 90% modulation index, respectively.
Plot them both (using subplot) in the time domain, with the envelope superimposed for each (in a di§erent color). Then plot the spectra on a decibel scale. Also
plot the zoomed-in spectra over the subrange 8 < f < 12Hz.
(e) Generate DSB-SC, USSB and LSSB. Plot the modulated signals in the time domain (use subplot). Also, for each signal, compute the envelope and superimpose
the envelope (in a di§erent color) on top of the modulated signal.
(f) It turns out the envelopes for USSB and LSSB have a simple formula. Find the
formula for the envelopes!
(g) Compute the spectra of DSB-SC, USSB and LSSB, and plot the magnitude spectra of each (use subplot) on a decibel scale, Örst over the full frequency range,
and then over the subrange 8 < f < 12Hz.
(h) Generate the FM signal using kf = 0:5. Plot the FM signal in the time domain.
It may be a little hard to see, so you may also want to provide a plot for a subset
of the time. It may not be immediately obvious that the frequency is shifting
since the frequency deviation is so small compared to the carrier frequency.
(i) Plot the FM magnitude spectrum in decibels, again over the full range and then
over 8 < f < 12Hz:
(j) In order to see the bandwidth expansion of FM, superimpose plots of the magnitude spectra of the DSB-SC and FM signals on a linear scale (i.e., not in dB),
normalized so each has peak magnitude 1, over the range 1 < f < 3Hz.
2