Problem Sheet Seven
1. Over an interval 0 t 1, an angle modulated signal is given by
(t) = 10 cos 13000t
The carrier frequency is c = 10000.
(a) If this were a PM signal with kp = 1000, determine m(t) over 0 t 1.
(b) If this were an FM signal with kf = 10
Communications I
Problem Sheet One
1. Find the mean value (time average) and the power for the following signals
(a) v(t) = 3 cos(2103 t),
[0, 9/2]
(b) v(t) = 3,
[3, 9]
(c) v(t) = 3 sin(40t) + 4 cos(104 t).
[0, 25/2]
2. (a) Find the energy of the signal
g
(a) If g(t) = g(t) (even symmetry), then all the sine terms in the trigonometric Fourier series vanish (bn = 0).
(b) If g(t) = g(t) (odd symmetry), then the dc and all the cosine terms
in the Fourier series vanish (a0 = an = 0).
4. Show that the trigonome
Problem Sheet Five
1. Consider the two baseband signals x1 (t) = cos 2000t and x2 (t) = cos 1900t.
Plot the magnitude spectrum of the signal s(t) = x1 (t) cos 10000t+x2 (t) cos 20000t.
(a) The signal s(t) is multiplied by cos 10000t and fed to the lter wi
Problem Sheet Two
1. Find the energy Ex and Ey of the signals x(t) and y(t) shown in Figure 1.
Find the correlation coecient between x(t) and y(t). Sketch the signal
x(t) + y(t) and show that the energy of this signal is equal to Ex + Ey .
(Why?)
y(t)
x(t
Problem Sheet Four
1. Find the Power Spectral Density Sg () of the power signal g(t) = cos 0 t.
(Hint: Compute the autocorrelation function rst, and then use the property Rg ( ) Sg ().
[ [( 0 ) + ( + 0 )]
2
2. Find the power of the output signal y(t) of t
Problem Sheet Six
1. Consider the amplitude modulated signal s(t) = (A + m(t) cos c t, where
A = 2, c = 10000rad/s and m(t) = cos 100t + sin 100t. Compute:
(a) the peak amplitude mp of m(t).
(Hint: use the identity a cos 0 t + b sin 0 t = c cos(0 t + ) wi
Communications I
Problem Sheet One
1. Find the mean value (time average) and the power for the following signals
(a) v(t) = 3 cos(2103 t),
[0, 9/2]
(b) v(t) = 3,
[3, 9]
(c) v(t) = 3 sin(40t) + 4 cos(104 t).
[0, 25/2]
2. (a) Find the energy of the signal
g
Problem Sheet Nine
1. A 50 line is connected to a 75 line with a matched termination, and a
sine wave of 1.0V amplitude propagating in the former is incident on the
junction. Find the voltage and the current amplitudes of the reected and
transmitted wave.
Problem Sheet Eight
1. Consider the FM signal
(t) = 10 cos[2f0 t + kf
t
x()d]
where kf = 10. The message x(t) is given by
2
mn (t)
x(t) =
n=0
with
mn (t) =
2n
sinc(t) cos(2nt).
(a) Sketch and dimension the Fourier transform of m1 (t).
(b) Sketch and dimen
Problem Sheet Three
1. Show that the Fourier transform of g(t) may be expressed as
G() =
g(t) cos tdt j
g(t) sin tdt
Hence, show that
if g(t) is a real and even function of t, then G() is real and even.
if g(t) is a real and odd function of t, then G()