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(ELEC211)[2009](sum)final~=_8udkjdm^_91941.pdf
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a. (3 pts) Sketch the discrete-time sequence h[n] =[n + 3] +[n . 3]
j j
b. (4 pts) Find H (e ) and sketch the magnitude of H (e ) for | | < 2 , the DTFS of h[n] =[n + 3] +[n . 3] . 2
j 2
c.
(3 pts) Find | H (e )| d
d.
(2 pts) Let x(t) be a bandlimited signal with Fourier Transform shown below: X ( j)
0
Now we sample x(t) at a sampling interval of T and let []n = x nT ) . What is required of T so that x(t)
xd (
can be recovered from xd []
n ?
e. (4 pts) Recall that if we multiply x(t) with the periodic impulse train p(t) =(t . nT ) such that
n =.
xp (t) = x(t) p(t) = x(nT ) (t . nT ),
n =.
the Fourier Transform of xp ()t is a Poisson sum (times 1/T) of the Fourier Transform of x(t):
1 .. 2n ..
X (j) = Xj .
p .
. ..
Tn =. .. T ..
For T = 1/15, sketch Xp( j) for . 30 30 . Carefully label your sketch, including the magnitude
of the spectrum.
j j
f. (4 pts) Determine and express Xd (e ) , the DTFT of n , in terms of X ) . Also, sketch Xd (e ) andxd []( j carefully label your sketch, including the magnitude of the spectrum.
The spectrum of a broadcast talk radio signal at baseband spans from 40 Hz to 5 kHz, as shown below.
X(j)
.10000 10000
a.
(2 pts) In many regions of the world, the medium wave frequency band of 500 kHz to 1,600 kHz is used for commercial AM radio broadcasting. If double sideband (DSB) system is used, how many commercial AM radio channels can be supported within the medium wave frequency band?
b.
(3 pts) Single sideband (SSB) transmission can be used to increase the number of channels that can be transmitted for a given frequency band. A SSB system transmitting the upper sideband of the signal can conceptually be implemented by passing the modulated signal through an Ideal Band Pass Filter (IBPF) with cutoff angular
frequency as below:
c
c(t) = 2cos ct
Sketch Y( j) , the spectrum of y(t). Note the factor of 2 in c(t) and, carefully label your sketch including the peak
magnitude of the spectrum. (Note that the carrier frequency is much larger than the bandwidth of the
c
baseband signal, and we can use the symbol // to indicate breaks on the frequency axis as shown above if preferred.)
c. (3 pts) To recover x(t) from y(t), consider what happens when we multiply y(t) by a sinusoid at the carrier frequency again:
c(t) = cos ct
Sketch R( j) , the spectrum of r(t). Carefully label your sketch, including the magnitude of each piece of the spectrum.
d. Actually implementation of SSB system is usually not done as described in b. Here we will work through an alternative way to implement SSB. First, we pass the baseband signal x(t) through the following filter:
j
.j
H( j)
..j <0
.j >0H( j) =.
The corresponding impulse response h(t) is in a simple form. For this problem we do not need to determine what h(