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(ELEC211)[2011](f)midterm~=_rrzkjdj^_50335.pdf
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a) (2 pts) Write down the magnitude and phase of the complex number z . 4ej 2.
1
1. jb) (2 pts) Find the magnitude and phase of the complex number z2 . j . .
2
e
(. 0.2 . j.) t
c) (3 pts) Sketch the continuous-time signal x (t) . Re{eu(t .1)} for . 5 . t . 5.
1
n. 0.2
d) (3 pts) Sketch the discrete-time signal x[n] . Re{zu[n .1]} for . 5 . n. 5 where | z| . e . 0.8 and . z ...
e) (3 pts) Plot the continuous-time signal x2(t) . u(t . 3) .u(t .1) . 0.5.(t . 2) .
f) (3 pts) Let x2(t) in (e) be the input to an LTI system with impulse response h1(t) . u(t) . u(t . 2) . Sketch the output y2(t) . Label all important dimensions.
g) (3 pts) Recall that the system function of an CT LTI system is its Laplace transform of its impulse response:
.
.
. st
H(s) . h(t) e dt . Determine the system function for h1(t) in part (f).
..
h) (2 pts) Determine H1( j.) , the frequency response for h1(t) in part (f).
i) (4 pts) Let x3(t). cos0.5.t be the input to the system h1(t) . Determine the output y3(t).

a) (2 pts) Let x[n] be a discrete-time periodic signal with period N = 7. What is the fundamental frequency . ?
o
b) (4 pts) The Fourier series coefficients of x[n] are known to be: a0 . 0, a . 0.5, a . 0, a .. j , a . j , a . 0, a . 0.5.
1 23 456
Express x[n] as a sum of complex sinusoids.
c) (3 pts) Express x[n] as a sum of real sinusoids.
1 N . 1
d) (3 pts) What is the average power in x[n]? (The average power of a DT periodic signal is . | x[n]|2 )Nn . 0
e) (3 pts) Determine what is the output y[n] if x[n] is the input to a DT filter with frequency response H (ej. ) as shown below:
j..
..e 1
2.
2. 14. 18..
.
2.
8
8 88
2.
7

f) (3 pts) Let x1[n] . x[n] ej n . Determine the Fourier series coefficients bk for x1[n].
g) (3 pts) Now let x1[n] be the input to the system with frequency response H (ej. ) as shown in (e). Determine the output signal y1[n].
h) (4 pts) Recall that the discrete-time Fourier transform (DTFT) of a discrete-time complex sinusoid is:
2.
.
jkn DTFT 2.
e .... 2( k . m2.)N . .. ..
m ... N
Sketch X (ej. ) , the DTFT of x[n] in part (a), for 0 ... 4. .

A CT signal x(t) is given as shown:

8 t
a) (2 pts) Just by considering x(t), can we say that its Fourier transform X ( j.) is real? Briefly explain.
b) (4 pts) Use tables 4.1 and 4.2 to find the Fourier transform of x(t).
c) (4 pts) Let x1(t) . x(t . 4).Sketch the magnitude and phase spectrum of x1(t). That is, sketch | X1( j.) | and

.X1( j.).
.
d) (2 pts) Sketch the signal x2(t) ..x(t .10k).
k ...
e) (2 pts) Let Y ( j.) below be the spectrum of the signal y(t). From the spectrum, can we tell whether y(t) is a periodic signal?

Y(j...
..

....
. .. f) (2 pts) Sketch the spectrum of y1(t) . y(3t)
g) (2 pts) In Amplitude Modulation, we multiply a signal with a sinusoid at a higher carrier frequency. Sketch the spectrum of y2(t) . y(t) cos 5t
h) (2