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(ELEC211)[2006](f)midterm~=tcs7dd^_32706.pdf
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1. (40 points)
. 1/ . 0 t .
a) (6 pt) . (t) is the narrow pulse function such that . (t)=.
0 otherwise
.
Let x(t) be an arbitrary signal as shown below . (t) and x(t) be as shown below: x(t) . (t)


Let r(t) = x(k.) . (t . k.) . . Sketch r(t).
k =.
b) (4 pt) The convolution integral is something that you should have remembered by heart. Write down the convolution integral between two CT signals x(t) and h(t).
x(t) * h(t) = ?
c) (6 pt) Let x[n] = 2 [n +1] + [n] + 3 [n .1] be the input to a DT LTI system with unit sample response h[n] =. [n +1] + [n] . The output of the LTI system is given by:

y[n] = x[k] h[n . k]
k =.
determine and plot y[n].
d) (6 pt) Let x.[n] be a periodic DT signal with period N = 8 and x.[n] is equal to x[n] for C3 < n < 4. Since x.[n] is
periodic, it can be represented by a discrete-time Fourier Series. Determine its Fourier Series coefficients a0 and
a1.
e) (6 pt) For each statement below, state if it is true or false. There is no need to explain. You get 2 points for a correct answer, 0 point for abstaining, and C1 point for an incorrect answer.
(i) a = a
.71
(ii) a = a
.11
(iii) a = a *
.11
2

N
f) (4 pt) For x.[n] with period N = 8 in part (d), let r[n] =ejM nx.[n] where M = 2. Determine the values of r[n] for 0 < n < 7.
g) (4 pt) Let bk be the Fourier Series coefficients of r[n]. Determine the value of b0.
h) (4 pt) Determine the value of t 3 (t . 2) dt . . 2 + t
r(t)
Solution : a)


b) x(t)* h(t) = x() h(t . ) d = x(t . ) h() d
. .

y[n]
c) x[k]
3 3
2
1


1 C2





C1 0 1 k C1 0
1 n
h[n . k]

1

C2
n +1 C2 n

C1
y[n] =.2 [n + 2] + [n +1] . 2 [n] + 3 [n .1] 141 3
d) a0 = x[n] = (2 +1+ 3) = 8 n =. 38 4
2 2 2
14 . jn 1 . j . j . 1 . j . j . 1 . 51 .
888 44
a = x[n] e =. 2e +1+ 3e ..1+ 2e + 3e ..1+
1 ..=..=
8 n =. 38 88 .
. ...
e) (i) True (ii) False (iii) True
2 2
jMn j(2) n jn
N 82
f) r[n] = ex.[n] = ex.[n] = ex.[n] = ( j)nx.[n]
01 .1
r[0] = ( j) x.[0] = 1 r[1] = ( j) x.[1] = j3 r[7] = r[.1] = ( j) x.[.1] =. j2
r[2] = r[3] = r[4] = r[5] = r[6] = 0

2
j (2) n
g) r[n] = e 8 x.[n] bk = ak . M = ak . 2 b0 = a. 2
2 2
4
11 . j(2) (. 1) j(2) (1) . 1 .. jj . 1
b0 =r[n] =. 2e 8 +1+ 3e 8 ..= . 2e 2 +1+ 3e 2 .. = (1+ j)

..
8 n =.38 .. 8 .. 8
2
14 . j(. 2) n 1 .. jj . 11
Alternatively, b0 = a. 2 = x.[n] e 8 =. 2e 2 +1+ 3e 2 ..= (. j2 +1+ j3)= (1+ j)

.
8 n =. 38 .. 88
t 21 1
h) (t . 2) dt = (t . 2) dt = (t . 2) dt =. 2 + t3 . 2 + (2)3 5 . 5
2. (40 points)
. 4t 4t . 4| t|
a) (6 pt) Sketch the following signals: (i) eu(), (ii) eu t. e
t (), (iii)
b) (6 pt) Use Table 4.1 and 4.2 to find the Fourier Transform of eu t.
4t ().
. 4| t|
c) (6 pt) Let h(t) =e be the impulse response of an LTI system. Determine the fr