Unformatted text preview: EECS 20. Final Exam Solution
May 15, 2000.
1. (a) Linear: all except S3 . S , S , and S .
Causal: S , S and S .
Since the system is causal, hn = 0 for n 0. In addition, h satisﬁes
hn = n + n , 1 , hn , 1 (b) Time invariant:
(c)
2. (a) 1 1 2 3 3 6 (just let the input be an impulse). Thus, h0
h1
h2
h3
h4
hn = 1 = 1 = ,
1 ,
, 1 , = =
= , , 1 2 3 , n,1 1 , so hn = , n, un , 1 + ,
where un is the unit step function.
1 n un; (b) Although we could calculate the DTFT of the impulse response, it is easier to just let
the input be a complex exponential,
xn = ei!n :
The output then will be
yn = H !ei!n :
Hence, the following equation must be satisﬁed,
H !ei!n + H !ei!n,1 = ei!n + ei!n,1 : We can factor out ei!n and divide through by it, getting
H !1 + e,i! = 1 + e,i! :
Hence, ,i! +
H ! = 11+ ee,i! : (c) The output will be zero if the frequency ! of the sinusoid is such that H ! = 0. This
occurs if e,i! = ,1, which occurs if ! = . Thus, the following input will yield zero
output: xn = cosn:
1 (d) omega = [0: pi/400: pi];
H = (1 + exp(i*omega))./(1 + alpha*exp(i * omega));
plot(omega, abs(H));
(e) A reasonable choice for the state s is
sn = xn , 1; yn , 1 T :
With this choice, A= (f)
3. " 0 , 1 ; b= " " ; c = , ; d = 1:
If = 1, the frequency response becomes H ! = 1 and the impulse response becomes
hn = n.
0 1 1 1 (a) False. The output frequency may not be the same as the input frequency.
(b) False. You only know the response to one frequency.
(c) True. The frequency response is the DTFT of the impulse response.
(d) True. The impulse response is
frequency response. yn , yn , 1, from which you can determine the (e) True. If the system were LTI, the response to the delayed impulse would be the delayed
impulse response. (f) False. The system might be LTI with impulse response given by hn = y n , y n ,
2 + y n , 4 , y n , 6 + . 4. (a) The fundamental frequency is ! 0 radians/second : = 10 (b) The Fourier series coefﬁcients are A ;A 0 =0 k =0 1 ;A =1 2 ;A =1 3 ; Ak = 0 for k =1 3 ; and
for all k: (c) The sampled signal is yn n=10 + cos20n=10 + cos30n=10
= 1 + 2 cosn:
The fundamental frequency is therefore ! = radians/sample.
= cos10 0 (d) The DFS coefﬁcients are A 0 ;A =1 1 ; =2 1 =0 : There are no more coefﬁcients, since the period is p = 2.
(e) The “smoothest” (lowest frequency content) interpolating signal is wt = 1 + 2 cos10t:
2 (f) Yes, there is aliasing distortion. The 10 Hz cosine has been aliased down to DC, and the
15 Hz cosine has been aliased down to 5 Hz, overlapping the 5 Hz cosine.
(g) Sampling at twice the highest frequency will work. The highest frequency is 15 Hz, so
sampling at 30 Hz will avoid aliasing distortion.
5. The sawtooth signal has period p = 1 second, so its fundamental frequency is 2 radians/second, considerably above the passband of the ﬁlter. Thus, only the DC term gets
through the ﬁlter. The DC term is the average over one period, which is 1/2, so the output is yn = 1=2:
6. (a) False.
(b) True.
(c) True.
(d) True.
(e) False. 7. The machine is shown below:
{0}/0 {1}/0 {1}/0 e f {0}/0 {0}/0 The simulation relation is fa; e; b; f ; c; g; d; gg: 3 {1}/1 g ...
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 Spring '08
 Ayazifar
 Electrical Engineering, Digital Signal Processing, Signal Processing, LTI system theory

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