EENGIINFE 420 Digital Signal Processing PI 1 25 P2 1 25 F3 1 25 P4 1 25 P5 1 25 Total 1 100 FALL 2016-2017 MIDTERM EXAM Name(fuU): __________ St. Number: _________ Lecture r: Prof. Dr. Erhan A. ince Date: 22 /1112016 Duration: 90 min Read the following instructions carefully: 1) Please put your name on both the question paper and the answer booklet. 2) Use / ront and back of each page on the anwers booklet to answer questions. 3) Please answer Any FOllR questions you like. Problem 1 For two discrete-time systems that are characterized by the input-output relationships depicted below, determine if the systems are (a) linear, (b) time-invariant and (c) causal. (i) y[n] = x[n] . cos (3n) (ii) y[n] = x[n-2] + x[2-n] Problem 2 For each of the linear time-invariant systems described by the input-output relationships shown below detennine the corresponding impulse response h[n], (i) y[n] = x[n] + 2x[n - 1] + x[n - 2] (ii) Y [n] - - 1] + - 2] = - 1] 6 6 3
Transcript
EENGIINFE 420
Digital Signal Processing
PI 125
P2 125
F3 125
P4 125
P5 125
Total 1100
FALL 2016-2017
MIDTERM EXAM
Name(fuU) __________
St Number _________
Lecturer Prof Dr Erhan A ince
Date 221112016
Duration 90 min
Read the following instructions carefully
1) Please put your name on both the question paper and the answer booklet
2) Use ront and back of each page on the anwers booklet to answer questions
3) Please answer Any FOllR questions you like
Problem 1
For two discrete-time systems that are characterized by the input-output relationships depicted below determine if the systems are (a) linear (b) time-invariant and (c) causal
(i) y[n] = x[n] cos (3n) (ii) y[n] = x[n-2] + x[2-n]
Problem 2
For each of the linear time-invariant systems described by the input-output relationships
shown below detennine the corresponding impulse response h[n]
(i) y[n] = x[n] + 2x[n - 1] + x[n - 2]
(ii) Y [n] - ~y[n - 1] + ~y[n - 2] = ~x[n - 1]6 6 3
Problem 3
Given a discrete time sequence x[n] = 2O111011
(i) Compute the fast Fourier Transform values showing all computations
(ii) Also draw and label the butterfly structures required
where t is in milliseconds This signal is pre-filtered by an analog pre-filter H(f) Then the outputy(t) of the pre-filter is sampled at a rate of 40 kHz and immediately reconstructed by an ideal analog reconstruction filter resulting in the final analog output Ya(t) as shown below
)x(t) prefilter yet)
40 kHz y(n1)
analog yaltt
alog an H(j)
analog sampler
digital reconstructor
anal0 g
Determine the output signals yet) and Ya(t) in the following cases
(a) When there is no pre-filter thatis H(f)= 1 for all f
(b) When H(f) is the ideal pre-filter with cutoffs2 = 20 kHz
Problem 5
Let x(t) be the sum of sinusoidal signals
x(t)= 4 + 3 cos(nt)+ 2 cos(2nt) + cos(3nt) where t is in milliseconds
(a) Determine the minimum sampling rate that will not cause any aliasing effects that is
the Nyquist rate
(b) To observe such aliasing effects suppose this signal is sampled at half its Nyquist rate
and determine the signal Xa(t) that would be aliased with x(t)
lt
B-o o k (yen) - I
)~ Q X I (Ij -gt JI L1 J Clx [1 J Cos lt3VL
bXZ Co)) 7 ~ 1- J= b ~L l f) (cgtS 3 rL
Lj [P1 0 ~ l () J+ b Xl l 1 7 ~ ((1J 0-x l3 -t ~l(l LY [053 VLshy
ct X LI1J (Or~ct11 + bXL lnj (~3r-L1
~ I lll ) + ~~ ltU
l Ltlj ~ X1LJ CO 3 ft
~ CA) C9-lbj -gt ch lflJ ~ x (l-noJ ioS3rlshy
- ~ In-roJ ~[n()1gtJ Co5 J(f-n)
~ ~ [I)J 4- ~ pound11-(1 J
01) ~ vJ X- (P - Z J +( ll-ilj c-~(~J gt ~ LA) ~~ [n-zJ +CCpound [2-J ~~ pound~J 01- CJ b -2 [11- 2j -t b~LlmiddotflJ
where t is in milliseconds This signal is pre-filtered by an analog pre-filter H(f) Then the outputy(t) of the pre-filter is sampled at a rate of 40 kHz and immediately reconstructed by an ideal analog reconstruction filter resulting in the final analog output Ya(t) as shown below
)x(t) prefilter yet)
40 kHz y(n1)
analog yaltt
alog an H(j)
analog sampler
digital reconstructor
anal0 g
Determine the output signals yet) and Ya(t) in the following cases
(a) When there is no pre-filter thatis H(f)= 1 for all f
(b) When H(f) is the ideal pre-filter with cutoffs2 = 20 kHz
Problem 5
Let x(t) be the sum of sinusoidal signals
x(t)= 4 + 3 cos(nt)+ 2 cos(2nt) + cos(3nt) where t is in milliseconds
(a) Determine the minimum sampling rate that will not cause any aliasing effects that is
the Nyquist rate
(b) To observe such aliasing effects suppose this signal is sampled at half its Nyquist rate
and determine the signal Xa(t) that would be aliased with x(t)
lt
B-o o k (yen) - I
)~ Q X I (Ij -gt JI L1 J Clx [1 J Cos lt3VL
bXZ Co)) 7 ~ 1- J= b ~L l f) (cgtS 3 rL
Lj [P1 0 ~ l () J+ b Xl l 1 7 ~ ((1J 0-x l3 -t ~l(l LY [053 VLshy
ct X LI1J (Or~ct11 + bXL lnj (~3r-L1
~ I lll ) + ~~ ltU
l Ltlj ~ X1LJ CO 3 ft
~ CA) C9-lbj -gt ch lflJ ~ x (l-noJ ioS3rlshy
- ~ In-roJ ~[n()1gtJ Co5 J(f-n)
~ ~ [I)J 4- ~ pound11-(1 J
01) ~ vJ X- (P - Z J +( ll-ilj c-~(~J gt ~ LA) ~~ [n-zJ +CCpound [2-J ~~ pound~J 01- CJ b -2 [11- 2j -t b~LlmiddotflJ
