Sample Mid-Semester Examsubjects.ee.unsw.edu.au/elec3104/Sample MidSession Exam.pdfSample MSemester...
Transcript of Sample Mid-Semester Examsubjects.ee.unsw.edu.au/elec3104/Sample MidSession Exam.pdfSample MSemester...
Sample Mid-Semester Exam
1
Sample Mid-Semester Exam Q1)
a) A digital filter structure is shown below. Determine its transfer function. [5 marks]
b) A digital filter structure is shown below. Write the difference equation for
the filter’s output in terms of its input. [5 marks]
Q2) A first-order digital filter is described by the system function:
11
11
)(
za
zbzH
Draw a canonic realisation of the transfer function H(z). [5 marks]
Q3) a) A difference equation for a particular filter is given by
y(n) = 0.12 x(n) – 0.1 x(n-2) + 0.82 x(n-3) – 0.1 x(n-4) + 0.12 x(n-6) Find the impulse response of the above filter. [5 marks]
b) Using minimum number of multiplications, draw an implementation the filter given in 3(a). [5 marks]
c) A first-order digital filter is described by the system function :
11
1)(
za
zaazH Assume a = ½.
Sample Mid-Semester Exam
2
Determine the impulse response of the above digital filter H(z). [5 marks]
Q4) a) Sketch an approximate magnitude response from the pole-zero map given
below: [5 marks]
b) Determine the transfer function H(z) of a discrete-time system with the
pole-zero map given below. [5 marks]
c) Sketch an approximate magnitude response from the pole-zero map given
below: [5 marks]
Q5) a) Determine and sketch the approximate magnitude response for each of the
following filters:
Sample Mid-Semester Exam
3
y(n) = x(n) + x(n-1) [5 marks]
b) Determine and sketch the approximate magnitude response for each of the following filters: y(n) = x(n) – x(n-1) [5 marks]
c) Determine the magnitude response of the following filter and show that it has an all-pass characteristic.
1))/1((
)/1(1)( 1
1
aza
zazH [5 marks]
Q6)
a) An analogue signal x(t) = 3cos(2000t) + 5sin (6000t) + 10cos(12000t) is sampled 5000 times per second. What is the discrete-time signal obtained after sampling? [5 marks]
Q7) a) A filter has the following transfer function:
)9.0)(9.0()2(3)(
22 jj
ezez
zzzH
Sketch the poles and zeros map for the above filter. [5 marks] b) Using part 7(a), state whether or not the above transfer function
corresponds to stable filter. Why? [5 marks]
Q8) Proof the following properties of the Z-Transform: [9 marks] a) ax[n]+by[n] aX(z)+bY(z) b) x[n-k] z-kX(z) c) x[-n] X(1/z)
Q9)
a) Compute the N-point DFT, H[k], of the sequence h[n] [4 marks]
.
otherwise0
2031
nnh
b) Find the value of H[3] when N = 8. [2 marks]
Q10) A digital oscillator has a unit impulse response given by: nunnh 01sin
a) Find the transfer function of this oscillator. [3 marks] b) Draw a structure for this oscillator using the transfer function obtained in
part (a) above. [4 marks] c) By setting the input in part (b) to zero and under certain initial conditions,
sinusoidal oscillation can be obtained. Find these initial conditions. [3 marks]
Model Answers for Sample Mid-Semester Exam
4
Model Answers for Sample Mid-Semester Exam Q1)
a) 11
220
1
zb
zaazH
b)
21 ...21
21
21
nybnyb
nxanxanxny
Q2)
Q3)
a) 7for0
0012.001.082.01.0012.0
nnh
nh
b) 421.0382.0612.0 nxnxnxnxnxny
c)
15.0215.0
21
111
1
1
1
1
1
nununh
nuaanuaanh
az
az
az
azH
nn
nn
Q4)
Model Answers for Sample Mid-Semester Exam
5
a)
b)
5.011
5.04
cos2
211
11
1
215.05.0
2
2
2
2
2442
2
44
22
zz
zz
zz
zzzH
rzeerz
zzzH
rezrez
jzjzzzH
r
jj
jj
c)
Model Answers for Sample Mid-Semester Exam
6
Q5)
a)
cos22sincos1
11
22
1
H
eH
zzH
j
b)
cos22sincos1
11
22
1
H
eH
zzH
j
c)
filter. pass all 1
11111
1111
1
11
1
11
2
2
2*
*
H
ea
eaa
ea
eaaHH
ea
eaH
ea
eaH
jj
jj
j
j
j
j
Model Answers for Sample Mid-Semester Exam
7
Q6)
nnnx
nnnnx
f
f
ff
fsf
fsf
fsf
fs
54sin5
52cos13
500010002cos10
500020002sin5
500010002cos3
Hz100050006000Hz200050003000aliasing.by affected are and
.2
Hz6000 ;2
Hz3000;2
Hz1000
Hz5000
3
2
21
321
Q7) a)
b) Stable filter. The poles are inside the unit circle.
Q8) a) { [ ] [ ]}
∑ ( [ ] [ ])
∑ ( [ ] [ ])
∑ [ ] ∑ [ ]
( ) ( )
b) { [ ]} ∑ ( [ ])
∑ ( [ ])
∑ ( [ ])
Model Answers for Sample Mid-Semester Exam
8
( )
c) { [ ]}
∑ ( [ ])
∑ ( [ ])
∑ ( [ ])
( )
( ) Q9)
a) [ ] ∑ ( ) (
)
[ ]
[ ]
[ (
)]
b)
[ ]
[ (
)]
[ ]
[ (
)]
[
√ ]
[ √ ]
Q10) a)
210
0
0
cos21sin
: tabletransform-z From1sin
zzzH
nunnh
b)
c)
01sin0:conditions critical Two
1sin
0
0
yandy
nny
z-1 z-1 +
y(n)
-1
y(n-1)
y(n-2)
2cos0
x(n)
sin0