Download - Sample Mid-Semester Exam - University of New South … MSemester Exam id-1 Sample Mid-Semester Exam Q1) a) A digital filter structure is shown below. Determine its transfer function.

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.


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:


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)


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)


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


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) { [ ]} ∑ ( [ ])

∑ ( [ ])

∑ ( [ ])


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