Sample Mid-Semester Exam - University of New South MSemester Exam id-1 Sample Mid-Semester Exam Q1)

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Transcript of Sample Mid-Semester Exam - University of New South MSemester Exam id-1 Sample Mid-Semester Exam Q1)

  • 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 filters 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 = .

  • 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)( 11

    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