Question Bank Digital Signal Processing

Digital Signal Processing
Question Bank
Department of Electronics Engineering
Dr. Nilesh Bhaskarrao Bahadure
Page 1

UNIT – I
1. What is Digital signal processing
2. What is signals
3. Explain differences between DSP and ASP
4. Explain the advantages of DSP
5. Explain the limitations of DSP
6. Explain the classification of signals
7. Explain elementary continuous time signals with suitable examples.
8. Explain elementary discrete – time signals with suitable examples.
9. Define system; explain classification of systems in details.
10. Explain in details the following with suitable example
1. Stable & Unstable system
2. Static & Dynamic System
3. Linear & non – linear system
4. Causal & Non – causal system
11. Explain time variant and time invariant system with suitable example.
12. Write short notes on the following
1. Energy and power signals
2. Even and odd signals
3. Periodic & Aperiodic signals
4. Causal & non – causal signals
13. Evaluate the following integrals
1. ∝
10
2. 3
3. sin 2
4. 3
5. 3 3
6. cos 1 sin
7.
14. Find the following summations
1. ∑ 2 sin 2
2. ∑
3. ∑ 1
4. ∑ 2 cos 2 1 sin 2
5. ∑ 1
15. Find the fundamental period T of the following continuous – time signals
1. X(t) =
2. X(t) = sin 50
Total Questions 370

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3. X(t) = 20 cos 10
16. Find whether the following signals are periodic or not also find period of the signals
1. X(t) = 2 cos 10 1 sin 4 1
2. X(t) = cos 60 sin 50
3. X(t) = 2 u(t) + 2 sin 2t
4. X(t) =  3 cos 4 2 sin 2
5. X(t) = u(t) – ½

6. X(t) =
3 cos 17 2 sin 19
7. X(t) = u(t) – u (t‐10)
8. X(t) = cos sin
9. X(t) = 12

9 ^ 3
17. Find whether the following signals are periodic or not
1. cos 2
2.
3. sin 1
4.

5.
6. 12 cos 20
18. Find the fundamental period of the following signals
1. X(t) = 2 sin 3 1
3 sin 4 1
2. X(t) = sin
3. X(n) = sin 2 sin 6
4. X(n) =
19. Find the even and odd components of the following signals.
1. X(t) = cos sin cos . sin
2. X(n) = {‐2, 1, 2, ‐1, 3}
3. X(t) = sin 2 sin 2 sin cos
4. X(n) = {1, 0, ‐1, 2, 3}
20. Find which of the following signals are causal & non – causal?
1. X(t) =
2. X(t) =
3. X(t) = sinc
4. X(n) = 2
21. Sketch the following signals
1. 1
2. X(t) = 2 1
3. X(t) = 3 1
4. X(t) = 2
5. X(t) = 2
6. X(t) = 3
7. 2 3
8. 2 .
22. Obtain energy for the signal x(n) = where a < 1.
23. Determine the values of energy & power of the following signals. Find whether the signals are
power, energy or neither energy nor power signals.
1. X(n) =
2. X(n) =
3. X(n) = sin
4. X(n) =
24. Determine the power and RMS value of the signal
X(t) = cos )
25. Determine the power and RMS value of the following signals
1. X(t) = 5 cos 50

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2. X(t) = 10 sin 50
16 cos 100
3. X(t) = 10 cos 5 cos 10
4. X(t) =
cos
5. X(t) =
26. Sketch the following signals and calculate their energies
1.
2. 15
3. cos 2 . . 2
27. Find which of the following signals are energy signals
1. 1
2. 2
3. 1
28. Find which of the following signals are energy signals, power signals, neither energy nor power
signals
1.
2.
3. cos
29. Find whether the following systems are dynamic or not
1. 2
2. 2
3.
30. Determine whether the system described by the difference equation is linear or not
1. 2
2. 4
31. Check whether the following systems are causal or non – causal
1.
2. 2
3. 2 2
4.
5.
6.
7. . 1
8.
9.
10. ∑
11. cos
32. Check whether the following systems are linear or not
1. 3
2. 2
3.
4. 2 .
5.
6. 2
7.
8.
9.
10.
11. 2

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12. . cos
33. Determine whether the following systems are time – invariant or not
1.
2. . cos 50
3.
4.
5.
6. 2
7. 1
8. 1
34. Check whether the following systems are time – invariant
1.
2. 5
3. 10
4.
5. 1
1
6. sin
35. Check whether the following systems are
1. Static or dynamic
2. Linear or non – linear
3. Causal or non – causal
4. Time – invariant or time – variant
(i)

(ii)
2
(iii)
(iv) 1
(v) cos
(vi) 10
(vii) ∑
(viii)
(ix)
36. Determine which of the following signals are periodic
1. 1 sin 15
2. 2 sin 20
3. 3 sin √2
4. 4 sin 5
5. 5 1 2
6. 6 2 4
37. Show that the complex exponential signal x(t) = is periodic and its fundamental period is
2 /
38. Check whether each of the following signals is periodic or not. If a signal is periodic, find its
fundamental period.
1. cos sin √2
2. sin
3.
4.
5. cos
6. cos sin
7. cos
39. Determine the fundamental period of
(a) x t cos sin
(b) cos 2
40. Determine the even & odd component of

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41. Show that the product of two even signals or of two odd signals is an even signal and that the
product of an even and an odd signal is an odd signal.
42. Determine the energy and power of the following signals
1. | |

2.
3.
43. Determine whether the following signals are energy signals, power signals or neither
1. , 0
2. cos
3.
4. 0.5
5. 2
44. Write short note on analog to digital converter.
45. Explain in details the sampling theorem, also explain nyquist rate criterion for the sampling of
the analog signal.
46. Construct the discrete time signal from the two analog sinusoidal signals shown below
1 cos 2 10
2 cos 2 50
Which are sampled at a rate Fs = 40 Hz.
47. If the continues – time signal
2 cos 400 5 sin 1200 6 cos 4400 2 sin 5200 is sampled at a 8
KHz rate generating the sequence x[n], also find x[n].
48. Determine the nyquist frequency and nyquist rate for the following sequence
(a) 50 cos 1000
(b) 20
(c) 5 5
49. Determine the nyquist rate for the analog signal given by
2 cos 50 5 sin 300 4 cos 100
50. The transfer of an ideal band pass filter is given by
1 45 60
           =    0 otherwise
Determine the minimum sampling frequency to avoid aliasing.
51. Consider the analog signal
3 cos 100
(a) Determine the minimum sampling rate required to avoid aliasing
(b) Suppose that the signal is sampled at the rate Fs = 200 Hz. What is the discrete time signal
obtained after sampling?
(c) Suppose that the signal is sampled at the rate Fs = 75 Hz. What is the discrete time signal
obtained after sampling?
(d) What is the frequency 0 < F < Fs/2 of a sinusoid that yields samples identical to those
obtained in part (c)?

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3 cos 50 10 sin 300 cos 100
What is the nyquist rate for this signal?
3 cos 2000 5 sin 6000 10 cos 12000
(a) What is the nyquist rate for this signal?
(b) Assume now that we sample this signal using a sampling rate Fs = 5000 samples/sec. what is
the discrete time signal obtains after sampling?
(c) What is the analog signal what we can reconstruct from the samples if we use ideal
interpolation?
54. Draw and explain analog to digital converter system, also explain in details the sampling process.
55. Explain the followings
(a) Sampling
(b) Quantization
(c) Encoding
56. Determine which of the following signals are periodic. If periodic determine the fundamental
period
1. 3
2. 1
3. 3 3 sin
4. ∑ 1 2
5. cos sin
57. Determine the energy and power of signal given by
1
2
0
                         3 n 0
58. Fine energy and power of the following signals
1.
2. 0
           =   0    n < 0
59. Determine whether the following Discrete time systems are stable or not
1.
2. 6
3. cos
4.
5. 2

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University Question Bank
(Electrical, EEE, Electronics)

1. (a) Name the different properties of the discrete time system.
(b) Differentiate between an energy and power signal. Also determine the energy and
power of the signal

(c) The discrete time system x(n) and y(n) are shown:

Sketch the signal 2 4
(d) State the sampling theorem.
A given analog signal is 2 cos 200 3 sin 600 8 cos 1200
1. Calculate the nyquist rate
2. If f(t) is sampled at a rate of fs = KHz. What is the discrete time signal?

1. (a) What are energy and power signals?
(b) Check whether the following systems are:
1. Static or dynamic
2. Causal or non – causal
3. Linear or non – linear
∑

1
(c) Consider the discrete time system is excited by following sequence:
1 0 3
           = 0    elsewhere
Find out the response y[n] if x[n] and y[n] are related by following relation:
i.
ii. 1
iii. 1
iv. 1 1

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(d) A discrete time signal is shown in figure – 1 carefully determines and sketch the
following signal

i. 2
ii. ∑
iii. 2 2

1. (a) Determine the Nyquist rate for the continuous time signal
6 cos 50 20 sin 300 10 cos 100
(b) (i) Decompose the discrete time signal x(n) shown into even and odd parts

(ii) A discrete time system is described by the following expression
1
Now a bounded input of x (n) = 2δ (n) is applied to this system. Assume that the system
is initially relaxed, check whether the system is stable or unstable.
(c) With illustration, explain shifting, folding and time scaling operations on the discrete
time signals.
(d) (i) Sketch the following signal
| |
0
Also determine the energy and power of the signal.
(ii) Explain if the following system described by sin 2
Is memoryless, causal, linear, time – invariant, stable

1. (a) Is the system described by the following equation stable or not? Why?

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(b) (i) Determine whether or not each of the following signals is periodic. If a signal is
periodic, specify its fundamental period
1. 1
2. 3
3. 4 3
(ii) Determine the fundamental period of the signal 2 cos 10 1
sin 4 1
(c) (i) Show that the signal 1 is neither an energy nor a power
Signal
(ii) Check the causality of the given system
(a)
(b) cos 1
(d) (i) A discrete time system is characterized by the following difference equation:
1 Check the system for BIBO stability
(ii) Consider the system shown in figure. Determine whether it is
(a) Memoryless
(b) Causal
(c) Linear
(d) Time – invariant
(e) Stable

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UNIT – II

1. Represent the sequence , , , , , , , as sum of shifted unit impulse.
2. Explain the properties of convolution
3. Explain causality of linear time ‐ invariant system
4. Explain the stability condition of linear time – invariant system
5. Finite duration sequence given as , , , resolve the sequence x(n) into sum of
weighted impulse.
6. Represent the sequence , , , , , as sum of shifted unit impulses.
7. Test the stability of the system whose impulse response
8. The impulse response of a linear time – invariant system is , , , determine
the response of the system to the input signal , , ,
9. Obtain linear convolution of following sequences
, , , , ,
10. Compute the convolution y(n) of the signals

,


,
,

11. Given
a. Plot x1(n) and x2(n)
b. Calculate and plot y(n) = x1(n) * x2(n) for all n
c. What are non – zero lengths of x1(n), x2(n) and y(n)
12. Prove and explain graphically the difference between relations:
a.
b. ∗
13. Obtain the expression for convolution of unit step sequence with finite duration sequence
14. Obtain linear convolution of two discrete time signals as
, 1
15. Use discrete convolution to find the response to the input of the LTI system
with impulse response
16. Impulse response of an LTI system is given as find the output y(n)
of the system to an input
17. Obtain linear convolution of the following sequence , , ,
, , ,
18. Obtain linear convolution of the discrete time signals

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19. Determine the convolution sum of two sequences , , , ; , , ,
20. Digital filter is characterized by the difference equation check the
filter for BIBO stability
21. Check whether the following digital systems are BIBO stable or not
(a)
(b)
(c)
(d) −1)
(e) −1)
(f) , ,
(g) , ,
(h)
22. Check the BIBO stability for the impulse response of a digital system given by
23. Determine the range of values of parameters ‘a’ for the LTI system with impulse response
,
= 0 , otherwise
To be stable
24. Test whether the following system are stable or not.
(a)
(b) ∑
(c) | |
(d)
25. From the impulse response h(n) of the system, find whether the following systems are causal
and stable
(a)
(b)
(c)
(d) | |

26. Determine the impulse response for the cascade of two linear time – invariant systems having
impulse responses.

27. Determine the range of values of a and b for which the linear time – invariant system with
impulse response

,
, 0

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Is stable also check causality?
28. Determine the homogeneous solution of the system described by the 1st
order difference
equation

29. Determine the zero input response of the system described by the homogeneous 2nd
order
difference equation

Also find the solution by assuming initial conditions

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UNIT – II and III

2 Number Questions
1. Represent the sequence , , , , , , , as sum of shifted unit impulse.
2. Explain the properties of convolution
3. Explain causality of linear time ‐ invariant system
4. Explain the stability condition of linear time – invariant system
5. Finite duration sequence given as , , , resolve the sequence x(n) into sum of
weighted impulse.
6. Represent the sequence , , , , , as sum of shifted unit impulses.
, , , , ,
8. What is convolution, explain with example
9. Prove that
10. Write short notes on homogeneous solution
11. Write short notes on particular solution
12. Write short notes on impulse response
13. Explain the importance of convolution
14. What is LTI System
15. Write the general difference equation for LTI system
16. Explain cascade and parallel interconnection method.
17. Show that the necessary and sufficient condition for a LTI system to be stable is
∑ | | ∞
Where is the impulse response?
18. The system shown in figure (a) is formed by connecting two systems in cascade. The impulse
responses of the system are given by and respectively and
, . Find the impulse response of the overall system shown in
figure (b)

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Long Answer Questions

1. Test the stability of the system whose impulse response
2. The impulse response of a linear time – invariant system is , , , determine
the response of the system to the input signal , , ,
, , , , ,
4. Compute the convolution y(n) of the signals

,


,
,

5. Given
a. Plot x1(n) and x2(n)
b. Calculate and plot y(n) = x1(n) * x2(n) for all n
c. What are non – zero lengths of x1(n), x2(n) and y(n)
6. Prove and explain graphically the difference between relations:
7.
8. ∗
9. Obtain the expression for convolution of unit step sequence with finite duration sequence
10. Obtain linear convolution of two discrete time signals as
, 1
11. Use discrete convolution to find the response to the input of the LTI system
with impulse response
12. Impulse response of an LTI system is given as find the output y(n)
of the system to an input
13. Obtain linear convolution of the following sequence , , ,
, , ,
14. Obtain linear convolution of the discrete time signals

15. Determine the convolution sum of two sequences , , , ; , , ,
16. Find the convolution of the signals
, , ,
,
,

17. Find the convolution of two finite duration sequence

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i. When a
ii. When a = b
18. Find y(n) if x(n) = n + 2 for

19. Determine the response of the relaxed system characterized by the impulse response
to the input signal
20. Compute convolution of following sequences
(a) , ,
(b) ;
(c) ;
21. Find the convolution of two finite duration sequences

,
,
   And
,
,

22. Find the convolution of the two signals shown below

23. Compute the convolution y(n) = x(n) * h(n) of the signals
, , , , , , ,
24. Digital filter is characterized by the difference equation check the
filter for BIBO stability
25. Check whether the following digital systems are BIBO stable or not
(a)
(b)
(c)
(d) −1)
(e) −1)
(f) , ,
(g) , ,
(h)
26. Check the BIBO stability for the impulse response of a digital system given by
27. Determine the range of values of parameters ‘a’ for the LTI system with impulse response
,
            = 0    , otherwise

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To be stable
28. Test whether the following system are stable or not.
(a)
(b) ∑
(c) | |
(d)
29. From the impulse response h(n) of the system, find whether the following systems are causal
and stable
(a)
(b)
(c)
(d) | |

30. Determine the impulse response for the cascade of two linear time – invariant systems having
impulse responses.

31. Determine the range of values of a and b for which the linear time – invariant system with
impulse response

,
, 0

Is stable also check causality?
32. Determine the homogeneous solution of the system described by the 1st
order difference
equation

33. Determine the particular solution of the difference equation

When the forcing function , and zero elsewhere.
34. Determine the total solution , , to the difference equation

When x (n) is a unit step sequence [i.e. is the initial condition.
35. Determine the response , , of the system described by the 2nd
order difference
equation.

When the input sequence

36. Determine the impulse response h(n) for the system described by the second – order
difference equation

37. Determine the zero input response of the system described by the homogeneous 2nd
order
difference equation

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Also find the solution by assuming initial conditions
38. Find the natural response of the system described by difference equation

With initial condition

With initial condition
40. Find the forced response of the system described by difference equation

When the input is
41. Find the response of the system described by the difference equation

For the input , with initial conditions
42. Find the natural response of the system described by the difference equation
. . ; ,
43. Find the forced response of the system described by the difference equation
. . ;
44. For a given system
. . ; ,
Find the response due to the input
45. Find natural response with initial conditions y(‐1) = y(‐2) = 1 for the difference equation

46. Find the forced response for input for the difference equation

47. Find the response with initial conditions y(‐1) = y(‐2) = 1

48. Determine the impulse response h(n) for the difference equation
. .
49. Determine the impulse response h(n) for the difference equation

50. Determine the impulse response of the difference equation
(a)
(b)
(c) . .
51. Determine the step response of the system described by

52. Determine the natural response

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(a) ; ,
(b) ; ,
53. Find the forced response
(a)
(b)
54. Determine the step response
(a)
(b) . .
(c)
55. Find the solution to the difference equation
(a)
With initial conditions
(b)
With initial conditions

When the input is with initial condition
57. Determine the impulse response h(n) for the system described by the second – order
difference equation
. .
58. Determine the impulse response h(n) for the system described by difference equation

59. Determine the impulse response h(n) for the system described by the 2nd
order difference
equation

60. Find the impulse response and step response of a discrete – time linear time – invariant
system whose difference equation is given by
.
61. Find the impulse response and step response for the given system

62. Consider a causal and stable LTI system whose input x(n) and output y(n) are related through
the 2nd
order difference equation

Determine the impulse response of the system.
63. Find the step response of the system

64. Solve the difference equation

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For the input sequence
;
; 0

Assume the initial condition y(n) = 0 for n < 0
65. Find the response of the following difference equation
,
66. A system is described by the difference equation
. Assuming that the system is initially relaxed, determine its unit sample response.
67. Determine the solution of the difference equation

68. Obtain the frequency response of the first order system with difference equation
with initial condition and sketch it. Comment about its
stability.
69. Determine the impulse response h(n) for the system described by the 2nd
order difference
equation

70. Find the magnitude and phase responses for the system characterized by the difference
equation.

71. Find the impulse response, frequency response, magnitude response and phase response of
the 2nd
order system

72. The output y(n) for an LTI system to the input x(n) is

Compute and sketch the magnitude and phase of the frequency response of the system for
| |
73. Determine the frequency response, magnitude response, phase response and time delay of
the system given by

74. Find the magnitude response and impulse response of a system described the difference
equation

75. LTI system is described by . plot magnitude response of system using
analytical method.
76. Find the magnitude and phase response of given system at frequency
.

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UNIT – IV to VI

1. Explain discrete time Fourier transform, also explain the difference between the Fourier
transform and DTFT
2. Write short note on existence of Fourier transform / DTFT
3. What is discrete Fourier transform, show the equation using twiddle factor
4. What do you mean by twiddle factor.
5. Explain the properties of Discrete Fourier transform
6. Define the Parseval’s theorem.
7. Define zero padding, also explain the use of zero padding in discrete Fourier transform
calculation.
8. Explain why number of elements (N) is greater than or equal to the length of sequence (L) in
the calculation of discrete Fourier transform
9. Explain with suitable example why linear convolution and circular convolution is produced
different result.
10. Explain with suitable example how to generate same result using linear and circular
convolution.
11. Explain the differences between linear and circular convolution.
12. What is meant by radix – 2 FFT
13. What is decimation – in – time algorithm?
14. What is decimation – in – time algorithm?
15. Explain the differences between DIF and DIT algorithms
16. Explain in details the comparison between DFT and FFT.
17. Explain in details how the FFT algorithms are efficient method or less complex compared to
DFT.
18. Obtain DTFT of unit impulse (n)
19. Obtain DTFT of unit step
20. Find DTFT of
21. Find DTFT of
22. Find DTFT of
23. Obtain DTFT of
24. Obtain DTFT of left handed exponential signal
25. Obtain DTFT of a double sided exponential signal
26. Find DTFT of , , ,
27. Find the DTFT of the following finite duration sequence of length L

28. Obtain DTFT of a signal , 1

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29. Obtain DTFT and sketch the magnitude spectrum for
30. Find DTFT of .
31. Find DTFT of | |

32. Find discrete time signal for
–

DISCRETE FOURIER TRANSFORM
33. Find DFT of unit impulse
34. Obtain DFT of delayed unit impulse
35. Obtain N – point DFT of exponential sequence
36. Find 4 – point DFT of the following sequence , , ,
37. Find 4 – point DFT of the sequence
38. Determine the DFT of the sequence
(a)
,
,

(b)

,

39. Derive the DFT of the sample data sequence , , , , , and compute the
corresponding amplitude and phase spectrum
40. Find the 4 – point DFT of the    function
41. Compute the DFT of

42. Compute N – point DFT of
43. Compute the DFT of the sequence given as
(a) N = 3
(b) N = 4
44. Obtain the value of for 8 – point DFT if , , , , , , ,
45. Determine and sketch the DFT of signal , , , ,
46. Determine 2 – point and 4 – point DFT of a sequence
Sketch the magnitude of DFT in the both the cases.
47. Obtain 2 – point and 4 – point DFT of
48. Calculate 8 – point DFT of , , ,
49. Find 8 – point DFT of , , ,
50. Find the sequence for which DFT given by , , ,
51. Find IDFT of the following sequence , , ,
52. Find IDFT of , , ,

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53. Find the inverse DFT of , , ,

CIRCULAR CONVOLUTION
54. Find the circular convolution of two sequences , , , , , ,
57. Find the linear and circular convolution of two sequences , , ,
, , ,
58. Using graphical method, obtain a 5 – point circular convolution of two DT signals defined as,
. ,
,
Does the circular convolution obtained is same to that of linear convolution?
59. Find circular convolution of two finite duration sequences
, , , , , , ,
60. Find the 8 point circular convolution for following sequences
, , , , , , ,
61. Compute the circular convolution of following sequences and compare the results with linear
convolution
, , , , , , , , , , , , , ,
62. Find the linear convolution of , , , ,
, , obtain the same result using circular convolution.
63. Let
,
,
      find  the circular convolution of the sequence
64. Compute (a) linear convolution (b) Circular convolution of the two sequences
, , , , , , . (c) also find the circular convolution using DFT & IDFT
65. Compute ∗   if
   and

66. Obtain the linear convolution of two sequences defined as,
,

67. Using circular convolution, find output of system if input x(n) and impulse response h(n) given
by

68. Use the four point DFT and IDFT to determine the circular convolution of sequences
, , , , , ,
69. Find the response of an FIR filter with the impulse response , , to the input
sequence ,

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70. Determine the response of FIR filter using DFT if , ,
71. The first five points of the 8 – point DFT of a real valued sequence are . ,
. , , . . , determine the remaining three points.
72. The first five points of real and even sequence x(n) of length eight are given below. Find
remaining three points , , , , , , … … .

FAST FOURIER TRANSFORM
73. Given , , , , find X(k) using DIT FFT algorithm.
74. Given , , , , , , , find X(k) using DIT FFT algorithm
75. Given , , , , , , , find X(k) using DIT FFT algorithm
76. Given , find X(k) using DIT FFT algorithm.
77. Given , , , , , , , find X(k) using DIF FFT algorithm
78. Given , find X(k) using DIF FFT algorithm.
79. Given , find X(k) using DIF FFT algorithm.
80. Compute the DFT’s of the sequence , where N = 4, using DIF FFT algorithm.
81. Use the 4 – point inverse FFT and verify the DFT results {6, ‐2+j2, ‐2, ‐2‐j2}
82. Given
, . . , , . . , , . . , , .
. ,
83. Given , . . , , . . , , .
. , , . . ,
84. Given , . , , . , , . , ,
. ,
85. Determine DFT (8 – point) for a continuous time signal,
.
86. Develop a radix – 3 DIT FFT algorithm for evaluating the DFT for N = 9
87. Develop DIT FFT algorithms for decomposing the DFT for N = 6 and draw the flow diagram for
(a) N = 2.3    and
(b) N = 3.2
(c) Also, by using the FFT algorithm develop in part (b), evaluate the DFT values for
, , , , ,
88. Develop the DFT FFT algorithm for decomposing the DFT for N = 12 and draw the flow
diagram.
89. Develop a radix – 4 DIT FFT algorithm for evaluating the DFT for N = 16 and hence determined
the 16 – point DFT of the sequence.
, , , , , , , , , , , , , , ,
90. Develop a DIT FFT algorithm for decomposing the DFT for N = 6 and draw the flow diagrams
for
(a) N = 3.2   and

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(b) N = 2.3
91. Develop a radix – 3 DIF FFT algorithm for evaluating the DFT for N = 9
92. An FIR digital filter has the unit impulse response sequence, , , , determine the
output sequence in response to the input sequence , , , , , , , , ,
using the overlap – add convolution method.

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Filter Design

1. Determine frequency response of FIR filter defined by .
. . Calculate the phase delay and group delay.
2. Explain Fourier series method of designing FIR filters.
3. Design an ideal low pass filter with a frequency response

| |
Find the values of h (n) for N=11. Find H (z). Plot the magnitude response.

4. Design an ideal high pass filter with a frequency response
| |
| |
Find the values of h (n) for N=11. Find H (z). Plot the magnitude response.
5. Design an ideal band pass filter with a frequency response
| |

Find the values of h(n) for N=11. Find H (z). Plot the magnitude response.
6. Design an ideal band reject filter with a frequency response
| | | |

Find the values of h(n) for N=11. Find H (z). Plot the magnitude response.
7. Explain design of FIR filters using windows.
8. Explain Gibb’s Phenomenon.
9. Explain rectangular, Hanning and Hamming windows.
| |
| |
Find the values of h (n) for N=11. Design using Hanning Window. Plot the magnitude
response.
| |
| |

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12. Find the values of h (n) for N=11.  Design using Hamming Window. Plot the magnitude
response
13. Design a filter with

Design using Hamming window with N = 7.

14. Design the following filters using Fourier series method. Assume N = 7
(a) Low pass filter
| |

(b) High pass filter
| |

(c) Band pass filter
| |

(d) Band reject filter
| |


| |
Design using Blackman window with N = 11.

16. Design an FIR filter approximate the ideal frequency response
| |
| |
Determine the filter coefficients for N = 13.
17. Repeat above (problem – 16) using Hamming window.

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Page 3

18. Using a rectangular window technique design a Lowpass filter with passband gain of unity,
cutoff frequency of 1000 Hz and working at a sampling frequency of 5 KHz. The length of the
impulse response should be 7.
19. What is Gibb’s phenomenon?
Or
What are Gibb’s oscillations?
20. What are the desirable characteristics of the window?
21. Draw the impulse response of an ideal low pass filter.
22. What is the need for employing window technique for FIR filter design?
23. Use Fourier series  method in conjunction with a Hamming window to design an
approximation to an ideal low pass filter with a magnitude response
| |

Compare the response with that obtained from an unwindowed design for N = 11.
24. Design an approximation  to an ideal high pass filter with magnitude response
| |

By the Fourier series method with N = 11.
25. Explain the advantages of FIR filters over IIR filters.
26. The following transfer function characterizes an FIR filter (M=11). Determine the magnitude
response and also show that the phase delay and group delay are constant.

27. Design a Finite impulse response low pass filter with a cut – off frequency of 1 KHz and
sampling rate of 4 KHz with eleven samples using Fourier series
28. Determine the unit sample response of the ideal low pass filter. Why it is not realizable?
29. A low pass filter is to be designed with the following desired frequency response.

| |
Determine the filter coefficients if the window function is defined as

,
,

Also, determine the frequency response of the designed filter.
30. A filter is to be designed with the following desired frequency response
,
| |
Determine the filter coefficients if the window function is defined as

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Page 4

,
,

Also, determine the frequency response of the designed filter.

| |
Determine for M = 7 using a Hamming window.
32. Repeat above using rectangular window
(Ans: . . . .
33. Design a high pass filter using Hamming window with a cut off frequency of 1.2 rad/sec and N
= 9
(Hint: , )
34. Design an FIR digital filter to approximate an ideal low pass filter with passband gain of unity,
cut – off frequency of 850 Hz and working at a sampling frequency of . The
length of the impulse response should be 5. Use a rectangular window.
35. An FIR linear phase filter has the following impulse response.

,
,

Use Bartlett’s window and compute the impulse response of the filter. Find its magnitude and
phase response as a function of frequency.
36. Design a bandpass filter to pass frequencies in the range 1  ‐ 2 rad/sec using Hanning window
N=5
(Hint: ,
37. Design a bandpass filter which approximates the ideal filter with cut off frequency at 0.2
rad/sec and 0.3 rad/sec. the filter order is M = 7. Use the Hanning window function.
(Ans: , . , . , . , . ,
. , .

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1
Filter Realization
1. Obtain Direct form I/II and cascade form realization of a system transfer
function described by
y(n)-3/4 y(n-1)+1/8y(n-2)=x(n)+1/2x(n-1) PU Dec’00 NU S-98
2. Consider a causal LTI system with TF
H (z) = (1 + 1/5 Z-1
) / (1-0.5 Z-1
+ 1/3 Z-2
) (1+0.25 Z-1
)
Draw the signal flow graph for implementation of the system using Direct
form-II.
3. The TF is given by
H (z) = (1 – Z-1
) / (1-0.2 Z-1
-0.15 Z-2
)
Draw 1st
and 2nd
order cascade realization and parallel realization.
4. Develop parallel form realization
H (z) = (1 + 0.2 Z-1
+ Z-2
) / (1 – 0.75 Z-1
+ 0.125 Z-2
)
5. The system function for FIR filter is given as follows
H(z) = 1 – 3/2 Z-1
+ 3/2 Z-2
– Z-3
draw cascade realization.
= (1 – Z-1
) (1 – ½ Z-1
+ Z-2
)
6. y(n) = 2rcosw0y(n-1) – r2
y(n-2) + x(n) – rcosw0x(n-1) Draw Direct form I/II
structure.
7. Draw Direct form I/II realization of
H (z) = (1 + 0.2 Z-1
-0.2 Z-2
) / (1 – 0.2 Z-1
+ 0.3 Z-2
+ Z-3
)
8. Draw cascade form realization
H (z) = (Z/3 + 5/12 + 5/12 Z-1
+ Z-2
/12) / (1 – ½ Z-1
+ ¼ Z-2
)
Z/3 (1 + ¼ Z-1
) (1+Z-1
+Z-2
)
9. y(n) – ¾ y(n-1) + 1/8 y(n-2) = x(n) + 1/3 x(n-1) Draw cascde realization.
10. Draw cascade canonical IIR filter
H (z) = (1+Z-1
) / (1- Z-1
+ ½ Z-2
) (1-Z-1
+Z-2
)

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2
Digital filter design
Part – A: Impulse Invariance Method
1. Find out H(z) using Impulse invariance method (IIM) at 5Hz sampling
frequency from H(s) as given below
H(s) = 2 /(S+1)(S+2) PU May’2001
2. Using IIM with T=1sec determine H(z) if H(s) = 1/ (s2
+ √2 s +1)
3. Apply IIM and find H (z) for H(s) = (s+a) / [(s+a)2
+ b2
)].
4. Apply IIM and find H(z) for H(s) = b / [(s+a)2
+ b2
)]
5. An analog filter has a transfer function H(s) = 10 / (s2
+ 7s + 10) Design a
digital filter equivalent to this using IIM for T = 0.2 sec.
+ 6s2
+11s + 6) Design a
digital filter equivalent to this using IIM for T = 1 sec.
7. An analog filter has a transfer function H(s) = (s + 3) / (s2
+ 6s + 25) Design a
digital filter equivalent to this using IIM for T = 1 sec.
8. Determine H(z) using IIM for the system function PU Dec’2000
H(s) = 1 / (S+0.5)(S2
+ 0.5S + 2)
Part – B: Bilinear Transformation Method
9. The transfer function of analog filter is:
H(s) = 3 / (s+2) (s+3)
With Ts=0.1Sec Design digital IIR filter using BLT
10. Find out H(z) using BLT from H(s) as given below
H(s) = 2 /(S+1)(S+2) with T =1 sec
11. Determine H(z) using BLT is applied to
H(s) = (s2
+ 4.525) / (s2
+ 0.692s + 0.504).
+ 6s + 9) Design a digital
filter using BLT.
13. Design a single pole low pass filter with a 3dB bandwidth of 0.2π by use of
bilinear transformation applied to the analog filter.
H(s) = Ωc / (s + Ωc) where Ωc is the 3dB bandwidth of analog filter

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3
14. The system transfer function of A/F is given by
H(s) = (s+0.1) / [(s+0.1)2
+ 16]
Obtain the system transfer function of digital filter using BLT which is
resonant at w = π/2.
15. Use the bilinear transformation to convert the analog filter with the system
function. H (s) = (s + 0.1) / [(s + 0.1)2
+9]
into a digital IIR filter. Select T =0.1 sec and compare the location of the
zeroes in H (z) with the location of zeroes obtained by applying the impulse
invariance method in the conversion of H(s).
16. The Analog transfer function of low pass filter is
H(s) = 1 / (s+2) and its BW is 1 rad/sec. design the digital filter using BLT
method whose cut-off frequency is 20π and sampling time is 0.0167sec by
considering the warping method.
Part – C: Butterworth Low Pass Filter
17. Given the specification αp = 1dB, αs = 30dB, Ωp = 200 rad/sec, Ωs = 600
rad/sec. determine the order of the filter.
18. Determine the order and the poles of the Butterworth filter have 3dB
attenuation at 500Hz and an attenuation of 40dB at 1000Hz.
19. Prove that Ωc = Ωp / (100.1Ap
– 1)1/2N
= Ωs / (100.1As
– 1)1/2N
20. Determine the order and the poles of low pass Butterworth filter that has a
-3 dB bandwidth of 500 Hz and an attenuation of 40 dB at 1000 Hz.
21. Design an analog Butterworth filter that has a -2dB passband attenuation at
frequency of 20 rad/sec and at least -10dB stopband attenuation at 30 rad/sec.
22. For the given specification design an analog Butterworth filter.
0.9 ≤ | H(jΩ)| ≤ 1 for 0 ≤ Ω ≤ 0.2π. | H(jΩ)| ≤ 0.2 for 0.4π ≤ Ω ≤ π.
23. For the given specification find the order of the Butterworth filter αp = 3dB,
αs = 18dB, fp = 1KHz, fs = 2KHz.
24. For the given specification find the order of the Butterworth filter αp = 0.5dB,
αs = 22dB, fp = 10KHz, fs = 25KHz

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4
25. Find the pole location of a 6th
order Butterworth filter with Ωc = 1 rad/sec.
26. Design a 2nd
order low pass Butterworth filter with a cut – off frequency of
1KHz and sampling frequency of 104
samples/sec, by bilinear transformation.
27. Using bilinear transformation, design a Butterworth filter which satisfies the
following conditions.
0.8 ≤ | H (ejω
)| ≤ 1 for 0 ≤ ω ≤ 0.2π.
| H (ejω
)| ≤ 0.2 for 0.6π ≤ ω ≤ π.
28. Design low pass IIR filter is to be designed with Butterworth approximation
using BLT technique. Find the order of the filter to meet the following
specifications.
(i) Passband magnitude is constant within 1dB for frequencies below 0.2π
(ii) Stopband attenuation is greater than 15dB for frequencies between
0.3π to π.
29. An IIR low pass filter is required to meet the following specifications:
Passband peak to peak ripple : ≤ 1 dB
Passband edge : 1.2 KHz
Stopband attenuation : ≥ 40 dB
Stopband edge : 2.5 KHz
Sample rate : 8 KHz
The filter is to be designed by performing BLT on an analog system function
of required order of Butterworth filter so as to meet the specifications in the
implementation.
Passband ripple : ≤ 1 dB
Passband edge : 4 KHz
Stopband edge : 6 KHz
Sampling rate : 24 KHz
implementation.

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5
Passband peak to peak ripple : ≤ 0.5 dB
Passband edge : 1.2 KHz
Stopband edge : 2.0 KHz
Sample rate : 8 KHz
implementation.
Part – D: Chebyshev Filter
32. Given the specification αp = 3 dB, αs = 16 dB, fp = 1KHz, fs = 2KHz.
Determine the order of the filter using Chebyshev approximation. Find H (s).
33. Obtain an analog Chebyshev filter transfer function that satisfies the
constraints
1/√2 ≤ | H (jΩ)| ≤ 1 for 0 ≤ Ω ≤ 2
| H (jΩ)| ≤ 0.1 for Ω ≥ 4.
34. Determine the order of the filter and the poles of a type I low pass Chebyshev
filter that has a 1 dB ripple in the passband and passband frequency Ωp =
1000π , a stopband frequency of 2000π and an attenuation of 40 dB or more.
35. Design a Chebyshev filter with a maximum passband attenuation of 2.5 dB, at
Ωp = 20 rad/sec and stopband attenuation of 30 dB at Ωs = 50 rad/sec.
36. For the given specification find the order of the chebyshev – I filter αp = 1.5
dB, αs = 10dB, Ωp = 2 rad/sec, Ωs = 30 rad/sec.
37. Find the pole locations of a normalized chebyshev filter of order 5.
38. Design a Chebyshev filter for the following specification: αp = 1 dB, αs = 25
dB, Ωp = 1 rad/sec, Ωs = 20 rad/sec.
39. Determine the system function H (z) of the lowest order chebyshev digital
filter that meets the following specification.
(i) 1 dB ripple in the passband 0 ≤ |ω| ≤ 0.3π

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6
(ii) At least 60 dB attenuation in the stopband 0.35π ≤ |ω| ≤ π. Use the
bilinear transformation.
40. Determine the system function H (z) of the lowest order chebyshev digital
filter that meet the following specification.
(i) ½ dB ripple in the passband 0 ≤ |ω| ≤ 0.24π
(ii) At least 50 dB attenuation in the stopband 0.35π ≤ |ω| ≤ π. Use the
bilinear transformation.
Part – E Additional Examples:
41. Design a digital Butterworth filter satisfying the constraints
0.707 ≤ | H (ejω
)| ≤ 1 for 0 ≤ ω ≤ π/2.
| H (ejω
)| ≤ 0.2 for 3π/4 ≤ ω ≤ π.
With T=1 sec using (a) The bilinear transformation (b) Impulse invariance.
Realize the filter in each case using the most convenient realization form.
42. Design a Chebyshev low pass filter with the specification αp = 1 dB ripple in
the pass band 0 ≤ ω ≤ 0.2π, αs = 15 dB ripple in the stop band 0.3 π ≤ ω ≤ π,
using (a) Bilinear transformation (b) Impulse invariance.
43. Design a digital Butterworth filter satisfying the constraints
0.8 ≤ | H (ejω
)| ≤ 1 for 0 ≤ ω ≤ 0.2π.
| H (ejω
)| ≤ 0.2 for 0.6π ≤ ω ≤ π.
44. Determine the system function H(z) of the lowest order Chebyshev and
Butterworth digital filter with the following specification
(a) 3 dB ripple in pass band 0 ≤ ω ≤ 0.2π.
(b) 25 dB attenuation in stop band 0.45π ≤ ω ≤ π.
45. Design a Chebyshev filter satisfying the constraints
0.8 ≤ | H (ejω
)| ≤ 1 for 0 ≤ ω ≤ 0.2π.
| H (ejω
)| ≤ 0.2 for 0.6π ≤ ω ≤ π.
46.

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7
47. Give the magnitude function of the Butterworth filter. What is the effect of
varying order of N on magnitude and phase response?
48. Give any two properties of Butterworth low pass filter
49. What are the properties of Chebyshev filter?
50. Give the equation for the order of N and cut off frequency ΩC of Butterworth
filter.
51. Give the Chebyshev filter transfer function and its magnitude response.
52. Distinguished between the frequency responses of Chebyshev type I filter for
N odd and N even.
53. Distinguished between the frequency response of Chebyshev type I and type
II filters.
54. Give the expression for the location of poles and zeros of a Chebyshev type II
filter.
55. Give the expression for the location of poles and zeros of a Chebyshev type I
filter.
56. Distinguished between Butterworth and Chebyshev type I filter.
57. Explain how one can design digital filters from analog filters?
58. Mention any two procedures for digitizing the transfer function of an analog
filter.
59. What are the properties that are maintained same in the transfer of analog
filter into digital filter?

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