Upload
vijay-sai
View
104
Download
10
Embed Size (px)
DESCRIPTION
DSP SOL
Citation preview
Digital Signal Processing 10EC52
SJBIT/ECE Page 1
UNIT 1
Discrete Fourier Transform
Question 1( Exam Dec 10/Jan11)
Solution :-
Question 2( Exam Dec7/Jan08)
Solution (a)
Solution (b)
Digital Signal Processing 10EC52
SJBIT/ECE Page 2
Solution (c)
Solution (d)
Question 3( Exam June/July 2009)
Solution
Question 4( Exam Dec 07/Jan08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 3
Solution
Question 5 ( Exam Dec 09/Jan10)
Solution
Question 6 ( Exam Dec 06/Jan7)
Digital Signal Processing 10EC52
SJBIT/ECE Page 4
Solution
Question 7 ( Exam June-July 2009)
Define DFT. Establish a relation between the Fourier series coefficients of a continuous time
signal and DFT
Solution
The DTFT representation for a finite duration sequence is
∞
X (jω) = ∑ x (n) -jωn
n= -∞
X (n) =1/2π ∫X (jω) e jωn
dω , Where ω 2πk/n
2π
Where x(n) is a finite duration sequence, X(jω) is periodic with period 2π.It is convenient
sample X(jω) with a sampling frequency equal an integer multiple of its period =m that is taking
N uniformly spaced samples between 0 and 2π.
Let ωk= 2πk/n, 0≤k≤N
∞
Therefore X(jω) = ∑ x(n) -j2πkn/N
n=−∞
Since X(jω) is sampled for one period and there are N samples X(jω) can be expressed as
N-1
Digital Signal Processing 10EC52
SJBIT/ECE Page 5
X(k) = X(jω) ω=2πkn/N ∑ x(n) -j2πkn/N
0≤k≤N-1
n=0
Question 8 ( Exam June-July 2009)
Digital Signal Processing 10EC52
SJBIT/ECE Page 6
Solution
Question 9 ( Exam Dec 08-Jan 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 7
Solution
Question 10 ( Exam Dec 05-Jan 06)
Solution
Question 11 ( Exam Jan/Feb 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 8
Ans:
Unit 2
Properties of DFT
Question 1 ( Exam Dec 10-Jan 11)
Digital Signal Processing 10EC52
SJBIT/ECE Page 9
State and Prove the Time shifting Property of DFT
Solution
The DFT and IDFT for an N-point sequence x(n) are given as
Time shift:
If x (n) X (k)
mk
Then x (n-m) WN X (k)
Question 2 ( Exam Dec 08-Jan 09, Dec 07-Jan 08, )
State and Prove the: (i) Circular convolution property of DFT; (ii) DFT of Real and even
sequence.
Solution
(i) Convolution theorem
Circular convolution in time domain corresponds to multiplication of the DFTs
If y(n) = x(n) h(n) then Y(k) = X(k) H(k)
Ex let x(n) = 1,2,2,1 and h(n) = 1,2,2,1 Then y (n) = x(n) h(n)
Y(n) = 9,10,9,8
N pt DFTs of 2 real sequences can be found using a single DFT
If g(n) & h(n) are two sequences then let x(n) = g(n) +j h(n)
G(k) = ½ (X(k) + X*(k))
H(k) = 1/2j (X(K) +X*(k))
2N pt DFT of a real sequence using a single N pt DFT
Let x(n) be a real sequence of length 2N with y(n) and g(n) denoting its N pt DFT
Let y(n) = x(2n) and g(2n+1)
k
Digital Signal Processing 10EC52
SJBIT/ECE Page 10
X (k) = Y (k) + WN G (k)
Using DFT to find IDFT
The DFT expression can be used to find IDFT
X(n) = 1/N [DFT(X*(k)]*
(ii)DFT of Real and even sequence.
For a real sequence, if x(n) X(k)
X (N-K) = X* (k)
For a complex sequence
DFT(x*(n)) = X*(N-K)
If x(n) then X(k)
Real and even real and even
Real and odd imaginary and odd
Odd and imaginary real odd
Even and imaginary imaginary and even
Question 3 ( Exam Dec 08-Jan 09)
Distinguish between circular and linear convolution
Solution
1) Circular convolution is used for periodic and finite signals while linear convolution is
used for aperiodic and infinite signals.
2) In linear convolution we convolved one signal with another signal where as in circular
convolution the same convolution is done but in circular pattern depending upon the
samples of the signal
3) Shifts are linear in linear in linear convolution, whereas it is circular in circular
convolution.
Question 4 ( Exam June – july 2009)
State and prove the following properties of the DFT
i) Symmetric property for real valued sequence x(n)
ii) Parseval’s theorem
Solution
Digital Signal Processing 10EC52
SJBIT/ECE Page 11
Digital Signal Processing 10EC52
SJBIT/ECE Page 12
Question 5 ( Exam June – July 2006)
Digital Signal Processing 10EC52
SJBIT/ECE Page 13
Solution
Question 6 ( Exam Dec 07– Jan 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 14
Solution(a)
Solution(b)
Solution(c)
Solution(d)
Digital Signal Processing 10EC52
SJBIT/ECE Page 15
Question 7 ( Exam Dec 06– Jan 07)
Solution
Question 8 ( Exam Dec 06– Jan 07)
Digital Signal Processing 10EC52
SJBIT/ECE Page 16
Solution
Question 9 ( Exam June –July 07)
Solution
Digital Signal Processing 10EC52
SJBIT/ECE Page 17
Question 10 ( Exam June –July 07)
Solution
Digital Signal Processing 10EC52
SJBIT/ECE Page 18
Digital Signal Processing 10EC52
SJBIT/ECE Page 19
Question 10 ( Exam June –July 06)
Solution
Question 11
Digital Signal Processing 10EC52
SJBIT/ECE Page 20
Solution
Question 12 ( Exam Dec –Jan 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 21
Question 13 ( Exam Dec –Jan 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 22
Question 14 ( Exam July –August 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 23
Question 15 ( Exam July –August 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 24
Question 16 ( Exam July –August 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 25
Question 17 ( Exam Jan –Feb 08) Using overlap-save method, compute yen), of a FIR filter with impulse response
H(n) = (3, 2, 1) and input x(n) = (2, 1, -1, -2, -3, 5, 6, -1, 2, 0, 2, 1J. Use only 8-point circular
convolution in your approach.
Digital Signal Processing 10EC52
SJBIT/ECE Page 26
Digital Signal Processing 10EC52
SJBIT/ECE Page 27
UNIT 3& 4
FAST-FOURIER-TRANSFORM (FFT) ALGORITHMS
Question 1( Exam July –August 10)
Question 2( Exam July –August 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 28
Digital Signal Processing 10EC52
SJBIT/ECE Page 29
Digital Signal Processing 10EC52
SJBIT/ECE Page 30
Question 3( Exam July –August 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 31
Question 4 ( Exam Dec10/Jan 11)
Digital Signal Processing 10EC52
SJBIT/ECE Page 32
Digital Signal Processing 10EC52
SJBIT/ECE Page 33
Digital Signal Processing 10EC52
SJBIT/ECE Page 34
Digital Signal Processing 10EC52
SJBIT/ECE Page 35
Question 5 ( Exam Dec08/Jan 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 36
Digital Signal Processing 10EC52
SJBIT/ECE Page 37
Question 6 ( Exam Dec08/Jan 09)
Question 7( Exam Dec08/Jan 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 38
Question 8 ( Exam June –July 06)
Digital Signal Processing 10EC52
SJBIT/ECE Page 39
Solution:
Question 9 ( Exam June –July 06)
Digital Signal Processing 10EC52
SJBIT/ECE Page 40
Question 10 ( Exam June –July 06)
Digital Signal Processing 10EC52
SJBIT/ECE Page 41
Question 11 ( Exam Jan –Feb 05)
Digital Signal Processing 10EC52
SJBIT/ECE Page 42
Question 12 ( Exam Jan –Feb 05)
Question 13 ( Exam June –July 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 43
Question 14 ( Exam June –July 09)
Question 15( Exam June –July 08)
Show that the impulse invariant transformation is not one-to-one form s-plane to
Z-plane
Digital Signal Processing 10EC52
SJBIT/ECE Page 44
Here pk are the poles of the analog filter and Ck are the coefficients of partial fraction
expansion.
Digital Signal Processing 10EC52
SJBIT/ECE Page 45
Digital Signal Processing 10EC52
SJBIT/ECE Page 46
Question 16( Exam Jan/Feb 08)
Tabulate the number of complex multiplication and complex additions required for
the direct computation of DFT and FFT algorithm for N = 16, 32, 128.
Question 17( Exam Jan/Feb 08)
Compute circular - convolution using DFT and IDFT formulae for the following
sequences, xl(n) = n and x2(n) = Cos(nπ/2) for 0<n<3.
The product of the DFT’s will be
Digital Signal Processing 10EC52
SJBIT/ECE Page 47
The circular convolution property states that
Hence the circular convolution of x1(n) and x2(n) can be obtained by taking the inverse IDFT of
the product X1(k) . X2(k)
x2(n)*x2(n) = -2,-2,2,2
Question 18( Exam Jan/Feb 08)
i) To obtain the DFT of X1(k):
The signal flow graph for calculation of DFT is shown in Fig
(ii) Calculations at intermediate steps:-
From above signal flow graph, the DFT is,
Digital Signal Processing 10EC52
SJBIT/ECE Page 48
Digital Signal Processing 10EC52
SJBIT/ECE Page 49
Question 19( Exam Jan/Feb 08)
Derive the signal flow graph for 8-point Radix-2 DIT-FFT algorithm.
Sol:-
Let f1 (n) contain even numbered samples of x(n) and f2(n) contain ordered samples of x(n). Thus
we can write,
Here time domain sequence x (n) is splitted into two sequences. This splitting action is called
decimation. Since it is done on time domain sequence it is called
decimation in Time (DIT).
We know that N-point DFT is given as,
Since the sequence x(n) is splitted into even numbered and odd numbered
samples, above equation can be written as,
From equation (1) we know that f1 (m) = x(2m) and f2(m) = x(2m + 1), hence above
equation becomes,
Digital Signal Processing 10EC52
SJBIT/ECE Page 50
In the above equation we have rearranged WN factors. We know that (WN)
2 = WN/2
Hence above equation can be written as,
Comparing above equation with the definition of DFT of equation (2), we find that
first summation represent N/2 point DFT of f1(m) and second summation representsN/2 point
DFT of f2(m)
Thus F1(k) is N - point DFT of f1(m) and F2(k) is N point-DFT of f2(m). Since
F1(k) and F2(k) are then N/2 - point DFTs, they are periodic with period N/2. i.e.,
Hence replacing 'k' by k +N/2 in equation (4) we get,
From equation (5) we know that WN
K+N/2 = - WN
K, and from equation (5) we can
write above equation is,
Here observe that X (k) is N-point DFT. By taking k = 0 to N/2 -1 we can calculate
F1(k) and F2 (k) since they are N/2 - point OPTs. The N-point DFT X (k) can be combinely
obtained from equation (4) and equation (6) above by taking k = 0 to N/2-1.
The above two equations show that N-point DFT can be obtained by two N/2-point DFTs.
Digital Signal Processing 10EC52
SJBIT/ECE Page 51
Fig. 2 shows that 8-point DFT can be computed directly and hence no reduction in computation.
It is shown symbolically as a single block which computes 8-point DFT directly. Fig. 3 shows
the symbolic diagram for this creation.
UNIT 5
Design of IIR Filters
Question 1 ( Exam Dec10 –Jan 11)
Digital Signal Processing 10EC52
SJBIT/ECE Page 52
Question 2 ( Exam Jan –Feb 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 53
Digital Signal Processing 10EC52
SJBIT/ECE Page 54
Question 3 ( Exam Jan –Feb 05) Starting from low pass butter worth prototype analog filter, design a fourth order
Butterworth band pass analog filter with upper and lower band edge frequencies 10 rad/sec and 5
rad/sec.
Question 4( Exam Jan –Feb 05)
Digital Signal Processing 10EC52
SJBIT/ECE Page 55
Transform the analog filter H (s) = s + 3 to a digital filter using the matched
(s+1)(s+2)
Z-transform. Let T == 0.5 sec.
Question 5 ( Exam Jan –Feb 05)
Using the impulse response technique design a low pass digital filter that is equiripple in the
passband, and monotone in the stopband. The filter passband edge is at 0.1 π rad and with a
ripple of 2.5 dB or less and the stopband edge at 0.2 π rad with attenuation of 40 dB or more. Use
T =1.
Digital Signal Processing 10EC52
SJBIT/ECE Page 56
Digital Signal Processing 10EC52
SJBIT/ECE Page 57
Digital Signal Processing 10EC52
SJBIT/ECE Page 58
Question 6( Exam Jan –Feb 05) Transform the analog filter H(s) =1/s+a, a> 0 to a digital filter using the
back-ward difference mapping. Comment on the stability of the digital filter.
Digital Signal Processing 10EC52
SJBIT/ECE Page 59
Question 7 ( Exam June –July 08)
An analog Chebyshev low pass filter with pass band attenuation of 2.5 dB at 200 Hz, stop band
attenuation of 30 dB at 500 Hz is required. Find the order of filter. Obtain the system transfer
function H(s). Sketch the magnitude response of the filter.
Digital Signal Processing 10EC52
SJBIT/ECE Page 60
iii) To obtain the magnitude response
The magnitude response can be obtained by putting s = jΩ, in Ha(s). i.e.
Question 8 ( Exam June –July 08)
Obtain the transfer function of the analog Butterworth low pass filter of third order with cut-off
frequency of 10 Hz. Find the transfer function of 3rd
order high pass Butterworth filter with 100
Hz cut-off frequency by frequency transformation of low pass filter.
Ans : Given
N = 3, Fe = 10 Hz (Low pass)
N = 3, Fe = 100 Hz (High pass)
Digital Signal Processing 10EC52
SJBIT/ECE Page 61
Digital Signal Processing 10EC52
SJBIT/ECE Page 62
Question 9 ( Exam June –July 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 63
Digital Signal Processing 10EC52
SJBIT/ECE Page 64
Digital Signal Processing 10EC52
SJBIT/ECE Page 65
Question 10( Exam June –July 09, Jan-Feb 09)
Question 11 ( Exam Jan –Feb 08)
Design a digital IIR low pass Butterworth filter that has a 2 dB pass band attenuation at a
frequency of 300n rad/sec and at least 60 dB stop band attenuation
at 4500π rad/sec. Use backward difference transformation.
(ii) Order of the filter
(iii)Cut off frequency
(iv)Poles of the filter
Digital Signal Processing 10EC52
SJBIT/ECE Page 66
(v) System function of analog filter
(vi) Back word difference transform
Digital Signal Processing 10EC52
SJBIT/ECE Page 67
Question 12( Exam Jan –Feb 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 68
This is the required system function of the digital filter. Since it is single pole function it can be
realized using direct form.
UNIT 6
Design of FIR Filters
Question 1 ( Exam Dec 10-Jan 11)
Digital Signal Processing 10EC52
SJBIT/ECE Page 69
Digital Signal Processing 10EC52
SJBIT/ECE Page 70
Question 2( Exam Jan –Feb 09)
b) What is Gibbs phenomenon? How it can be reduced?
Digital Signal Processing 10EC52
SJBIT/ECE Page 71
Question 3( Exam Jan –Feb 09) c) Show that the roots of H (z) occur in reciprocal pair for a linear phase FIR filter.
Digital Signal Processing 10EC52
SJBIT/ECE Page 72
Digital Signal Processing 10EC52
SJBIT/ECE Page 73
Question 4 ( Exam July –Aug 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 74
Question 5 ( Exam July –Aug 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 75
Question 6 ( Exam Jan –Feb 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 76
Digital Signal Processing 10EC52
SJBIT/ECE Page 77
Digital Signal Processing 10EC52
SJBIT/ECE Page 78
Question 7( Exam Jan –Feb 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 79
Digital Signal Processing 10EC52
SJBIT/ECE Page 80
Putting for k = 0 in equation (4), we get H (0) = 1 and with M = 17, above equation
becomes,
UNIT 7
Digital Signal Processing 10EC52
SJBIT/ECE Page 81
Design of IIR Filters from Analog Filters
Question 1 ( Exam Dec 10-Jan 11)
Digital Signal Processing 10EC52
SJBIT/ECE Page 82
Show that the impulse invariant transformation is not one-to-one form s-plane to
Question 2 ( Exam June –July 09)
Question 3 ( Exam June –July 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 83
Question 4 ( Exam June –July 09)
Question 5 ( Exam June –July 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 84
Digital Signal Processing 10EC52
SJBIT/ECE Page 85
Question 6 ( Exam June –July 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 86
Question 7 ( Exam June –July 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 87
Digital Signal Processing 10EC52
SJBIT/ECE Page 88
UNIT 8
Implementation of Discrete-Time Systems
Question 1 ( Exam Dec 10 –Jan 11)
Consider a FIR filter with system function:
H(z) = 1+2.82 Z-1
+3.4048z-2 +1.74z
- 3. Sketch the direct form and lattice
realizations of the filter.
Digital Signal Processing 10EC52
SJBIT/ECE Page 89
Digital Signal Processing 10EC52
SJBIT/ECE Page 90
Question 2 ( Exam Jan –Feb 09)
For y(n) = -0.1y(n-1)+0.2y(n-2) +3x(n) +3.6x(n-1)+0.6x(n-2), obtain form I and II, cascade form
and parallel form with single pole-zero subsystems.
Digital Signal Processing 10EC52
SJBIT/ECE Page 91
Digital Signal Processing 10EC52
SJBIT/ECE Page 92
Question 3 ( Exam Jan –Feb 05)
Consider the transfer function
Question 4 ( Exam Jan –Feb 05)
Digital Signal Processing 10EC52
SJBIT/ECE Page 93
Digital Signal Processing 10EC52
SJBIT/ECE Page 94
Question 5 ( Exam July–Aug 08)
Digital Signal Processing 10EC52
SJBIT/ECE Page 95
Digital Signal Processing 10EC52
SJBIT/ECE Page 96
Digital Signal Processing 10EC52
SJBIT/ECE Page 97
Question 6 ( Exam July–Aug 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 98
Question 7 ( Exam July–Aug 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 99
Digital Signal Processing 10EC52
SJBIT/ECE Page 100
Question 8 ( Exam July–Aug 09)
Digital Signal Processing 10EC52
SJBIT/ECE Page 101