1ME606 Digital Signal Processing: A Practical Analysis using MATLAB

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ME606 Digital Signal Processing
Student name:
Student id:

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Contents
1. Introduction:.........................................................................................................................................................4
2. Part A: System Analysis.......................................................................................................................................5
1)...............................................................................................................................................................................5
2)...............................................................................................................................................................................6
3)...............................................................................................................................................................................6
4)...............................................................................................................................................................................6
3. Part B: Analyzing a system in Z and time domain...............................................................................................7
Given,...........................................................................................................................................................................7
1)...............................................................................................................................................................................7
2)...............................................................................................................................................................................7
3)...............................................................................................................................................................................7
4)...............................................................................................................................................................................8
5).............................................................................................................................................................................10
6).............................................................................................................................................................................10
8).............................................................................................................................................................................10
4. Part C: image decimation Using MATLAB.......................................................................................................11
1).............................................................................................................................................................................12
5. Part D: Signal decimation and interpolation in MATLAB................................................................................13
1).............................................................................................................................................................................13
2).............................................................................................................................................................................13
3).............................................................................................................................................................................13
5).............................................................................................................................................................................14
6).............................................................................................................................................................................15
7).............................................................................................................................................................................15
8).............................................................................................................................................................................15
9).............................................................................................................................................................................15
10)...........................................................................................................................................................................15
11)...........................................................................................................................................................................15
12)...........................................................................................................................................................................17
13)...........................................................................................................................................................................18
14)...........................................................................................................................................................................18
15)...........................................................................................................................................................................19
6. Part E: System Eigen function and Eigen Value................................................................................................19
1).............................................................................................................................................................................19
2).............................................................................................................................................................................19
3).............................................................................................................................................................................19
4).............................................................................................................................................................................20

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5).............................................................................................................................................................................20
6).............................................................................................................................................................................20
References:.................................................................................................................................................................22
Table of Figures
Figure 1: ROC in z-plane.............................................................................................................................................7
Figure 2: Magnitude and phase frequency response.................................................................................................8
Figure 3: Impulse response......................................................................................................................................10
Figure 4: Original image...........................................................................................................................................11
Figure 5: Decimated with drv =4∧drh=4............................................................................................................11
Figure 6: Decimation with drv =8∧drh=8............................................................................................................11
Figure 7: First 200 samples......................................................................................................................................12
Figure 8: Power spectrum using periodogram.........................................................................................................13
Figure 9: Power spectrum using pwelch..................................................................................................................13
Figure 10: Power spectrum of convolve image........................................................................................................14
Figure 11: Cubic spline interpolation signal.............................................................................................................16
Figure 12: Power spectrum of cubic spline signal....................................................................................................16
Figure 13: Spectrum of filter signal..........................................................................................................................18
Figure 14: Amplitude of system...............................................................................................................................19
Figure 15: Real and imaginary parts of complex distribytion...................................................................................20

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1. Introduction:
The purpose of this study is to evaluate different signal processing techniques by practical implementation on
Matlab. For the course of this assignment, the concept of interpolation and decimation were focused. In first
part, system analysis is done by finding ROC of given system and then proving main signal processing properties
for this system. Second section included analyzing of system in both time and z-domain. Third part included the
practical analysis of image decimation technique by using different values. Fourth part consist of techniques to
verify both interpolation and decimation of signals, this section included detail analysis on both techniques. Final
part consists of Eigen values and system Eigen functions.

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2. Part A: System Analysis
1)
h(n)=an u(n) ,∨a∨¿ 1
Taking z-transform
h(z )= ∑
n=−∞
∞
x [ n ] z−n
h( z )= ∑
n=−∞
∞
an z−n
h ( z )= ∑
n=−∞
∞
¿ ¿
h( z )= 1
1−a z−1
h( z )= z
z−a
For convergence
¿ a z−1∨¿ 1
So ROC is
¿ z∨¿∨α∨¿
If
¿ a z−1∨≥ 1
h(z ) doesn’t converge
From this diagram ROC can be seen, that is ¿ z∨¿∨a∨.
All poles lie inside unit circle, so this system is stable.
-1 1

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2)
Prove:
Let ud is the delay.
ud ( t ) =u (t+ δ)
h( t)=at u(t)
h1 (t )=at ud (t )
h1 (t )=at u (t+ δ)
h2 ( t )=h ( t+δ )
h2 ( t ) =at + δ u ( t+ δ )
Since h1 ( t ) ≠ h2 ( t )
So this is not time invariant system
3)
The system is said to be memoryless if at any value n1 the value of output at n1 depends only on current
value of input.
h(n)=an u(n) ,∨a∨¿ 1
This is memory less system as output at any given point only depend on current input.
4)
h ( n ) = {0. 2n , 0 ≤n ≤ 4
0 , otherwise
h ( n ) =∑
n=0
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( 1
4 )n
z−n
¿ ( 1
4 ) 0
z0 +( 1
4 )
−1
z−1 +( 1
4 )
−2
z−2 +( 1
4 )
−3
z−3
( 1
4 ) 4
z−4
¿ 1+ 1
4 z−1+ 1
16 z−2+ 1
64 z−3 + 1
256 z−4

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ROC : z −plane excluding z=0
This is not stable system, its poles lies outside the unity circle.
3. Part B: Analyzing a system in Z and time domain
Given,
h ( n ) = 1−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1) + 2
(1−2 z−1 )
1)
Zeros: [ −3
0.2 ]
Poles: [ 1
0.5 , 1
0.3 , 1
2 ]
The ROC of above transfer function is shown
Figure 1: ROC in z-plane
As not all poles lies inside unit circle, so this system is not stable.
2)

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As not all poles lies inside unit circle, so this system is not stable.
3)
∑
k=0
N
ak y [ n−k ]=¿∑
k=0
N
bk x [ n−k ] ¿
Sol:
h ( n ) = 1−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1) + 2
(1−2 z−1 )
h ( n ) = 3−0.2 z−1
(1−0.5 z−1−0.3 z−1 +0.15 z−2 )(1−2 z−1)
h ( n ) = 3−0.2 z−1
( 1−0.8 z−1+ 0.15 z−2 ) ( 1−2 z−1 )
h ( n ) = 3−0.2 z−1
(1−2.8 z−1 +1.75 z−2−0.3 z−3)
2 x [ n ] +0.2 x [ n−1 ]= y [n ] −2.8 y [ n−1 ]+1.75 [ n−2 ] −0.3 y [n−3 ]
Where a & b are the coefficients.
4)

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Figure 2: Magnitude and phase frequency response

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5)
The above transfer function is all pass filter.
6)
Using Partial fraction decomposition
h ( n ) = 1−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1) + 2
(1−2 z−1 )
h ( n ) = 3−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1)(1−2 z−1)
3−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1 )(1−2 z−1) = A
(1−0.5 z−1)+ B
(1−0.3 z−1 ) + C
(1−2 z−1 )
There are three poles:
z−1= 1
0.5 , 1
0.3 , 1
2
Putting z−1= 1
0.5
3−0.2 ( 1
0.5 )=B ( 1−2 ( 1
0.5 ) )( 1−0.3 ( 1
0.5 ) )
B=−2.166
Putting z−1= 1
0.3
3−0.2 ( 1
0.3 )=B ( 1−0.5 ( 1
0.3 ))( 1−2 ( 1
0.3 ) )
C=0.654
Putting z−1= 1
2
3−0.2 ( 1
2 )=B ( 1−0.5 ( 1
2 ) )( 1−0.3 ( 1
2 ))
A=2.169
3−0.2 z−1
(1−0.5 z−1 )(1−0.3 z−1 )(1−2 z−1) = 2.169
(1−0.5 z−1)+ −2.166
(1−0.3 z−1 ) + 0.654
(1−2 z−1 )

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8)
h ( n ) = z +1 .2−1 .15 z−1 +0 . 85 z−2 −0 . 8 z−3 −0 .15 z− 4
1−0 . 8 z−1+0 . 15 z−2
Matlab Code
nominator=[1 1.2−1.15 0.85−0.8−0.15 ];
denominator=[1−0.8−0.15];
H=tf (nominator , denominator);
impz (nominator , denominator );
Figure 3: Impulse response
4. Part C: image decimation Using MATLAB
In this section, you will see the effect of down sampling on a picture.