Digital Signal P Ramesh Babu

Scilab Textbook Companion for Digital Signal Processing by P. Ramesh Babu1 Created by Mohammad Faisal Siddiqui B.Tech (pursuing) Electronics Engineering Jamia Milia Islamia College Teacher Dr. Sajad A. Loan, JMI, New D Cross-Checked by Santosh Kumar, IITB July 12, 2011

1 Funded

by a grant from the National Mission on Education through ICT, http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilab codes written in it can be downloaded from the ”Textbook Companion Project” section at the website http://scilab.in

Book Description Title: Digital Signal Processing Author: P. Ramesh Babu Publisher: Scitech Publications (INDIA) Pvt. Ltd. Edition: 4 Year: 2010 ISBN: 81-8371-081-7

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Scilab numbering policy used in this document and the relation to the above book. Exa Example (Solved example) Eqn Equation (Particular equation of the above book) AP Appendix to Example(Scilab Code that is an Appednix to a particular Example of the above book) For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 means a scilab code whose theory is explained in Section 2.3 of the book.

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Contents List of Scilab Codes

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1 DISCRETE TIME SIGNALS AND LINEAR SYSTEMS

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2 THE Z TRANSFORM

34

3 THE DISCRETE FOURIER TRANSFORM

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4 THE FAST FOURIER TRANSFORM

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5 INFINITE IMPULSE RESPONSE FILTERS

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6 FINITE IMPULSE RESPONSE FILTERS

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7 FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS 139 8 MULTIRATE SIGNAL PROCESSING

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9 STATISTICAL DIGITAL SIGNAL PROCESSING

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11 DIGITAL SIGNAL PROCESSORS

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List of Scilab Codes Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa

1.1 1.2 1.3.a 1.3.b 1.4.a 1.4.d 1.5.a 1.5.c 1.5.d 1.11 1.12 1.13 1.18 1.19 1.32.a 1.37 1.38 1.45 1.57.a 1.61 1.62 1.64.a 1.64.c 2.1 2.2 2.3 2.4 2.5

Continuous Time Plot and Discrete Time Plot Continuous Time Plot and Discrete Time Plot Evaluate the Summations . . . . . . . . . . . . Evaluate the Summations . . . . . . . . . . . . Check for Energy or Power Signals . . . . . . . Check for Energy or Power Signals . . . . . . . Determining Periodicity of Signal . . . . . . . . Determining Periodicity of Signal . . . . . . . . Determining Periodicity of Signal . . . . . . . . Stability of the System . . . . . . . . . . . . . Convolution Sum of Two Sequences . . . . . . Convolution of Two Signals . . . . . . . . . . . Cross Correlation of Two Sequences . . . . . . Determination of Input Sequence . . . . . . . . Plot Magnitude and Phase Response . . . . . . Sketch Magnitude and Phase Response . . . . . Plot Magnitude and Phase Response . . . . . . Filter to Eliminate High Frequency Component Discrete Convolution of Sequences . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . Frequency Response of LTI System . . . . . . . Frequency Response of LTI System . . . . . . . z Transform and ROC of Causal Sequence . . . z Transform and ROC of Anticausal Sequence . z Transform of the Sequence . . . . . . . . . . z Transform and ROC of the Signal . . . . . . z Transform and ROC of the Signal . . . . . . 4

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11 13 14 14 15 15 16 18 19 21 21 22 22 23 24 24 26 27 29 29 30 31 31 34 34 35 36 36

Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa

2.6 2.7 2.8.a 2.9 2.10 2.11 2.13.a 2.13.b 2.13.c 2.13.d 2.16 2.17 2.19 2.20.a 2.22 2.23 2.34 2.35.a 2.35.b 2.38 2.40 2.41.a 2.41.b 2.41.c 2.45 2.53.a 2.53.b 2.53.c 2.53.d 2.54 2.58 3.1 3.2 3.3 3.4 3.7 3.9 3.11

Stability of the System . . . . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Sequence . . . . . . . z Transform Computation . . . . . . . . . z Transform of the Sequence . . . . . . . z Transform of Discrete Time Signals . . . z Transform of Discrete Time Signals . . . z Transform of Discrete Time Signals . . . z Transform of Discrete Time Signals . . . Impulse Response of the System . . . . . Pole Zero Plot of the Difference Equation Frequency Response of the System . . . . Inverse z Transform Computation . . . . Inverse z Transform Computation . . . . Causal Sequence Determination . . . . . . Impulse Response of the System . . . . . Pole Zero Plot of the System . . . . . . . Unit Sample Response of the System . . . Determine Output Response . . . . . . . Input Sequence Computation . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Signal . . . . . . . . . Pole Zero Pattern of the System . . . . . z Transform of the Sequence . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of the Signal . . . . . . . . . z Transform of Cosine Signal . . . . . . . Impulse Response of the System . . . . . DFT and IDFT . . . . . . . . . . . . . . DFT of the Sequence . . . . . . . . . . . 8 Point DFT . . . . . . . . . . . . . . . . IDFT of the given Sequence . . . . . . . . Plot the Sequence . . . . . . . . . . . . . Remaining Samples . . . . . . . . . . . . DFT Computation . . . . . . . . . . . . . 5

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37 37 37 38 38 39 39 40 40 41 41 42 44 44 45 45 46 47 49 50 52 53 53 54 55 55 55 56 56 56 58 59 60 62 62 63 64 65


3.13 3.14 3.15 3.16 3.17 3.18 3.20 3.21 3.23.a 3.23.b 3.23.c 3.23.d 3.23.e 3.23.f 3.24 3.25 3.26 3.27.a 3.27.b 3.30 3.32 3.36 4.3 4.4 4.6 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16.a 4.16.b 4.17 4.18 4.19

Circular Convolution . . . . . . . . . . . . . Circular Convolution . . . . . . . . . . . . . Determine Sequence x3 . . . . . . . . . . . Circular Convolution . . . . . . . . . . . . . Circular Convolution . . . . . . . . . . . . . Output Response . . . . . . . . . . . . . . . Output Response . . . . . . . . . . . . . . . Linear Convolution . . . . . . . . . . . . . . N Point DFT Computation . . . . . . . . . N Point DFT Computation . . . . . . . . . N Point DFT Computation . . . . . . . . . N Point DFT Computation . . . . . . . . . N Point DFT Computation . . . . . . . . . N Point DFT Computation . . . . . . . . . DFT of the Sequence . . . . . . . . . . . . 8 Point Circular Convolution . . . . . . . . Linear Convolution using DFT . . . . . . . Circular Convolution Computation . . . . . Circular Convolution Computation . . . . . Calculate value of N . . . . . . . . . . . . . Sketch Sequence . . . . . . . . . . . . . . . Determine IDFT . . . . . . . . . . . . . . . Shortest Sequence N Computation . . . . . Twiddle Factor Exponents Calculation . . . DFT using DIT Algorithm . . . . . . . . . DFT using DIF Algorithm . . . . . . . . . 8 Point DFT of the Sequence . . . . . . . . 4 Point DFT of the Sequence . . . . . . . . IDFT of the Sequence using DIT Algorithm 8 Point DFT of the Sequence . . . . . . . . 8 Point DFT of the Sequence . . . . . . . . DFT using DIT Algorithm . . . . . . . . . DFT using DIF Algorithm . . . . . . . . . 8 Point DFT using DIT FFT . . . . . . . . 8 Point DFT using DIT FFT . . . . . . . . IDFT using DIF Algorithm . . . . . . . . . IDFT using DIT Algorithm . . . . . . . . . FFT Computation of the Sequence . . . . . 6

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65 66 67 67 68 69 70 70 71 71 72 72 72 73 73 73 74 75 75 76 76 78 79 80 80 81 81 82 82 82 83 83 84 84 85 85 86 86


4.20 4.21 4.22 4.23 4.24 5.1 5.2 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.15 5.16 5.17 5.18 5.19 5.29 6.1 6.5 6.6 6.7 6.8 6.9.a 6.9.b 6.10 6.11 6.12 6.13.a 6.13.b 6.14.a 6.14.b 6.15

8 Point DFT by Radix 2 DIT FFT . . . . . . . . . . DFT using DIT FFT Algorithm . . . . . . . . . . . Compute X using DIT FFT . . . . . . . . . . . . . . DFT using DIF FFT Algorithm . . . . . . . . . . . 8 Point DFT of the Sequence . . . . . . . . . . . . . Order of the Filter Determination . . . . . . . . . . Order of Low Pass Butterworth Filter . . . . . . . . Analog Butterworth Filter Design . . . . . . . . . . Analog Butterworth Filter Design . . . . . . . . . . Order of Chebyshev Filter . . . . . . . . . . . . . . . Chebyshev Filter Design . . . . . . . . . . . . . . . . Order of Type 1 Low Pass Chebyshev Filter . . . . . Chebyshev Filter Design . . . . . . . . . . . . . . . . HPF Filter Design with given Specifications . . . . . Impulse Invariant Method Filter Design . . . . . . . Impulse Invariant Method Filter Design . . . . . . . Impulse Invariant Method Filter Design . . . . . . . Impulse Invariant Method Filter Design . . . . . . . Bilinear Transformation Method Filter Design . . . . HPF Design using Bilinear Transform . . . . . . . . Bilinear Transformation Method Filter Design . . . . Single Pole LPF into BPF Conversion . . . . . . . . Pole Zero IIR Filter into Lattice Ladder Structure . Group Delay and Phase Delay . . . . . . . . . . . . LPF Magnitude Response . . . . . . . . . . . . . . . HPF Magnitude Response . . . . . . . . . . . . . . . BPF Magnitude Response . . . . . . . . . . . . . . . BRF Magnitude Response . . . . . . . . . . . . . . . HPF Magnitude Response using Hanning Window . HPF Magnitude Response using Hamming Window . Hanning Window Filter Design . . . . . . . . . . . . LPF Filter Design using Kaiser Window . . . . . . . BPF Filter Design using Kaiser Window . . . . . . . Digital Differentiator using Rectangular Window . . Digital Differentiator using Hamming Window . . . Hilbert Transformer using Rectangular Window . . . Hilbert Transformer using Blackman Window . . . . Filter Coefficients obtained by Sampling . . . . . . 7

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86 87 87 88 88 90 90 91 92 92 93 93 94 94 95 96 96 97 97 98 99 99 100 102 104 104 106 108 110 112 114 116 118 122 124 124 126 127

Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa Exa

6.16 6.17 6.18.a 6.18.b 6.19 6.20 6.21 6.28 6.29 7.2 7.14 8.9 8.10 9.7.a 9.7.b 9.8.a 9.8.b 11.3 11.5

Coefficients of Linear phase FIR Filter . . . . . . . BPF Filter Design using Sampling Method . . . . Frequency Sampling Method FIR LPF Filter . . . Frequency Sampling Method FIR LPF Filter . . . Filter Coefficients Determination . . . . . . . . . . Filter Coefficients using Hamming Window . . . . LPF Filter using Rectangular Window . . . . . . . Filter Coefficients for Direct Form Structure . . . . Lattice Filter Coefficients Determination . . . . . . Subtraction Computation . . . . . . . . . . . . . . Variance of Output due to AD Conversion Process Two Component Decomposition . . . . . . . . . . Two Band Polyphase Decomposition . . . . . . . . Frequency Resolution Determination . . . . . . . . Record Length Determination . . . . . . . . . . . . Smallest Record Length Computation . . . . . . . Quality Factor Computation . . . . . . . . . . . . Program for Integer Multiplication . . . . . . . . . Function Value Calculation . . . . . . . . . . . . .

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128 128 129 130 131 133 135 137 138 139 139 141 142 143 144 144 145 146 146

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

Continuous Time Plot and Discrete Time Plot . Continuous Time Plot and Discrete Time Plot . Determining Periodicity of Signal . . . . . . . . Determining Periodicity of Signal . . . . . . . . Determining Periodicity of Signal . . . . . . . . Plot Magnitude and Phase Response . . . . . . Sketch Magnitude and Phase Response . . . . . Plot Magnitude and Phase Response . . . . . . Filter to Eliminate High Frequency Component Frequency Response of LTI System . . . . . . . Frequency Response of LTI System . . . . . . .

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12 13 17 18 20 23 25 26 28 30 32

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Pole Zero Plot of the Difference Equation Frequency Response of the System . . . Impulse Response of the System . . . . . Pole Zero Plot of the System . . . . . . . Unit Sample Response of the System . . Determine Output Response . . . . . . . Pole Zero Pattern of the System . . . . . Impulse Response of the System . . . . .

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3.1 3.2 3.3

DFT of the Sequence . . . . . . . . . . . . . . . . . . . . . . Plot the Sequence . . . . . . . . . . . . . . . . . . . . . . . . Sketch Sequence . . . . . . . . . . . . . . . . . . . . . . . . .

60 63 77

6.1 6.2 6.3 6.4

LPF Magnitude Response HPF Magnitude Response BPF Magnitude Response BRF Magnitude Response

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103 105 107 109

6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18

HPF Magnitude Response using Hanning Window . HPF Magnitude Response using Hamming Window Hanning Window Filter Design . . . . . . . . . . . LPF Filter Design using Kaiser Window . . . . . . BPF Filter Design using Kaiser Window . . . . . . Digital Differentiator using Rectangular Window . . Digital Differentiator using Hamming Window . . . Hilbert Transformer using Rectangular Window . . Hilbert Transformer using Blackman Window . . . Frequency Sampling Method FIR LPF Filter . . . . Frequency Sampling Method FIR LPF Filter . . . . Filter Coefficients Determination . . . . . . . . . . Filter Coefficients using Hamming Window . . . . . LPF Filter using Rectangular Window . . . . . . .

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111 113 115 117 119 121 123 125 126 129 131 132 134 136

Chapter 1 DISCRETE TIME SIGNALS AND LINEAR SYSTEMS

Scilab code Exa 1.1 Continuous Time Plot and Discrete Time Plot 1 2

3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 1 . 1 // S k e t c h t h e c o n t i n u o u s t i m e s i g n a l x ( t ) =2∗ exp (−2 t ) and a l s o i t s d i s c r e t e t i m e e q u i v a l e n t s i g n a l w i t h a sampling period T = 0.2 sec clear all ; clc ; close ; t =0:0.01:2; x1 =2* exp ( -2* t ) ; subplot (1 ,2 ,1) ; plot (t , x1 ) ; xlabel ( ’ t ’ ) ; ylabel ( ’ x ( t ) ’ ) ; title ( ’CONTINUOUS TIME PLOT ’ ) ; n =0:0.2:2; x2 =2* exp ( -2* n ) ; subplot (1 ,2 ,2) ; 11

Figure 1.1: Continuous Time Plot and Discrete Time Plot

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Figure 1.2: Continuous Time Plot and Discrete Time Plot 16 17 18 19

plot2d3 (n , x2 ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’DISCRETE TIME PLOT ’ ) ;

Scilab code Exa 1.2 Continuous Time Plot and Discrete Time Plot 1 2

// Example 1 . 2 // S k e t c h t h e c o n t i n u o u s t i m e s i g n a l x=s i n ( 7 ∗ t )+s i n ( 1 0 ∗ t ) and a l s o i t s d i s c r e t e t i m e e q u i v a l e n t s i g n a l with a sampling p e r i o d T = 0 . 2 s e c 13

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

clear all ; clc ; close ; t =0:0.01:2; x1 = sin (7* t ) + sin (10* t ) ; subplot (1 ,2 ,1) ; plot (t , x1 ) ; xlabel ( ’ t ’ ) ; ylabel ( ’ x ( t ) ’ ) ; title ( ’CONTINUOUS TIME PLOT ’ ) ; n =0:0.2:2; x2 = sin (7* n ) + sin (10* n ) ; subplot (1 ,2 ,2) ; plot2d3 (n , x2 ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’DISCRETE TIME PLOT ’ ) ;

Scilab code Exa 1.3.a Evaluate the Summations 1 // Example 1 . 3 ( a ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // C a l c u l a t e F o l l o w i n g Summations 4 clear all ; 5 clc ; 6 close ; 7 syms n ; 8 X = symsum ( sin (2* n ) ,n ,2 , 2) ; 9 // D i s p l a y t h e r e s u l t i n command window 10 disp (X , ” The V a l u e o f summation comes o u t t o be : ” ) ;

Scilab code Exa 1.3.b Evaluate the Summations 14

1 // Example 1 . 3 ( b ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // C a l c u l a t e F o l l o w i n g Summations 4 clear all ; 5 clc ; 6 close ; 7 syms n ; 8 X = symsum ( %e ^(2* n ) ,n ,0 , 0) ; 9 // D i s p l a y t h e r e s u l t i n command window 10 disp (X , ” The V a l u e o f summation comes o u t t o be : ” ) ;

Scilab code Exa 1.4.a Check for Energy or Power Signals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 1 . 4 ( a ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM // Find Energy and Power o f Given S i g n a l s clear all ; clc ; close ; syms n N ; x =(1/3) ^ n ; E = symsum ( x ^2 , n ,0 , %inf ) ; // D i s p l a y t h e r e s u l t i n command window disp (E , ” Energy : ” ) ; p =(1/(2* N +1) ) * symsum ( x ^2 , n ,0 , N ) ; P = limit (p ,N , %inf ) ; disp (P , ” Power : ” ) ; // The Energy i s F i n i t e and Power i s 0 . T h e r e f o r e t h e g i v e n s i g n a l i s an Energy S i g n a l

Scilab code Exa 1.4.d Check for Energy or Power Signals 1

// Example 1 . 4 ( d ) 15

2 3 4 5 6 7 8 9 10 11 12 13 14 15

//MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM // Find Energy and Power o f Given S i g n a l s clear all ; clc ; close ; syms n N ; x = %e ^(2* n ) ; E = symsum ( x ^2 , n ,0 , %inf ) ; // D i s p l a y t h e r e s u l t i n command window disp (E , ” Energy : ” ) ; p =(1/(2* N +1) ) * symsum ( x ^2 , n ,0 , N ) ; P = limit (p ,N , %inf ) ; disp (P , ” Power : ” ) ; // The Energy andPower i s i n f i n i t e . T h e r e f o r e t h e g i v e n s i g n a l i s an n e i t h e r Energy S i g n a l n o r Power S i g n a l

Scilab code Exa 1.5.a Determining Periodicity of Signal 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 1 . 5 ( a ) //To D e t e r m i n e Whether Given S i g n a l i s P e r i o d i c o r not clear all ; clc ; close ; t =0:0.01:2; x1 = exp ( %i *6* %pi * t ) ; subplot (1 ,2 ,1) ; plot (t , x1 ) ; xlabel ( ’ t ’ ) ; ylabel ( ’ x ( t ) ’ ) ; title ( ’CONTINUOUS TIME PLOT ’ ) ; n =0:0.2:2; 16

Figure 1.3: Determining Periodicity of Signal

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Figure 1.4: Determining Periodicity of Signal 14 x2 = exp ( %i *6* %pi * n ) ; 15 subplot (1 ,2 ,2) ; 16 plot2d3 (n , x2 ) ; 17 xlabel ( ’ n ’ ) ; 18 ylabel ( ’ x ( n ) ’ ) ; 19 title ( ’DISCRETE TIME PLOT ’ ) ; 20 // Hence Given S i g n a l i s P e r i o d i c w i t h N=1

Scilab code Exa 1.5.c Determining Periodicity of Signal 1

// Example 1 . 5 ( c ) 18

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

//To D e t e r m i n e Whether Given S i g n a l i s P e r i o d i c o r not clear all ; clc ; close ; t =0:0.01:10; x1 = cos (2* %pi * t /3) ; subplot (1 ,2 ,1) ; plot (t , x1 ) ; xlabel ( ’ t ’ ) ; ylabel ( ’ x ( t ) ’ ) ; title ( ’CONTINUOUS TIME PLOT ’ ) ; n =0:0.2:10; x2 = cos (2* %pi * n /3) ; subplot (1 ,2 ,2) ; plot2d3 (n , x2 ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’DISCRETE TIME PLOT ’ ) ; // Hence Given S i g n a l i s P e r i o d i c w i t h N=3

Scilab code Exa 1.5.d Determining Periodicity of Signal 1 2 3 4 5 6 7 8 9

// Example 1 . 5 ( d ) //To D e t e r m i n e Whether Given S i g n a l i s P e r i o d i c o r not clear all ; clc ; close ; t =0:0.01:50; x1 = cos ( %pi * t /3) + cos (3* %pi * t /4) ; subplot (1 ,2 ,1) ; plot (t , x1 ) ; 19

Figure 1.5: Determining Periodicity of Signal

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10 xlabel ( ’ t ’ ) ; 11 ylabel ( ’ x ( t ) ’ ) ; 12 title ( ’CONTINUOUS TIME PLOT ’ ) ; 13 n =0:0.2:50; 14 x2 = cos ( %pi * n /3) + cos (3* %pi * n /4) ; 15 subplot (1 ,2 ,2) ; 16 plot2d3 (n , x2 ) ; 17 xlabel ( ’ n ’ ) ; 18 ylabel ( ’ x ( n ) ’ ) ; 19 title ( ’DISCRETE TIME PLOT ’ ) ; 20 // Hence Given S i g n a l i s P e r i o d i c w i t h N=24

Scilab code Exa 1.11 Stability of the System 1 // Example 1 . 1 1 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // T e s t i n g S t a b i l i t y o f Given System 4 clear all ; 5 clc ; 6 close ; 7 syms n ; 8 x =(1/2) ^ n 9 X = symsum (x , n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ” Summation i s : ” ) ; 12 disp ( ’ Hence Summation < i n f i n i t y . Given System i s

S t a b l e ’ );

Scilab code Exa 1.12 Convolution Sum of Two Sequences 1 2 3

// Example 1 . 1 2 // Program t o Compute c o n v o l u t i o n o f g i v e n s e q u e n c e s // x ( n ) =[3 2 1 2 ] , h ( n ) =[1 2 1 2 ] ; 21

4 5 6 7 8 9 10

clear all ; clc ; close ; x =[3 2 1 2]; h =[1 2 1 2]; y = convol (x , h ) ; disp ( y ) ;

Scilab code Exa 1.13 Convolution of Two Signals 1 2 3 4 5 6 7 8 9 10

// Example 1 . 1 3 // Program t o Compute c o n v o l u t i o n o f g i v e n s e q u e n c e s // x ( n ) =[1 2 1 1 ] , h ( n ) =[1 −1 1 − 1 ] ; clear all ; clc ; close ; x =[1 2 1 1]; h =[1 -1 1 -1]; y = convol (x , h ) ; disp ( round ( y ) ) ;

Scilab code Exa 1.18 Cross Correlation of Two Sequences 1 2 3 4 5 6 7 8 9

// Example 1 . 1 8 // Program t o Compute C r o s s − c o r r e l a t i o n sequences // x ( n ) =[1 2 1 1 ] , h ( n ) =[1 1 2 1 ] ; clear all ; clc ; close ; x =[1 2 1 1]; h =[1 1 2 1]; h1 =[1 2 1 1]; 22

of given

Figure 1.6: Plot Magnitude and Phase Response 10 y = convol (x , h1 ) ; 11 disp ( round ( y ) ) ;

Scilab code Exa 1.19 Determination of Input Sequence 1 // Example 1 . 1 9 2 //To f i n d i n p u t x ( n ) 3 // h ( n ) =[1 2 1 ] , y ( n ) =[1 5 10 11 8 4 1 ] 4 clear all ; 5 clc ; 6 close ; 7 z = %z ; 8 a = z ^6+5*( z ^(5) ) +10*( z ^(4) ) +11*( z ^(3) ) +8*( z ^(2) ) +4*( z

^(1) ) +1; 9 b = z ^6+2* z ^(5) +1* z ^(4) ; 10 x = ldiv (a ,b ,5) ; 11 disp (x , ” x ( n )=” ) ;

23

Scilab code Exa 1.32.a Plot Magnitude and Phase Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 1 . 3 2 // Program t o P l o t Magnitude and Phase R e s p o n c e clear all ; clc ; close ; w = - %pi :0.01: %pi ; H =1+2* cos ( w ) +2* cos (2* w ) ; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

Scilab code Exa 1.37 Sketch Magnitude and Phase Response

24

Figure 1.7: Sketch Magnitude and Phase Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

// Example 1 . 3 7 // Program t o P l o t Magnitude and Phase R e s p o n c e // y ( n ) =1/2[ x ( n )+x ( n−2) ] clear all ; clc ; close ; w =0:0.01: %pi ; H =(1+ cos (2* w ) - %i * sin (2* w ) ) /2; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; 25

Figure 1.8: Plot Magnitude and Phase Response 24 25 26

xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

Scilab code Exa 1.38 Plot Magnitude and Phase Response 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 1 . 3 8 // Program t o P l o t Magnitude and Phase R e s p o n c e // 0 . 5 d e l t a ( n )+d e l t a ( n−1) +0.5 d e l t a ( n−2) clear all ; clc ; close ; w = - %pi :0.01: %pi ; H =0.5+ exp ( - %i * w ) +0.5* exp ( - %i * w ) ; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; 26

14 15 16 17 18 19 20 21 22 23 24 25 26

a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

Scilab code Exa 1.45 Filter to Eliminate High Frequency Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

// Example 1 . 4 5 // clear all ; clc ; close ; t =0:0.01:10; x =2* cos (5* t ) + cos (300* t ) ; x1 =2* cos (5* t ) ; b =[0.05 0.05]; a =[1 -0.9]; y = filter (b ,a , x ) ; subplot (2 ,1 ,1) ; plot (t , x ) ; xlabel ( ’ Time i n S e c ’ ) ; ylabel ( ’ A m p l i t u d e ’ ) ; subplot (2 ,1 ,2) ; plot (t , y ) ; 27

Figure 1.9: Filter to Eliminate High Frequency Component

28

18 19 20

subplot (2 ,1 ,2) ; plot (t , x1 , ’ : ’ ) ; title ( ’ x : SIGNAL WITHOUT NOISE ; 21 xlabel ( ’ Time i n S e c ’ ) ; 22 ylabel ( ’ A m p l i t u d e ’ ) ;

y : SIGNAL WITH NOISE ’ )

Scilab code Exa 1.57.a Discrete Convolution of Sequences 1 2 3 4 5 6 7 8 9 10

// Example 1 . 5 7 ( a ) // Program t o Compute d i s c r e t e c o n v o l u t i o n o f g i v e n sequences // x ( n ) =[1 2 −1 1 ] , h ( n ) =[1 0 1 1 ] ; clear all ; clc ; close ; x =[1 2 -1 1]; h =[1 0 1 1]; y = convol (x , h ) ; disp ( round ( y ) ) ;

Scilab code Exa 1.61 Fourier Transform 1 // Example 1 . 6 1 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // F o u r i e r t r a n s f o r m o f ( 3 ) ˆ n u ( n ) 4 clear all ; 5 clc ; 6 close ; 7 syms n ; 8 x =(3) ^ n ; 9 X = symsum (x , n ,0 , %inf ) 10 // D i s p l a y t h e r e s u l t i n command window

29

Figure 1.10: Frequency Response of LTI System 11

disp (X , ” The F o u r i e r T r a n s f o r m d o e s n o t e x i t a s x ( n ) i s n o t a b s o l u t e l y summable and a p p r o a c h e s i n f i n i t y i . e . ”);

Scilab code Exa 1.62 Fourier Transform 1 // Example 1 . 6 2 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // F o u r i e r t r a n s f o r m o f ( 0 . 8 ) ˆ | n | u ( n ) 4 clear all ; 5 clc ; 6 close ; 7 syms w n ; 8 X = symsum ((0.8) ^ n * %e ^( %i * w * n ) ,n ,1 , %inf ) + symsum 9 10

((0.8) ^ n * %e ^( - %i * w * n ) ,n ,0 , %inf ) // D i s p l a y t h e r e s u l t i n command window disp (X , ” The F o u r i e r T r a n s f o r m comes o u t t o be : ” ) ;

30

Scilab code Exa 1.64.a Frequency Response of LTI System 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 1 . 6 4 ( a ) // Program t o C a l c u l a t e P l o t Magnitude and Phase Responce clear all ; clc ; close ; w =0:0.01: %pi ; H =1/(1 -0.5* %e ^( - %i * w ) ) ; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

Scilab code Exa 1.64.c Frequency Response of LTI System 31

Figure 1.11: Frequency Response of LTI System

32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 1 . 6 4 ( c ) // Program t o C a l c u l a t e P l o t Magnitude and Phase Responce clear all ; clc ; close ; w =0:0.01: %pi ; H =1/(1 -0.9* %i * %e ^( - %i * w ) ) ; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

33

Chapter 2 THE Z TRANSFORM

Scilab code Exa 2.1 z Transform and ROC of Causal Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 2 . 1 //Z− t r a n s f o r m o f [ 1 0 3 −1 2 ] clear all ; clc ; close ; function [ za ]= ztransfer ( sequence , n ) z = poly (0 , ’ z ’ , ’ r ’ ) za = sequence *(1/ z ) ^n ’ endfunction x1 =[1 0 3 -1 2]; n =0: length ( x1 ) -1; zz = ztransfer ( x1 , n ) ; // D i s p l a y t h e r e s u l t i n command window disp ( zz , ”Z−t r a n s f o r m o f s e q u e n c e i s : ” ) ; disp ( ’ROC i s t h e e n t i r e p l a n e e x c e p t z = 0 ’ ) ;

Scilab code Exa 2.2 z Transform and ROC of Anticausal Sequence

34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 2 . 2 //Z− t r a n s f o r m o f [ −3 −2 −1 0 1 ] clear all ; clc ; close ; function [ za ]= ztransfer ( sequence , n ) z = poly (0 , ’ z ’ , ’ r ’ ) za = sequence *(1/ z ) ^n ’ endfunction x1 =[ -3 -2 -1 0 1]; n = -( length ( x1 ) -1) :0; zz = ztransfer ( x1 , n ) ; // D i s p l a y t h e r e s u l t i n command window disp ( zz , ”Z−t r a n s f o r m o f s e q u e n c e i s : ” ) ; disp ( ’ROC i s t h e e n t i r e p l a n e e x c e p t z = % i n f ’ ) ;

Scilab code Exa 2.3 z Transform of the Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 2 . 3 //Z− t r a n s f o r m o f [ 2 −1 3 2 1 0 2 3 −1] clear all ; clc ; close ; function [ za ]= ztransfer ( sequence , n ) z = poly (0 , ’ z ’ , ’ r ’ ) za = sequence *(1/ z ) ^n ’ endfunction x1 =[2 -1 3 2 1 0 2 3 -1]; n = -4:4; zz = ztransfer ( x1 , n ) ; // D i s p l a y t h e r e s u l t i n command window disp ( zz , ”Z−t r a n s f o r m o f s e q u e n c e i s : ” ) ; disp ( ’ROC i s t h e e n t i r e p l a n e e x c e p t z = 0 and z = %inf ’ );

35

Scilab code Exa 2.4 z Transform and ROC of the Signal 1 // Example 2 . 4 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f a ˆn u ( n ) 4 clear all ; 5 clc ; 6 close ; 7 syms a n z ; 8 x =a^n 9 X = symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ”Z−t r a n s f o r m o f a ˆn u ( n ) w i t h i s : ” ) ; 12 disp ( ’ROC i s t h e R e g i o n mod ( z ) > a ’ )

Scilab code Exa 2.5 z Transform and ROC of the Signal 1 // Example 2 . 5 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f −b ˆn u(−n −1) 4 clear all ; 5 clc ; 6 close ; 7 syms b n z ; 8 x =b^n 9 X = symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ”Z−t r a n s f o r m o f b ˆ n u ( n ) w i t h i s : ” ) ; 12 disp ( ’ROC i s t h e R e g i o n mod ( z ) < b ’ )

36

Scilab code Exa 2.6 Stability of the System 1 // Example 2 . 6 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f 2ˆ n u ( n ) 4 clear all ; 5 clc ; 6 close ; 7 syms n z ; 8 x =(2) ^ n 9 X = symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ”Z−t r a n s f o r m o f 2ˆ n u ( n ) i s : ” ) ; 12 disp ( ’ROC i s t h e R e g i o n mod ( z ) > 2 ’ ) ;

Scilab code Exa 2.7 z Transform of the Signal 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 2 . 7 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z− t r a n s f o r m o f [ 3 ( 3 ˆ n ) −4(2) ˆ n ] u ( n ) clear all ; clc ; close ; syms n z ; x1 =(3) ^( n ) ; X1 = symsum (3* x1 *( z ^( - n ) ) ,n ,0 , %inf ) ; x2 =(4) ^( n ) ; X2 = symsum (4* x2 *( z ^( - n ) ) ,n ,0 , %inf ) ; X = ( X1 - X2 ) ; // D i s p l a y t h e r e s u l t i n command window disp (X , ”Z−t r a n s f o r m o f [ 3 ( 3 ˆ n ) −4(2) ˆ n ] u ( n ) i s : ”);

Scilab code Exa 2.8.a z Transform of the Signal 37

1 2 3 4 5 6 7 8 9 10 11

// Example 2 . 8 ( a ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z t r a n s f o r m o f c o s (Wo∗n ) clc ; syms Wo n z ; x1 = exp ( sqrt ( -1) * Wo * n ) ; X1 = symsum ( x1 *( z ^ - n ) ,n ,0 , %inf ) ; x2 = exp ( - sqrt ( -1) * Wo * n ) ; X2 = symsum ( x2 *( z ^ - n ) ,n ,0 , %inf ) ; X =( X1 + X2 ) /2; disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.9 z Transform of the Sequence 1 // Example 2 . 9 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f ( 1 / 3 ) ˆn u ( n −1) 4 clear all ; 5 clc ; 6 close ; 7 syms n z ; 8 x =(1/3) ^ n ; 9 X = (1/ z ) * symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ”Z−t r a n s f o r m o f ( 1 / 3 ) ˆn u ( n −1) i s : ” ) ;

Scilab code Exa 2.10 z Transform Computation 1 // Example 2 . 1 0 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z t r a n s f o r m o f r ˆn . c o s (Wo∗n ) 4 clc ; 5 syms r Wo n z ;

38

6 7 8 9 10 11

x1 =( r ^ n ) * exp ( sqrt ( -1) * Wo * n ) ; X1 = symsum ( x1 *( z ^ - n ) ,n ,0 , %inf ) ; x2 =( r ^ n ) * exp ( - sqrt ( -1) * Wo * n ) ; X2 = symsum ( x2 *( z ^ - n ) ,n ,0 , %inf ) ; X =( X1 + X2 ) /2; disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.11 z Transform of the Sequence 1 2 3 4 5 6 7 8 9 10 11 12

// Example 2 . 1 1 //MAXIMA SCILAB TOOLBOX //Z− t r a n s f o r m o f n . a ˆn clear all ; clc ; close ; syms a n z ; x =( a ) ^ n ; X = symsum ( x *( z ^( - n ) ) ,n Y = diff (X , z ) ; // D i s p l a y t h e r e s u l t i n disp (Y , ”Z−t r a n s f o r m o f

REQUIRED FOR THIS PROGRAM u(n)

,0 , %inf ) command window n . a ˆn u ( n ) i s : ”);

Scilab code Exa 2.13.a z Transform of Discrete Time Signals 1 // Example 2 . 1 3 ( a ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f ( −1/5) ˆ n u ( n ) + 5 ( 1 / 2 ) ˆ(−n ) u(−n −1) 4 clear all ; 5 clc ; 6 close ; 7 syms n z ; 8 x1 =( -1/5) ^ n ; 9 X1 = symsum ( x1 *( z ^( - n ) ) ,n ,0 , %inf ) ;

39

10 x2 =(1/2) ^( - n ) ; 11 X2 = symsum (5* x2 *( z ^( - n ) ) ,n ,0 , %inf ) ; 12 X = ( X1 - X2 ) ; 13 // D i s p l a y t h e r e s u l t i n command window 14 disp (X , ”Z−t r a n s f o r m o f [ 3 ( 3 ˆ n ) −4(2) ˆ n ] u ( n ) 15 disp ( ’ROC i s t h e R e g i o n 1/5 < mod ( z ) < 2 ’ ) ;

i s : ”);

Scilab code Exa 2.13.b z Transform of Discrete Time Signals 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 2 . 1 3 ( b ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z t r a n s f o r m clc ; syms n z k ; x1 =1; X1 = symsum ( x1 * z ^( - n ) ,n ,0 ,0) ; x2 =1; X2 = symsum ( x2 * z ^( - n ) ,n ,1 ,1) ; x3 =1; X3 = symsum ( x3 * z ^( - n ) ,n ,2 ,2) ; X =0.5* X1 + X2 -1/3* X3 ; disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.13.c z Transform of Discrete Time Signals 1 // Example 2 . 1 3 ( c ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z− t r a n s f o r m o f u ( n −2) 4 clear all ; 5 clc ; 6 close ; 7 syms n z ; 8 x =1;

40

9 X = (1/( z ^2) ) * symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) ; 10 // D i s p l a y t h e r e s u l t i n command window 11 disp (X , ”Z−t r a n s f o r m o f u ( n −2) i s : ” ) ;

Scilab code Exa 2.13.d z Transform of Discrete Time Signals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 2 . 1 3 ( d ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z− t r a n s f o r m o f ( n + 0 . 5 ) ( ( 1 / 3 ) ˆn ) u ( n ) clear all ; clc ; close ; syms n z ; x1 =(1/3) ^ n ; X11 = symsum ( x1 *( z ^( - n ) ) ,n ,0 , %inf ) X1 = diff ( X11 , z ) ; x2 =(1/3) ^( n ) ; X2 = symsum (0.5* x2 *( z ^( - n ) ) ,n ,0 , %inf ) ; X = ( X1 + X2 ) ; // D i s p l a y t h e r e s u l t i n command window disp (X , ”Z−t r a n s f o r m o f ( n + 0 . 5 ) ( ( 1 / 3 ) ˆn ) u ( n ) i s : ”);

Scilab code Exa 2.16 Impulse Response of the System 1 2 3 4 5 6 7 8 9

// Example 2 . 1 6 //To f i n d i n p u t h ( n ) // a =[1 2 −4 1 ] , b = [ 1 ] clear all ; clc ; close ; z = %z ; a = z ^3+2*( z ^(2) ) -4*( z ) +1; b = z ^3; 41

Figure 2.1: Pole Zero Plot of the Difference Equation 10 h = ldiv (a ,b ,4) ; 11 disp (h , ” h ( n )=” ) ;

Scilab code Exa 2.17 Pole Zero Plot of the Difference Equation 1 // Example 2 . 1 7 2 //To draw t h e p o l e −z e r o p l o t 3 clear all ; 4 clc ; 5 close ; 6 z = %z 7 H1Z =(( z ) *( z -1) ) /(( z -0.25) *( z -0.5) ) ; 8 xset ( ’ window ’ ,1) ; 9 plzr ( H1Z ) ;

42

Figure 2.2: Frequency Response of the System

43

Scilab code Exa 2.19 Frequency Response of the System 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 2 . 1 9 // Program t o P l o t Magnitude and Phase R e s p o n c e clear all ; clc ; close ; w = - %pi :0.01: %pi ; H =1/(1 -0.5*( cos ( w ) - %i * sin ( w ) ) ) ; // c a l u c u l a t i o n o f Phase and Magnitude o f H [ phase_H , m ]= phasemag ( H ) ; Hm = abs ( H ) ; a = gca () ; subplot (2 ,1 ,1) ; a . y_location = ” o r i g i n ” ; plot2d ( w / %pi , Hm ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ a b s (Hm) ’ ) ; title ( ’MAGNITUDE RESPONSE ’ ) ; subplot (2 ,1 ,2) ; a = gca () ; a . x_location = ” o r i g i n ” ; a . y_location = ” o r i g i n ” ; plot2d ( w /(2* %pi ) , phase_H ) ; xlabel ( ’ F r e q u e n c y i n R a d i a n s ’ ) ; ylabel ( ’ <(H) ’ ) ; title ( ’PHASE RESPONSE ’ ) ;

Scilab code Exa 2.20.a Inverse z Transform Computation 1 // Example 2 . 1 0 ( a ) 2 //To f i n d i n p u t h ( n ) 3 //X( z ) =( z + 0 . 2 ) / ( ( z + 0 . 5 ) ( z −1) ; 4 clear all ; 5 clc ;

44

6 7 8 9 10 11

close ; z = %z ; a =( z +0.5) *( z -1) ; b = z +0.2; h = ldiv (b ,a ,4) ; disp (h , ” h ( n )=” ) ;

Scilab code Exa 2.22 Inverse z Transform Computation 1 2 3 4 5 6 7 8 9 10 11

// Example 2 . 2 2 //To f i n d i n p u t x ( n ) //X( z ) =1/(2∗ z ˆ( −2) +2∗ z ˆ( −1) +1) ; clear all ; clc ; close ; z = %z ; a =(2+2* z + z ^2) ; b = z ^2; h = ldiv (b ,a ,6) ; disp (h , ” F i r s t s i x v a l u e s o f h ( n )=” ) ;

Scilab code Exa 2.23 Causal Sequence Determination 1 2 3 4 5 6 7 8 9 10

// Example 2 . 2 3 //To f i n d i n p u t x ( n ) //X( z ) =1/(1−2 z ˆ( −1) ) (1− z ˆ( −1) ) ˆ 2 ; clear all ; clc ; close ; z = %z ; a =( z -2) *( z -1) ^2; b = z ^3; h = ldiv (b ,a ,6) ; 45

Figure 2.3: Impulse Response of the System 11

disp (h , ” F i r s t s i x v a l u e s o f h ( n )=” ) ;

Scilab code Exa 2.34 Impulse Response of the System 1 2 3 4 5 6

// Example 2 . 3 4 //To p l o t t h e i m p u l s e r e s p o n c e o f t h e s y s t e m a n a l y i c a l l y and u s i n g s c i l a b clear all ; clc ; close ; n =0:1:50; 46

7 x =[1 , zeros (1 ,50) ]; 8 b =[1 2]; 9 a =[1 -3 -4]; 10 yanaly =6/5*4.^ n -1/5*( -1) .^ n ; // A n a l y t i c a l S o l u t i o n 11 ymat = filter (b ,a , x ) ; 12 subplot (3 ,1 ,1) ; 13 plot2d3 (n , x ) ; 14 xlabel ( ’ n ’ ) ; 15 ylabel ( ’ x ( n ) ’ ) ; 16 title ( ’INPUT SEQUENCE (IMPULSE FUNCTION) ’ ) ; 17 subplot (3 ,1 ,2) ; 18 plot2d3 (n , yanaly ) ; 19 xlabel ( ’ n ’ ) ; 20 ylabel ( ’ y ( n ) ’ ) ; 21 title ( ’OUTPUT SEQUENCE y a n a l y ’ ) ; 22 subplot (3 ,1 ,3) ; 23 plot2d3 (n , ymat ) ; 24 xlabel ( ’ n ’ ) ; 25 ylabel ( ’ y ( n ) ’ ) ; 26 title ( ’OUTPUT SEQUENCE ymat ’ ) ; 27 // As t h e A n a l t i c a l P l o t m a t c h e s t h e S c i l a b P l o t

hence i t

i s the Responce o f the system

Scilab code Exa 2.35.a Pole Zero Plot of the System 1 // Example 2 . 3 5 ( a ) 2 //To draw t h e p o l e −z e r o 3 clear all ; 4 clc ; 5 close ; 6 z = %z 7 H1Z =( z ) /( z ^2 -z -1) ; 8 xset ( ’ window ’ ,1) ;

plot

47

Figure 2.4: Pole Zero Plot of the System

48

Figure 2.5: Unit Sample Response of the System 9

plzr ( H1Z ) ;

Scilab code Exa 2.35.b Unit Sample Response of the System 1 2 3 4 5 6

// Example 2 . 3 5 ( b ) //To p l o t t h e r e s p o n c e o f t h e s y s t e m a n a l y i c a l l y and using s c i l a b clear all ; clc ; close ; n =0:1:20; 49

7 x = ones (1 , length ( n ) ) ; 8 b =[0 1]; 9 a =[1 -1 -1]; 10 yanaly =0.447*(1.618) .^ n -0.447*( -0.618) .^ n ; // 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Analytical Solution [ ymat , zf ]= filter (b , a , x ) ; subplot (3 ,1 ,1) ; plot2d3 (n , x ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’INPUT SEQUENCE (STEP FUNCTION) ’ ) ; subplot (3 ,1 ,2) ; plot2d3 (n , yanaly ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE y a n a l y ’ ) ; subplot (3 ,1 ,3) ; plot2d3 (n , ymat , zf ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE ymat ’ ) ; // As t h e A n a l t i c a l P l o t m a t c h e s t h e S c i l a b P l o t hence i t i s the Responce o f the system

Scilab code Exa 2.38 Determine Output Response 1 2 3 4 5 6

// Example 2 . 3 8 //To p l o t t h e r e s p o n c e o f t h e s y s t e m a n a l y i c a l l y and using s c i l a b clear all ; clc ; close ; n =0:1:20; 50

Figure 2.6: Determine Output Response

51

7 x=n; 8 b =[0 1 1]; 9 a =[1 -0.7 0.12]; 10 yanaly =38.89*(0.4) .^ n -26.53*(0.3) .^ n -12.36+4.76* n ; // 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Analytical Solution ymat = filter (b ,a , x ) ; subplot (3 ,1 ,1) ; plot2d3 (n , x ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’INPUT SEQUENCE (RAMP FUNCTION) ’ ) ; subplot (3 ,1 ,2) ; plot2d3 (n , yanaly ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE y a n a l y ’ ) ; subplot (3 ,1 ,3) ; plot2d3 (n , ymat ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE ymat ’ ) ; // As t h e A n a l t i c a l P l o t m a t c h e s t h e S c i l a b P l o t hence i t i s the Responce o f the system

Scilab code Exa 2.40 Input Sequence Computation 1 // Example 2 . 4 0 2 //To f i n d i n p u t x ( n ) 3 // h ( n ) =1 2 3 2 , y ( n ) =[1 3 7 10 10 7 2 ] 4 clear all ; 5 clc ; 6 close ; 7 z = %z ; 8 a = z ^6+3*( z ^(5) ) +7*( z ^(4) ) +10*( z ^(3) ) +10*( z ^(2) ) +7*( z

^(1) ) +2; 52

9 b = z ^6+2* z ^(5) +3* z ^(4) +2* z ^(3) ; 10 x = ldiv (a ,b ,4) ; 11 disp (x , ” x ( n )=” ) ;

Scilab code Exa 2.41.a z Transform of the Signal 1 2 3 4 5 6 7 8 9 10 11 12

// Example 2 . 4 1 ( a ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z− t r a n s f o r m o f n . ( − 1 ) ˆn u ( n ) clear all ; clc ; close ; syms a n z ; x =( -1) ^ n ; X = symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) Y = diff (X , z ) ; // D i s p l a y t h e r e s u l t i n command window disp (Y , ”Z−t r a n s f o r m o f n . ( − 1 ) ˆn u ( n ) i s : ”);

Scilab code Exa 2.41.b z Transform of the Signal 1 2 3 4 5 6 7 8 9 10 11 12

// Example 2 . 4 1 ( b ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z− t r a n s f o r m o f n ˆ2 u ( n ) clear all ; clc ; close ; syms n z ; x =1; X = symsum ( x *( z ^( - n ) ) ,n ,0 , %inf ) Y = diff ( diff (X , z ) ,z ) ; // D i s p l a y t h e r e s u l t i n command window disp (Y , ”Z−t r a n s f o r m o f n ˆ2 u ( n ) i s : ”); 53

Figure 2.7: Pole Zero Pattern of the System

Scilab code Exa 2.41.c z Transform of the Signal 1 2 3 4 5 6 7 8 9 10 11 12

// Example 2 . 4 1 ( c ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z t r a n s f o r m o f ( −1) ˆ n . c o s ( %pi /3∗ n ) clc ; syms n z ; Wo = %pi /3; x1 = exp ( sqrt ( -1) * Wo * n ) ; X1 =( -1) ^ n * symsum ( x1 *( z ^ - n ) ,n ,0 , %inf ) ; x2 = exp ( - sqrt ( -1) * Wo * n ) ; X2 =( -1) ^ n * symsum ( x2 *( z ^ - n ) ,n ,0 , %inf ) ; X =( X1 + X2 ) /2; disp (X , ’X( z )= ’ ) ;

54

Scilab code Exa 2.45 Pole Zero Pattern of the System 1 // Example 2 . 4 5 2 //To draw t h e p o l e −z e r o p l o t 3 clear all ; 4 clc ; 5 close ; 6 z = %z 7 H1Z =(( z ) *( z +1) ) /( z ^2 - z +0.5) ; 8 xset ( ’ window ’ ,1) ; 9 plzr ( H1Z ) ;

Scilab code Exa 2.53.a z Transform of the Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 2 . 5 3 ( a ) //Z− t r a n s f o r m o f [ 3 1 2 5 7 0 1 ] clear all ; clc ; close ; function [ za ]= ztransfer ( sequence , n ) z = poly (0 , ’ z ’ , ’ r ’ ) za = sequence *(1/ z ) ^n ’ endfunction x1 =[3 1 2 5 7 0 1]; n = -3:3; zz = ztransfer ( x1 , n ) ; // D i s p l a y t h e r e s u l t i n command window disp ( zz , ”Z−t r a n s f o r m o f s e q u e n c e i s : ” ) ;

Scilab code Exa 2.53.b z Transform of the Signal 1 2

// Example 2 . 5 3 ( b ) //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 55

3 //Z t r a n s f o r m o f d e l t a ( n ) 4 clc ; 5 syms n z ; 6 x =1; 7 X = symsum ( x * z ^( - n ) ,n ,0 ,0) ; 8 disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.53.c z Transform of the Signal 1 // Example 2 . 5 3 ( c ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z t r a n s f o r m o f d e l t a ( n ) 4 clc ; 5 syms n z k ; 6 x =1; 7 X = symsum ( x * z ^( - n ) ,n ,k , k ) ; 8 disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.53.d z Transform of the Signal 1 // Example 2 . 5 3 ( d ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //Z t r a n s f o r m o f d e l t a ( n ) 4 clc ; 5 syms n z kc ; 6 x =1; 7 X = symsum ( x * z ^( - n ) ,n , -k , - k ) ; 8 disp (X , ’X( z )= ’ ) ;

Scilab code Exa 2.54 z Transform of Cosine Signal 56

Figure 2.8: Impulse Response of the System 1 2 3 4 5 6 7 8 9 10 11

// Example 2 . 5 4 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //Z t r a n s f o r m o f c o s (Wo∗n ) clc ; syms Wo n z ; x1 = exp ( sqrt ( -1) * Wo * n ) ; X1 = symsum ( x1 *( z ^ - n ) ,n ,0 , %inf ) ; x2 = exp ( - sqrt ( -1) * Wo * n ) ; X2 = symsum ( x2 *( z ^ - n ) ,n ,0 , %inf ) ; X =( X1 + X2 ) /2; disp (X , ’X( z )= ’ ) ;

57

Scilab code Exa 2.58 Impulse Response of the System 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

// Example 2 . 5 8 //To p l o t t h e r e s p o n c e o f t h e s y s t e m a n a l y i c a l l y and using s c i l a b clear all ; clc ; close ; n =0:1:20; x =[1 zeros (1 ,20) ]; b =[1 -0.5]; a =[1 -1 3/16]; yanaly =0.5*(0.75) .^ n +0.5*(0.25) .^ n ; // A n a l y t i c a l Solution ymat = filter (b ,a , x ) ; subplot (3 ,1 ,1) ; plot2d3 (n , x ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ x ( n ) ’ ) ; title ( ’INPUT SEQUENCE (IMPULSE FUNCTION) ’ ) ; subplot (3 ,1 ,2) ; plot2d3 (n , yanaly ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE y a n a l y ’ ) ; subplot (3 ,1 ,3) ; plot2d3 (n , ymat ) ; xlabel ( ’ n ’ ) ; ylabel ( ’ y ( n ) ’ ) ; title ( ’OUTPUT SEQUENCE ymat ’ ) ; // As t h e A n a l t i c a l P l o t m a t c h e s t h e S c i l a b P l o t hence i t i s the Responce o f the system

58

Chapter 3 THE DISCRETE FOURIER TRANSFORM

Scilab code Exa 3.1 DFT and IDFT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 3 . 1 // Program t o Compute t h e DFT o f a S e q u e n c e x [ n ]=[1 ,1 ,0 ,0] // and IDFT o f a S e q u e n c e Y [ k ] = [ 1 , 0 , 1 , 0 ] clear all ; clc ; close ; x = [1 ,1 ,0 ,0]; //DFT Computation X = fft ( x , -1) ; Y = [1 ,0 ,1 ,0]; //IDFT Computation y = fft ( Y , 1) ; // D i s p l a y s e q u e n c e X [ k ] and y [ n ] i n command window disp (X , ”X [ k ]= ” ) ; disp (y , ” y [ n ]= ” ) ;

59

Figure 3.1: DFT of the Sequence

Scilab code Exa 3.2 DFT of the Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 3 . 2 // Program t o Compute t h e DFT o f a S e q u e n c e x [ n ] = 1 , 0<=n<=2; and 0 o t h e r w i s e // f o r N=4 and N=8. P l o t Magnitude and p h a s e p l o t s o f each . clear all ; clc ; close ; //N=4 x1 = [1 ,1 ,1 ,0]; //DFT Computation X1 = fft ( x1 , -1) ; //N=8 x2 = [1 ,1 ,1 ,0 ,0 ,0 ,0 ,0]; //DFT Computation X2 = fft ( x2 , -1) ; // D i s p l a y s e q u e n c e X1 [ k ] and X2 [ k ] i n command window disp ( X1 , ”X1 [ k ]= ” ) ; 60

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

disp ( X2 , ”X2 [ k ]= ” ) ; // P l o t s f o r N=4 n1 =0:1:3; subplot (2 ,2 ,1) ; a = gca () ; a . y_location = ” o r i g i n ” ; a . x_location = ” o r i g i n ” ; plot2d3 ( n1 , abs ( X1 ) ,2) ; poly1 = a . children (1) . children (1) ; poly1 . thickness =2; xtitle ( ’N=4 ’ , ’ k ’ , ’ | X1 ( k ) | ’ ) ; subplot (2 ,2 ,2) ; a = gca () ; a . y_location = ” o r i g i n ” ; a . x_location = ” o r i g i n ” ; plot2d3 ( n1 , atan ( imag ( X1 ) , real ( X1 ) ) ,5) ; poly1 = a . children (1) . children (1) ; poly1 . thickness =2; xtitle ( ’N=4 ’ , ’ k ’ , ’
61

Scilab code Exa 3.3 8 Point DFT 1 2 3 4 5 6 7 8 9 10

// Example 3 . 3 // Program t o Compute t h e 8− p o i n t DFT o f t h e S e q u e n c e x [ n]=[1 ,1 ,1 ,1 ,1 ,1 ,0 ,0] clear all ; clc ; close ; x = [1 ,1 ,1 ,1 ,1 ,1 ,0 ,0]; //DFT Computation X = fft ( x , -1) ; // D i s p l a y s e q u e n c e X [ k ] i n command window disp (X , ”X [ k ]= ” ) ;

Scilab code Exa 3.4 IDFT of the given Sequence 1 2 3 4 5 6 7 8 9 10 11

// Example 3 . 4 // Program t o Compute t h e IDFT o f t h e S e q u e n c e X [ k ]=[5 ,0 ,1 − j ,0 ,1 ,0 ,1+ j , 0 ] clear all ; clc ; close ; j = sqrt ( -1) ; X = [5 ,0 ,1 -j ,0 ,1 ,0 ,1+ j ,0] //IDFT Computation x = fft ( X , 1) ; // D i s p l a y s e q u e n c e s x [ n ] i n command window disp (x , ” x [ n ]= ” ) ;

62

Figure 3.2: Plot the Sequence

Scilab code Exa 3.7 Plot the Sequence 1 2 3 4 5 6 7 8 9 10 11 12

// Example 3 . 7 // Program t o Compute c i r c u l a r c o n v o l u t i o n o f following sequences // x [ n ] = [ 1 , 2 , 2 , 1 , 0 ] //Y [ k ]= exp (− j ∗4∗ p i ∗ k / 5 ) . X [ k ] clear all ; clc ; close ; x =[1 ,2 ,2 ,1 ,0]; X = fft (x , -1) ; k =0:1:4; j = sqrt ( -1) ; pi =22/7; 63

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

H = exp ( - j *4* pi * k /5) ; Y = H .* X ; //IDFT Computation y = fft (Y ,1) ; // D i s p l a y s e q u e n c e y [ n ] i n command window disp ( round ( y ) ,” y [ n ]= ” ) ; // P l o t s n =0:1:4; a = gca () ; a . y_location = ” o r i g i n ” ; a . x_location = ” o r i g i n ” ; plot2d3 (n , round ( y ) ,5) ; poly1 = a . children (1) . children (1) ; poly1 . thickness =2; xtitle ( ’ P l o t o f s e q u e n c e y [ n ] ’ , ’ n ’ , ’ y [ n ] ’ ) ;

Scilab code Exa 3.9 Remaining Samples 1 2 3 4 5 6 7 8 9

10 11 12

// Example 3 . 9 // Program t o r e m a i n i n g s a m p l e s o f t h e s e q u e n c e //X( 0 ) =12 ,X( 1 )=−1+j 3 , X( 2 ) =3+j 4 , X( 3 ) =1−j 5 , X( 4 )=−2+j 2 , X( 5 ) =6+j 3 , X( 6 ) =−2−j 3 , X( 7 ) =10 clear all ; clc ; close ; j = sqrt ( -1) ; z =1; X (0+ z ) =12 , X (1+ z ) = -1+ j *3 , X (2+ z ) =3+ j *4 , X (3+ z ) =1 - j *5 , X (4+ z ) = -2+ j *2 , X (5+ z ) =6+ j *3 , X (6+ z ) = -2 - j *3 , X (7+ z ) =10; for a =9:1:14 do X ( a ) = conj ( X (16 - a ) ) , end ; // D i s p l a y t h e c o m p l e t e s e q u e n c e X [ k ] i n command window disp (X , ”X [ k ]= ” ) ;

64

Scilab code Exa 3.11 DFT Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 3 . 1 1 // Program t o Compute t h e 8− p o i n t DFT o f t h e following sequences // x1 [ n ] = [ 1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 ] // x2 [ n ] = [ 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 ] clear all ; clc ; close ; x1 =[1 ,0 ,0 ,0 ,0 ,1 ,1 ,1]; x2 =[0 ,0 ,1 ,1 ,1 ,1 ,0 ,0]; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; // D i s p l a y s e q u e n c e s X1 [ k ] and X2 [ k ] i n command window disp ( X1 , ”X1 [ k ]= ” ) ; disp ( X2 , ”X2 [ k ]= ” ) ;

Scilab code Exa 3.13 Circular Convolution 1 2 3 4 5 6 7 8 9

// Example 3 . 1 3 // Program t o Compute c i r c u l a r c o n v o l u t i o n o f following sequences // x1 [ n ] = [ 1 , − 1 , − 2 , 3 , − 1 ] // x2 [ n ] = [ 1 , 2 , 3 ] clear all ; clc ; close ; x1 =[1 , -1 , -2 ,3 , -1]; x2 =[1 ,2 ,3]; 65

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

// Loop f o r z e r o p a d d i n g t h e s m a l l e r s e q u e n c e o u t o f t h e two n1 = length ( x1 ) ; n2 = length ( x2 ) ; n3 = n2 - n1 ; if ( n3 >=0) then x1 =[ x1 , zeros (1 , n3 ) ]; else x2 =[ x2 , zeros (1 , - n3 ) ]; end //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; Y = X1 .* X2 ; //IDFT Computation y = fft (Y ,1) ; // D i s p l a y s e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.14 Circular Convolution 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 3 . 1 4 // Program t o Compute c i r c u l a r c o n v o l u t i o n o f following sequences // x1 [ n ] = [ 1 , 2 , 2 , 1 ] // x2 [ n ] = [ 1 , 2 , 3 , 1 ] clear all ; clc ; close ; x1 =[1 ,2 ,2 ,1]; x2 =[1 ,2 ,3 ,1]; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; Y = X1 .* X2 ; 66

14 //IDFT Computation 15 y = fft (Y ,1) ; 16 // D i s p l a y s e q u e n c e y [ n ] 17 disp (y , ” y [ n ]= ” ) ;

i n command window

Scilab code Exa 3.15 Determine Sequence x3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

// Example 3 . 1 5 // Program t o Compute x3 [ n ] where X3 [ k ]=X1 [ k ] . X2 [ k ] // x1 [ n ] = [ 1 , 2 , 3 , 4 ] // x2 [ n ] = [ 1 , 1 , 2 , 2 ] clear all ; clc ; close ; x1 =[1 ,2 ,3 ,4]; x2 =[1 ,1 ,2 ,2]; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; X3 = X1 .* X2 ; //IDFT Computation x3 = fft ( X3 ,1) ; // D i s p l a y s e q u e n c e x3 [ n ] i n command window disp ( x3 , ” x3 [ n ]= ” ) ;

Scilab code Exa 3.16 Circular Convolution // Example 3 . 1 6 // Program t o Compute c i r c u l a r c o n v o l u t i o n o f following sequences 3 // x1 [ n ] = [ 1 , 1 , 2 , 1 ] 4 // x2 [ n ] = [ 1 , 2 , 3 , 4 ] 5 clear all ; 1 2

67

6 7 8 9 10 11 12 13 14 15 16 17

clc ; close ; x1 =[1 ,1 ,2 ,1]; x2 =[1 ,2 ,3 ,4]; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; X3 = X1 .* X2 ; //IDFT Computation x3 = fft ( X3 ,1) ; // D i s p l a y s e q u e n c e x3 [ n ] i n command window disp ( x3 , ” x3 [ n ]= ” ) ;

Scilab code Exa 3.17 Circular Convolution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

// Example 3 . 1 7 // Program t o Compute y [ n ] where Y [ k ]=X1 [ k ] . X2 [ k ] // x1 [ n ] = [ 0 , 1 , 2 , 3 , 4 ] // x2 [ n ] = [ 0 , 1 , 0 , 0 , 0 ] clear all ; clc ; close ; x1 =[0 ,1 ,2 ,3 ,4]; x2 =[0 ,1 ,0 ,0 ,0]; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; Y = X1 .* X2 ; //IDFT Computation y = round ( fft (Y ,1) ) ; // D i s p l a y s e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

68

Scilab code Exa 3.18 Output Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

// Example 3 . 1 8 // Program t o Compute o u t p u t r e s p o n c e o f f o l l o w i n g sequences // x [ n ] = [ 1 , 2 , 3 , 1 ] // h [ n ] = [ 1 , 1 , 1 ] // ( 1 ) L i n e a r C o n v o l u t i o n // ( 2 ) C i r c u l a r C o n v o l u t i o n // ( 3 ) C i r c u l a r C o n v o l u t i o n w i t h z e r o p a d d i n g clear all ; clc ; close ; x =[1 ,2 ,3 ,1]; h =[1 ,1 ,1]; // ( 1 ) L i n e a r C o n v o l u t i o n Computation ylinear = convol (x , h ) ; // D i s p l a y L i n e a r C o n v o l u t e d S e q u e n c e y [ n ] i n command window disp ( ylinear , ” y l i n e a r [ n ]= ” ) ; // ( 2 ) C i r c u l a r C o n v o l u t i o n Computation //Now z e r o p a d d i n g i n h [ n ] s e q u e n c e t o make l e n g t h o f x [ n ] and h [ n ] e q u a l h1 =[ h , zeros (1 ,1) ]; //Now P e r f o r m i n g C i r c u l a r C o n v o l u t i o n by DFT method X = fft (x , -1) ; H = fft ( h1 , -1) ; Y = X .* H ; ycircular = fft (Y ,1) ; // D i s p l a y C i r c u l a r C o n v o l u t e d S e q u e n c e y [ n ] i n command window disp ( ycircular , ” y c i r c u l a r [ n ]= ” ) ; // ( 3 ) C i r c u l a r C o n v o l u t i o n Computation w i t h z e r o Padding x2 =[ x , zeros (1 ,2) ]; h2 =[ h , zeros (1 ,3) ]; //Now P e r f o r m i n g C i r c u l a r C o n v o l u t i o n by DFT method X2 = fft ( x2 , -1) ; 69

32 H2 = fft ( h2 , -1) ; 33 Y2 = X2 .* H2 ; 34 ycircularp = fft ( Y2 ,1) ; 35 // D i s p l a y C i r c u l a r C o n v o l u t e d S e q u e n c e w i t h z e r o

Padding y [ n ] i n command window 36 disp ( ycircularp , ” y c i r c u l a r p [ n ]= ” ) ;

Scilab code Exa 3.20 Output Response 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 3 . 2 0 // Program t o Compute L i n e a r C o n v o l u t i o n o f f o l l o w i n g sequences // x [ n ] = [ 3 , − 1 , 0 , 1 , 3 , 2 , 0 , 1 , 2 , 1 ] // h [ n ] = [ 1 , 1 , 1 ] clear all ; clc ; close ; x =[3 , -1 ,0 ,1 ,3 ,2 ,0 ,1 ,2 ,1]; h =[1 ,1 ,1]; // L i n e a r C o n v o l u t i o n Computation y = convol (x , h ) ; // D i s p l a y S e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.21 Linear Convolution 1 2 3 4 5 6

// Example 3 . 2 1 // Program t o Compute L i n e a r C o n v o l u t i o n o f f o l l o w i n g sequences // x [ n ] = [ 1 , 2 , − 1 , 2 , 3 , − 2 , − 3 , − 1 , 1 , 1 , 2 , − 1 ] // h [ n ] = [ 1 , 2 ] clear all ; clc ; 70

7 close ; 8 x =[1 ,2 , -1 ,2 ,3 , -2 , -3 , -1 ,1 ,1 ,2 , -1]; 9 h =[1 ,2]; 10 // L i n e a r C o n v o l u t i o n Computation 11 y = convol (x , h ) ; 12 // D i s p l a y S e q u e n c e y [ n ] i n command window 13 disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.23.a N Point DFT Computation 1 // Example 3 . 2 3 ( a ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f d e l t a ( n ) 4 clc ; 5 syms n k N ; 6 x =1; 7 X = symsum ( x * exp ( - %i *2* %pi * n * k / N ) ,n ,0 ,0) ; 8 disp (X , ’X( k )= ’ ) ;

Scilab code Exa 3.23.b N Point DFT Computation 1 // Example 3 . 2 3 ( b ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f d e l t a ( n−no ) 4 clc ; 5 syms n k N no ; 6 x =1; 7 X = symsum ( x * exp ( - %i *2* %pi * n * k / N ) ,n , - no , - no ) ; 8 disp (X , ’X( k )= ’ ) ;

71

Scilab code Exa 3.23.c N Point DFT Computation 1 // Example 3 . 2 3 ( c ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f a ˆn 4 clc ; 5 syms a n k N ; 6 x=a^n; 7 X = symsum ( x * exp ( - %i *2* %pi * n * k / N ) ,n ,0 ,N -1) ; 8 disp (X , ’X( k )= ’ ) ;

Scilab code Exa 3.23.d N Point DFT Computation 1 // Example 3 . 2 3 ( d ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f x ( n ) =1 , 0<=n<=N/2−1 4 clc ; 5 syms n k N ; 6 x =1; 7 X = symsum ( x * exp ( - %i *2* %pi * n * k / N ) ,n ,0 ,( N /2) -1) ; 8 disp (X , ’X( k )= ’ ) ;

Scilab code Exa 3.23.e N Point DFT Computation 1 // Example 3 . 2 3 ( e ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f x ( n )=exp ( %i ∗2∗ %pi ∗ ko ∗n /N) ; 4 clc ; 5 syms n k N ko ; 6 x = exp ( %i *2* %pi * ko * n / N ) ; 7 X = symsum ( x * exp ( - %i *2* %pi * n * k / N ) ,n ,0 ,( N /2) -1) ; 8 disp (X , ’X( k )= ’ ) ;

72

Scilab code Exa 3.23.f N Point DFT Computation 1 // Example 3 . 2 3 ( f ) 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 //N p o i n t DFT o f x ( n ) =1 , f o r n=e v e n and 0 , f o r n=odd 4 clc ; 5 syms n k N ; 6 x =1; // x ( 2 n ) =1 , f o r a l l n 7 X = symsum ( x * exp ( - %i *4* %pi * n * k / N ) ,n ,0 , N /2 -1) ; 8 disp (X , ’X( k )= ’ ) ;

Scilab code Exa 3.24 DFT of the Sequence 1 2 3 4 5 6 7 8 9 10 11 12

// Example 3 . 2 4 // Program t o Compute t h e DFT o f t h e S e q u e n c e x [ n ]=( −1) ˆn , f o r N=4 clear all ; clc ; close ; N =4; n =0:1: N -1; x =( -1) ^ n ; //DFT Computation X = fft (x , -1) ; // D i s p l a y S e q u e n c e X [ k ] i n command window disp (X , ”X [ k ]= ” ) ;

Scilab code Exa 3.25 8 Point Circular Convolution

73

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

// Example 3 . 2 5 // Program t o Compute t h e 8− p o i n t C i r c u l a r Convolution of the Sequences // x1 [ n ] = [ 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 ] // x2 [ n ]= s i n ( 3 ∗ p i ∗n / 8 ) clear all ; clc ; close ; x1 =[1 ,1 ,1 ,1 ,0 ,0 ,0 ,0]; n =0:1:7; pi =22/7; x2 = sin (3* pi * n /8) ; //DFT Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; // C i r c u l a r C o n v o l u t i o n u s i n g DFT Y = X1 .* X2 ; //IDFT Computation y = fft (Y ,1) ; // D i s p l a y s e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.26 Linear Convolution using DFT 1 2 3 4 5 6 7 8 9 10

// Example 3 . 2 6 // Program t o Compute t h e L i n e a r C o n v o l u t i o n o f t h e f o l l o w i n g Sequences // x [ n ] = [ 1 , − 1 , 1 ] // h [ n ] = [ 2 , 2 , 1 ] clear all ; clc ; close ; x =[1 , -1 ,1]; h =[2 ,2 ,1]; // C o n v o l u t i o n Computation 74

11 y = convol (x , h ) ; 12 // D i s p l a y s e q u e n c e y [ n ] 13 disp (y , ” y [ n ]= ” ) ;

i n command window

Scilab code Exa 3.27.a Circular Convolution Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 3 . 2 7 ( a ) // Program t o Compute t h e C o n v o l u t i o n o f t h e f o l l o w i n g Sequences // x1 [ n ] = [ 1 , 1 , 1 ] // x2 [ n ] = [ 2 , − 1 , 2 ] clear all ; clc ; close ; x1 =[1 ,1 ,1]; x2 =[2 , -1 ,2]; // C o n v o l u t i o n Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; Y = X1 .* X2 ; y = fft (Y ,1) ; // D i s p l a y S e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.27.b Circular Convolution Computation 1 2 3 4 5 6

// Example 3 . 2 7 ( b ) // Program t o Compute t h e C o n v o l u t i o n o f t h e f o l l o w i n g Sequences // x1 [ n ] = [ 1 , 1 , − 1 , − 1 , 0 ] // x2 [ n ] = [ 1 , 0 , − 1 , 0 , 1 ] clear all ; clc ; 75

7 8 9 10 11 12 13 14 15 16

close ; x1 =[1 ,1 , -1 , -1 ,0]; x2 =[1 ,0 , -1 ,0 ,1]; // C o n v o l u t i o n Computation X1 = fft ( x1 , -1) ; X2 = fft ( x2 , -1) ; Y = X1 .* X2 ; y = fft (Y ,1) ; // D i s p l a y S e q u e n c e y [ n ] i n command window disp (y , ” y [ n ]= ” ) ;

Scilab code Exa 3.30 Calculate value of N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 3 . 3 0 // Program t o C a l c u l a t e N from g i v e n d a t a // fm =5000Hz // d f =50Hz // t =0.5 s e c clear all ; clc ; close ; fm =5000 // Hz df =50 // Hz t =0.5 // s e c N1 =2* fm / df ; N =2; while N <= N1 , N = N *2 , end // D i s p l a y i n g t h e v a l u e o f N i n command window disp (N , ”N=” ) ;

Scilab code Exa 3.32 Sketch Sequence 76

Figure 3.3: Sketch Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

// Example 3 . 3 2 // Program t o p l o t t h e r e s u l t o f t h e g i v e n s e q u e n c e //X [ k ] = [ 1 , 2 , 2 , 1 , 0 , 2 , 1 , 2 ] // y [ n ]= x [ n / 2 ] f o r n=even , 0 f o r n=odd clear all ; clc ; close ; X =[1 ,2 ,2 ,1 ,0 ,2 ,1 ,2]; x = fft ( X , 1) ; y =[ x (1) ,0 , x (2) ,0 , x (3) ,0 , x (4) ,0 , x (5) ,0 , x (6) ,0 , x (7) ,0 , x (8) ,0]; Y = fft ( y , -1) ; // D i s p l a y s e q u e n c e Y [ k ] and i n command window disp (Y , ”Y [ k ]= ” ) ; // P l o t t i n g t h e s e q u e n c e Y [ k ] k =0:1:15; a = gca () ; a . y_location = ” o r i g i n ” ; a . x_location = ” o r i g i n ” ; plot2d3 (k ,Y ,2) ; poly1 = a . children (1) . children (1) ; poly1 . thickness =2; xtitle ( ’ P l o t o f Y( k ) ’ , ’ k ’ , ’Y( k ) ’ ) ; 77

Scilab code Exa 3.36 Determine IDFT 1 2 3 4 5 6 7 8 9 10 11 12

// Example 3 . 3 6 // Program t o Compute t h e IDFT o f t h e f o l l o w i n g Sequence //X [ k ] = [ 1 2 , − 1 . 5 + j 2 . 5 9 8 , − 1 . 5 + j 0 . 8 6 6 , 0 , − 1 . 5 − j 0 .866 , −1.5 − j 2 . 5 9 8 ] clear all ; clc ; close ; j = sqrt ( -1) ; X =[12 , -1.5+ j *2.598 , -1.5+ j *0.866 ,0 , -1.5 - j *0.866 , -1.5 j *2.598]; //IDFT Computation x = fft ( X , 1) ; // D i s p l a y S e q u e n c e x [ n ] i n command window disp ( round ( x ) ,” x [ n ]= ” ) ;

78

Chapter 4 THE FAST FOURIER TRANSFORM

Scilab code Exa 4.3 Shortest Sequence N Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

// Example 4 . 3 // Program t o c a l c u l a t e s h o r t e s t s e q u e n c e N s u c h t h a t a l g o r i t h m B r u n s // f a s t e r t h a n A clear all ; clc ; close ; i =0; N =32; // Given // C a l c u l a t i o n o f T w i d d l e f a c t o r e x p o n e n t s f o r e a c h stage while 1==1 i = i +1; N =2^ i ; A = N ^2; B =5* N * log2 ( N ) ; if A > B then break ; end ; end disp (N , ’SHORTEST SEQUENCE N = ’ ) ; 79

Scilab code Exa 4.4 Twiddle Factor Exponents Calculation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 4 . 4 // Program t o c a l c u l a t e T w i d d l e f a c t o r e x p o n e n t s f o r each s t a g e clear all ; clc ; close ; N =32; // Given // C a l c u l a t i o n o f T w i d d l e f a c t o r e x p o n e n t s f o r e a c h stage for m =1:5 disp (m , ’ S t a g e : m = ’ ) ; disp ( ’ k = ’ ) ; for t =0:(2^( m -1) -1) k = N * t /2^ m ; disp ( k ) ; end end

Scilab code Exa 4.6 DFT using DIT Algorithm 1 2 3 4 5 6 7 8

// Example 4 . 6 // Program t o f i n d t h e DFT o f a S e q u e n c e x [ n ]=[1 ,2 ,3 ,4 ,4 ,3 ,2 ,1] // u s i n g DIT A l g o r i t h m . clear all ; clc ; close ; x = [1 ,2 ,3 ,4 ,4 ,3 ,2 ,1]; //FFT Computation 80

9 X = fft ( x , -1) ; 10 disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.8 DFT using DIF Algorithm 1 2 3 4 5 6 7 8 9 10

// Example 4 . 8 // Program t o f i n d t h e DFT o f a S e q u e n c e x [ n ]=[1 ,2 ,3 ,4 ,4 ,3 ,2 ,1] // u s i n g DIF A l g o r i t h m . clear all ; clc ; close ; x = [1 ,2 ,3 ,4 ,4 ,3 ,2 ,1]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.9 8 Point DFT of the Sequence 1 2 3 4 5 6 7 8 9 10

// Example 4 . 9 // Program t o f i n d t h e 8− p o i n t DFT o f a S e q u e n c e x [ n ] = 1 , 0<=n<=7 // u s i n g DIT , DIF A l g o r i t h m . clear all ; clc ; close ; x = [1 ,1 ,1 ,1 ,1 ,1 ,1 ,1]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

81


// Example 4 . 1 0 // Program t o Compute t h e 4− p o i n t DFT o f a S e q u e n c e x [n]=[0 ,1 ,2 ,3] // u s i n g DIT−DIF A l g o r i t h m . clear all ; clc ; close ; x = [0 ,1 ,2 ,3]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.11 IDFT of the Sequence using DIT Algorithm 1 2 3 4 5 6 7 8 9 10 11

// Example 4 . 1 1 // Program t o Compute t h e IDFT o f a S e q u e n c e u s i n g DIT A l g o r i t h m . //X [ k ] = [7 , −0.707 − j 0 . 7 0 7 , − j , 0 . 7 0 7 − j 0 . 7 0 7 , 1 , 0 . 7 0 7 + j 0 . 7 0 7 , j , −0.707+ j 0 . 7 0 7 ] clear all ; clc ; close ; j = sqrt ( -1) ; X = [7 , -0.707 - j *0.707 , - j ,0.707 - j *0.707 ,1 ,0.707+ j *0.707 , j , -0.707+ j *0.707]; // I n v e r s e FFT Computation x = fft ( X , 1) ; disp (x , ’ x ( n ) = ’ ) ;

Scilab code Exa 4.12 8 Point DFT of the Sequence 82

1 2 3 4 5 6 7 8 9 10

// Example 4 . 1 2 // Program t o Compute t h e 8− p o i n t DFT o f a S e q u e n c e // x [ n ] = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] u s i n g r a d i x −2 DIT Algorithm . clear all ; clc ; close ; x =[0.5 ,0.5 ,0.5 ,0.5 ,0 ,0 ,0 ,0]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;


// Example 4 . 1 3 // Program t o Compute t h e 8− p o i n t DFT o f a S e q u e n c e // x [ n ] = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] u s i n g r a d i x −2 DIF Algorithm . clear all ; clc ; close ; x =[0.5 ,0.5 ,0.5 ,0.5 ,0 ,0 ,0 ,0]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.14 DFT using DIT Algorithm // Example 4 . 1 4 // Program t o Compute t h e 4− p o i n t DFT o f a S e q u e n c e x [ n ]=[1 , −1 ,1 , −1] 3 // u s i n g DIT A l g o r i t h m . 4 clear all ;

1 2

83

5 clc ; 6 close ; 7 x =[1 , -1 ,1 , -1]; 8 //FFT Computation 9 X = fft ( x , -1) ; 10 disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.15 DFT using DIF Algorithm 1 2 3 4 5 6 7 8 9 10

// Example 4 . 1 5 // Program t o Compute t h e 4− p o i n t DFT o f a S e q u e n c e x [n]=[1 ,0 ,0 ,1] // u s i n g DIF A l g o r i t h m . clear all ; clc ; close ; x =[1 ,0 ,0 ,1]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.16.a 8 Point DFT using DIT FFT 1 2 3 4 5 6 7 8 9

// Example 4 . 1 6 ( a ) // Program t o E v a l u a t e and Compare t h e 8− p o i n t DFT o f the given Sequence // x1 [ n ] = 1 , −3<=n<=3 u s i n g DIT−FFT A l g o r i t h m . clear all ; clc ; close ; x1 =[1 ,1 ,1 ,1 ,0 ,1 ,1 ,1]; //FFT Computation X1 = fft ( x1 , -1) ; 84

10

disp ( X1 , ’ X1 ( k ) = ’ ) ;

Scilab code Exa 4.16.b 8 Point DFT using DIT FFT 1 2 3 4 5 6 7 8 9 10

// Example 4 . 1 6 ( b ) // Program t o E v a l u a t e and Compare t h e 8− p o i n t DFT o f the given Sequence // x2 [ n ] = 1 , 0<=n<=6 u s i n g DIT−FFT A l g o r i t h m . clear all ; clc ; close ; x2 =[1 ,1 ,1 ,1 ,1 ,1 ,1 ,0]; //FFT Computation X2 = fft ( x2 , -1) ; disp ( X2 , ’ X2 ( k ) = ’ ) ;

Scilab code Exa 4.17 IDFT using DIF Algorithm 1 2 3 4 5 6 7 8 9 10 11

// Example 4 . 1 7 // Program t o f i n d t h e IDFT o f t h e S e q u e n c e u s i n g DIF Algorithm . //X [ k ]= [4 ,1 − j 2 . 4 1 4 , 0 , 1 − j 0 . 4 1 4 , 0 , 1 + j 0 . 4 1 4 , 0 , 1 + j 2 .414] clear all ; clc ; close ; j = sqrt ( -1) ; X = [4 ,1 - j *2.414 ,0 ,1 - j *0.414 ,0 ,1+ j *0.414 ,0 ,1+ j *2.414]; // I n v e r s e FFT Computation x = fft ( X , 1) ; disp (x , ’ x ( n ) = ’ ) ; 85

Scilab code Exa 4.18 IDFT using DIT Algorithm 1 2 3 4 5 6 7 8 9 10 11

// Example 4 . 1 8 // Program t o f i n d t h e IDFT o f t h e S e q u e n c e X [ k ]= [10 , −2+ j 2 ,−2,−2− j 2 ] // u s i n g DIT A l g o r i t h m . clear all ; clc ; close ; j = sqrt ( -1) ; X = [10 , -2+ j *2 , -2 , -2 - j *2]; // I n v e r s e FFT Computation x = fft ( X , 1) ; disp (x , ’ x ( n ) = ’ ) ;

Scilab code Exa 4.19 FFT Computation of the Sequence 1 2 3 4 5 6 7 8 9

// Example 4 . 1 9 // Program t o Compute t h e FFT o f g i v e n S e q u e n c e x [ n ]=[1 ,0 ,0 ,0 ,0 ,0 ,0 ,0]. clear all ; clc ; close ; x = [1 ,0 ,0 ,0 ,0 ,0 ,0 ,0]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.20 8 Point DFT by Radix 2 DIT FFT 86

1 2 3 4 5 6 7 8 9 10

// Example 4 . 2 0 // Program t o Compute t h e 8− p o i n t DFT o f g i v e n Sequence // x [ n ] = [ 2 , 2 , 2 , 2 , 1 , 1 , 1 , 1 ] u s i n g DIT , r a d i x −2 ,FFT Algorithm . clear all ; clc ; close ; x = [2 ,2 ,2 ,2 ,1 ,1 ,1 ,1]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.21 DFT using DIT FFT Algorithm 1 2 3 4 5 6 7 8 9 10

// Example 4 . 2 1 // Program t o Compute t h e DFT o f g i v e n S e q u e n c e // x [ n ] = [ 1 , − 1 , − 1 , − 1 , 1 , 1 , 1 , − 1 ] u s i n g DIT−FFT A l g o r i t h m . clear all ; clc ; close ; x = [1 , -1 , -1 , -1 ,1 ,1 ,1 , -1]; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.22 Compute X using DIT FFT 1 2 3 4

// Example 4 . 2 2 // Program t o Compute t h e DFT o f g i v e n S e q u e n c e // x [ n ]=2ˆ n and N=8 u s i n g DIT−FFT A l g o r i t h m . clear all ; 87

5 6 7 8 9 10 11 12

clc ; close ; N =8; n =0:1: N -1; x =2^ n ; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.23 DFT using DIF FFT Algorithm 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 4 . 2 3 // Program t o Compute t h e DFT o f g i v e n S e q u e n c e // x [ n ]= c o s ( n∗ p i / 2 ) , and N=4 u s i n g DIF−FFT A l g o r i t h m . clear all ; clc ; close ; N =4; pi =22/7; n =0:1: N -1; x = cos ( n * pi /2) ; //FFT Computation X = fft ( x , -1) ; disp (X , ’X( z ) = ’ ) ;

Scilab code Exa 4.24 8 Point DFT of the Sequence // Example 4 . 2 4 // Program t o Compute t h e 8− p o i n t DFT o f g i v e n Sequence 3 // x [ n ] = [ 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 ] u s i n g DIF , r a d i x −2 ,FFT Algorithm . 4 clear all ;

1 2

88

5 clc ; 6 close ; 7 x = [0 ,1 ,2 ,3 ,4 ,5 ,6 ,7]; 8 //FFT Computation 9 X = fft ( x , -1) ; 10 disp (X , ’X( z ) = ’ ) ;

89

Chapter 5 INFINITE IMPULSE RESPONSE FILTERS

Scilab code Exa 5.1 Order of the Filter Determination // Example 5 . 1 //To Find o u t t h e o r d e r o f t h e f i l t e r clear all ; clc ; close ; ap =1; // db as =30; // db op =200; // r a d / s e c os =600; // r a d / s e c N = log ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / log ( os / op ) ; 11 disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ;

1 2 3 4 5 6 7 8 9 10

Scilab code Exa 5.2 Order of Low Pass Butterworth Filter 1

// Example 5 . 2 90

2 3 4 5 6 7 8 9 10 11 12 13

//To Find o u t t h e o r d e r o f a Low P a s s B u t t e r w o r t h Filter clear all ; clc ; close ; ap =3; // db as =40; // db fp =500; // Hz fs =1000; // Hz op =2* %pi * fp ; os =2* %pi * fs ; N = log ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / log ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ;

Scilab code Exa 5.4 Analog Butterworth Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 5 . 4 //To D e s i g n an Analog B u t t e r w o r t h F i l t e r clear all ; clc ; close ; ap =2; // db as =10; // db op =20; // r a d / s e c os =30; // r a d / s e c N = log ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / log ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ; s = %s ; HS =1/(( s ^2+0.76537* s +1) *( s ^2+1.8477* s +1) ) ; // T r a n s f e r F u n c t i o n f o r N=4 oc = op /(10^(0.1* ap ) -1) ^(1/(2* ceil ( N ) ) ) ; HS1 = horner ( HS , s / oc ) ; disp ( HS1 , ’ N o r m a l i z e d T r a n s f e r F u n c t i o n , H( s ) = ’ ) ; 91

Scilab code Exa 5.5 Analog Butterworth Filter Design // Example 5 . 5 //To D e s i g n an Analog B u t t e r w o r t h F i l t e r clear all ; clc ; close ; op =0.2* %pi ; os =0.4* %pi ; e1 =0.9; l1 =0.2; epsilon = sqrt (1/( e1 ^2) -1) ; lambda = sqrt (1/( l1 ^2) -1) ; N = log ( lambda / epsilon ) / log ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ; s = %s ; HS =1/(( s ^2+0.76537* s +1) *( s ^2+1.8477* s +1) ) ; // T r a n s f e r F u n c t i o n f o r N=4 16 oc = op / epsilon ^(1/ ceil ( N ) ) ; 17 HS1 = horner ( HS , s / oc ) ; 18 disp ( HS1 , ’ N o r m a l i z e d T r a n s f e r F u n c t i o n , H( s ) = ’ ) ; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Scilab code Exa 5.6 Order of Chebyshev Filter 1 2 3 4 5 6

// Example 5 . 6 //To Find o u t t h e o r d e r o f t h e F i l t e r u s i n g Chebyshev A p p r o x i m a t i o n clear all ; clc ; close ; ap =3; // db 92

as =16; // db fp =1000; // Hz fs =2000; // Hz op =2* %pi * fp ; os =2* %pi * fs ; N = acosh ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / acosh ( os / op ) ; 13 disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ; 7 8 9 10 11 12

Scilab code Exa 5.7 Chebyshev Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 5 . 7 //To D e s i g n an a n a l o g Chebyshev F i l t e r w i t h Given Specifications clear all ; clc ; close ; os =2; op =1; ap =3; // db as =16; // db e1 =1/ sqrt (2) ; l1 =0.1; epsilon = sqrt (1/( e1 ^2) -1) ; lambda = sqrt (1/( l1 ^2) -1) ; N = acosh ( lambda / epsilon ) / acosh ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ;

Scilab code Exa 5.8 Order of Type 1 Low Pass Chebyshev Filter 1 2

// Example 5 . 8 //To Find o u t t h e o r d e r o f t h e p o l e s o f t h e Type 1 Lowpass Chebyshev F i l t e r 93

3 4 5 6 7 8 9 10

clear all ; clc ; close ; ap =1; //dB as =40; //dB op =1000* %pi ; os =2000* %pi ; N = acosh ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / acosh ( os / op ) ; 11 disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ;

Scilab code Exa 5.9 Chebyshev Filter Design 1 2 3 4 5 6 7 8 9 10 11

// Example 5 . 9 //To D e s i g n a Chebyshev F i l t e r w i t h Given Specifications clear all ; clc ; close ; ap =2.5; // db as =30; // db op =20; // r a d / s e c os =50; // r a d / s e c N = acosh ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / acosh ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ;

Scilab code Exa 5.10 HPF Filter Design with given Specifications 1 // Example 5 . 1 0 2 //To D e s i g n a H . P . F . w i t h g i v e n 3 clear all ; 4 clc ;

94

specifications

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close ; ap =3; // db as =15; // db op =500; // r a d / s e c os =1000; // r a d / s e c N = log ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / log ( os / op ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ; s = %s ; HS =1/(( s +1) *( s ^2+ s +1) ) ; // T r a n s f e r F u n c t i o n f o r N=3 oc =1000 // r a d / s e c HS1 = horner ( HS , oc / s ) ; disp ( HS1 , ’ N o r m a l i z e d T r a n s f e r F u n c t i o n , H( s ) = ’ ) ;

Scilab code Exa 5.11 Impulse Invariant Method Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 5 . 1 1 //To D e s i g n t h e F i l t e r u s i n g I m p u l s e I n v a r i e n t Method clear all ; clc ; close ; s = %s ; T =1; HS =(2) /( s ^2+3* s +2) ; elts = pfss ( HS ) ; disp ( elts , ’ F a c t o r i z e d HS = ’ ) ; // The p o l e s comes o u t t o be a t −2 and −1 p1 = -2; p2 = -1; z = %z ; HZ =(2/(1 - %e ^( p2 * T ) * z ^( -1) ) ) -(2/(1 - %e ^( p1 * T ) * z ^( -1) ) ) disp ( HZ , ’HZ = ’ ) ;

95

Scilab code Exa 5.12 Impulse Invariant Method Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 5 . 1 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //To D e s i g n t h e F i l t e r u s i n g I m p u l s e I n v a r i e n t Method clear all ; clc ; close ; s = %s ; HS =1/( s ^2+ sqrt (2) * s +1) ; pp = ilaplace ( HS ) ; syms n z ; t =1; X = symsum ( pp *( z ^( - n ) ) ,n ,0 , %inf ) ; disp (X , ’ F a c t o r i z e d HS = ’ ) ;

Scilab code Exa 5.13 Impulse Invariant Method Filter Design 1 2 3 4 5 6 7 8 9 10 11 12

// Example 5 . 1 3 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM //To D e s i g n t h e 3 r d Order B u t t e r w o r t h F i l t e r u s i n g I m p u l s e I n v a r i e n t Method clear all ; clc ; close ; s = %s ; HS =1/(( s +1) *( s ^2+ s +1) ) ; pp = ilaplace ( HS ) ; // I n v e r s e L a p l a c e syms n z ; t =1; X = symsum ( pp *( z ^( - n ) ) ,n ,0 , %inf ) ; //Z T r a n s f o r m 96

13

disp (X , ’H( z )= ’ ) ;

Scilab code Exa 5.15 Impulse Invariant Method Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 5 . 1 5 //To D e s i g n t h e F i l t e r u s i n g I m p u l s e I n v a r i e n t Method clear all ; clc ; close ; s = %s ; T =0.2; HS =10/( s ^2+7* s +10) ; elts = pfss ( HS ) ; disp ( elts , ’ F a c t o r i z e d HS = ’ ) ; // The p o l e s comes o u t t o be a t −5 and −2 p1 = -5; p2 = -2; z = %z ; HZ = T *(( -3.33/(1 - %e ^( p1 * T ) * z ^( -1) ) ) +(3.33/(1 - %e ^( p2 * T ) * z ^( -1) ) ) ) disp ( HZ , ’HZ = ’ ) ;

Scilab code Exa 5.16 Bilinear Transformation Method Filter Design 1 2 3 4 5 6 7

// Example 5 . 1 6 //To Find o u t B i l i n e a r T r a n s f o r m a t i o n o f HS=2/(( s +1) ∗ ( s +2) ) clear all ; clc ; close ; s = %s ; z = %z ; 97

8 HS =2/(( s +1) *( s +2) ) ; 9 T =1; 10 HZ = horner ( HS ,(2/ T ) *( z -1) /( z +1) ) ; 11 disp ( HZ , ’H( z ) = ’ ) ;

Scilab code Exa 5.17 HPF Design using Bilinear Transform 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 5 . 1 7 //To D e s i g n an H . P . F . m o n o t o n i c i n p a s s b a n d u s i n g B i l i n e a r Transform clear all ; clc ; close ; ap =3; // db as =10; // db fp =1000; // Hz fs =350; // Hz f =5000; T =1/ f ; wp =2* %pi * fp ; ws =2* %pi * fs ; op =2/ T * tan ( wp * T /2) ; os =2/ T * tan ( ws * T /2) ; N = log ( sqrt ((10^(0.1* as ) -1) /(10^(0.1* ap ) -1) ) ) / log ( op / os ) ; disp ( ceil ( N ) , ’ Order o f t h e f i l t e r , N = ’ ) ; s = %s ; HS =1/( s +1) // T r a n s f e r F u n c t i o n f o r N=1 oc = op // r a d / s e c HS1 = horner ( HS , oc / s ) ; disp ( HS1 , ’ N o r m a l i z e d T r a n s f e r F u n c t i o n , H( s ) = ’ ) ; z = %z ; HZ = horner ( HS ,(2/ T ) *( z -1) /( z +1) ) ; disp ( HZ , ’H( z ) = ’ ) ;

98

Scilab code Exa 5.18 Bilinear Transformation Method Filter Design 1 2 3 4 5 6 7 8 9 10 11

// Example 5 . 1 8 //To Find o u t B i l i n e a r T r a n s f o r m a t i o n o f H( s ) =( s ˆ2+4.525) /( s ˆ2+0.692∗ s +0.504) clear all ; clc ; close ; s = %s ; z = %z ; HS =( s ^2+4.525) /( s ^2+0.692* s +0.504) ; T =1; HZ = horner ( HS ,(2/ T ) *( z -1) /( z +1) ) ; disp ( HZ , ’H( z ) = ’ ) ;

Scilab code Exa 5.19 Single Pole LPF into BPF Conversion 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 5 . 1 9 //To C o n v e r t a s i n g l e P o l e LPF i n t o BPF clear all ; clc ; close ; s = %s ; z = %z ; HZ =(0.5*(1+ z ^( -1) ) ) /(1 -0.302* z ^( -2) ) ; T =1; wu =3* %pi /4; wl = %pi /4; wp = %pi /6; k = tan ( wp /2) / tan (( wu - wl ) /2) ; a = cos (( wu + wl ) /2) / cos (( wu - wl ) /2) ;

99

15

transf = -(((( k -1) /( k +1) ) *( z ^( -2) ) ) -((2* a * k /( k +1) ) *( z ^( -1) ) ) +1) /( z ^( -2) -(2* a * k /(1+ k ) * z ^( -1) ) +(( k -1) /( k +1) ) ) ; 16 HZ1 = horner ( HZ , transf ) ; 17 disp ( HZ1 , ’H( z ) o f B . P . F = ’ ) ;

Scilab code Exa 5.29 Pole Zero IIR Filter into Lattice Ladder Structure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

// Example 5 . 2 9 // Program t o c o n v e r t g i v e n I I R p o l e −z e r o F i l t e r i n t o L a t t i c e Ladder S t r u c t u r e . clear all ; clc ; close ; U =1; // Z e r o A d j u s t a (3+ U ,0+ U ) =1; a (3+ U ,1+ U ) =13/24; a (3+ U ,2+ U ) =5/8; a (3+ U ,3+ U ) =1/3; a (2+ U ,0+ U ) =1; // a (m, 0 ) =1 a (2+ U ,3+ U ) =1/3; m =3 , k =1; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; m =3 , k =2; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; m =2 , k =1; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; disp ( ’LATTICE COEFFICIENTS ’ ) ; disp ( a (1+ U ,1+ U ) , ’ k1 ’ ) ; disp ( a (2+ U ,2+ U ) , ’ k2 ’ ) ; disp ( a (3+ U ,3+ U ) , ’ k3 ’ ) ; b0 =1; 100

24 25 26 27 28 29 30 31 32 33 34 35

b1 =2; b2 =2; b3 =1; c3 = b3 ; c2 = b2 - c3 * a (3+ U ,1+ U ) ; c1 = b1 -( c2 * a (2+ U ,1+ U ) + c3 * a (3+ U ,2+ U ) ) ; c0 = b0 -( c1 * a (1+ U ,1+ U ) + c2 * a (2+ U ,2+ U ) + c3 * a (3+ U ,3+ U ) ) ; disp ( ’LADDER COEFFICIENTS ’ ) ; disp ( c0 , ’ c 0 = ’ ) ; disp ( c1 , ’ c 1 = ’ ) ; disp ( c2 , ’ c 2 = ’ ) ; disp ( c3 , ’ c 3 = ’ ) ;

101

Chapter 6 FINITE IMPULSE RESPONSE FILTERS

Scilab code Exa 6.1 Group Delay and Phase Delay 1 // Example 6 . 1 2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM 3 // Program t o C a l c u l a t e Group D e l a y and Phase D e l a y 4 // y ( n ) =0.25 x ( n )+x ( n−1) +0.25 x ( n −2) 5 clear all ; 6 clc ; 7 close ; 8 //w=p o l y ( 0 , ”w” ) ; 9 syms w ; 10 theeta = - w ; 11 gd = - diff ( theeta , w ) ; // Group D e l a y 12 pd = - theeta / w ; // Phase D e l a y 13 disp ( gd , ’GROUP DELAY = ’ ) ; 14 disp ( pd , ’PHASE DELAY = ’ ) ;

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Figure 6.1: LPF Magnitude Response

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Scilab code Exa 6.5 LPF Magnitude Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

// Example 6 . 5 // Program t o P l o t Magnitude R e s p o n c e o f i d e a l L . P . F . w i t h wc =0.5∗ p i //N=11 clear all ; clc ; close ; N =11; U =6; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0.5; else hd ( n ) =( sin ( %pi *( n - U ) /2) ) /( %pi *( n - U ) ) ; end end [ hzm , fr ]= frmag ( hd ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure ; plot (2* fr , hzm_dB ) ; a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR LPF w i t h N=11 ’ ) ; xgrid (2) ;

Scilab code Exa 6.6 HPF Magnitude Response 1 2

// Example 6 . 6 // Program t o P l o t Magnitude R e s p o n c e o f i d e a l H . P . F . w i t h wc = 0.2 5 ∗ p i 104

Figure 6.2: HPF Magnitude Response

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//N=11 clear all ; clc ; close ; N =11; U =6; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0.5; else hd ( n ) =( sin ( %pi *( n - U ) ) - sin ( %pi *( n - U ) /4) ) /( %pi *( n - U ) ) ; end end [ hzm , fr ]= frmag ( hd ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR HPF w i t h N=11 ’ ) ; xgrid (2) ;

Scilab code Exa 6.7 BPF Magnitude Response 1 2 3 4 5 6 7

// Example 6 . 7 // Program t o P l o t Magnitude R e s p o n c e o f i d e a l B . P . F . with // wc1 = 0. 25 ∗ p i and wc2 = 0.7 5 ∗ p i //N=11 clear all ; clc ; close ; 106

Figure 6.3: BPF Magnitude Response

107

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N =11; U =6; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0.5; else hd ( n ) =( sin ( %pi *3*( n - U ) /4) - sin ( %pi *( n - U ) /4) ) /( %pi *( n U)); end end [ hzm , fr ]= frmag ( hd ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR BPF w i t h N=11 ’ ) ; xgrid (2) ;

Scilab code Exa 6.8 BRF Magnitude Response 1 2 3 4 5 6 7 8 9 10

// Example 6 . 8 // Program t o P l o t Magnitude R e s p o n c e o f i d e a l B . R . F . with // wc1 = 0. 33 ∗ p i and wc2 = 0.6 7 ∗ p i //N=11 clear all ; clc ; close ; N =11; U =6; for n = -5+ U :1:5+ U 108

Figure 6.4: BRF Magnitude Response

109

11 if n ==6 12 hd ( n ) =0.5; 13 else 14 hd ( n ) =( sin ( %pi *( n - U ) ) + sin ( %pi *( n - U ) /3) - sin ( %pi *2*( n 15 16 17 18 19 20 21 22 23 24 25

U ) /3) ) /( %pi *( n - U ) ) ; end end [ hzm , fr ]= frmag ( hd ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR BRF w i t h N=11 ’ ) ; xgrid (2) ;

Scilab code Exa 6.9.a HPF Magnitude Response using Hanning Window 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 6 . 9 a // Program t o P l o t Magnitude R e s p o n c e o f i d e a l H . P . F . // u s i n g Hanning Window // wc1 = 0. 25 ∗ p i //N=11 clear all ; clc ; close ; N =11; U =6; h_hann = window ( ’ hn ’ ,N ) ; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0.75; 110

Figure 6.5: HPF Magnitude Response using Hanning Window

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else hd ( n ) =( sin ( %pi *( n - U ) ) - sin ( %pi *( n - U ) /4) ) /( %pi *( n - U ) ) ; end h ( n ) = h_hann ( n ) * hd ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR HRF w i t h N=11 u s i n g Hanning Window ’ ) ; 28 xgrid (2) ;

Scilab code Exa 6.9.b HPF Magnitude Response using Hamming Window 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 6 . 9 b // Program t o P l o t Magnitude R e s p o n c e o f i d e a l H . P . F . // u s i n g Hamming Window // wc1 = 0. 25 ∗ p i //N=11 clear all ; clc ; close ; N =11; U =6; h_hamm = window ( ’hm ’ ,N ) ; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0.75; 112

Figure 6.6: HPF Magnitude Response using Hamming Window

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else hd ( n ) =( sin ( %pi *( n - U ) ) - sin ( %pi *( n - U ) /4) ) /( %pi *( n - U ) ) ; end h ( n ) = h_hamm ( n ) * hd ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR HRF w i t h N=11 u s i n g Hamming Window ’ ) ; 28 xgrid (2) ;

Scilab code Exa 6.10 Hanning Window Filter Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 6 . 1 0 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n L . P . F . with s p e c i f i c a t i o n s : //N=7 ,w=p i /4 // U s i n g Hanning Window clear all ; clc ; close ; N =7; alpha =3; U =1; h_hann = window ( ’ hn ’ ,N ) ; for n =0+ U :1:6+ U if n ==4 hd ( n ) =0.25; 114

Figure 6.7: Hanning Window Filter Design

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else hd ( n ) =( sin ( %pi *( n -U - alpha ) /4) ) /( %pi *( n -U - alpha ) ) ; end h ( n ) = hd ( n ) * h_hann ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n LPF w i t h N=7 ’ ) ; xgrid (2) ;

Scilab code Exa 6.11 LPF Filter Design using Kaiser Window 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

// Example 6 . 1 1 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n L . P . F . with s p e c i f i c a t i o n s : //wp=20 r a d / s e c , ws=30 r a d / s e c , w s f =100 r a d / s e c // a s =44.0dB , ap =0.1dB // U s i n g K a i s e r Window clear all ; clc ; close ; wsf =100 // r a d / s e c ws =30; // r a d / s e c wp =20; // r a d / s e c as =44.0 //dB ap =0.1 //dB B = ws - wp ; wc =0.5*( ws + wp ) ; 116

Figure 6.8: LPF Filter Design using Kaiser Window

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wc1 = wc *2* %pi / wsf ; delta1 =10^( -0.05* as ) ; delta2 =(10^(0.05* as ) -1) /(10^(0.05* as ) +1) ; delta = min ( delta1 , delta2 ) ; alphas = -20* log10 ( delta ) ; alpha =0.5842*( alphas -21) ^0.4+0.07886*( alphas -21) D =( alphas -7.95) /14.36; N1 = wsf * D / B +1; N = ceil ( N1 ) ; U = ceil ( N /2) ; win_l = window ( ’ k r ’ ,N , alpha ) ; for n = - floor ( N /2) + U :1: floor ( N /2) + U if n == ceil ( N /2) ; hd ( n ) =0.5; else hd ( n ) =( sin ( %pi *( n - U ) /2) ) /( %pi *( n - U ) ) ; end h ( n ) = hd ( n ) * win_l ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n LPF u s i n g K a i s e r Window ’ ) ; 43 xgrid (2) ; 44 disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.12 BPF Filter Design using Kaiser Window

118

Figure 6.9: BPF Filter Design using Kaiser Window

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// Example 6 . 1 2 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n B . P . F . with s p e c i f i c a t i o n s : // wp1=40 p i r a d / s e c , wp2=60 p i r a d / s e c // ws1=20 p i r a d / s e c , ws2=80 p i r a d / s e c // a s =30dB , ap =0.5dB //F=100 Hz // U s i n g K a i s e r Window clear all ; clc ; close ; wsf =200* %pi ; // r a d / s e c ws1 =20* %pi ; // r a d / s e c ws2 =80* %pi ; // r a d / s e c wp1 =40* %pi ; // r a d / s e c wp2 =60* %pi ; // r a d / s e c as =30 //dB ap =0.5 //dB B = min ( wp1 - ws1 , ws2 - wp2 ) ; wc1 = wp1 - B /2; wc2 = wp2 + B /2; wc1 = wc1 *2* %pi / wsf ; wc2 = wc2 *2* %pi / wsf ; delta1 =10^( -0.05* as ) ; delta2 =(10^(0.05* as ) -1) /(10^(0.05* as ) +1) ; delta = min ( delta1 , delta2 ) ; alphas = -20* log10 ( delta ) ; alpha =0.5842*( alphas -21) ^0.4+0.07886*( alphas -21) D =( alphas -7.95) /14.36; N1 = wsf * D / B +1; N = ceil ( N1 ) ; U = ceil ( N /2) ; win_l = window ( ’ k r ’ ,N , alpha ) ; for n = - floor ( N /2) + U :1: floor ( N /2) + U if n == ceil ( N /2) ; hd ( n ) =0.4; else hd ( n ) =( sin (0.7* %pi *( n - U ) ) - sin (0.3* %pi *( n - U ) ) ) /( %pi *( 120

Figure 6.10: Digital Differentiator using Rectangular Window n-U)); end h ( n ) = hd ( n ) * win_l ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n LPF u s i n g K a i s e r Window ’ ) ; 49 xgrid (2) ; 50 disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

38 39 40 41 42 43 44 45 46 47 48

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Scilab code Exa 6.13.a Digital Differentiator using Rectangular Window 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

// Example 6 . 1 3 a // Program t o P l o t Magnitude R e s p o n c e o f i d e a l d i f f e r e n t i a t o r with s p e c i f i c a t i o n s : //N=8 ,w=p i // u s i n g R e c t a n g u l a r window clear all ; clc ; close ; N =8; alpha =7/2; U =1; h_Rect = window ( ’ r e ’ ,N ) ; for n =0+ U :1:7+ U hd ( n ) = -( sin ( %pi *( n -U - alpha ) ) ) /( %pi *( n -U - alpha ) *( n -U alpha ) ) ; h ( n ) = hd ( n ) * h_Rect ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n i d e a l d i f f e r e n t i a t o r u s i n g R e c t a n g u l a r Window , N=8 ’ ) ; xgrid (2)

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Figure 6.11: Digital Differentiator using Hamming Window

123

Scilab code Exa 6.13.b Digital Differentiator using Hamming Window 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

// Example 6 . 1 3 b // Program t o P l o t Magnitude R e s p o n c e o f i d e a l d i f f e r e n t i a t o r with s p e c i f i c a t i o n s : //N=8 ,w=p i // u s i n g Hamming window clear all ; clc ; close ; N =8; alpha =7/2; U =1; // Z e r o A d j u s t h_hamm = window ( ’hm ’ ,N ) ; for n =0+ U :1:7+ U hd ( n ) = -( sin ( %pi *( n -U - alpha ) ) ) /( %pi *( n -U - alpha ) *( n -U alpha ) ) ; h ( n ) = hd ( n ) * h_hamm ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n i d e a l d i f f e r e n t i a t o r u s i n g Hamming Window , N=8 ’ ) ; xgrid (2)

Scilab code Exa 6.14.a Hilbert Transformer using Rectangular Window 1

// Example 6 . 1 4 a 124

Figure 6.12: Hilbert Transformer using Rectangular Window 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

// Program t o P l o t Magnitude R e s p o n c e o f i d e a l H i l b e r t Transformer // u s i n g R e c t a n g u l a r Window //N=11 clear all ; clc ; close ; N =11; U =6; h_Rect = window ( ’ r e ’ ,N ) ; for n = -5+ U :1:5+ U if n ==6 hd ( n ) =0; else hd ( n ) =(1 - cos ( %pi *( n - U ) ) ) /( %pi *( n - U ) ) ; end h ( n ) = hd ( n ) * h_Rect ( n ) ; end [ hzm , fr ]= frmag (h ,256) ; 125

Figure 6.13: Hilbert Transformer using Blackman Window 20 figure 21 plot (2* fr ,- hzm ) ; 22 a = gca () ; 23 xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; 24 ylabel ( ’H( exp ( j ∗w) ) ’ ) ; 25 title ( ’ F r e q u e n c y R e s p o n s e o f

H i l b e r t Transformer w i t h N=11 u s i n g R e c t a n g u l a r Window ’ ) ; 26 xgrid (2) ;

Scilab code Exa 6.14.b Hilbert Transformer using Blackman Window 1 2

// Example 6 . 1 4 b // Program t o P l o t Magnitude R e s p o n c e o f i d e a l H i l b e r t Transformer 126

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

// u s i n g Blackman Window //N=11 clear all ; clc ; close ; N =11; U =6; for n = -5+ U :1:5+ U h_balckmann ( n ) = 0.42+0.5* cos (2* %pi *( n - U ) /( N -1) ) +0.08* cos (4* %pi *( n - U ) /( N -1) ) ; if n ==6 hd ( n ) =0; else hd ( n ) =(1 - cos ( %pi *( n - U ) ) ) /( %pi *( n - U ) ) ; end h ( n ) = hd ( n ) * h_balckmann ( n ) ; end [ hzm , fr ]= frmag (h ,256) ; figure plot (2* fr ,- hzm ) ; a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’H( exp ( j ∗w) ) ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f H i l b e r t T r a n s f o r m e r w i t h N=11 u s i n g Blackman Window ’ ) ; xgrid (2) ;

Scilab code Exa 6.15 Filter Coefficients obtained by Sampling // Example 6 . 1 5 // Program t o d e t e r m i n e f i l t e r by s a m p l i n g : 3 //N=7 ,w=p i /2 4 clear all ; 5 clc ; 1 2

127

c o e f f i c i e n t s obtained

6 7 8 9 10 11 12

close ; N =7; U =1; // Z e r o A d j u s t for n =0+ U :1: N -1+ U h ( n ) =(1+2* cos (2* %pi *( n -U -3) /7) ) / N end disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” )

Scilab code Exa 6.16 Coefficients of Linear phase FIR Filter 1 2 3 4 5 6 7 8 9 10 11 12

// Example 6 . 1 6 // Program t o d e t e r m i n e f i l t e r c o e f f i c i e n t s o b t a i n e d by s a m p l i n g : //N=15 clear all ; clc ; close ; N =15; U =1; // Z e r o A d j u s t for n =0:1: N -1 h ( n + U ) =(1+2* cos (2* %pi *(7 - n ) / N ) +2* cos (4* %pi *(7 - n ) / N ) +2* cos (6* %pi *(7 - n ) / N ) ) / N ; end disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.17 BPF Filter Design using Sampling Method // Example 6 . 1 7 // Program t o d e s i g n b a n d p a s s f i l t e r w i t h f o l l o w i n g specifications : 3 //N=7 , f c 1 =1000Hz , f c 2 =3000Hz , F=8000Hz 4 clear all ; 5 clc ; 1 2

128

Figure 6.14: Frequency Sampling Method FIR LPF Filter 6 7 8 9 10 11 12

close ; N =7; U =1; // Z e r o A d j u s t for n =0:1: N -1 h ( n + U ) =2*( cos (2* %pi *(3 - n ) / N ) + cos (4* %pi *(3 - n ) / N ) ) / N ; end disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.18.a Frequency Sampling Method FIR LPF Filter 1

// Example 6 . 1 8 a 129

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

// Program t o d e s i g n L . P . F . f i l t e r w i t h f o l l o w i n g specifications : //N=15 , wc=p i /4 clear all ; clc ; close ; N =15; U =1; for n =0+ U :1: N -1+ U h ( n ) =(1+ cos (2* %pi *(7 - n ) / N ) ) / N ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure ; plot (2* fr , hzm_dB ) ; a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR LPF w i t h N=15 ’ ) ; xgrid (2)

Scilab code Exa 6.18.b Frequency Sampling Method FIR LPF Filter 1 2 3 4 5 6 7 8 9

// Example 6 . 1 8 b // Program t o d e s i g n L . P . F . specifications : //N=15 , wc=p i /4 clear all ; clc ; close ; N =15; U =1; for n =0+ U :1: N -1+ U

f i l t e r with f o l l o w i n g

130

Figure 6.15: Frequency Sampling Method FIR LPF Filter 10 11 12 13 14 15 16 17 18 19 20

h ( n ) =(1+ cos (2* %pi *(7 - n ) / N ) + cos (4* %pi *(7 - n ) / N ) ) / N ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure ; plot (2* fr , hzm_dB ) ; a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f FIR LPF w i t h N=11 ’ ) ; xgrid (2)

Scilab code Exa 6.19 Filter Coefficients Determination 131

Figure 6.16: Filter Coefficients Determination

132

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

// Example 6 . 1 9 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n L . P . F . with s p e c i f i c a t i o n s : //N=13 ,w=p i /6 clear all ; clc ; close ; alpha =6; U =1; for n =0+ U :1:12+ U if n ==7 hd ( n ) =0.167; else hd ( n ) =( sin ( %pi *( n -U - alpha ) /6) ) /( %pi *( n -U - alpha ) ) ; end end [ hzm , fr ]= frmag ( hd ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n LPF w i t h N=13 ’ ) ; xgrid (2) disp ( hd , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.20 Filter Coefficients using Hamming Window // Example 6 . 2 0 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n L . P . F . with s p e c i f i c a t i o n s : 3 //N=13 ,w=p i /6 1 2

133

Figure 6.17: Filter Coefficients using Hamming Window

134

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

// U s i n g Hamming Window clear all ; clc ; close ; N =13; alpha =6; U =1; h_hamm = window ( ’hm ’ ,N ) ; for n =0+ U :1:12+ U if n ==7 hd ( n ) =0.167; else hd ( n ) =( sin ( %pi *( n -U - alpha ) /6) ) /( %pi *( n -U - alpha ) ) ; end h ( n ) = hd ( n ) * h_hamm ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) a = gca () ; xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; ylabel ( ’ Magnitude i n dB ’ ) ; title ( ’ F r e q u e n c y R e s p o n s e o f g i v e n LPF w i t h N=13 ’ ) ; xgrid (2) disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ; disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.21 LPF Filter using Rectangular Window 1 2

// Example 6 . 2 1 // Program t o P l o t Magnitude R e s p o n c e o f g i v e n L . P . F . with s p e c i f i c a t i o n s : 135

Figure 6.18: LPF Filter using Rectangular Window 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

//N=7 , f c =1000Hz , F=5000Hz clear all ; clc ; close ; N =7; U =4; h_Rect = window ( ’ r e ’ ,N ) ; for n = -3+ U :1:3+ U if n ==4 hd ( n ) =0.4; else hd ( n ) =( sin (2* %pi *( n - U ) /5) ) /( %pi *( n - U ) ) ; end h ( n ) = hd ( n ) * h_Rect ( n ) ; end [ hzm , fr ]= frmag ( h ,256) ; hzm_dB = 20* log10 ( hzm ) ./ max ( hzm ) ; figure plot (2* fr , hzm_dB ) 136

22 a = gca () ; 23 xlabel ( ’ F r e q u e n c y w∗ p i ’ ) ; 24 ylabel ( ’ Magnitude i n dB ’ ) ; 25 title ( ’ F r e q u e n c y R e s p o n s e o f FIR LPF w i t h N=7 ’ ) ; 26 xgrid (2) 27 disp (h , ” F i l t e r C o e f f i c i e n t s , h ( n )=” ) ;

Scilab code Exa 6.28 Filter Coefficients for Direct Form Structure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

// Example 6 . 2 8 // Program t o c a l c u l a t e FIR F i l t e r c o e f f i c i e n t s f o r t h e d i r e c t form s t r u c t u r e // k1 =1/2 , k2 =1/3 , k3 =1/4 clear all ; clc ; close ; U =1; k1 =1/2; k2 =1/3; k3 =1/4; a (3+ U ,0+ U ) =1; a (1+ U ,1+ U ) = k1 ; a (2+ U ,2+ U ) = k2 ; a (3+ U ,3+ U ) = k3 ; m =2 , k =1; a ( m +U , k + U ) = a (m -1+ U , k + U ) + a ( m +U , m + U ) * a (m -1+ U ,m - k + U ) ; m =3 , k =1; a ( m +U , k + U ) = a (m -1+ U , k + U ) + a ( m +U , m + U ) * a (m -1+ U ,m - k + U ) ; m =3 , k =2; a ( m +U , k + U ) = a (m -1+ U , k + U ) + a ( m +U , m + U ) * a (m -1+ U ,m - k + U ) ; disp ( a (3+ U ,0+ U ) , ’ a ( 3 , 0 ) ’ ) ; disp ( a (3+ U ,1+ U ) , ’ a ( 3 , 1 ) ’ ) ; disp ( a (3+ U ,2+ U ) , ’ a ( 3 , 2 ) ’ ) ; disp ( a (3+ U ,3+ U ) , ’ a ( 3 , 3 ) ’ ) ;

137

Scilab code Exa 6.29 Lattice Filter Coefficients Determination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

// Example 6 . 2 9 // Program t o c a l c u l a t e g i v e n FIR F i l t e r ’ s L a t t i c e form c o e f f i c i e n t s . clear all ; clc ; close ; U =1; // Z e r o A d j u s t a (3+ U ,0+ U ) =1; a (3+ U ,1+ U ) =2/5; a (3+ U ,2+ U ) =3/4; a (3+ U ,3+ U ) =1/3; a (2+ U ,0+ U ) =1; // a (m, 0 ) =1 a (2+ U ,3+ U ) =1/3; m =3 , k =1; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; m =3 , k =2; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; m =2 , k =1; a (m -1+ U , k + U ) =( a ( m +U , k + U ) -a ( m +U , m + U ) * a ( m +U ,m - k + U ) ) /(1 - a ( m +U , m + U ) * a ( m +U , m + U ) ) ; disp ( a (1+ U ,1+ U ) , ’ k1 ’ ) ; disp ( a (2+ U ,2+ U ) , ’ k2 ’ ) ; disp ( a (3+ U ,3+ U ) , ’ k3 ’ ) ;

138

Chapter 7 FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS

Scilab code Exa 7.2 Subtraction Computation 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 7 . 2 //To Compute S u b t r a c t i o n // ( a ) 0 . 2 5 from 0 . 5 clear all ; clc ; close ; a =0.5; b =0.25; c =a - b ; disp (c , ’= ’ ,b , ’− ’ ,a , ’PART 1 ’ ) ; // ( a ) 0 . 5 from 0 . 2 5 d =b - a ; disp (d , ’= ’ ,a , ’− ’ ,b , ’PART 2 ’ ) ;

139

Scilab code Exa 7.14 Variance of Output due to AD Conversion Process 1 2 3 4 5 6 7 8 9 10 11 12 13 14

// Example 7 . 1 4 //To Compare t h e V a r i e n c e o f Output due t o A/D Conversion process // y ( n ) =0.8 y ( n −1)+x ( n ) clear all ; clc ; close ; n =8; // B i t s r =100; // Range Q =2* r /(2^ n ) ; // Q u a n t i z a t i o n S t e p S i z e Ve =( Q ^2) /12; Vo = Ve *(1/(1 -0.8^2) ) ; disp (Q , ’QUANTIZATION STEP SIZE = ’ ) ; disp ( Ve , ’VARIANCE OF ERROR SIGNAL = ’ ) ; disp ( Vo , ’VARIANCE OF OUTPUT = ’ ) ;

140

Chapter 8 MULTIRATE SIGNAL PROCESSING

Scilab code Exa 8.9 Two Component Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

// Example 8 . 9 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM // D e v e l o p a two component d e c o m p o s i t i o n f o r t h e transfer function // and d e t e r m i n e P0 ( z ) and P1 ( z ) clear all ; clc ; close ; syms z a n ; HZ =( z ) /( z - a ) ; hn = a ^ n ; // I n v e r s e Z T r a n s f o r m o f HZ h2n = a ^(2* n ) ; P0 = symsum ( h2n * z ^( - n ) ,n ,0 , %inf ) ; h2n1 = a ^(2* n +1) ; P1 = symsum ( h2n1 * z ^( - n ) ,n ,0 , %inf ) ; disp ( P0 , ’ P0 ( Z ) = ’ ) ; disp ( P1 , ’ P1 ( Z ) = ’ ) ;

141

Scilab code Exa 8.10 Two Band Polyphase Decomposition 1 2 3 4 5 6 7 8 9 10 11 12

// Example 8 . 1 0 // D e v e l o p a two band p o l y p h a s e d e c o m p o s i t i o n f o r t h e transfer function //H( z )=z ˆ2+ z +2/ z ˆ 2 + 0 . 8 z +0.6 clear all ; clc ; close ; z = %z ; HZ =( z ^2+ z +2) /( z ^2+0.8* z +0.6) ; HZa = horner ( HZ , - z ) ; P0 =0.5*( HZ + HZa ) ; P1 =0.5*( HZ - HZa ) ; disp ( P1 /z , ’+ ’ ,P0 , ’H( z ) = ’ )

142

Chapter 9 STATISTICAL DIGITAL SIGNAL PROCESSING

Scilab code Exa 9.7.a Frequency Resolution Determination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

// Example 9 . 7 ( a ) // Program To D e t e r m i n e F r e q u e n c y R e s o l u t i o n o f Bartlett , // Welch ( 5 0% O v e r l a p ) and Blackmann−Tukey Methods clear all ; clc ; close ; // Data Q =10; // Q u a l i t y F a c t o r N =1000; // S a m p l e s //FREQUENCY RESOLUTION CALCULATION K=Q; rb =0.89*(2* %pi * K / N ) ; rw =1.28*(2* %pi *9* Q ) /(16* N ) ; rbt =0.64*(2* %pi *2* Q ) /(3* N ) ; // D i s p l a y t h e r e s u l t i n command window disp ( rb , ” R e s o l u t i o n o f B a r t l e t t Method ” ) ; disp ( rw , ” R e s o l u t i o n o f Welch ( 5 0% o v e r l a p ) Method ” ) ; disp ( rbt , ” R e s o l u t i o n o f Blackmann−Tukey Method ” ) ; 143

Scilab code Exa 9.7.b Record Length Determination // Example 9 . 7 ( b ) // Program To D e t e r m i n e Record Length o f B a r t l e t t , // Welch ( 5 0% O v e r l a p ) and Blackmann−Tukey Methods clear all ; clc ; close ; // Data Q =10; // Q u a l i t y F a c t o r N =1000; // S a m p l e s //RECORD LENGTH CALCULATION lb = N / Q ; lw =16* N /(9* Q ) ; lbt =3* N /(2* Q ) ; // D i s p l a y t h e r e s u l t i n command window disp ( lb , ” Record Length o f B a r t l e t t Method ” ) ; disp ( lw , ” Record Length o f Welch ( 5 0% o v e r l a p ) Method ” ); 17 disp ( lbt , ” Record Length o f Blackmann−Tukey Method ” ) ; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Scilab code Exa 9.8.a Smallest Record Length Computation 1 2 3 4 5 6 7

// Example 9 . 8 ( a ) // Program To D e t e r m i n e S m a l l e s t Record Length o f B a r t l e t t Method clear all ; clc ; close ; // Data fr =0.01; // F r e q u e n c y R e s o l u t i o n 144

8 N =2400; // S a m p l e s 9 //RECORD LENGTH CALCULATION 10 lb =0.89/ fr ; 11 // D i s p l a y t h e r e s u l t i n command window 12 disp ( lb , ” Record Length o f B a r t l e t t Method ” ) ;

Scilab code Exa 9.8.b Quality Factor Computation 1 2 3 4 5 6 7 8 9 10 11 12 13

// Example 9 . 8 ( b ) // Program To D e t e r m i n e Q u a l i t y F a c t o r o f B a r t l e t t Method clear all ; clc ; close ; // Data fr =0.01; // F r e q u e n c y R e s o l u t i o n N =2400; // S a m p l e s lb =0.89/ fr ; //QUALITY FACTOR CALCULATION Q = N / lb ; // D i s p l a y t h e r e s u l t i n command window disp (Q , ” Q u a l i t y F a c t o r o f B a r t l e t t Method ” ) ;

145

Chapter 11 DIGITAL SIGNAL PROCESSORS

Scilab code Exa 11.3 Program for Integer Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13

// Program 1 1 . 3 // Program To C a l c u l a t e t h e v a l u e o f t h e f u n c t i o n //Y=A∗B clear all ; clc ; close ; // I n p u t Data A = input ( ’ E n t e r I n t e g e r Number A = ’ ) ; B = input ( ’ E n t e r I n t e g e r Number B = ’ ) ; // M u l t i p l i c a t i o n Computation Y=A*B; // D i s p l a y t h e r e s u l t i n command window disp (Y , ”Y = A∗B = ” ) ;

Scilab code Exa 11.5 Function Value Calculation

146

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

// Program 1 1 . 5 // Program To C a l c u l a t e t h e v a l u e o f t h e f u n c t i o n //Y=A∗X1+B∗X2+C∗X3 clear all ; clc ; close ; // Data A =1; B =2; C =3; X1 =4; X2 =5; X3 =6; // Compute t h e f u n c t i o n Y = A * X1 + B * X2 + C * X3 ; // D i s p l a y t h e r e s u l t i n command window disp (Y , ”Y = A∗X1+B∗X2+C∗X3 = ” ) ;

147

Digital Signal P Ramesh Babu

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