Digital Signal Processing_R. Babu

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Scilab Textbook Companion for

Digital Signal Processing

by R. Babu1

Created byMohammad Faisal Siddiqui

B.Tech (pursuing)Electronics Engineering

Jamia Milia IslamiaCollege Teacher

Dr. Sajad A. Loan, JMI, New DCross-Checked by

Santosh Kumar, IITB

August 10, 2013

1Funded by a grant from the National Mission on Education through ICT,http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilabcodes written in it can be downloaded from the Textbook Companion Projectsection at the website http://scilab.in


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Book Description

Title: Digital Signal Processing

Author: R. Babu

Publisher: Scitech Publications

Edition: 4

Year: 2010

ISBN: 81-8371-081-7

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Scilab numbering policy used in this document and the relation to theabove 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 particularExample of the above book)

For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 meansa scilab code whose theory is explained in Section 2.3 of the book.

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Contents

List of Scilab Codes 4

1 DISCRETE TIME SIGNALS AND LINEAR SYSTEMS 11

2 THE Z TRANSFORM 34

3 THE DISCRETE FOURIER TRANSFORM 59

4 THE FAST FOURIER TRANSFORM 79

5 INFINITE IMPULSE RESPONSE FILTERS 90

6 FINITE IMPULSE RESPONSE FILTERS 102

7 FINITE WORD LENGTH EFFECTS IN DIGITAL FIL-

TERS 139

8 MULTIRATE SIGNAL PROCESSING 141

9 STATISTICAL DIGITAL SIGNAL PROCESSING 143

11 DIGITAL SIGNAL PROCESSORS 146

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List of Scilab Codes

Exa 1.1 Continuous Time Plot and Discrete Time Plot . . . . 11Exa 1.2 Continuous Time Plot and Discrete Time Plot . . . . 13Exa 1.3.a Evaluate the Summations . . . . . . . . . . . . . . . . 14Exa 1.3.b Evaluate the Summations . . . . . . . . . . . . . . . . 14Exa 1.4.a Check for Energy or Power Signals . . . . . . . . . . . 15Exa 1.4.d Check for Energy or Power Signals . . . . . . . . . . . 15Exa 1.5.a Determining Periodicity of Signal . . . . . . . . . . . . 16Exa 1.5.c Determining Periodicity of Signal . . . . . . . . . . . . 18Exa 1.5.d Determining Periodicity of Signal . . . . . . . . . . . . 19Exa 1.11 Stability of the System . . . . . . . . . . . . . . . . . 21Exa 1.12 Convolution Sum of Two Sequences . . . . . . . . . . 21Exa 1.13 Convolution of Two Signals . . . . . . . . . . . . . . . 22Exa 1.18 Cross Correlation of Two Sequences . . . . . . . . . . 22

Exa 1.19 Determination of Input Sequence . . . . . . . . . . . . 23Exa 1.32.a Plot Magnitude and Phase Response . . . . . . . . . . 24Exa 1.37 Sketch Magnitude and Phase Response. . . . . . . . . 24Exa 1.38 Plot Magnitude and Phase Response . . . . . . . . . . 26Exa 1.45 Filter to Eliminate High Frequency Component . . . . 27Exa 1.57.a Discrete Convolution of Sequences . . . . . . . . . . . 29Exa 1.61 Fourier Transform . . . . . . . . . . . . . . . . . . . . 29Exa 1.62 Fourier Transform . . . . . . . . . . . . . . . . . . . . 30Exa 1.64.a Frequency Response of LTI System . . . . . . . . . . . 31Exa 1.64.c Frequency Response of LTI System . . . . . . . . . . . 31Exa 2.1 z Transform and ROC of Causal Sequence . . . . . . . 34

Exa 2.2 z Transform and ROC of Anticausal Sequence . . . . . 34Exa 2.3 z Transform of the Sequence . . . . . . . . . . . . . . 35Exa 2.4 z Transform and ROC of the Signal . . . . . . . . . . 36Exa 2.5 z Transform and ROC of the Signal . . . . . . . . . . 36

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Exa 2.6 Stability of the System . . . . . . . . . . . . . . . . . 37

Exa 2.7 z Transform of the Signal . . . . . . . . . . . . . . . . 37Exa 2.8.a z Transform of the Signal . . . . . . . . . . . . . . . . 37Exa 2.9 z Transform of the Sequence . . . . . . . . . . . . . . 38Exa 2.10 z Transform Computation . . . . . . . . . . . . . . . . 38Exa 2.11 z Transform of the Sequence . . . . . . . . . . . . . . 39Exa 2.13.a z Transform of Discrete Time Signals . . . . . . . . . . 39Exa 2.13.b z Transform of Discrete Time Signals. . . . . . . . . . 40Exa 2.13.c z Transform of Discrete Time Signals . . . . . . . . . . 40Exa 2.13.d z Transform of Discrete Time Signals. . . . . . . . . . 41Exa 2.16 Impulse Response of the System . . . . . . . . . . . . 41Exa 2.17 Pole Zero Plot of the Difference Equation . . . . . . . 42

Exa 2.19 Frequency Response of the System . . . . . . . . . . . 44Exa 2.20.a Inverse z Transform Computation . . . . . . . . . . . 44Exa 2.22 Inverse z Transform Computation . . . . . . . . . . . 45Exa 2.23 Causal Sequence Determination. . . . . . . . . . . . . 45Exa 2.34 Impulse Response of the System . . . . . . . . . . . . 46Exa 2.35.a Pole Zero Plot of the System . . . . . . . . . . . . . . 47Exa 2.35.b Unit Sample Response of the System . . . . . . . . . . 49Exa 2.38 Determine Output Response . . . . . . . . . . . . . . 50Exa 2.40 Input Sequence Computation . . . . . . . . . . . . . . 52Exa 2.41.a z Transform of the Signal . . . . . . . . . . . . . . . . 53

Exa 2.41.b z Transform of the Signal . . . . . . . . . . . . . . . . 53Exa 2.41.c z Transform of the Signal . . . . . . . . . . . . . . . . 54Exa 2.45 Pole Zero Pattern of the System . . . . . . . . . . . . 55Exa 2.53.a z Transform of the Sequence . . . . . . . . . . . . . . 55Exa 2.53.b z Transform of the Signal . . . . . . . . . . . . . . . . 55Exa 2.53.c z Transform of the Signal . . . . . . . . . . . . . . . . 56Exa 2.53.d z Transform of the Signal . . . . . . . . . . . . . . . . 56Exa 2.54 z Transform of Cosine Signal . . . . . . . . . . . . . . 56Exa 2.58 Impulse Response of the System . . . . . . . . . . . . 58Exa 3.1 DFT and IDFT . . . . . . . . . . . . . . . . . . . . . 59Exa 3.2 DFT of the Sequence . . . . . . . . . . . . . . . . . . 60

Exa 3.3 8 Point DFT . . . . . . . . . . . . . . . . . . . . . . . 62Exa 3.4 IDFT of the given Sequence . . . . . . . . . . . . . . . 62Exa 3.7 Plot the Sequence . . . . . . . . . . . . . . . . . . . . 63Exa 3.9 Remaining Samples . . . . . . . . . . . . . . . . . . . 64Exa 3.11 DFT Computation . . . . . . . . . . . . . . . . . . . . 65

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Exa 3.13 Circular Convolution. . . . . . . . . . . . . . . . . . . 65

Exa 3.14 Circular Convolution. . . . . . . . . . . . . . . . . . . 66Exa 3.15 Determine Sequence x3 . . . . . . . . . . . . . . . . . 67Exa 3.16 Circular Convolution. . . . . . . . . . . . . . . . . . . 67Exa 3.17 Circular Convolution. . . . . . . . . . . . . . . . . . . 68Exa 3.18 Output Response. . . . . . . . . . . . . . . . . . . . . 69Exa 3.20 Output Response. . . . . . . . . . . . . . . . . . . . . 70Exa 3.21 Linear Convolution. . . . . . . . . . . . . . . . . . . . 70Exa 3.23.a N Point DFT Computation . . . . . . . . . . . . . . . 71Exa 3.23.b N Point DFT Computation . . . . . . . . . . . . . . . 71Exa 3.23.c N Point DFT Computation . . . . . . . . . . . . . . . 72Exa 3.23.d N Point DFT Computation . . . . . . . . . . . . . . . 72

Exa 3.23.e N Point DFT Computation . . . . . . . . . . . . . . . 72Exa 3.23.f N Point DFT Computation . . . . . . . . . . . . . . . 73Exa 3.24 DFT of the Sequence . . . . . . . . . . . . . . . . . . 73Exa 3.25 8 Point Circular Convolution . . . . . . . . . . . . . . 73Exa 3.26 Linear Convolution using DFT . . . . . . . . . . . . . 74Exa 3.27.a Circular Convolution Computation . . . . . . . . . . . 75Exa 3.27.b Circular Convolution Computation . . . . . . . . . . . 75Exa 3.30 Calculate value of N . . . . . . . . . . . . . . . . . . . 76Exa 3.32 Sketch Sequence . . . . . . . . . . . . . . . . . . . . . 76Exa 3.36 Determine IDFT . . . . . . . . . . . . . . . . . . . . . 78

Exa 4.3 Shortest Sequence N Computation . . . . . . . . . . . 79Exa 4.4 Twiddle Factor Exponents Calculation . . . . . . . . . 80Exa 4.6 DFT using DIT Algorithm . . . . . . . . . . . . . . . 80Exa 4.8 DFT using DIF Algorithm . . . . . . . . . . . . . . . 81Exa 4.9 8 Point DFT of the Sequence . . . . . . . . . . . . . . 81Exa 4.10 4 Point DFT of the Sequence . . . . . . . . . . . . . . 82Exa 4.11 IDFT of the Sequence using DIT Algorithm . . . . . . 82Exa 4.12 8 Point DFT of the Sequence . . . . . . . . . . . . . . 82Exa 4.13 8 Point DFT of the Sequence . . . . . . . . . . . . . . 83Exa 4.14 DFT using DIT Algorithm . . . . . . . . . . . . . . . 83Exa 4.15 DFT using DIF Algorithm . . . . . . . . . . . . . . . 84

Exa 4.16.a 8 Point DFT using DIT FFT . . . . . . . . . . . . . . 84Exa 4.16.b 8 Point DFT using DIT FFT . . . . . . . . . . . . . . 85Exa 4.17 IDFT using DIF Algorithm . . . . . . . . . . . . . . . 85Exa 4.18 IDFT using DIT Algorithm . . . . . . . . . . . . . . . 86Exa 4.19 FFT Computation of the Sequence . . . . . . . . . . . 86

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Exa 4.20 8 Point DFT by Radix 2 DIT FFT . . . . . . . . . . . 86

Exa 4.21 DFT using DIT FFT Algorithm . . . . . . . . . . . . 87Exa 4.22 Compute X using DIT FFT . . . . . . . . . . . . . . . 87Exa 4.23 DFT using DIF FFT Algorithm . . . . . . . . . . . . 88Exa 4.24 8 Point DFT of the Sequence . . . . . . . . . . . . . . 88Exa 5.1 Order of the Filter Determination . . . . . . . . . . . 90Exa 5.2 Order of Low Pass Butterworth Filter . . . . . . . . . 90Exa 5.4 Analog Butterworth Filter Design . . . . . . . . . . . 91Exa 5.5 Analog Butterworth Filter Design . . . . . . . . . . . 92Exa 5.6 Order of Chebyshev Filter . . . . . . . . . . . . . . . . 92Exa 5.7 Chebyshev Filter Design. . . . . . . . . . . . . . . . . 93Exa 5.8 Order of Type 1 Low Pass Chebyshev Filter . . . . . . 93

Exa 5.9 Chebyshev Filter Design. . . . . . . . . . . . . . . . . 94Exa 5.10 HPF Filter Design with given Specifications . . . . . . 94Exa 5.11 Impulse Invariant Method Filter Design . . . . . . . . 95Exa 5.12 Impulse Invariant Method Filter Design . . . . . . . . 96Exa 5.13 Impulse Invariant Method Filter Design . . . . . . . . 96Exa 5.15 Impulse Invariant Method Filter Design . . . . . . . . 97Exa 5.16 Bilinear Transformation Method Filter Design. . . . . 97Exa 5.17 HPF Design using Bilinear Transform . . . . . . . . . 98Exa 5.18 Bilinear Transformation Method Filter Design. . . . . 99Exa 5.19 Single Pole LPF into BPF Conversion . . . . . . . . . 99

Exa 5.29 Pole Zero IIR Filter into Lattice Ladder Structure . . 100Exa 6.1 Group Delay and Phase Delay . . . . . . . . . . . . . 102Exa 6.5 LPF Magnitude Response . . . . . . . . . . . . . . . . 104Exa 6.6 HPF Magnitude Response . . . . . . . . . . . . . . . . 104Exa 6.7 BPF Magnitude Response . . . . . . . . . . . . . . . . 106Exa 6.8 BRF Magnitude Response. . . . . . . . . . . . . . . . 108Exa 6.9.a HPF Magnitude Response using Hanning Window . . 110Exa 6.9.b HPF Magnitude Response using Hamming Window. . 112Exa 6.10 Hanning Window Filter Design . . . . . . . . . . . . . 114Exa 6.11 LPF Filter Design using Kaiser Window . . . . . . . . 116Exa 6.12 BPF Filter Design using Kaiser Window . . . . . . . . 118

Exa 6.13.a Digital Differentiator using Rectangular Window . . . 122Exa 6.13.b Digital Differentiator using Hamming Window . . . . 124Exa 6.14.a Hilbert Transformer using Rectangular Window. . . . 124Exa 6.14.b Hilbert Transformer using Blackman Window . . . . . 126Exa 6.15 Filter Coefficients obtained by Sampling . . . . . . . . 127

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Exa 6.16 Coefficients of Linear phase FIR Filter . . . . . . . . . 128

Exa 6.17 BPF Filter Design using Sampling Method . . . . . . 128Exa 6.18.a Frequency Sampling Method FIR LPF Filter . . . . . 129Exa 6.18.b Frequency Sampling Method FIR LPF Filter . . . . . 130Exa 6.19 Filter Coefficients Determination . . . . . . . . . . . . 131Exa 6.20 Filter Coefficients using Hamming Window . . . . . . 133Exa 6.21 LPF Filter using Rectangular Window . . . . . . . . . 135Exa 6.28 Filter Coefficients for Direct Form Structure. . . . . . 137Exa 6.29 Lattice Filter Coefficients Determination. . . . . . . . 138Exa 7.2 Subtraction Computation . . . . . . . . . . . . . . . . 139Exa 7.14 Variance of Output due to AD Conversion Process . . 139Exa 8.9 Two Component Decomposition . . . . . . . . . . . . 141

Exa 8.10 Two Band Polyphase Decomposition . . . . . . . . . . 142Exa 9.7.a Frequency Resolution Determination . . . . . . . . . . 143Exa 9.7.b Record Length Determination. . . . . . . . . . . . . . 144Exa 9.8.a Smallest Record Length Computation . . . . . . . . . 144Exa 9.8.b Quality Factor Computation . . . . . . . . . . . . . . 145Exa 11.3 Program for Integer Multiplication . . . . . . . . . . . 146Exa 11.5 Function Value Calculation . . . . . . . . . . . . . . . 146

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List of Figures

1.1 Continuous Time Plot and Discrete Time Plot . . . . . . . . 121.2 Continuous Time Plot and Discrete Time Plot . . . . . . . . 131.3 Determining Periodicity of Signal . . . . . . . . . . . . . . . 171.4 Determining Periodicity of Signal . . . . . . . . . . . . . . . 181.5 Determining Periodicity of Signal . . . . . . . . . . . . . . . 201.6 Plot Magnitude and Phase Response . . . . . . . . . . . . . 231.7 Sketch Magnitude and Phase Response . . . . . . . . . . . . 251.8 Plot Magnitude and Phase Response . . . . . . . . . . . . . 261.9 Filter to Eliminate High Frequency Component . . . . . . . 281.10 Frequency Response of LTI System . . . . . . . . . . . . . . 301.11 Frequency Response of LTI System . . . . . . . . . . . . . . 32

2.1 Pole Zero Plot of the Difference Equation. . . . . . . . . . . 422.2 Frequency Response of the System . . . . . . . . . . . . . . 432.3 Impulse Response of the System . . . . . . . . . . . . . . . . 462.4 Pole Zero Plot of the System. . . . . . . . . . . . . . . . . . 482.5 Unit Sample Response of the System . . . . . . . . . . . . . 492.6 Determine Output Response . . . . . . . . . . . . . . . . . . 512.7 Pole Zero Pattern of the System . . . . . . . . . . . . . . . . 542.8 Impulse Response of the System . . . . . . . . . . . . . . . . 57

3.1 DFT of the Sequence . . . . . . . . . . . . . . . . . . . . . . 603.2 Plot the Sequence . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Sketch Sequence. . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1 LPF Magnitude Response . . . . . . . . . . . . . . . . . . . 1036.2 HPF Magnitude Response . . . . . . . . . . . . . . . . . . . 1056.3 BPF Magnitude Response . . . . . . . . . . . . . . . . . . . 1076.4 BRF Magnitude Response . . . . . . . . . . . . . . . . . . . 109

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6.5 HPF Magnitude Response using Hanning Window. . . . . . 111

6.6 HPF Magnitude Response using Hamming Window . . . . . 1136.7 Hanning Window Filter Design . . . . . . . . . . . . . . . . 1156.8 LPF Filter Design using Kaiser Window . . . . . . . . . . . 1176.9 BPF Filter Design using Kaiser Window . . . . . . . . . . . 1196.10 Digital Differentiator using Rectangular Window. . . . . . . 1216.11 Digital Differentiator using Hamming Window . . . . . . . . 1236.12 Hilbert Transformer using Rectangular Window . . . . . . . 1256.13 Hilbert Transformer using Blackman Window . . . . . . . . 1266.14 Frequency Sampling Method FIR LPF Filter . . . . . . . . . 1296.15 Frequency Sampling Method FIR LPF Filter . . . . . . . . . 1316.16 Filter Coefficients Determination . . . . . . . . . . . . . . . 132

6.17 Filter Coefficients using Hamming Window . . . . . . . . . . 1346.18 LPF Filter using Rectangular Window . . . . . . . . . . . . 136

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Chapter 1

DISCRETE TIME SIGNALS

AND LINEAR SYSTEMS

Scilab code Exa 1.1 Continuous Time Plot and Discrete Time Plot

1 / / E xa mp le 1 . 12 // S k et ch t he c o n ti n uo u s t im e s i g n a l x ( t ) =2e x p(2 t )

and a l s o i t s d i s c r e t e t i me e q u i v a l e n t s i g n a l w i t ha s am pl in g p e r i od T = 0 . 2 s e c

3 c l e ar a ll ;

4 clc ;

5 close ;

6 t = 0 : 0 . 0 1 : 2 ;

7 x 1 = 2 * exp ( - 2 * t ) ;

8 subplot ( 1 , 2 , 1 ) ;

9 plot ( t , x 1 ) ;

10 x l a b e l ( t ) ;

11 y l a b e l ( x ( t ) ) ;12 t i t l e (CONTINUOUS TIME PLOT ) ;13 n = 0 : 0 . 2 : 2 ;

14 x 2 = 2 * exp ( - 2 * n ) ;

15 subplot ( 1 , 2 , 2 ) ;

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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 plot2d3 ( n , x 2 ) ;17 x l a b e l ( n ) ;18 y l a b e l ( x ( n ) ) ;19 t i t l e ( DISCRETE TIME PLOT ) ;

Scilab code Exa 1.2 Continuous Time Plot and Discrete Time Plot

1 / / E xa mp le 1 . 22 // S k et ch t he c o n ti n uo u s t im e s i g n a l x=s i n ( 7t )+si n

( 1 0t ) and a l s o i t s d i s c r e t e t i m e e q ui v a l e n ts i g n a l w i th a s am pl in g p e ri o d T = 0 . 2 s e c

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3 c l e ar a ll ;

4 clc ;5 close ;

6 t = 0 : 0 . 0 1 : 2 ;

7 x1 = sin ( 7 * t ) + sin ( 1 0 * t ) ;

8 subplot ( 1 , 2 , 1 ) ;

9 plot ( t , x 1 ) ;

10 x l a b e l ( t ) ;11 y l a b e l ( x ( t ) ) ;12 t i t l e (CONTINUOUS TIME PLOT ) ;13 n = 0 : 0 . 2 : 2 ;

14 x2 = sin ( 7 * n ) + sin ( 1 0 * n ) ;

15 subplot ( 1 , 2 , 2 ) ;16 plot2d3 ( n , x 2 ) ;

17 x l a b e l ( n ) ;18 y l a b e l ( x ( n ) ) ;19 t i t l e ( DISCRETE TIME PLOT ) ;

Scilab code Exa 1.3.a Evaluate the Summations

1 / / Example 1 . 3 ( a )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / / C a l c u l a t e F o l l o w i n g S um ma ti on s4 c l e ar a ll ;

5 clc ;

6 close ;

7 s y ms n ;

8 X = s ym su m ( 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 window10 disp (X , The V al ue o f s um ma ti on c om es o u t t o b e : ) ;

Scilab code Exa 1.3.b Evaluate the Summations

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1 / / Exam ple 1 . 3 ( b )

2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / / C a l c u l a t e F o l l o w i n g S um ma ti on s4 c l e ar a ll ;

5 clc ;

6 close ;

7 s y ms n ;

8 X = s ym su m ( %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 window10 disp (X , The V al ue o f s um ma ti on c om es o u t t o b e : ) ;

Scilab code Exa 1.4.a Check for Energy or Power Signals

1 / / Example 1 . 4 ( a )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / / Fi nd E ne rg y and Power o f G iv en S i g n a l s4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms n N ;

8 x = ( 1 / 3 ) ^ n ;9 E = s ym su m ( x ^2 , n ,0 , % in f );

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (E , Energy : ) ;12 p = ( 1/ (2 * N + 1) ) * s y ms um ( x ^2 , n ,0 , N ) ;

13 P = l i m i t ( p , N , % i n f ) ;

14 disp (P , Power : ) ;15 // The E nergy i s F i n i t e and P ower i s 0 . T h e re f o re t he

g iv en s i g n a l i s an Energy S i gn a l

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

1 / / Exam ple 1 . 4 ( d )

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2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM

3 / / Fi nd E ne rg y and Power o f G iv en S i g n a l s4 c l e ar a ll ;5 clc ;

6 close ;

7 syms n N ;

8 x = % e ^ ( 2 * n ) ;

9 E = s ym su m ( x ^2 , n ,0 , % in f );

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (E , Energy : ) ;12 p = ( 1/ (2 * N + 1) ) * s y ms um ( x ^2 , n ,0 , N ) ;

13 P = l i m i t ( p , N , % i n f ) ;

14 disp (P , Power : ) ;15 / / The E n er gy a nd Po we r 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 en s i g n a l i s an n e i t h e r Energy S i gn a l n o rPower S i g n a l

Scilab code Exa 1.5.a Determining Periodicity of Signal

1 / / Example 1 . 5 ( a )2 / /To D e te rm in e Whether G ive n S i g n a l i s P e r i o d i c o r

n ot3 c l e ar a ll ;

4 clc ;

5 close ;

6 t = 0 : 0 . 0 1 : 2 ;

7 x1 = exp ( % i * 6 * % p i * t ) ;

8 subplot ( 1 , 2 , 1 ) ;

9 plot ( t , x 1 ) ;10 x l a b e l ( t ) ;11 y l a b e l ( x ( t ) ) ;12 t i t l e (CONTINUOUS TIME PLOT ) ;13 n = 0 : 0 . 2 : 2 ;

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Figure 1.3: Determining Periodicity of Signal

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2 / /To D e te rm in e Whether G ive n S i g n a l i s P e r i o d i c o r

n ot3 c l e ar a ll ;4 clc ;

5 close ;

6 t = 0 : 0 . 0 1 : 1 0 ;

7 x1 = cos ( 2 * % p i * t / 3 ) ;

8 subplot ( 1 , 2 , 1 ) ;

9 plot ( t , x 1 ) ;

10 x l a b e l ( t ) ;11 y l a b e l ( x ( t ) ) ;12 t i t l e (CONTINUOUS TIME PLOT ) ;

13 n = 0 : 0 . 2 : 1 0 ;14 x2 = cos ( 2 * % p i * n / 3 ) ;

15 subplot ( 1 , 2 , 2 ) ;

16 plot2d3 ( n , x 2 ) ;

17 x l a b e l ( n ) ;18 y l a b e l ( x ( n ) ) ;19 t i t l e ( DISCRETE TIME PLOT ) ;20 / / Hen ce G ive n S i g n a l i s P e r i o d i c w it h N=3

Scilab code Exa 1.5.d Determining Periodicity of Signal

1 / / Exam ple 1 . 5 ( d )2 / /To D e te rm in e Whether G ive n S i g n a l i s P e r i o d i c o r

n ot3 c l e ar a ll ;

4 clc ;

5 close ;6 t = 0 : 0 . 0 1 : 5 0 ;

7 x1 = cos ( % p i * t / 3 ) + cos ( 3 * % p i * t / 4 ) ;

8 subplot ( 1 , 2 , 1 ) ;

9 plot ( t , x 1 ) ;

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Figure 1.5: Determining Periodicity of Signal

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10 x l a b e l ( t ) ;

11 y l a b e l ( x ( t ) ) ;12 t i t l e (CONTINUOUS TIME PLOT ) ;13 n = 0 : 0 . 2 : 5 0 ;

14 x2 = cos ( % p i * n / 3 ) + cos ( 3 * % p i * n / 4 ) ;

15 subplot ( 1 , 2 , 2 ) ;

16 plot2d3 ( n , x 2 ) ;

17 x l a b e l ( n ) ;18 y l a b e l ( x ( n ) ) ;19 t i t l e ( DISCRETE TIME PLOT ) ;20 / / Hen ce G ive n S i g n a l i s P e r i o d i c w it h N=24

Scilab code Exa 1.11 Stability of the System

1 / / E x am pl e 1 . 1 12 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 // T e st i ng S t a b i l i t y o f Given System4 c l e ar a ll ;

5 clc ;

6 close ;

7 s y ms n ;8 x = (1 /2 ) ^ n

9 X = s ym su m ( x, n ,0 , % in f ) ;

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (X , S um ma ti on 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 / / E x am pl e 1 . 1 22 / / 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 es3 // x ( n ) =[3 2 1 2 ] , h ( n) =[1 2 1 2 ] ;

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Figure 1.6: Plot Magnitude and Phase Response

10 y = convol ( x , h 1 ) ;

11 disp ( round ( y ) ) ;

Scilab code Exa 1.19 Determination of Input Sequence

1 / / E x am pl e 1 . 1 92 / /To f i n d i n pu t x ( n )3 // h ( n) =[1 2 1 ] , y ( n ) =[1 5 10 11 8 4 1 ]4 c l e ar a ll ;

5 clc ;

6 close ;

7 z = % z ;

8 a = z ^ 6 + 5 * ( z ^ ( 5 ) ) + 1 0 * ( z ^ ( 4 ) ) + 1 1 * ( 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) = ) ;

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Scilab code Exa 1.32.a Plot Magnitude and Phase Response

1 / / E x am pl e 1 . 3 22 / / P ro gr am t o P l o t M ag n it ud e a nd P ha s e R e sp o n ce3 c l e ar a ll ;

4 clc ;

5 close ;

6 w = - % p i : 0 . 0 1 : % p i ;

7 H = 1 + 2 * cos ( w ) + 2 * cos ( 2 * w ) ;

8 // c a l u c u l a t i o n o f Pha se and Mag ni tud e o f H9 [phase_H ,m]= phasemag ( H ) ;

10 Hm = abs ( H ) ;

11 a = gca () ;

12 subplot ( 2 , 1 , 1 ) ;

13 a . y _ l o c a t i o n = o r i g i n ;14 plot2d ( w / % p i , H m ) ;

15 x l a b e l ( F r e qu e n cy i n R a d ia n s )16 y l a b e l ( abs (Hm) ) ;17 t i t l e ( MAGNITUDE RESPONSE ) ;

18 subplot ( 2 , 1 , 2 ) ;19 a = gca () ;

20 a . x _ l o c a t i o n = o r i g i n ;21 a . y _ l o c a t i o n = o r i g i n ;22 plot2d ( w / ( 2 * % p i ) , p h a s e _ H ) ;

23 x l a b e l ( F r e qu e n cy i n R a d ia n s ) ;24 y l a b e l (


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Figure 1.7: Sketch Magnitude and Phase Response

1 / / E x am pl e 1 . 3 72 / / P ro gr am t o P l o t M ag n it ud e a nd P ha s e R e sp o n ce3 / / y ( n ) = 1 / 2[ x ( n )+x ( n2) ]4 c l e ar a ll ;

5 clc ;

6 close ;

7 w = 0 : 0 . 0 1 : % p i ;

8 H = ( 1 + cos ( 2 * w ) - % i * sin ( 2 * w ) ) / 2 ;


11 Hm = abs ( H ) ;

12 a = gca () ;

13 subplot ( 2 , 1 , 1 ) ;


16 x l a b e l ( F r e qu e n cy i n R a d ia n s )17 y l a b e l ( abs (Hm) ) ;18 t i t l e ( MAGNITUDE RESPONSE ) ;19 subplot ( 2 , 1 , 2 ) ;

20 a = gca () ;


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Figure 1.8: Plot Magnitude and Phase Response



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14 a . y _ l o c a t i o n = o r i g i n ;

15 plot2d ( w / % p i , H m ) ;16 x l a b e l ( F r e qu e n cy i n R a d ia n s )17 y l a b e l ( abs (Hm) ) ;18 t i t l e ( MAGNITUDE RESPONSE ) ;19 subplot ( 2 , 1 , 2 ) ;

20 a = gca () ;




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Figure 1.9: Filter to Eliminate High Frequency Component

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18 subplot ( 2 , 1 , 2 ) ;

19 plot ( t , x 1 , : ) ;20 t i t l e ( x : SIGNAL WITHOUT NOISE y : SIGNAL WITH NOISE );

21 x l a b e l ( Time i n S e c ) ;22 y l a b e l ( Amplitud e ) ;

Scilab code Exa 1.57.a Discrete Convolution of Sequences

1 / / Example 1 . 5 7 ( a )2 // Program t o C ompute 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 ns e q u e n c e s

3 / / x ( n ) = [1 2 1 1 ] , h ( n ) =[1 0 1 1 ] ;4 c l e ar a ll ;

5 clc ;

6 close ;

7 x =[1 2 -1 1];

8 h = [1 0 1 1];

9 y = convol ( x , h ) ;

10 disp ( round ( y ) ) ;

Scilab code Exa 1.61 Fourier Transform

1 / / E x am pl e 1 . 6 12 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 // F o u r ie r t r an s fo r m o f ( 3 ) n u ( n )4 c l e ar a ll ;

5 clc ;

6 close ;7 s y ms n ;

8 x = (3) ^ n;

9 X = s ym su m ( x, n ,0 , % in f )

10 / / D i s p l a y t h e r e s u l t i n command window

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Figure 1.10: Frequency Response of LTI System

11 disp (X , The F o u r ie r T ra ns fo rm d oe s n ot e x i t a s x ( n )i s n ot a b s o l u t e l y summable a nd a p p ro a c he s

i n f i n i t y i . e . ) ;

Scilab code Exa 1.62 Fourier Transform

1 / / E x am pl e 1 . 6 22 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 // F o u r ie r t r an s fo r m o f ( 0 . 8 ) | n | u ( n )4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms w n ;

8 X = s y ms um ( (0 . 8) ^ n * %e ^ ( %i * w * n) , n ,1 , % in f ) + s ym su m

( (0 .8 ) ^ n * %e ^ ( - %i * w * n) , n ,0 , % in f )


10 disp (X , The F o u r i e r T ra n sf or m c om es o u t t o b e : ) ;

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Figure 1.11: Frequency Response of LTI System

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1 / / E xa mp le 1 . 6 4 ( c )

2 / / Pr ogr am t o C a l c u l a t e P l o t M ag ni tu de and P ha seR e sponc e3 c l e ar a ll ;

4 clc ;

5 close ;

6 w = 0 : 0 . 0 1 : % p i ;

7 H = 1 / ( 1 - 0 . 9 * % i * % e ^ ( - % i * w ) ) ;


10 Hm = abs ( H ) ;

11 a = gca () ;

12 subplot ( 2 , 1 , 1 ) ;13 a . y _ l o c a t i o n = o r i g i n ;14 plot2d ( w / % p i , H m ) ;

15 x l a b e l ( F r e qu e n cy i n R a d ia n s )16 y l a b e l ( abs (Hm) ) ;17 t i t l e ( MAGNITUDE RESPONSE ) ;18 subplot ( 2 , 1 , 2 ) ;

19 a = gca () ;




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Chapter 2

THE Z TRANSFORM

Scilab code Exa 2.1 z Transform and ROC of Causal Sequence

1 / / E xa mp le 2 . 12 //Z tr an sf or m o f [ 1 0 3 1 2 ]3 c l e ar a ll ;

4 clc ;

5 close ;

6 function [ z a ] = z t r a n s f e r ( s e q u e n ce , n )

7 z = poly (0 , z , r )8 z a = s e q u e n c e * ( 1 / z ) ^ n

9 e n d f u n c t i o n

10 x 1 =[1 0 3 -1 2];

11 n = 0 : length ( x 1 ) - 1 ;

12 z z = z t r a n s f e r ( x 1 , n ) ;

13 / / D i s p l a y t h e r e s u l t i n command window14 disp ( z z , Zt r an s f or m o f s e qu e nc e i s : ) ;15 disp (ROC i s t he e n t i r e p la ne e xc ep t z = 0 ) ;

Scilab code Exa 2.2 z Transform and ROC of Anticausal Sequence

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1 / / E xa mp le 2 . 2

2 //Z t ra n sf or m o f [3 2 1 0 1 ]3 c l e ar a ll ;4 clc ;

5 close ;




10 x 1 =[ -3 -2 -1 0 1] ;

11 n = - ( length ( x 1 ) - 1 ) : 0 ;

12 z z = z t r a n s f e r ( x 1 , n ) ;

13 / / D i s p l a y t h e r e s u l t i n command window14 disp ( z z , Zt r an s f or m o f s e qu e nc e i s : ) ;15 disp (ROC i s t he e n t i r e p l an e e x ce p t z = % i nf ) ;

Scilab code Exa 2.3 z Transform of the Sequence

1 / / E xa mp le 2 . 32 //Z t ra n sf or m o f [ 2 1 3 2 1 0 2 3 1]

3 c l e ar a ll ;4 clc ;

5 close ;




10 x1 =[2 -1 3 2 1 0 2 3 -1];

11 n = - 4 : 4 ;

12 z z = z t r a n s f e r ( x 1 , n ) ;

13 / / D i s p l a y t h e r e s u l t i n command window14 disp ( z z , Zt r an s f or m o f s e qu e nc e i s : ) ;15 disp (ROC i s t he e n t i r e p la ne e xc ep t z = 0 and z =

%inf ) ;

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Scilab code Exa 2.4 z Transform and ROC of the Signal

1 / / E xa mp le 2 . 42 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r a ns f o r m o f a n u ( n )4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms a n z ;

8 x = a^ n

9 X = s ym su m ( x *( z ^( - n )) , n ,0 , % in f ) ;

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (X , Zt r an s fo r m o f a n u ( n ) w it h i s : ) ;12 disp (ROC i s t he R e gi on mod( z ) > a )

Scilab code Exa 2.5 z Transform and ROC of the Signal

1 / / E xa mp le 2 . 52 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z tr a n sf o rm o f b n u (n1)4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms b n z ;

8 x = b^ n

9 X = s ym su m ( x *( z ^( - n )) , n ,0 , % in f ) ;


11 disp (X , Zt r an s fo r m o f b n u ( n ) w it h i s : ) ;12 disp (ROC i s t he R e gi on mod( z ) < b )

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1 / / Example 2 . 8 ( a )

2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /Z t r a n s f o r m o f c o s (Won )4 clc ;

5 syms Wo n z ;

6 x1 = exp ( sqrt ( - 1 ) * W o * n ) ;

7 X 1 = s y m s u m ( x 1 * ( z ^ - n ) , n , 0 , % i n f ) ;

8 x2 = exp ( - sqrt ( - 1 ) * W o * n ) ;

9 X 2 = s y m s u m ( x 2 * ( z ^ - n ) , n , 0 , % i n f ) ;

10 X = ( X 1 + X 2 ) / 2 ;

11 disp (X ,X( z )= ) ;


1 / / E xa mp le 2 . 92 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r a ns f or m o f ( 1 / 3 ) n u ( n1)4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms n z ;8 x = (1 /3 ) ^ n ;

9 X = ( 1/ z ) * s ym su m ( x *( z ^( - n ) ) ,n ,0 , % in f ) ;

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (X , Zt r an s fo r m o f ( 1 / 3) n u ( n1) i s : ) ;

Scilab code Exa 2.10 z Transform Computation

1 / / E x am pl e 2 . 1 02 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /Z t r a n s f o r m o f r n . c o s (Won )4 clc ;

5 syms r Wo n z ;

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6 x 1 = ( r ^ n ) * exp ( sqrt ( - 1 ) * W o * n ) ;

7 X 1 = s y m s u m ( x 1 * ( z ^ - n ) , n , 0 , % i n f ) ;8 x 2 = ( r ^ n ) * exp ( - sqrt ( - 1 ) * W o * n ) ;

9 X 2 = s y m s u m ( x 2 * ( z ^ - n ) , n , 0 , % i n f ) ;

10 X = ( X 1 + X 2 ) / 2 ;

11 disp (X ,X( z )= ) ;


1 / / E x am pl e 2 . 1 12 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r a n s f o r m o f n . a n u ( n )4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms a n z ;

8 x =( a) ^ n;

9 X = s ym su m ( x *( z ^( - n )) , n ,0 , % in f )

10 Y = diff ( X , z ) ;


12 disp (Y , Zt r an s fo r m o f n . a n 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 PROGRAM3 //Z t ra n sf or m o f ( 1 / 5) n u ( n ) + 5 ( 1 / 2 ) (n ) u (n1)4 c l e ar a ll ;

5 clc ;6 close ;

7 syms n z ;

8 x 1 = ( -1 /5 ) ^n ;

9 X 1= s ym su m ( x1 * (z ^( - n )) , n ,0 , % in f ) ;

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10 x 2 = ( 1 /2 ) ^ ( - n ) ;

11 X 2 = s ym su m ( 5* x 2 *( z ^( - n )) , n ,0 , % in f ) ;12 X = ( X1 - X2 );

13 / / D i s p l a y t h e r e s u l t i n command window14 disp (X , Zt r a ns f o r m o f [ 3 ( 3 n )4(2) n ] u ( n ) i s : ) ;15 disp (ROC i s t h e R eg i on 1 /5 < mod( z ) < 2 ) ;

Scilab code Exa 2.13.b z Transform of Discrete Time Signals

1 / / Exam ple 2 . 1 3 ( b )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /Z t r a n s fo r m4 clc ;

5 syms n z k ;

6 x 1 = 1 ;

7 X 1 = s y m s u m ( x 1 * z ^ ( - n ) , n , 0 , 0 ) ;

8 x 2 = 1 ;

9 X 2 = s y m s u m ( x 2 * z ^ ( - n ) , n , 1 , 1 ) ;

10 x 3 = 1 ;

11 X 3 = s y m s u m ( x 3 * z ^ ( - n ) , n , 2 , 2 ) ;

12 X = 0 . 5 * X 1 + X 2 - 1 / 3 * X 3 ;13 disp (X ,X( z )= ) ;

Scilab code Exa 2.13.c z Transform of Discrete Time Signals

1 / / E xa mp le 2 . 1 3 ( c )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z tr a ns f o r m o f u ( n2)


6 close ;

7 syms n z ;

8 x =1;

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9 X = ( 1/ ( z ^ 2) ) * s y ms um ( x *( z ^( - n ) ) ,n ,0 , % in f ) ;

10 / / D i s p l a y t h e r e s u l t i n command window11 disp (X , Zt r a ns f o r m o f u ( n2) i s : ) ;

Scilab code Exa 2.13.d z Transform of Discrete Time Signals

1 / / Exam ple 2 . 1 3 ( d )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r a n s fo r m o f ( n + 0. 5 ) ( ( 1 / 3 ) n ) u ( n )


6 close ;

7 syms n z ;

8 x 1 = ( 1/ 3) ^ n ;

9 X 1 1 = s ym s um ( x 1 *( z ^( - n ) ) ,n ,0 , % in f )

10 X1 = diff ( X 1 1 , z ) ;

11 x 2 = ( 1/ 3) ^ ( n) ;

12 X 2 = s ym su m ( 0. 5* x 2 * (z ^( - n )) , n ,0 , % in f ) ;

13 X = ( X1 + X2 );


15 disp (X , Zt r an s fo r m o f ( n +0 .5 ) ( ( 1 / 3 ) n )u ( n ) i s : ) ;

Scilab code Exa 2.16 Impulse Response of the System

1 / / E x am pl e 2 . 1 62 / /To f i n d i n pu t h ( n )3 / / a = [1 2 4 1 ] , b = [1 ]4 c l e ar a ll ;

5 clc ;6 close ;

7 z = % z ;

8 a = z ^ 3 + 2 * ( z ^ ( 2 ) ) - 4 * ( z ) + 1 ;

9 b = z ^ 3 ;

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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 / / E x am pl e 2 . 1 72 / / To dra w t h e p o l ez er o p l ot3 c l e ar a ll ;

4 clc ;

5 close ;

6 z = % z

7 H 1 Z = ( ( z ) * ( z - 1 ) ) / ( ( z - 0 . 2 5 ) * ( z - 0 . 5 ) ) ;

8 xset ( window ,1);9 plzr ( H 1 Z ) ;

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Figure 2.2: Frequency Response of the System

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Scilab code Exa 2.19 Frequency Response of the System

1 / / E x am pl e 2 . 1 92 / / P ro gr am t o P l o t M ag n it ud e a nd P ha s e R e sp o n ce3 c l e ar a ll ;

4 clc ;

5 close ;

6 w = - % p i : 0 . 0 1 : % p i ;

7 H = 1 / ( 1 - 0 . 5 * ( cos ( w ) - % i * sin ( w ) ) ) ;


10 Hm = abs ( H ) ;

11 a = gca () ;12 subplot ( 2 , 1 , 1 ) ;


15 x l a b e l ( F r e qu e n cy i n R a d ia n s ) ;16 y l a b e l ( abs (Hm) ) ;17 t i t l e ( MAGNITUDE RESPONSE ) ;18 subplot ( 2 , 1 , 2 ) ;

19 a = gca () ;

20 a . x _ l o c a t i o n = o r i g i n ;21 a . y _ l o c a t i o n =

o r i g i n ;

22 plot2d ( w / ( 2 * % p i ) , p h a s e _ H ) ;



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6 close ;

7 z = % z ;8 a = ( z + 0 . 5 ) * ( z - 1 ) ;

9 b = z + 0 . 2 ;

10 h = ldiv ( b , a , 4 ) ;

11 disp (h , h( n) = ) ;

Scilab code Exa 2.22 Inverse z Transform Computation

1 / / E x am pl e 2 . 2 22 / /To f i n d i n pu t x ( n )3 //X( z ) =1/(2 z ( 2)+2z ( 1) +1) ;4 c l e ar a ll ;

5 clc ;

6 close ;

7 z = % z ;

8 a = ( 2 + 2 * z + z ^ 2 ) ;

9 b = z ^ 2 ;

10 h = ldiv ( b , a , 6 ) ;

11 disp (h , F i r s t s i x v a l ue s o f h ( n )= ) ;

Scilab code Exa 2.23 Causal Sequence Determination

1 / / E x am pl e 2 . 2 32 / /To f i n d i n pu t x ( n )3 //X( z ) =1/(12 z ( 1 ) ) ( 1z (1 ) ) 2 ;4 c l e ar a ll ;

5 clc ;

6 close ;7 z = % z ;

8 a = ( z - 2 ) * ( z - 1 ) ^ 2 ;

9 b = z ^ 3 ;

10 h = ldiv ( b , a , 6 ) ;

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Figure 2.3: Impulse Response of the System

11 disp (h , F i r s t s i x v a l ue s o f h ( n )= ) ;


1 / / E x am pl e 2 . 3 42 //To p l o t t he i m pu l se r e s po n c e o f t he s ys te m

a n a l y i c a l l y and u si ng s c i l a b3 c l e ar a ll ;4 clc ;

5 close ;

6 n = 0 : 1 : 5 0 ;

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7 x = [ 1 , zeros ( 1 , 5 0 ) ] ;

8 b = [ 1 2 ];9 a = [1 -3 -4] ;

10 y a n a l y = 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 n11 y m a t = filter ( b , a , x ) ;

12 subplot ( 3 , 1 , 1 ) ;

13 plot2d3 ( n , x ) ;

14 x l a b e l ( n ) ;15 y l a b e l ( x ( n ) ) ;16 t i t l e ( INPUT SEQUENCE (IMPULSE FUNCTION) ) ;17 subplot ( 3 , 1 , 2 ) ;

18 plot2d3 ( n , y a n a l y ) ;

19 x l a b e l ( n ) ;20 y l a b e l ( y ( n ) ) ;21 t i t l e (OUTPUT SEQUENCE y a n a l y ) ;22 subplot ( 3 , 1 , 3 ) ;

23 plot2d3 ( n , y m a t ) ;

24 x l a b e l ( n ) ;25 y l a b e l ( y ( n ) ) ;26 t i t l e (OUTPUT SEQUENCE ymat ) ;27 / /As t he A n a l t i c a l P lo t ma tc he s t he S c i l a b P lo t

h en ce i t i s t he R es pon ce o f t he s ys te m

Scilab code Exa 2.35.a Pole Zero Plot of the System

1 / / Example 2 . 3 5 ( a )2 / / To dra w t h e p o l ez er o p l ot3 c l e ar a ll ;

4 clc ;5 close ;

6 z = % z

7 H 1 Z = ( z ) / ( z ^ 2 - z - 1 ) ;

8 xset ( window ,1);

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Figure 2.4: Pole Zero Plot of the System

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Figure 2.5: Unit Sample Response of the System

9 plzr ( H 1 Z ) ;

Scilab code Exa 2.35.b Unit Sample Response of the System

1 / / Exam ple 2 . 3 5 ( b )2 //To p l o t t he r e s p on c e o f t he s ys te m a n a l y i c a l l y and

u si ng s c i l a b3 c l e ar a ll ;4 clc ;

5 close ;

6 n = 0 : 1 : 2 0 ;

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7 x = ones (1 , length ( n ) ) ;

8 b = [ 0 1 ];9 a = [1 -1 -1] ;

10 y a n a l y = 0 . 4 4 7 * ( 1 . 6 1 8 ) . ^ n - 0 . 4 4 7 * ( - 0 . 6 1 8 ) . ^ n ; //A n a l y t i c a l S o l u t i o n

11 [ y m a t , z f ] = filter ( b , a , x ) ;

12 subplot ( 3 , 1 , 1 ) ;

13 plot2d3 ( n , x ) ;

14 x l a b e l ( n ) ;15 y l a b e l ( x ( n ) ) ;16 t i t l e ( INPUT SEQUENCE (STEP FUNCTION) ) ;17 subplot ( 3 , 1 , 2 ) ;

18 plot2d3 ( n , y a n a l y ) ;19 x l a b e l ( n ) ;20 y l a b e l ( y ( n ) ) ;21 t i t l e (OUTPUT SEQUENCE y a n a l y ) ;22 subplot ( 3 , 1 , 3 ) ;

23 plot2d3 ( n , y m a t , z f ) ;



Scilab code Exa 2.38 Determine Output Response

1 / / E x am pl e 2 . 3 82 //To p l o t t he r e s p on c e o f t he s ys te m a n a l y i c a l l y and

u si ng s c i l a b3 c l e ar a ll ;4 clc ;

5 close ;

6 n = 0 : 1 : 2 0 ;

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Figure 2.6: Determine Output Response

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7 x = n ;

8 b = [0 1 1];9 a = [ 1 - 0. 7 0 .1 2 ];

10 y a n a l y = 3 8 . 8 9 * ( 0 . 4 ) . ^ n - 2 6 . 5 3 * ( 0 . 3 ) . ^ n - 1 2 . 3 6 + 4 . 7 6 * n ; //A n a l y t i c a l S o l u t i o n

11 y m a t = filter ( b , a , x ) ;

12 subplot ( 3 , 1 , 1 ) ;

13 plot2d3 ( n , x ) ;

14 x l a b e l ( n ) ;15 y l a b e l ( x ( n ) ) ;16 t i t l e ( INPUT SEQUENCE (RAMP FUNCTION) ) ;17 subplot ( 3 , 1 , 2 ) ;

18 plot2d3 ( n , y a n a l y ) ;19 x l a b e l ( n ) ;20 y l a b e l ( y ( n ) ) ;21 t i t l e (OUTPUT SEQUENCE y a n a l y ) ;22 subplot ( 3 , 1 , 3 ) ;

23 plot2d3 ( n , y m a t ) ;



Scilab code Exa 2.40 Input Sequence Computation

1 / / E x am pl e 2 . 4 02 / /To f i n d i n pu t x ( n )3 // h ( n) =1 2 3 2 , y ( n) =[1 3 7 10 10 7 2 ]4 c l e ar a ll ;

5 clc ;6 close ;

7 z = % z ;

8 a = z ^ 6 + 3 * ( z ^ ( 5 ) ) + 7 * ( z ^ ( 4 ) ) + 1 0 * ( z ^ ( 3 ) ) + 1 0 * ( z ^ ( 2 ) ) + 7 * ( z

^ ( 1 ) ) + 2 ;

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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 / / Example 2 . 4 1 ( a )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z tr a ns f o r m o f n .(1 ) n u ( n )


6 close ;

7 syms a n z ;

8 x =( -1) ^ n;

9 X = s ym su m ( x *( z ^( - n )) , n ,0 , % in f )

10 Y = diff ( X , z ) ;

11 / / D i s p l a y t h e r e s u l t i n command window12 disp (Y , Zt r a ns f o r m o f n .(1 ) n u ( n ) i s : ) ;

Scilab code Exa 2.41.b z Transform of the Signal

1 / / Exam ple 2 . 4 1 ( b )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r a ns f o r m o f n 2 u ( n )4 c l e ar a ll ;

5 clc ;

6 close ;

7 syms n z ;

8 x =1;9 X = s ym su m ( x *( z ^( - n )) , n ,0 , % in f )

10 Y = diff ( diff ( X , z ) , z ) ;

11 / / D i s p l a y t h e r e s u l t i n command window12 disp (Y , Zt ra ns fo rm o f n 2 u ( n) i s : ) ;

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Figure 2.7: Pole Zero Pattern of the System

Scilab code Exa 2.41.c z Transform of the Signal

1 / / E xa mp le 2 . 4 1 ( c )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM

3 / /Z t r a ns f o r m o f (1)n . co s (%pi/3n )4 clc ;

5 syms n z ;

6 W o = % p i / 3 ;

7 x1 = exp ( sqrt ( - 1 ) * W o * n ) ;

8 X 1 = ( - 1 ) ^ n * s y m s u m ( x 1 * ( z ^ - n ) , n , 0 , % i n f ) ;

9 x2 = exp ( - sqrt ( - 1 ) * W o * n ) ;

10 X 2 = ( - 1 ) ^ n * s y m s u m ( x 2 * ( z ^ - n ) , n , 0 , % i n f ) ;

11 X = ( X 1 + X 2 ) / 2 ;

12 disp (X ,X( z )= ) ;

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Scilab code Exa 2.45 Pole Zero Pattern of the System

1 / / E x am pl e 2 . 4 52 / / To dra w t h e p o l ez er o p l ot3 c l e ar a ll ;

4 clc ;

5 close ;

6 z = % z

7 H 1 Z = ( ( z ) * ( z + 1 ) ) / ( z ^ 2 - z + 0 . 5 ) ;

8 xset ( window ,1);9 plzr ( H 1 Z ) ;

Scilab code Exa 2.53.a z Transform of the Sequence

1 / / Example 2 . 5 3 ( a )2 //Z t r a ns f o r m o f [ 3 1 2 5 7 0 1 ]3 c l e ar a ll ;

4 clc ;

5 close ;




10 x1 =[3 1 2 5 7 0 1];

11 n = - 3 : 3 ;

12 z z = z t r a n s f e r ( x 1 , n ) ;

13 / / D i s p l a y t h e r e s u l t i n command window14 disp ( z z , Zt r an s f or m o f s e qu e nc e i s : ) ;

Scilab code Exa 2.53.b z Transform of the Signal

1 / / Exam ple 2 . 5 3 ( b )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM

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3 //Z t r an s fo r m o f d e l t a ( n )

4 clc ;5 syms n z ;

6 x = 1 ;

7 X = s y m s u m ( x * z ^ ( - n ) , n , 0 , 0 ) ;

8 disp (X ,X( z )= ) ;

Scilab code Exa 2.53.c z Transform of the Signal

1 / / E xa mp le 2 . 5 3 ( c )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r an s fo r m o f d e l t a ( n )4 clc ;

5 syms n z k ;

6 x = 1 ;

7 X = s y m s u m ( x * z ^ ( - n ) , n , k , k ) ;

8 disp (X ,X( z )= ) ;

Scilab code Exa 2.53.d z Transform of the Signal

1 / / Exam ple 2 . 5 3 ( d )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 //Z t r an s fo r m o f d e l t a ( n )4 clc ;

5 syms n z kc ;

6 x = 1 ;

7 X = s y m s u m ( x * z ^ ( - n ) , n , - k , - k ) ;

8 disp (X ,X( z )= ) ;

Scilab code Exa 2.54 z Transform of Cosine Signal

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Figure 2.8: Impulse Response of the System

1 / / E x am pl e 2 . 5 42 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /Z t r a n s f o r m o f c o s (Won )4 clc ;

5 syms Wo n z ;

6 x1 = exp ( sqrt ( - 1 ) * W o * n ) ;

7 X 1 = s y m s u m ( x 1 * ( z ^ - n ) , n , 0 , % i n f ) ;

8 x2 = exp ( - sqrt ( - 1 ) * W o * n ) ;

9 X 2 = s y m s u m ( x 2 * ( z ^ - n ) , n , 0 , % i n f ) ;

10 X = ( X 1 + X 2 ) / 2 ;

11 disp (X ,X( z )= ) ;

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1 / / E x am pl e 2 . 5 82 //To p l o t t he r e s p on c e o f t he s ys te m a n a l y i c a l l y and

u si ng s c i l a b3 c l e ar a ll ;

4 clc ;

5 close ;

6 n = 0 : 1 : 2 0 ;

7 x = [ 1 zeros ( 1 , 2 0 ) ] ;

8 b = [ 1 - 0. 5 ];

9 a = [ 1 -1 3 / 16 ];

10 y a n a l y = 0 . 5 * ( 0 . 7 5 ) . ^ n + 0 . 5 * ( 0 . 2 5 ) . ^ n ; / / A n a l y t i c a lS o l u t i o n

11 y m a t = filter ( b , a , x ) ;

12 subplot ( 3 , 1 , 1 ) ;

13 plot2d3 ( n , x ) ;

14 x l a b e l ( n ) ;15 y l a b e l ( x ( n ) ) ;16 t i t l e ( INPUT SEQUENCE (IMPULSE FUNCTION) ) ;17 subplot ( 3 , 1 , 2 ) ;

18 plot2d3 ( n , y a n a l y ) ;

19 x l a b e l ( n

) ;

20 y l a b e l ( y ( n ) ) ;21 t i t l e (OUTPUT SEQUENCE y a n a l y ) ;22 subplot ( 3 , 1 , 3 ) ;

23 plot2d3 ( n , y m a t ) ;



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Figure 3.1: DFT of the Sequence

Scilab code Exa 3.2 DFT of the Sequence

1 / / E xa mp le 3 . 22 / / P r og ra m t o Com pu te t h e DFT o f a S e q u e n c e x [ n ] = 1 ,

0


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17 disp ( X 2 , X2 [ k]= ) ;

18 / / P l o t s f o r N=419 n 1 = 0 : 1 : 3 ;20 subplot ( 2 , 2 , 1 ) ;

21 a = gca () ;

22 a . y _ l o c at i o n = o r i g i n ;23 a . x _ l o c at i o n = o r i g i n ;24 plot2d3 ( n 1 , abs ( X 1 ) , 2 ) ;

25 p o ly 1 = a . c h i l dr e n ( 1 ) . c h i ld r e n ( 1) ;

26 p o l y 1 . t h i c k n e s s = 2 ;

27 xtitle (N=4 , k , | X 1 ( k )| ) ;28 subplot ( 2 , 2 , 2 ) ;

29 a = gca () ;30 a . y _ l o c at i o n = o r i g i n ;31 a . x _ l o c at i o n = o r i g i n ;32 plot2d3 ( n 1 , atan ( imag ( X 1 ) , real ( X 1 ) ) , 5 ) ;


34 p o l y 1 . t h i c k n e s s = 2 ;

35 xtitle (N=4 , k ,


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Scilab code Exa 3.3 8 Point DFT

1 / / E xa mp le 3 . 32 / / P r o gr am t o Com pu te t h e 8p o i n t DFT o f t h e S e qu e nc e

x [ n ] = [ 1 , 1 , 1 , 1 , 1 , 1 , 0 , 0 ]3 c l e ar a ll ;

4 clc ;

5 close ;

6 x = [ 1 ,1 , 1 , 1 ,1 , 1 , 0 ,0 ];

7 //DFT Compu tatio n8 X = fft ( x , -1) ;

9 / / D i s p l a y s e q u e n c e X [ k ] i n command w in do w10 disp (X , X[ k]= ) ;

Scilab code Exa 3.4 IDFT of the given Sequence

1 / / E xa mp le 3 . 42 / / P ro gr am t o Compute t h e IDFT o f t h e S e q u en c e X [ k]=[ 5 ,0 ,1j , 0 , 1 , 0 , 1 + j , 0 ]

3 c l e ar a ll ;

4 clc ;

5 close ;

6 j = sqrt ( - 1 ) ;

7 X = [ 5 ,0 , 1 - j ,0 , 1 , 0 ,1 +j , 0]

8 //IDFT Comput ation9 x = fft ( X , 1 ) ;

10 // Di sp l a y se q u e n c e s x [ n ] i n command w indow

11 disp (x , x [ n]= ) ;

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Figure 3.2: Plot the Sequence

Scilab code Exa 3.7 Plot the Sequence

1 / / E xa mp le 3 . 72 / / Program t o C ompute c i r c u l a r c o n v o l u t i o n o f

f o l l o w i n g s e qu e nc e s3 / / x [ n ] = [ 1 , 2 , 2 , 1 , 0 ]4 //Y[ k]= exp(j4p i k /5 ) .X[ k ]5 c l e ar a ll ;

6 clc ;

7 close ;

8 x = [ 1 , 2 , 2 , 1 , 0 ] ;

9 X = fft ( x , - 1 ) ;

10 k = 0 : 1 : 4 ;

11 j = sqrt ( - 1 ) ;

12 p i = 2 2 / 7 ;

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13 H = exp ( - j * 4 * p i * k / 5 ) ;

14 Y = H . * X ;15 //IDFT Comput ation16 y = fft ( Y , 1 ) ;

17 / / D i s p l a y s e q u e n c e y [ n ] i n command w in do w18 disp ( round ( y ) , y [ n]= ) ;19 / / P l o t s20 n = 0 : 1 : 4 ;

21 a = gca () ;

22 a . y _ l o c at i o n = o r i g i n ;23 a . x _ l o c at i o n = o r i g i n ;24 plot2d3 (n , round ( y ) , 5 ) ;

25 p o ly 1 = a . c h i l dr e n ( 1 ) . c h i ld r e n ( 1) ;26 p o l y 1 . t h i c k n e s s = 2 ;

27 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 / / E xa mp le 3 . 92 / / Program t o r e m ai n i n g s a mp l es o f t h e s e q ue n c e

3 //X(0 ) =12,X(1)=1+j3 , X( 2 )=3+j4 , X( 3 )=1j 5 , X( 4 )=2+j2 ,X(5 )=6+j3 ,X( 6) =2j 3 , X( 7 ) =10

4 c l e ar a ll ;

5 clc ;

6 close ;

7 j = sqrt ( - 1 ) ;

8 z = 1 ;

9 X ( 0 + z ) = 1 2 , 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;

10 for a = 9 :1 : 14 d o X ( a )= conj ( X ( 1 6 - a ) ) , end ;11 / / 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 commandwindow

12 disp (X , X[ k]= ) ;

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Scilab code Exa 3.11 DFT Computation

1 / / E x am pl e 3 . 1 12 / / P r o gr am t o Com pu te t h e 8p o i n t DFT o f t h e

f o l l o w i n g s e qu e nc e s3 / / x 1 [ n ] = [ 1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 ]4 / / x 2 [ n ] = [ 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 ]5 c l e ar a ll ;

6 clc ;

7 close ;

8 x 1 = [ 1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 ] ;

9 x 2 = [ 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 ] ;

10 //DFT Compu tatio n11 X1 = fft ( x1 , -1) ;

12 X2 = fft ( x2 , -1) ;

13 / / D i s p l a y s e q u e n c e s X1 [ k ] a nd X2 [ k ] i n commandwindow

14 disp ( X 1 , X1 [ k]= ) ;15 disp ( X 2 , X2 [ k]= ) ;

Scilab code Exa 3.13 Circular Convolution

1 / / E x am pl e 3 . 1 32 / / Program t o C ompute c i r c u l a r c o n v o l u t i o n o f

f o l l o w i n g s e qu e nc e s3 //x 1 [ n] = [ 1 , 1 , 2 , 3 , 1]4 / / x 2 [ n ] = [ 1 , 2 , 3 ]


7 close ;

8 x 1 = [ 1 , - 1 , - 2 , 3 , - 1 ] ;

9 x 2 = [ 1 , 2 , 3 ] ;

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10 / /Loop f o r z e r o p add in g t he s m a l l er s eq ue nc e o ut o f

t h e two11 n1 = length ( x 1 ) ;12 n2 = length ( x 2 ) ;

13 n 3 = n 2 - n 1 ;

14 if ( n 3 > = 0 ) then

15 x 1 = [ x 1 , zeros ( 1 , n 3 ) ] ;

16 else

17 x 2 = [ x 2 , zeros ( 1 , - n 3 ) ] ;

18 end

19 //DFT Compu tatio n20 X1 = fft ( x 1 , - 1 ) ;

21 X2 = fft ( x 2 , - 1 ) ;22 Y = X 1 . * X 2 ;

23 //IDFT Comput ation24 y = fft ( Y , 1 ) ;

25 / / D i s p l a y s e q u e n c e y [ n ] i n command w in do w26 disp (y , y [ n]= ) ;


1 / / E x am pl e 3 . 1 42 / / Program t o C ompute c i r c u l a r c o n v o l u t i o n o f

f o l l o w i n g s e qu e nc e s3 / / x 1 [ n ] = [ 1 , 2 , 2 , 1 ]4 / / x 2 [ n ] = [ 1 , 2 , 3 , 1 ]5 c l e ar a ll ;

6 clc ;

7 close ;

8 x 1 = [ 1 , 2 , 2 , 1 ] ;

9 x 2 = [ 1 , 2 , 3 , 1 ] ;10 //DFT Compu tatio n11 X1 = fft ( x 1 , - 1 ) ;

12 X2 = fft ( x 2 , - 1 ) ;

13 Y = X 1 . * X 2 ;

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6 clc ;

7 close ;8 x 1 = [ 1 , 1 , 2 , 1 ] ;

9 x 2 = [ 1 , 2 , 3 , 4 ] ;


12 X2 = fft ( x 2 , - 1 ) ;

13 X 3 = X 1 . * X 2 ;

14 //IDFT Comput ation15 x3 = fft ( X 3 , 1 ) ;

16 / / D i s p l a y s e q u e n c e x 3 [ n ] i n command wi nd ow17 disp ( x 3 , x3 [ n]= ) ;


1 / / E x am pl e 3 . 1 72 // Program to Compute y [ n ] where Y[ k]=X1 [ k ] . X2 [ k ]3 / / x 1 [ n ] = [ 0 , 1 , 2 , 3 , 4 ]4 / / x 2 [ n ] = [ 0 , 1 , 0 , 0 , 0 ]5 c l e ar a ll ;

6 clc ;7 close ;

8 x 1 = [ 0 , 1 , 2 , 3 , 4 ] ;

9 x 2 = [ 0 , 1 , 0 , 0 , 0 ] ;


12 X2 = fft ( x 2 , - 1 ) ;

13 Y = X 1 . * X 2 ;

14 //IDFT Comput ation15 y = round ( fft ( Y , 1 ) ) ;


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Scilab code Exa 3.18 Output Response

1 / / E x am pl e 3 . 1 82 / / Program t o Compute o u tp ut r e s p o n c e o f f o l l o w i n g

s e q u e n c e s3 / / x [ n ] = [ 1 , 2 , 3 , 1 ]4 / / h [ n ] = [ 1 , 1 , 1 ]5 / / ( 1 ) L i n e a r C o n v o l u t i o n6 / / ( 2 ) C i r c u l a r C o n v o l u ti o n7 / / ( 3 ) C i r c u l a r C o n vo l u ti o n w it h z e r o p ad di ng8 c l e ar a ll ;

9 clc ;

10 close ;11 x = [ 1 , 2 , 3 , 1 ] ;

12 h = [ 1 , 1 , 1 ] ;

13 / / ( 1 ) L i n e a r C o n v o l u t i o n C o mp ut a ti on14 y l i n e a r = convol ( x , h ) ;

15 / / D i s p l a y L i n e a r C o n v ol u t e d S e q u en c e y [ n ] i n commandwindow

16 disp (yli near , y l i n e a r [ n ]= ) ;17 / / ( 2 ) C i r c u l a r C o n v o l u ti o n C om pu ta ti on18 / /Now z e r o p a dd i ng 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 ] a nd h [ n ] e q u a l19 h 1 = [ h , zeros ( 1 , 1 ) ] ;20 / / 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 m et ho d21 X = fft ( x , - 1 ) ;

22 H = fft ( h 1 , - 1 ) ;

23 Y = X . * H ;

24 y c i r c u l a r = fft ( Y , 1 ) ;

25 / / D i s p l a y C i r c u l a r C o nv o lu t ed S e qu e nc e y [ n ] i ncommand window

26 disp (ycircular , y c i r c u l a r [ n ]= ) ;27 / / ( 3 ) C i r c u l a r C o n v o l u ti o n C om pu ta ti on w i th z e r o

Padding28 x 2 = [ x , zeros ( 1 , 2 ) ] ;

29 h 2 = [ h , zeros ( 1 , 3 ) ] ;

30 / / 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 m et ho d31 X2 = fft ( x 2 , - 1 ) ;

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32 H2 = fft ( h 2 , - 1 ) ;

33 Y 2 = X 2 . * H 2 ;34 y c i r c u l a r p = fft ( Y 2 , 1 ) ;

35 / / D i s p l ay C i r c u l a r C on vo lu te d S eq ue nc e w it h z e r oPadding y [ n ] in command window

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

Scilab code Exa 3.20 Output Response

1 / / E x am pl e 3 . 2 02 / / Program t o Compute L i n e a r C o nv o l ut i o n o f f o l l o w i n gs e q u e n c e s

3 //x [ n ] = [ 3 , 1 , 0 , 1 , 3 , 2 , 0 , 1 , 2 , 1 ]4 / / h [ n ] = [ 1 , 1 , 1 ]5 c l e ar a ll ;

6 clc ;

7 close ;

8 x = [ 3 , - 1 , 0 , 1 , 3 , 2 , 0 , 1 , 2 , 1 ] ;

9 h = [ 1 , 1 , 1 ] ;

10 / / L i n e a r C o n v o l ut i o n C om pu ta ti on

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 w in do w13 disp (y , y [ n]= ) ;

Scilab code Exa 3.21 Linear Convolution

1 / / E x am pl e 3 . 2 12 / / Program t o Compute L i n e a r C o nv o l ut i o n o f f o l l o w i n g

s e q u e n c e s3 // x [ n ]=[ 1 ,2 ,1 ,2 ,3 ,2 ,3 ,1 ,1 ,1 ,2 ,1 ]4 / / h [ n ] = [ 1 , 2 ]5 c l e ar a ll ;

6 clc ;

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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 ut i o n C om pu ta ti on11 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 w in do w13 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 PROGRAM3 / /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 = s y m s u m ( x * exp ( - % i * 2 * % p i * n * k / N ) , n , 0 , 0 ) ;

8 disp (X , X( k ) = ) ;

Scilab code Exa 3.23.b N Point DFT Computation

1 / / Exam ple 3 . 2 3 ( b )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /N p o i n t DFT o f d e l t a ( nno )4 clc ;

5 syms n k N no ;

6 x = 1 ;

7 X = s y m s u m ( x * exp ( - % i * 2 * % p i * n * k / N ) , n , - n o , - n o ) ;

8 disp (X , X( k ) = ) ;

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Scilab code Exa 3.23.c N Point DFT Computation

1 / / E xa mp le 3 . 2 3 ( c )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /N p o i n t DFT o f a n4 clc ;

5 syms a n k N ;

6 x = a ^ n ;

7 X = s y m s u m ( x * exp ( - % i * 2 * % p i * n * k / N ) , n , 0 , N - 1 ) ;

8 disp (X , X( k ) = ) ;

Scilab code Exa 3.23.d N Point DFT Computation

1 / / Exam ple 3 . 2 3 ( d )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /N p o i n t DFT o f x ( n ) =1 , 0


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Scilab code Exa 3.23.f N Point DFT Computation

1 / / E xa mp le 3 . 2 3 ( f )2 //MAXIMA SCILAB TOOLBOX REQUIRED FOR THIS PROGRAM3 / /N p o i n t DFT o f x ( n ) =1 , f o r n=e ve n and 0 , f o r n=odd4 clc ;

5 syms n k N ;

6 x = 1 ; // x (2 n ) =1 , f o r a l l n7 X = s y m s u m ( x * exp ( - % i * 4 * % p i * n * k / N ) , n , 0 , N / 2 - 1 ) ;

8 disp (X , X( k ) = ) ;

Scilab code Exa 3.24 DFT of the Sequence

1 / / E x am pl e 3 . 2 42 / / P ro gr am t o Compute t h e DFT o f t h e S e q u e nc e x [ n

]=(1 ) n , f o r N=4


5 close ;

6 N = 4 ;

7 n = 0 : 1 : N - 1 ;

8 x = ( - 1 ) ^ n ;

9 //DFT Compu tatio n10 X = fft ( x , - 1 ) ;

11 / / D i s p l a y S e q u e n c e X [ k ] i n command w in do w12 disp (X , X[ k]= ) ;

Scilab code Exa 3.25 8 Point Circular Convolution

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1 / / E x am pl e 3 . 2 5

2 / / P r o gr am t o Com pu te t h e 8p o in t C i r c ul a rC o nv o l ut i on o f t h e S e qu e nc e s3 / / x 1 [ n ] = [ 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 ]4 // x2 [ n]= s i n (3 p i n / 8 )5 c l e ar a ll ;

6 clc ;

7 close ;

8 x 1 = [ 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 ] ;

9 n = 0 : 1 : 7 ;

10 p i = 2 2 / 7 ;

11 x2 = sin ( 3 * p i * n / 8 ) ;


14 X2 = fft ( x 2 , - 1 ) ;

15 / / C i r c u l a r C o n v o l ut i o n u s i n g DFT16 Y = X 1 . * X 2 ;

17 //IDFT Comput ation18 y = fft ( Y , 1 ) ;


Scilab code Exa 3.26 Linear Convolution using DFT

1 / / E x am pl e 3 . 2 62 / / Pr ogr am t o Compute t h e L i n e a r C o n v o l ut i o n o f t h e

f o l l o w i n g S eq ue nc es3 //x [ n ]= [1 ,1 , 1]4 / / h [ n ] = [ 2 , 2 , 1 ]5 c l e ar a ll ;

6 clc ;7 close ;

8 x = [ 1 , - 1 , 1 ] ;

9 h = [ 2 , 2 , 1 ] ;

10 / / C o n v o l u t i o n C o m pu t a ti o n

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11 y = convol ( x , h ) ;


Scilab code Exa 3.27.a Circular Convolution Computation

1 / / Example 3 . 2 7 ( a )2 / / P ro gr am t o Compute t h e C o n v o l ut i o n o f t h e

f o l l o w i n g S eq ue nc es

3 / / x 1 [ n ] = [ 1 , 1 , 1 ]4 // x2 [ n ]= [2 ,1 ,2 ]5 c l e ar a ll ;

6 clc ;

7 close ;

8 x 1 = [ 1 , 1 , 1 ] ;

9 x 2 = [ 2 , - 1 , 2 ] ;

10 / / C o n v o l u t i o n C o m pu t a ti o n11 X1 = fft ( x 1 , - 1 ) ;

12 X2 = fft ( x 2 , - 1 ) ;

13 Y = X 1 . * X 2 ;

14 y = fft ( Y , 1 ) ;15 / / D i s p l a y S e q u e n c e y [ n ] i n command w in do w16 disp (y , y [ n]= ) ;

Scilab code Exa 3.27.b Circular Convolution Computation

1 / / Exam ple 3 . 2 7 ( b )2 / / P ro gr am t o Compute t h e C o n v o l ut i o n o f t h e

f o l l o w i n g S eq ue nc es3 // x1 [ n ]= [1 ,1 ,1 ,1 ,0 ]4 // x2 [ n ]= [1 ,0 , 1 , 0 , 1 ]5 c l e ar a ll ;

6 clc ;

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7 close ;

8 x 1 = [ 1 , 1 , - 1 , - 1 , 0 ] ;9 x 2 = [ 1 , 0 , - 1 , 0 , 1 ] ;

10 / / C o n v o l u t i o n C o m pu t a ti o n11 X1 = fft ( x 1 , - 1 ) ;

12 X2 = fft ( x 2 , - 1 ) ;

13 Y = X 1 . * X 2 ;

14 y = fft ( Y , 1 ) ;

15 / / D i s p l a y S e q u e n c e y [ n ] i n command w in do w16 disp (y , y [ n]= ) ;

Scilab code Exa 3.30 Calculate value of N

1 / / E x am pl e 3 . 3 02 / / Program t o C a l c u l a t e N fr om g i v e n d at a3 //fm=5000Hz4 // d f =50Hz5 // t = 0. 5 se c6 c l e ar a ll ;

7 clc ;

8 close ;9 f m = 5 0 0 0 //Hz

10 d f = 5 0 //Hz11 t = 0 . 5 // se c12 N 1 = 2 * f m / d f ;

13 N = 2 ;

14 while N < = N1 , N = N *2 , end

15 / / D i s p l a y i n g t h e v a l u e o f N i n command window16 disp (N , N= ) ;

Scilab code Exa 3.32 Sketch Sequence

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Figure 3.3: Sketch Sequence

1 / / E x am pl e 3 . 3 22 // Program t o p l o t t he r e s u l t o f t he g i ve n s eq ue nc e3 //X [ k ] = [ 1 , 2 , 2 , 1 , 0 , 2 , 1 , 2 ]4 / / y [ n ] = x [ n / 2 ] f o r n=e v en , 0 f o r n=o dd5 c l e ar a ll ;

6 clc ;

7 close ;

8 X = [ 1 , 2 , 2 , 1 , 0 , 2 , 1 , 2 ] ;

9 x = fft ( X , 1 ) ;10 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];

11 Y = fft ( y , -1) ;

12 / / D i s p l a y s e q u e n c e Y [ k ] a nd i n command w in do w13 disp (Y , Y[ k]= ) ;14 / / P l o t t i n g t h e s e q ue n c e Y [ k ]15 k = 0 : 1 : 1 5 ;

16 a = gca () ;

17 a . y _ l o c at i o n = o r i g i n ;18 a . x _ l o c at i o n = o r i g i n ;19 plot2d3 ( k , Y , 2 ) ;


21 p o l y 1 . t h i c k n e s s = 2 ;

22 xtitle ( P l o t o f Y( k ) , k ,Y ( k ) ) ;

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Scilab code Exa 3.36 Determine IDFT

1 / / E x am pl e 3 . 3 62 / / P ro gr am t o Compute t h e IDFT o f t h e f o l l o w i n g

S e q u e n c e3 //X [ k ] = [ 12 , 1.5 + j2 .5 98 ,1.5 + j0 .86 6 ,0 ,1 . 5j 0

. 8 6 6 , 1.5 j 2 . 5 9 8 ]4 c l e ar a ll ;

5 clc ;6 close ;

7 j = sqrt ( - 1 ) ;

8 X = [ 1 2 , - 1 . 5 + j * 2 . 5 9 8 , - 1 . 5 + j * 0 . 8 6 6 , 0 , - 1 . 5 - j * 0 . 8 6 6 , - 1 . 5 -

j * 2 . 5 9 8 ] ;

9 //IDFT Comput ation10 x = fft ( X , 1 ) ;

11 / / D i s p l a y S e q u e n c e x [ n ] i n command w in do w12 disp ( round ( x ) , x [ n]= ) ;

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Chapter 4

THE FAST FOURIER

TRANSFORM

Scilab code Exa 4.3 Shortest Sequence N Computation

1 / / E xa mp le 4 . 32 // Program t o c a l c u l a t e s h o r t e s t s eq ue nc e N su ch t ha t

a l go r it h m B r un s // f a s t e r t han A3 c l e ar a ll ;

4 clc ;

5 close ;

6 i = 0 ;

7 N = 3 2 ; //Gi v e n8 / / C a l c u l a t i o n o f T wi dd le f a c t o r e xp o ne nt s f o r e ac h

s t a g e9 while 1==1

10 i = i + 1 ;

11 N = 2 ^ i ;

12 A = N ^ 2 ;

13 B = 5 * N * log2 ( N ) ;14 if A > B t he n b r ea k ;

15 end ;

16 end

17 disp (N ,SHORTEST SEQUENCE N = ) ;

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Scilab code Exa 4.4 Twiddle Factor Exponents Calculation

1 / / E xa mp le 4 . 42 // Program t o c a l c u l a t e T wi dd le f a c t o r e xp o ne nt s f o r

e ac h s t a g e3 c l e ar a ll ;

4 clc ;

5 close ;

6 N = 3 2 ; //Gi v e n7 / / C a l c u l a t i o n o f T wi dd le f a c t o r e xp o ne nt s f o r e ac h

s t a g e8 for m = 1 : 5

9 disp (m , S t a g e : m = ) ;10 disp ( k = ) ;11 for t = 0 : ( 2 ^ ( m - 1 ) - 1 )

12 k = N * t / 2 ^ m ;

13 disp ( k ) ;

14 end

15 end

Scilab code Exa 4.6 DFT using DIT Algorithm

1 / / E xa mp le 4 . 62 / / Pr ogr am t o f i n d t h e DFT o f a S e qu e nc e x [ n

] = [ 1 , 2 , 3 , 4 , 4 , 3 , 2 , 1 ]3 / / u s i n g DIT A l g o r i t h m .4 c l e ar a ll ;

5 clc ;

6 close ;

7 x = [ 1 ,2 , 3 , 4 ,4 , 3 , 2 ,1 ];

8 //FFT Compu tatio n

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9 X = fft ( x , -1) ;

10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.8 DFT using DIF Algorithm

1 / / E xa mp le 4 . 82 / / Pr ogr am t o f i n d t h e DFT o f a S e qu e nc e x [ n

] = [ 1 , 2 , 3 , 4 , 4 , 3 , 2 , 1 ]3 / / u s i n g DIF A l g o r i t h m .4 c l e ar a ll ;

5 clc ;

6 close ;

7 x = [ 1 ,2 , 3 , 4 ,4 , 3 , 2 ,1 ];

8 //FFT Compu tatio n9 X = fft ( x , -1) ;

10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.9 8 Point DFT of the Sequence

1 / / E xa mp le 4 . 92 / / Program t o f i n d t he 8p o i n t DFT o f a S e qu e nc e x [ n

] = 1 , 0


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1 / / E x am pl e 4 . 1 02 / / P r o gr am t o Com pu te t h e 4p o i n t DFT o f a S eq ue nc e x

[ n ] = [ 0 , 1 , 2 , 3 ]3 / / u s i n g DITDIF A l g o r i t h m .4 c l e ar a ll ;

5 clc ;

6 close ;

7 x = [0 , 1 ,2 , 3] ;


10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.11 IDFT of the Sequence using DIT Algorithm

1 / / E x am pl e 4 . 1 12 / / P ro gr am t o Compute t h e IDFT o f a S e q u en c e u s i n g

DIT A l go r i t hm .3 / /X [ k ] = [ 7 , 0. 707 j 0 . 70 7 , j , 0 . 707j 0 . 7 0 7 , 1 , 0 . 7 0 7 + j 0

.7 0 7 , j ,0 . 70 7 + j 0 . 7 0 7 ]4 c l e ar a ll ;

5 clc ;

6 close ;

7 j = sqrt ( - 1 ) ;

8 X = [ 7 , - 0 .7 07 - j * 0. 70 7 , - j , 0 .7 0 7 - j * 0 . 7 07 , 1 , 0 . 7 07 + j

* 0 . 7 0 7 , j , - 0 . 7 0 7 + j * 0 . 7 0 7 ] ;

9 / / I n v e r s e FFT C o m pu t a ti o n10 x = fft ( X , 1 ) ;

11 disp (x , x ( n ) = ) ;


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1 / / E x am pl e 4 . 1 2

2 / / P r o gr am t o Com pu te t h e 8p o i n t DFT o f a S e qu e nc e3 / /x [ n ] = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] u s i n g r a di x2 DITA l gor i t hm .

4 c l e ar a ll ;

5 clc ;

6 close ;

7 x = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] ;


10 disp (X , X ( z ) = ) ;


1 / / E x am pl e 4 . 1 32 / / P r o gr am t o Com pu te t h e 8p o i n t DFT o f a S e qu e nc e3 / /x [ n ] = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] u s i n g r a di x2 DIF

A l gor i t hm .4 c l e ar a ll ;

5 clc ;

6 close ;7 x = [ 0 . 5 , 0 . 5 , 0 . 5 , 0 . 5 , 0 , 0 , 0 , 0 ] ;


10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.14 DFT using DIT Algorithm

1 / / E x am pl e 4 . 1 42 / / P r o gr am t o Com pu te t h e 4p o i n t DFT o f a S eq ue nc e x

[ n ]=[ 1 ,1 ,1 ,1 ]3 / / u s i n g DIT A l g o r i t h m .4 c l e ar a ll ;

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5 clc ;

6 close ;7 x = [ 1 , - 1 , 1 , - 1 ] ;


10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.15 DFT using DIF Algorithm

1 / / E x am pl e 4 . 1 52 / / P r o gr am t o Com pu te t h e 4p o i n t DFT o f a S eq ue nc e x[ n ] = [ 1 , 0 , 0 , 1 ]

3 / / u s i n g DIF A l g o r i t h m .4 c l e ar a ll ;

5 clc ;

6 close ;

7 x = [ 1 , 0 , 0 , 1 ] ;


10 disp (X , X ( z ) = ) ;

Scilab code Exa 4.16.a 8 Point DFT using DIT FFT

1 / / Example 4 . 1 6 ( a )2 / / P ro gr am t o E v a l u a t e a nd Co mpa re t h e 8p o i n t DFT o f

t h e g i v e n S eq ue nc e3 // x1 [ n ]=1 , 3


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10 disp ( X 1 , X1 ( k ) = ) ;

Scilab code Exa 4.16.b 8 Point DFT using DIT FFT

1 / / Exam ple 4 . 1 6 ( b )2 / / P ro gr am t o E v a l u a t e a nd Co mpa re t h e 8p o i n t DFT o f

t h e g i v e n S eq ue nc e3 //x 2 [ n] = 1 , 0


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5 clc ;

6 close ;7 N = 8 ;

8 n = 0 : 1 : N - 1 ;

9 x = 2^ n ;


12 disp (X , X ( z ) = ) ;

Scilab code Exa 4.23 DFT using DIF FFT Algorithm

1 / / E x am pl e 4 . 2 32 / / P r og ra m t o Compute t h e DFT o f g i v e n S e q u e nc e3 //x [ n]= co s (n p i / 2 ) , a nd N=4 u s i n g DIFFFT Alg ori thm .4 c l e ar a ll ;

5 clc ;

6 close ;

7 N = 4 ;

8 p i = 2 2 / 7 ;

9 n = 0 : 1 : N - 1 ;

10 x = cos ( n * p i / 2 ) ;11 //FFT Compu tatio n12 X = fft ( x , -1) ;

13 disp (X , X ( z ) = ) ;


1 / / E x am pl e 4 . 2 4

2 / / P r o gr am t o Com pu te t h e 8p o i n t DFT o f g i v e nS e q u e n c e

3 / / x [ n ] = [ 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 ] u s i n g DIF , r a d i x2,FFTA l gor i t hm .

4 c l e ar a ll ;

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5 clc ;

6 close ;7 x = [ 0 ,1 , 2 , 3 ,4 , 5 , 6 ,7 ];


10 disp (X , X ( z ) = ) ;

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Chapter 5

INFINITE IMPULSE

RESPONSE FILTERS

Scilab code Exa 5.1 Order of the Filter Determination

1 / / E xa mp le 5 . 12 / /To Fi nd o ut t h e o r d e r o f t h e f i l t e r3 c l e ar a ll ;

4 clc ;

5 close ;

6 a p = 1 ; //db7 a s = 3 0 ; //db8 o p = 2 0 0 ; / / r a d / s e c9 o s = 6 0 0 ; / / r a d / s e c

10 N = log ( sqrt ( ( 1 0 ^ ( 0 . 1 * a s ) - 1 ) / ( 1 0 ^ ( 0 . 1 * a p ) - 1 ) ) ) / log ( o s /

o p ) ;

11 disp ( ceil ( N ) , O rd er o f t h e f i l t e r , N = ) ;

Scilab code Exa 5.2 Order of Low Pass Butterworth Filter

1 / / E xa mp le 5 . 2

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2 / /To F i nd o ut t h e o r d e r o f a Low P a s s B u tt er w or th

F i l t e r3 c l e ar a ll ;4 clc ;

5 close ;

6 a p = 3 ; //db7 a s = 4 0 ; //db8 f p = 5 0 0 ; //Hz9 f s = 1 0 0 0 ; //Hz

10 o p = 2 * % p i * f p ;

11 o s = 2 * % p i * f s ;


o p ) ;13 disp ( ceil ( N ) , O rd er o f t h e f i l t e r , N = ) ;

Scilab code Exa 5.4 Analog Butterworth Filter Design

1 / / E xa mp le 5 . 42 / /To D es i gn an A na lo g B u tt e rw o rt h F i l t e r3 c l e ar a ll ;

4 clc ;5 close ;

6 a p = 2 ; //db7 a s = 1 0 ; //db8 o p = 2 0 ; / / r a d / s e c9 o s = 3 0 ; / / r a d / s e c


o p ) ;

11 disp ( ceil ( N ) , O rd er o f t h e f i l t e r , N = ) ;12 s = % s ;

13 H S = 1 / ( ( s ^ 2 + 0 . 7 6 5 3 7 * s + 1 ) * ( s ^ 2 + 1 . 8 4 7 7 * s + 1 ) ) ; / / T r a n s f e rF u n c t i o n f o r N=414 o c = o p / ( 1 0 ^ ( 0 . 1 * a p ) - 1 ) ^ ( 1 / ( 2 * ceil ( N ) ) ) ;

15 H S 1 = horner ( H S , s / o c ) ;

16 disp ( H S 1 , N o r m a l i ze d T r a n s f e r F u n ct i o n , H( s ) = ) ;

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Scilab code Exa 5.5 Analog Butterworth Filter Design

1 / / E xa mp le 5 . 52 / /To D es i gn an A na lo g B u tt e rw o rt h F i l t e r3 c l e ar a ll ;

4 clc ;

5 close ;

6 o p = 0 . 2 * % p i ;

7 o s = 0 . 4 * % p i ;8 e 1 = 0 . 9 ;

9 l 1 = 0 . 2 ;

10 e p s i l o n = sqrt ( 1 / ( e 1 ^ 2 ) - 1 ) ;

11 l a m b d a = sqrt ( 1 / ( l 1 ^ 2 ) - 1 ) ;

12 N = log ( l a m b d a / e p s i l o n ) / log ( o s / o p ) ;

13 disp ( ceil ( N ) , O rd er o f t h e f i l t e r , N = ) ;14 s = % s ;

15 H S = 1 / ( ( s ^ 2 + 0 . 7 6 5 3 7 * s + 1 ) * ( s ^ 2 + 1 . 8 4 7 7 * s + 1 ) ) ; / / T r a n s f e rF u n c t i o n f o r N=4

16 o c = o p / e p s i l o n ^ ( 1 / ceil ( N ) ) ;

17 H S 1 = horner ( H S , s / o c ) ;

18 disp ( H S 1 , N o r m a l i ze d T r a n s f e r F u n ct i o n , H( s ) = ) ;

Scilab code Exa 5.6 Order of Chebyshev Filter

1 / / E xa mp le 5 . 62 / /To F i nd o ut t h e o r d e r o f t he F i l t e r u si ng

C h eb y s he v A p p r o x i m a t i o n3 c l e ar a ll ;4 clc ;

5 close ;

6 a p = 3 ; //db

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7 a s = 1 6 ; //db

8 f p = 1 0 0 0 ; //Hz9 f s = 2 0 0 0 ; //Hz10 o p = 2 * % p i * f p ;

11 o s = 2 * % p i * f s ;

12 N = acosh ( sqrt ( ( 1 0 ^ ( 0 . 1 * a s ) - 1 ) / ( 1 0 ^ ( 0 . 1 * a p ) - 1 ) ) ) / acosh

( o s / o p ) ;


Scilab code Exa 5.7 Chebyshev Filter Design

1 / / E xa mp le 5 . 72 / /To D es i gn an a n a l og C he bys hev F i l t e r w it h G iv en

S p e c i f i c a t i o n s3 c l e ar a ll ;

4 clc ;

5 close ;

6 o s = 2 ;

7 o p = 1 ;

8 a p = 3 ; //db

9 a s = 1 6 ; //db10 e 1 = 1 / sqrt ( 2 ) ;

11 l 1 = 0 . 1 ;

12 e p s i l o n = sqrt ( 1 / ( e 1 ^ 2 ) - 1 ) ;

13 l a m b d a = sqrt ( 1 / ( l 1 ^ 2 ) - 1 ) ;

14 N = acosh ( l a m b d a / e p s i l o n ) / acosh ( o s / o p ) ;


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

1 / / E xa mp le 5 . 82 / /To F i nd o ut t he o r de r o f t he p o l e s o f t he Type 1

L ow pa ss C he by sh ev F i l t e r

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3 c l e ar a ll ;

4 clc ;5 close ;

6 a p = 1 ; //dB7 a s = 4 0 ; //dB8 o p = 1 0 0 0 * % p i ;

9 o s = 2 0 0 0 * % p i ;

10 N = acosh ( sqrt ( ( 1 0 ^ ( 0 . 1 * a s ) - 1 ) / ( 1 0 ^ ( 0 . 1 * a p ) - 1 ) ) ) / acosh

( o s / o p ) ;


Scilab

Digital Signal Processing_R. Babu

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