Scilab Textbook Companion for Digital Signal Processing: A Computer Based Approach by S. K. Mitra1 Created by Sanjeev Irny B.Tech Instrumentation Instrumentation Engineering Shri Gurugobind Singhji Institute of Engg. & Tech. College Teacher Dr. B. M. Patre Cross-Checked by
November 3, 2014
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
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Book Description Title: Digital Signal Processing: A Computer Based Approach Author: S. K. Mitra Publisher: Tata McGraw - Hill Education Edition: 3 Year: 2008 ISBN: 978-0-07-066756-3
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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 Listt of Scilab Codes Lis
4
2 Dis Discree creett Time Time Signal Signalss and System Systemss
8
3 Dis Discree creett TIme Four Fourier ier Trans Transfor form m
22
4 Dig Digita itall Processing Processing of Contin Continous ous TIme TIme Systems Systems
29
5 Fin Finite ite Lengt Length h Discreet Discreet Trans Transfor form m
33
6 z Tra rans nsfo form rm
45
7 LTI Discreet Discreet Time Time sys system temss in the Tran Transfo sform rm Domain Domain
61
8 Dig Digita itall Filter Filter Stru Structu ctures res
64
9 IIR dig digit ital al filt filter er desi design gn
69
10 FIR digital filter design
74
11 DSP Algorithm implementation
80
12 Analysis of Finite Wordlength Effects
83
13 Multirate DIgital Signal Processing Findamentals
84
14 Applications of Digital Signal Processing
87
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List of Scilab Codes Exa 2. Exa 2.11 Exaa 2. Ex 2.22 Exaa 2. Ex 2.33 Exaa 2. Ex 2.55 Exaa 2. Ex 2.66 Exaa 2. Ex 2.77 Exaa 2. Ex 2.99 Exa 2.1 2.166 Exa 2.2 2.200 Exa 2.2 2.222 Exa 2.2 2.266 Exa 2.2 2.277 Exa 2.2 2.288 Exa 2.2 2.299 Exa 2.3 2.300 Exa 2.3 2.311 Exa 2.3 2.322 Exa 2.3 2.333 Exa 2.4 2.466 Exaa 3. Ex 3.55 Exaa 3. Ex 3.66 Exaa 3.7 Ex 3.7 Exa 3.10 3.10 Exa 3.12 3.12 Exa 3.1 3.133 Exaa 3.1 Ex 3.144 Exa 3.1 3.155
Ensemb Ense mble le Av Aver eragi aging ng . . . . . . . . . . . . . . Basi Ba sicc ope opera rati tion onss . . . . . . . . . . . . . . . . Uneq Un equa uall le leng ngth th se sequ quen ence ce . . . . . . . . . . . . Gene Ge nera rati ting ng sy symm mmet etri ricc pa part rtss . . . . . . . . . . Ener En ergy gy Si Sign gnal al . . . . . . . . . . . . . . . . . . Power Si Sign gnal al . . . . . . . . . . . . . . . . . . Squa Sq uare re wav avee ge gene nera rati tion on . . . . . . . . . . . . Linear Lin earit ity y of of accum accumula ulator tor . . . . . . . . . . . . Passi Pa ssive ve sys system tem . . . . . . . . . . . . . . . . . Impuls Imp ulsee respon response se of Acc Accum umulat ulator or . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . Stabbi Sta bbilit lity y of of caus causal al sys system tem . . . . . . . . . . Stabil Sta bilit ity y of An Anti ti caus causal al syste system m . . . . . . . . Stabil Sta bilit ity y of a syst system em . . . . . . . . . . . . . . Cross Cro ss core coreati ation on comp computa utatio tion n . . . . . . . . . DTFT DT FT co comp mput utat atio ion n . . . . . . . . . . . . . . DTFT DT FT co comp mput utat atio ion n . . . . . . . . . . . . . . Plot Pl otti ting ng re real al an and d im imag agin inar ary y pa part rt . . . . . . . DTFT DTF T of finite finite lengt length h exponen exponentia tiall sequen sequence ce . Plotti Plo tting ng DTFT DTFT of expon exponen entia tiall sequen sequence ce . . . DTFT DTF T com comput putati ation on . . . . . . . . . . . . . . Ener En ergy gy of si sign gnal al . . . . . . . . . . . . . . . . Energy Ene rgy of expon exponen entia tiall seque sequence nce . . . . . . . .
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8 9 9 10 11 12 12 13 13 14 15 15 16 17 18 19 19 20 21 22 23 24 25 25 26 27 28
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Exa 4. Exa 4.66 Exa 4.7 Exa 4.8 Exa 4.16 4.16 Exaa 5. Ex 5.11 Exaa 5. Ex 5.22 Exaa 5. Ex 5.33 Exaa 5. Ex 5.44 Exaa 5. Ex 5.55 Exa 5.7 Exa 5.8 Exa 5.1 5.100 Exa 5.11 Exa 5.1 5.122 Exa 5.1 5.144 Exa 5.15 5.15 Exa 6.1 Exa 6.2 Exaa 6. Ex 6.33 Exaa 6. Ex 6.44 Exaa 6.5 Ex 6.5 Exaa 6. Ex 6.99 Exaa 6.1 Ex 6.100 Exa 6.1 6.111 Exa 6.1 6.122 Exa 6.13 6.13 Exa 6.14 6.14 Exa 6.15 Exa 6.16 6.16 Exa 6.1 6.177 Exa 6.18 Exa 6.19 Exa 6.2 6.200 Exaa 6. Ex 6.22 22 Exaa 6. Ex 6.23 23
Orderr of An Orde Anal alog og fil filte terr . . . . . . . . . . . . . . . . . . Order Ord er of Ana Analog log Che Cheby byshe shev v Fil Filter ter . . . . . . . . . . . Order Ord er of Ana Analog log Lo Lowpa wpass ss Ell Ellipt iptic ic Fil Filter ter . . . . . . . . Design Des ign of Anal Analof of Butte Butterw rwort orth h HP Filte Filterr . . . . . . . . DFT DF T co comp mput utat atio ion n . . . . . . . . . . . . . . . . . . . . DFT DF T of si sin nus usoi oida dall se sequ quen ence ce . . . . . . . . . . . . . . . DFT DF T co comp mput utat atio ion n . . . . . . . . . . . . . . . . . . . . IDFT ID FT Co Comp mput utat atio ion n . . . . . . . . . . . . . . . . . . . DFT DF T co comp mput utat atio ion n . . . . . . . . . . . . . . . . . . . . Cicula Cic ularr con convo volut lution ion com comput putati ation on . . . . . . . . . . . . Cicula Cic ularr con convo volut lution ion com comput putati ation on . . . . . . . . . . . . Genera Gen eratin tingg symm symmetr etric ic par parts ts . . . . . . . . . . . . . . . Cicular Cicul ar con convol volution ution compu computatio tation n . . . . . . . . . . . . Linear Lin ear Con Convo volut lution ion usi using ng DFT DFT . . . . . . . . . . . . . DFT com comput putati ation onusi using ng singl singlee DFT DFT . . . . . . . . . . . DFT comput computation ationusing using single DFT of shorter shorter lengt length h . z Tran Transfo sform rm of cau causal sal expo exponen nentia tiall sequ sequenc encee . . . . . . z tra transf nsform orm of an antic ticaus ausal al seq sequen uence ce . . . . . . . . . . . z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . . z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . . Z tra trans nsfo form rm of ca caus usal al se sequ quen ence ce . . . . . . . . . . . . . z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . . Rati Ra tion onal al form of z Tra rans nsfo form rm from from it itss ze zero ro and pol polee locations . . . . . . . . . . . . . . . . . . . . . . . . . Inve In verse rse z Tran Transfo sform rm . . . . . . . . . . . . . . . . . . . Inve In verse rse z Tran Transfo sform rm . . . . . . . . . . . . . . . . . . . Proper Pro per fract fraction ion of Ratio Rational nal z Tran Transfo sform rm . . . . . . . . Inverse Inv erse z Transf ransform orm by by partial partial fraction fraction expansi expansion on . . . residue resid ue comput computation ation using coeffici coefficient ent matc matching hing approa approach ch Inverse Inv erse z Transf ransform orm by by power power series series expansio expansion n. . . . . Coefficie Coeffi cient ntss of rat ration ional al form form . . . . . . . . . . . . . . . Inverse Inv erse z Transf Transform orm using long divis division ion . . . . . . . . Inverse Inv erse z Transf Transform orm using long divis division ion . . . . . . . . Inve In verse rse z Tran Transfo sform rm . . . . . . . . . . . . . . . . . . . z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . . z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . .
29 30 31 31 33 33 34 35 37 37 38 39 40 41 42 42 45 45 46 46 46 47 47 48 48 49 50 50 51 52 52 53 53 53 54
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Exa 6. Exa 6.26 26 Exa 6.2 6.277 Exa 6.28 6.28 Exa 6.3 6.300 Exa 6.3 6.311 Exa 6.33 6.33 Exa 6.34 Exaa 7. Ex 7.11 Exa 7.2 Exaa 7. Ex 7.66 Exa 8.1 Exa 8.6 Exa 8.7 Exa 8.10 Exa 8.12 8.12 Exa 8.1 8.188 Exaa 9. Ex 9.11 Exa 9.2 Exa 9. Exa 9.33 Exa 9.6 Exa 9.7 Exaa 9.1 Ex 9.122 Exaa 10 Ex 10.1 .1
z Tra rans nsfo form rm . . . . . . . . . . . . . . . . . . . . . . . . Inve In verse rse z Tran Transfo sform rm . . . . . . . . . . . . . . . . . . . Enlarg Enl argeme ement nt of ROC ROC by by pole zero zero cancel cancellat lation ion . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . . . . . . Convo Con volut lution ion . . . . . . . . . . . . . . . . . . . . . . . . Transf ransfer er Functi Function on of Moving Moving Aver Average age Filter Filter . . . . . . Transf ransfer er func function tion dete determina rmination tion . . . . . . . . . . . . . Boun Bo unde ded d re real al fu func ncti tion on . . . . . . . . . . . . . . . . . . Tran ransfe sferr fun functi ction on det deter ermin minati ation on . . . . . . . . . . . . . FIR FI R Tra rans nsfe ferr fu func ncti tion on . . . . . . . . . . . . . . . . . . Analys Ana lysis is of of Casca Cascaded ded lat lattic ticee digit digital al filte filterr struc structur turee . . Fact actori orizat zation ion of FIR FIR Tran Transfe sferr Func Functio tion n. . . . . . . . . Fact actori orizat zation ion of IIR Tran Transfe sferr Func Functio tion n . . . . . . . . . Cascaded Casca ded lattice lattice reali realizatio zation n of IIR digital digital Transf ransfer er FuncFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gray Gra y Marke Markell method method of real realiza izatio tion n . . . . . . . . . . . Cascad Cas caded ed latt lattice ice rea realiz lizati ation on . . . . . . . . . . . . . . . Comp Co mput utat atin ingg ri ripp pple le valu alues es . . . . . . . . . . . . . . . . conver con version sion of bande bandedged dged freq frequenc uencies ies to Norma Normalized lized digital frequencies . . . . . . . . . . . . . . . . . . . . . . Desi De sign gn of HP Di Digi gita tall Fi Filt lter er . . . . . . . . . . . . . . . Changi Cha nging ng pass passban band d edge edge freq frequen uencie ciess to LP IIR IIR digit digital al frequencies . . . . . . . . . . . . . . . . . . . . . . . . Design Des ign of HP HP IIR Dig Digita itall Filte Filterr from from LP Digi Digital tal Fil Filter ter Mini Mi nimu mum m or orde derr of Type 2 Ch Cheb ebys yshe hev v HP IIR di digi gita tall filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kais Ka iser er fo form rmul ulaa . . . . . . . . . . . . . . . . . . . . . .
55 56 57 57 58 59 60 61 62 63 64 64 65 65 66 67 69 69 70 71 72 73 74
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Exa 13. 13.22 Exa 13.6 Exaa 14.1 Ex 14.1 Exaa 14.2 Ex 14.2
Down sam Down sampli pling ng oper operati ation on . . . . . . Decimator Decim ator Compu Computatio tation n compl complexit exity y Effec Eff ectt of of DFT DFT le leng ngth th . . . . . . . . . Effec Eff ectt of of DFT DFT le leng ngth th . . . . . . . . .
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85 85 87 88
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Chapter 2 Discreet Time Signals and Systems
Scilab code Exa 2.1 Ensemble Averaging
1 2 3 4 5 6 7 8 9
/ /E /EX XAMPLE 2 . 1 / / E n se s e mb m b le le a v e r a g i n g clear ; clc ; n = 1 : 50 50 ; clf(); figure ( 0 ) a = gca () ; a . x _ l o c a t i o n =”
ig in ”
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19 20 21 22 23 24 25 26 27 28
d(i)=(-0.1)^i; S= sum (d);
end Eav=(s+S/M)’; / / E n se s e mb m b le le a v e r a g e . Th h e o u tp t p u t o f E ns n s em e m bl bl e a v e r a g i n g i s ’ ) disp ( E a v , ’ T plot2d3 ( n , s ) plot ( n , s , ’ r . ’ ) xtitle ( ’ E ns n s em e m bl b l e a v e r a g i n g ’ , ’ n ’ , ’ Ea Eav v−s ’ ) ; a.children.children.thickness=2; a.children.children.foreground=2;
Scilab code Exa 2.2 Basic operations
1 2 3 4 5 6 7 8 9 10 11 12
/ /E /EX XAMPLE 2 . 2 , BASIC OP OPERATIONS . clear ; clc ; c = [ 3. 3 . 2 4 1 3 6 - 9 .5 .5 0 ]; ]; disp ( c , ’ c = ’ ) ; d = [ 1. 1 . 7 - 0. 0. 5 0 0 .8 .8 1 ]; ]; disp ( d , ’ d = ’ ) ; w1=c.*d; // M u l t i p l i c a t i o n Th e p ro r o du d u ct c t o f t w o i n p ut ut v e c t o rs i s = ’ ); disp ( w 1 , ’ Th w2=c+d; // addi tion Th e a d d i t io i o n o f t w o i np n p ut ut v e ct o rs i s = ’ ); disp ( w 2 , ’ Th w3=3.5*c;
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5 6 7 8 9 10 11 12 13 14 15 16 17 18
disp ( c , ’ c = ’ ) ; g = [ - 21 2 1 1 .5 . 5 3 ]; ]; disp ( g , ’ g = ’ ) ; a = length ( g ) ; b = length ( c ) ; i=0; while ( i < b - a ) g(b-i)=0; i=i+1; end w4=g.*c; Th h e p ro r o du d u ct c t o f t w o s e q ue u e n c es es i s = ’ ) ; disp ( w 4 , ’ T w5=c+g; Th e a d d i t io io n o f t w o s e q u e nc n c es es i s = ’ ); disp ( w 5 , ’ Th
Scilab code Exa 2.5 Generating symmetric parts
1 //E //EX XAMPLE 2 . 5 , Co nj ug a te − A n t i s y m m e t r i c & C o n j u g a t e −
s ym y m me m e tr t r ic i c p a r t s o f S eq e q ue u e nc nc e 2 3 4 5 6 7
clc ; clear ; g = [ 0 , 1 + %i % i * 4 , - 2+ 2+ % i *3 * 3 , 4 - %i % i * 2 , - 5 - %i % i * 6 , - %i % i * 2 , 3 ]; ]; disp ( g , ’ g = ’ ) g 1 = conj ( g ) ; / / C on o n ju j u g at at e o f g ; disp ( g 1 , conj ( g ) ) ;
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Scilab code Exa 2.6 Energy Signal
1 2 3 4 5 6 7 8 9 10 11 12 13
//E // EXAMPLE 2 . 6 , Energy S ig n a l clear ; clc ; n = -5:5; f o r i = 1 : length ( n ) if ( n ( i ) > = 1 ) h(i)=1/n(i); else h(i)=0; end end
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Scilab code Exa 2.7 Power Signal
1 2 3 4 5 6 7 8 9 10 11 12 13
//E // EXAMPLE 2 . 7 , Exa Example mple o f Pow Power er s i g n a l clear ; clc ; Sum=0; N=1:10000; f o r i = 1 : length ( N ) h 1 = 3 * (( (( - 1 ) ^ i ) ; h=h1^2; end E n e rg rg y = s u m ( h ) ; i f ( E n e r g y / ( 2 * ( length ( N ) ) +1 + 1 ) < %i % i nf n f ) then
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10 11 12 13 14 15 16
y2 = sin(x*.15*%pi); y3 = sin(x*.25*%pi); y4=y1+y2/3+y3/5; plot2d3 ( x , y 4 , 2 ) plot ( x , y 4 , ’ r . ’ ) xtitle ( ’ A p p r o x i m a t e S q u a r e w av av e ’ , ’ x ’ , ’ y 4 ’ ) ; a.children.children.thickness=3;
Scilab code Exa 2.16 Linearity of accumulator
1 2 3 4 5 6
/ /E /EX XAMPLE 2 . 1 6 , clear ; clc ;
/ / Gi G i v e n i n p ut u t s eq e q ue u e nc nc e = [ 3 4 5 ] x = [0 3 4 5 0]; p u t s e q ue ue n c e = ’ ) disp ( [ 3 4 5 ] , ’ I n pu
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3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
clc ; a = input ( ” an a n y v al a l ue ue o f a l e s s n=-10:1:10; x=n; y=a*n; S=0; f o r i = 1 : length ( n ) S=S+y^2; end i f a < 1 then h e s ys y s te te m i s disp ( ’ t he else end
a==1 disp ( ’ t h e s y s t e m i s
t h a n o r e qu q u al a l t o o ne ne ” )
passive ’)
lossless ’)
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15 16 17 18 19 20
h=0; else h=d; plot2d3 ( i - 1 0 1 , h ) plot ( i - 1 0 1 , h , ’ . r ’ ) xtitle ( ’ I m p ul u l s e R es e s p on o n s e o f a c c u mu m u l a t o r ’ , ’ t ’ ’ Y ’ ) ; , ’Y a.children.children.thickness=1; a.children.children.foreground=2; end
21 22 23 24 e n d 25 disp ( h , ’ T Th h e i m pu p u l se s e r e s p on o n s e o f A cc c c um u m ul u l aatt o r i s = ’ )
Scilab code Exa 2.26 Convolution
1 / /E /EXAMPLE 2 . 2 6 , c o n v o l u t i o n o f x & h
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7 8 9 10 11 12 13 14 15 16
clf(); figure ( 0 ) ; a = gca () ; a . x _ l o c a t i o n =” o r i g i n ” ; x = c^n ; subplot ( 2 , 2 , 1 ) ; plot2d3 ( n , x , 2 ) ; plot ( n , x , ’ . r ’ ) ; xtitle ( ’ ’ , ’ n ’ , ’ x ’ ) ; h = b^
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10 11 12 13 14 15 16 17 18 19
ut i o n = ’ ); disp ( y , ’ c o n v o l ut clf(); figure ( 0 ) ; a = gca () ; a . x _ l o c a t i o n =” o r i g i n ” ; a . y _ l o c a t i o n =” o r i g i n ” ; plot2d3 ( n , y ) plot ( n , y , ’ r . ’ ) ’ Y ’ ) ; xtitle ( ’ c o n v o l u t i o n ’ , ’ n ’ , ’Y a.children.children.thickness=2;
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22 end 23 e n d 24 disp ( y ’ , ’ T Th h e C on o n vo v o l ut u t io i o n o f t he h e t w o s e q ue u e n c es es
Scilab code Exa 2.30 Convolution
1 / /E /EX XAMPLE 2 . 3 0
is =’)
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30 31 32 33
a . x _ l o c a t i o n =” o r i g i n ” ; plot2d3 ( n , y , 2 ) ; plot ( n , y , ’ r . ’ ) ; xtitle ( ’ c o n v o l u t i o n ’ , ’ n ’ , ’ y ’ ) ;
Scilab code Exa 2.31 Stabbility of causal system
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2 3 4 5 6 7 8 9 10 11
/ /h / h [ i ] = i m p u l s e r e s p o n s e o f L T I s y st st e m . clear ; clc ; n = - 5 :1 : 1 / 1 00 0 0 0 :5 :5 ; a=5; f o r i=1: length ( n ) if ( n ( i ) > - 1 ) h(i)=0; else h(i)=a^n(i);
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12
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20 xtitle ( ’ M a g ni n i t u de d e p l o t ’ ’W
>
’ ’X mag mag
>
’ );
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29 a = g c a ( ) ;
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23 a = g c a ( ) ;
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Chapter 7 LTI Discreet Time systems in the Transform Domain
Scilab code Exa 7.1 Bounded real function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
/ /E /EX XAMPLE 7 . 1 / /P /PR ROGRAM REQUIRES MAXI XIM MA SC SCIL ILAB AB TOO OOLB LBOX OX clc ; clear ; syms K a z w ; h zd z d en e n = ( 1 -a - a * ( z ^ -1 - 1 ) ); ); / / 0 <| a | < 1 ; H z = K / hz h z de de n ; (w) ’ ) ; disp ( ’ | H( e ˆ( jw ) ) | ˆ 2 = K ˆ 2 / ( ( 1 + a ) ˆ 2 − 2 ∗ co s (w)
/ / c o n s i d e r i n g a >0 pi ) , Kˆ Kˆ 2 / (1 ( 1 + a ) ˆ 2 < | H | ˆ 2 < Kˆ2/(1 − a ) disp ( ’ ( a t w = % pi ˆ2 ,( at w = 0) ’ ); / / c o n s i d e r i n g a <0 2 / (1 ( 1 + a ) ˆ 2 < | H | ˆ 2 < Kˆ2/(1 − a ) disp ( ’ ( a t w = 0 ) , Kˆ 2/ ˆ 2 , ( a t w = %p i ) ’ ) ; disp ( ’ i f K = +/ −(1− a ) , o b s e r v e . . . . ’ ) ; disp ( ’ | H( e ˆ( jw ) ) | <= 1 H e n c e a Bounded r e a l
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17 //w=0:%pi ; 18 // [ a , b]= f r e q (hznum (hznum , hzden , w) ; 19 disp ( a b s ( H z ) )
Scilab code Exa 7.2 Transfer function determination
1 / /E /EX XAMPLE 7 . 2 2 / / | H( e ˆ( jw ) ) | ˆ 2 = 4 ∗ ( ( 1 . 0 9 + 0 . 6 ∗ cos w ) ∗ ( 1 . 1 6 − 0 . 8 ∗
cosw ) ) /((1 .0 4 − 0 . 2 ∗ cos w ) ∗ (1 .2 5+ cosw ) ) 3 //R //RE EPLACING cos w = ( z + z (ˆ − 1 ) ) / 2 4 5 6 7
8 9 10 11 12 13 14 15
16 17 18 19 20 21 22
clc ; clear ; z=%z; H 1 = 4 * ( (1 ( 1 . 0 9 + ( 0 .3 .3 ) * ( z + 1 / z ) ) * ( 1. 1 . 1 6 - ( 0 .4 . 4 ) * ( z + 1 / z ) )) )) ; H 2 = (( ( ( 1. 1 . 0 4 - ( 0. 0 . 2) 2 ) * ( z + 1/ 1 / z ) ) * (1 ( 1 .2 . 2 5 + ( 0. 0 . 5) 5 ) * ( z + 1/ 1/ z ) ) ); ); H=H1/H2; Th e t r a n s f e r f u n c t i o n i s , H = ’ ) disp ( H , ’ Th e l t s 1 = factors ( numer ( H ) ) ; Th e f a c t o r s o f n um u m er e r a to to r a r e : ’ ) ; disp ( e l t s 1 , ’ Th e l t s 2 = factors ( denom ( H ) ) ; Th e f a c t o r s o f d en e n om o m in i n at a t or or a r e : ’ ) ; disp ( e l t s 2 , ’ Th Th e F o u r p o s i b l e s t a b l e t r a n s f e r f u n c t i o n w i t h disp ( ’ Th s a m e s q ua u a r e m aagg ni n i tu t u de de f u n c t i o n a r e : ’ ) ; h 1 = 2 * ( ( 1+ 1 + ( 0 . 3) 3 ) / z ) * ( 1 - ( 0 .4 . 4 ) / z ) ) / ( (1 ( 1 - ( 0. 0. 2 ) / z ) *(1+(0.5)/z)); nc t i o n , h1 = ’ ) ; disp ( h 1 , ’ s t a b l e t r a n s f e r f u nc h 2 = 2 * ( ( 1+ 1 + ( 0 . 3) 3) / z ) * ( ( 0 . 4 ) - ( 1) 1 ) / z ) ) / ( ( 1 - ( 0 .2 .2 ) / z ) *(1+(0.5)/z)); disp ( h 2 , ’ s t a b l e t r a n s f e r f u nc nc ti o n , h2 s = ’ ) ; h 3 = 2 * ( (( ( ( 0 . 3 ) + 1/ 1 / z ) * ( ( 1 ) - ( 0 .4 .4 ) / z ) ) / ( ( 1 - ( 0. 0. 2 ) / z ) *(1+(0.5)/z)); nc t i o n , h3 = ’ ) ; disp ( h 3 , ’ s t a b l e t r a n s f e r f u nc h 4 = 2 * ( (( ( ( 0 . 3 ) + 1/ 1 / z ) * ( ( 0 . 4) 4 ) - ( 1) 1 ) / z ) ) / ( ( 1 - ( 0 .2 .2 ) / z )
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23 disp ( h 4 , ’ s t a b l e
t r a n s f e r f u nc nc t i o n , h4 = ’ ) ;
Scilab code Exa 7.6 FIR Transfer function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
/ /E /EX XAMPLE 7 . 6 / /F / F I R T r a sn s n f e r f u n c t i o n s w it i t h d i f f e r e n t P h a se se .
clc ; clear ; z = %z ; W = 0 :( : ( 1/ 1 / 40 4 0 0) 0 ) : 1; 1; z = exp (%i*2*%pi*W); for i=1:401 H 1z 1 z ( i ) = -1 -1 + 2 / z( z ( i ) - 3 /( / ( z ( i ) ^2 ^ 2 ) + 6/ 6 / ( z (i ( i ) ^ 3) 3) - 3/ 3 / ( z ( i ) ^4 ^ 4 ) + 2 /( / ( z ( i ) ^ 5 ) - 1/ 1/ z ( i ) ^ ( 6) 6) ; end H 1 z _ ph p h a s e = phasemag ( H 1 z ) ; clf(); figure ( 0 ) ; plot2d ( W / ( 2 * % p i ) , H 1 z _ p ha ha s e , 1 ) ; xtitle ( ’ p h as a s e r e s p o n s e ’ , ’W/(2 ∗ %pi %pi ) ’ , ’ H 2 z p h a s e i n d e g r e e s ’ );
17 18 f o r i = 1 : 4 0 1 19 H 2z 2z ( i )= ) = +1 + 1 - 2/ 2 / z (i ( i ) + 3/ 3 / ( z( z ( i )^ ) ^ 2) 2 ) - 6 /( /( z ( i) i ) ^ 3) 3) + 3 /( / ( z ( i ) ^4 ^ 4 ) - 2 /( / ( z ( i ) ^5 ^ 5 ) + 1 / z( z ( i ) ^ (6 (6 ) ; 20 e n d p h a s e = phasemag ( H 2 z ) ; 21 H 2 z _ ph 22 23 plot2d ( W / ( 2 * % p i ) , H 2 z _ p ha ha s e , 2 ) ; 24 xtitle ( ’ p h as a s e r e s p o n s e ’ , ’W/(2 ∗ %pi %pi ) ’ , ’ H 2 z p h a s e i n d e g r e e s ’ );
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Chapter 8 Digital Filter Structures
Scilab code Exa 8.1 Analysis of Cascaded lattice digital filter structure
1 2 3 4 5 6 7 8 9 10 11 12 13
/ /E /EX XAMPLE 8 . 1 / /M /MA AXIMA SCI SCILAB LAB TOOLBO BOX X REQ REQUIR UIRED ED FO FOR R TH THIS IS EX EXAM AMPL PLE E // Dig ita l f i l t e r stru ctur e clear ; clc ; sy ms W1 W2 W3 X Y a d B y E z ;
/ / E qu q u at a t io i o n s o b ta t a i n ed ed a r e a s f o l l o w s : W1 W2 W3 Y
= = = =
X - a*W3/z; W 1 - d *W * W 2 /z /z ; W 2 /z / z + E *W *W2 ; B *W * W 1 + y* y * W3 W3 /z /z ;
/ / S o l v i n g t h e a bo b o ve ve e q u a t i o n s :
H z =( =( B + ( B *d * d + y* y * E) E ) / z + y /( / ( z ^2 ^ 2 ) )/ ) / (1 ( 1 + ( d +a + a * E) E ) / z + a /( /( z ^2)) 14 disp ( H z , ’ Hz Hz = ’ ) ;
Scilab code Exa 8.6 Factorization of FIR Transfer Function
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1 2 3 4 5 6
/ / E xa x a mp m p le le 8 . 6 / / F a c t o r i z a t i o n o f F I R T r a ns n s f e r F un u n ct c t io io n
clear ; clc ; z=%z; H z = 5 0 . 4 + 2 8 . 0 2 / z + 1 3 . 8 9 / z ^ 2 + 7 . 4 2 / z ^ 3 + 6 . 0 9 / z ^ 4 + 3 / z ^ 5 +1 / z^6; 7 disp ( factors ( numer ( H z ) ) , ’ T Th h e F a ct c t o r s o f t he h e FIR T r a n sf s f e r F un u n ct c t io io n a r e = ’ );
Scilab code Exa 8.7 Factorization of IIR Transfer Function
1 2 3 4 5 6 7
/ / E xa x a mp m p le le 8 . 7 / / F a c t o r i z a t i o n o f I IR I R T r a ns n s f e r F un u n ct c t io io n clear ; clc ; z=%z;
/ / Nu N u m e r a t o r o f t he he t r a n s f e r f u n c t i o n N u m z = 6 + 1 7 . 1 / z + 3 3 . 0 5 / z ^ 2 + 2 4 . 7 2 / z ^ 3 + 1 9 . 9 0 8 / z ^ 4 - 5 . 2 9 2/ z ^5+18.144/z^6;
8 // / / D en e n om o m i na n a to t o r o f t he he t r a n s f e r f u n c t i o n 9 Denz=1+2.2/z+2.56/z^2+1.372/z^3+0.118/z^4-0.332/z ^5-0.168/z^6; 10 F n = factors ( numer ( N u m z ) ) ; 11 disp ( F n , ’ F a ct c t o rs r s o f t he h e n um u m er e r at a t or o r o f t he h e T r a ns ns f e r F u nc n c ti ti o n = ’ ) ; 12 F d = factors ( numer ( D e n z ) ) ; 13 disp ( F d , ’ F a ct c t o rs r s o f t he h e d en e n om o m in i n at a t or o r o f t he h e T r a ns ns f e r F u nc n c ti ti o n = ’ ) ;
Scilab code Exa 8.10 Cascaded lattice realization of IIR digital Transfer
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1 / / E x am am pl pl e 8 . 1 0 2 / / C as a s ca c a de de d l a t t i c e
r e a l i z a t i o n o f I IR I R T r a ns ns f e r
Function 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
clear ; clc ; z=%z; P 3z 3z = - 0 . 2 + 0 .1 . 1 8/ 8 / z + 0 .4 . 4 /( / ( z ^ 2) 2) + 1 /( /( z ^3 ^3 ) ; D 3z 3 z = 1 + 0 . 4/ 4 / z + 0 .1 . 1 8/ 8 / ( z^ z ^ 2 ) - 0 .2 . 2 /( / ( z ^3 ^3 ) ; A3z=P3z/D3z; p 1 = coeff ( numer ( P 3 z ) ) ; p=mtlb_fliplr(p1); disp ( m t l b _ f l i p l r ( p ) , ’The c o e f f i c i e n t s of nu me ra to r a re re = ’ ) ; d 1 = coeff ( numer ( D 3 z - 1 ) ) ; d=mtlb_fliplr(d1) he c o e f f i c i e n t s o f n u m e r a t o r a r e = ’ ) ; disp ( ( d ) , ’ T he d1_1dash=(d(1)-d(3)*d(2))/(1-d(3)*d(3)); as h = ” ) ; disp (d1 _1dash , ” d 1 1 d as d2_1dash=(d(2)-d(3)*d(1))/(1-d(3)*d(3)); ”) ; disp (d2 _1dash , ” d 2 1 d a s h d1_2dash=(d1_1dash)/(1+d2_1dash); as h = ” ) ; disp (d1 _2dash , ” d 1 2 d as A 1z 1 z = ( d 1 _2 _ 2 da d a s h + 1 / z) z ) / (1 ( 1 + d 1 _2 _ 2 d as as h / z ) ; disp ( A 1 z , ’ A1 A 1z = ’ ) ; A 2z 2 z = ( d 2 _1 _ 1 da d a s h + d 1 _1 _ 1 d as a s h * 1/ 1 / z + 1 / z ^2 ^ 2 ) / (1 (1 + d 1 _1 _ 1 d as as h / z - d 2 _1 _ 1 d as a s h / z ^ 2) 2) ; A 2z = ’ ) ; disp ( A 2 z , ’ A2
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6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
P 3z 3z = 0 + 0 .4 . 4 4/ 4 / z + 0 .3 . 3 62 6 2 /( / ( z ^ 2) 2 ) + 0. 0 . 02 0 2 /( / ( z ^ 3) 3) ; D 3z 3z = 0 .4 . 4 / z + 0 .1 . 1 8/ 8 / ( z ^2 ^ 2 ) - . 2/ 2 / ( z ^3 ^3 ) ; Hz=P3z/D3z; p 1 = coeff ( numer ( P 3 z ) ) ; p=mtlb_fliplr(p1) disp ( m t l b _ f l i p l r ( p ) , ’The c o e f f i c i e n t s of nu me ra to r a re re = ’ ) ; d 1 = coeff ( numer ( D 3 z ) ) ; d=mtlb_fliplr(d1) disp ( m t l b _ f l i p l r ( d ) , ’The c o e f f i c i e n t s of nu me ra to r a re re = ’ ) ; d1_1dash=(d(1)-d(3)*d(2))/(1-d(3)*d(3)); disp (d1 _1dash , ” d 1 1 d as as h = ” ) ; d2_1dash=(d(2)-d(3)*d(1))/(1-d(3)*d(3)); disp (d2 _1dash , ” d 2 1 d a s h ”) ; d1_2dash=(d1_1dash)/(1+d2_1dash); disp (d1 _2dash , ” d 1 2 d as as h = ” ) ; a1=p(3); disp ( p ( 3 ) , ’ a1 a1 = ’ ); a2=p(2)-a1*d(1); disp ( p ( 2 ) - a 1 * d ( 1 ) , ’ a2 a2 = ’ ); a3=p(1)-a1*d(2)-a2*d1_1dash; disp ( p ( 1 ) - a 1 * d ( 2 ) - a 2 * d 1 _ 1 d a sh sh , ’ a3 a3 = ’ ); disp (0-a1 *d(3) -a3* d1_2dash -a2* d2_1dash , ’ a a44 = ’ ) ;
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8 9 10 11 12 13 14 15 16 17 18 19
0.2/z^5); disp ( H 5 z , ’ F IR IR f i l t e r = ’ ) ; G 5 = horner ( H 5 z , - 1 / z ) ; G5z=G5/z^5; IR f i l t e r = ’ ) ; disp ( G 5 z , ’ F IR k5=0.2; H 3 z = ( 1 / (1 ( 1 + k 5 ^ 2 ) ) * ( H 5z 5 z - k 5 * G 5z 5z ) ; disp ( H 3 z , ’ S y n t h e s i s e q n , H 3 z = ’ ) ; G 3 z = ( 1 / (1 ( 1 + k 5 ^ 2 ) ) * ( k5 k 5 * H 5 z + G 5z 5z ) ; disp ( G 3 z , ’ S y n t h e s i s e q n , G 3 z = ’ ) ; k = coeff ( numer ( G 3 z ) ) ; k3 = ’ ) ; disp ( k ( 4 ) , ’ k3 disp ( k ( 2 ) , ’ k1 k1 = ’ ) ;
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Chapter 9 IIR digital filter design
Scilab code Exa 9.1 Computating ripple values
1 2 3 4 5 6 7 8 9 10 11
/ /E /EX XAMPLE 9 . 1 / / p a s s band & s t o p band r i p p l e clc ; clear ; a p = 0 . 1 ; / / p ea e a k p as a s sb s b an a n d r i p p l e i n dB dB mi n . s to t o pb p b an a n d a t t e n u a t i o n i n dB a s = 3 5 ; / / mi
/ / c a l c u l a t i o n o f p e a k r i p p l e v a lu lu es dp=1-10^-(ap/20); disp ( d p , ’ dp dp = ’ ) ; ds=10^-(as/20);
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3 4 5 6 7 8 9 10 11 12 13
clc ; clear ; ap=7; as=3;
/ / S a m p l in i n g f r e q u e n c y ( i n KHz ) : FT=25;
// d i g i t a l f r e q u e n c i e s :
wp=2*%pi*ap/FT; wp = ’ ) ; disp ( w p , ’ wp ws=2*%pi*as/FT; ws = ’ ) ; disp ( w s , ’ ws
Scilab code Exa 9.3 Design of HP Digital Filter
1 2 3 4 5 6 7 8 9 10 11
/ / E xa x a mp m p le le 9 . 3 / / D e s i g n o f HP I I R f i l t e r
clc ; clear ; F p = 7 0 0 //Hz F s = 5 0 0 //Hz a p = 1 //dB a s = 3 2 //dB F T = 2 0 0 0 //Hz
/ / n o rm r m a li l i z ed e d a n g ul u l a r e dg d g e f r e q u e n c i e s i n r ad ad / s e c wp=2*%pi*Fp/FT;
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22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
disp ( 1 / k , ’ 1 / k = ’ ) ; k 1 = 1 / ( sqrt ( ( ( 10 1 0 ^ ( 0 .1 . 1 * a s ) ) - 1) 1 ) / ( ( 1 0 ^ ( 0. 0. 1 * a p ) ) - 1 ) ) ) ; disp ( 1 / k 1 , ’ 1 / k 1 = ’ ) ; N = ceil ( acosh ( 1 / k 1 ) / acosh ( 1 / k ) ) ; disp ( N , ’ N = ’ ) ; disp ( N , ’ O rd r d er er o f t h e f i l t e r i s , N = ’ ) ; e = sqrt ( 1 0 ^ ( 0 . 1 * a p ) - 1 ) ; u = 1/ 1 / e + sqrt ( 1 + ( 1 / ( e * e ) ) ) ; a = A p * ( u ^ ( 1/ 1/ N ) - u ^ ( - 1 / N ) ) /2 /2 ; b = Ap A p * ( u ^( ^ ( 1/ 1 / N ) + u ^ ( - 1/ 1/ N ) ) /2 /2 ; for i=1:N p hi h i ( i ) = % pi p i / 2 + ( 2* 2 * i -1 -1 ) *( *( % p i ) /( / ( 2* 2* N ) ; p ( i ) = a * c o s ( ph p h i (i ( i )) ) ) + % i * b* b*s i n ( p h i ( i ) ) ; end s=%s; z=%z; H1=1;
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1 2 3 4 5 6 7 8 9 10 11 12 13 14
/ /E /EX XAMPLE 9 . 6 / /L / LP TO TO LP LP T r a n s f o r m a t i o n
clc ; clear ; z=%z; w=0:0.001*%pi:%pi; Glz=(0.0662272*(1+1/z)^3)/((1-0.2593284/z) *(1-0.6762858/z+0.3917468/(z^2))) w c = 0 . 2 5 * % p i ; / / O r i n g i n a l p a ss s s b an a n d e d ge ge i r e d p a ss s s b an a n d e d ge ge W c = 0 . 3 5 * % p i ; / / R e q u ir l= sin ((wc-Wc)/2)/ sin ((wc+Wc)/2) disp ( l , ’ l a mb mb da da = ’ ) ; G d z = horner ( G l z , ( ( 1 - l / z ) / ( 1 / z - l ) ) ) ; disp ( G d z , ’ Th Th e t r a n s f e r f u n c t i o n i s G d z = ’ ) ;
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16 disp ( G h z , ’ Th Th e t r a n s f e r f u n c t i o n i s G d z = ’ ) ; 17 disp ( d e n , ’ t he h e f a c o r s o f t he h e d en e n om o m in i n at a t or o r a re re = ’ );
Scilab code Exa 9.12 Minimum order of Type 2 Chebyshev HP IIR digital
filter 1 / /E /EX XAMPLE 9 . 1 2 2 / / Mi Minimum o r d e r
filter 3 4 5 6 7 8 9
clc ; clear ; a p = 1 //dB a s = 4 0 //dB F p = 1 0 00 00 //Hz F s = 6 0 0 //Hz W p = F p * 2* 2 * % pi pi ;
o f t y p e −2 C he h e b y s h e v h i g h pa pa s s d i g i t a l
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Chapter 10 FIR digital filter design
Scilab code Exa 10.1 Kaiser formula
1 / / E x am am pl pl e 1 0 . 0 1 2 / / Or O r de de r e s t i m a t i o n
u s i n g K a is i s e r ’ s f o r mu mu l a
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Scilab code Exa 10.2 Bellenger formula
1 2 3 4 5 6 7 8 9 10 11 12 13
/ / E x am am pl pl e 1 0 . 0 1 / / O r d er e r e s t i m a t i o n u s i ng n g B e l la l a n g er e r ’ s f or o r mu mu l a clear ; clc ; F p = 1 8 0 0 ; / / Pa P a ss s s ba b a nd n d e dg d g e f r e q . i n Hz o p ba b a nd n d e dg d g e f r e q . i n Hz F s = 2 0 0 0 ; / / s t op a p = 0 . 1 ; / / p ea e a k p as a s sb s b an a n d r i p p l e i n dB dB mi n . s to t o pb p b an a n d a t t e n u a t i o n i n dB a s = 3 5 ; / / mi m p li l i ng n g f r e q . i n Hz F T = 1 2 0 0 0 ; / / S a mp
/ / c a l c u l a t i o n o f p e a k r i p p l e v a lu lu es dp=1-10^-(ap/20); dp = ’ ) ; disp ( d p , ’ dp ds=10^-(as/20);
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9 10 11 12 13 14 15 16 17 18
m p li l i ng n g f r e q . i n Hz F T = 1 2 0 0 0 ; / / S a mp / / c a l c u l a t i o n o f p e a k r i p p l e v a lu lu es
dp=1-10^-(ap/20); dp = ’ ) ; disp ( d p , ’ dp ds=10^-(as/20); disp ( d s , ’ d s = ’ ) ;
a1=0.005309; a2=0.07114;
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8 9 10 11 12 13 14
o p ba b a nd n d e dg d g e f r e q . i n Hz F s 2 = 1 1 0 0 ; / / s t op s s ba b a nd n d r i p p l e i n dB dB d p = 0 . 0 0 4 ; / / p a ss d s = 0 . 0 1 ; / / s t op o p ba b a nd n d r i p p l e i n dB dB m p li l i ng n g f r e q . i n Hz F T = 1 0 0 0 0 ; / / S a mp / / S i n c e ( Fp F p 1−F s 1 ) <(Fs2 −F p 2 ) , b a nd n d w it i t h u s e d i s ( Fp F p 1− Fs1)
15 16 / / O r de d e r o f t h e FI R f i l t e r
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20 disp ( M 3 , ’ V al a l ue ue o f M = ’ )
Scilab code Exa 10.7 Order Estimation
1 / /E /EX XAMPLE 1 0 . 6 2 / / Or O r de de r e s t i m a t i o n 3 clear ;
u s i n g D o l p h −Cebyshev wi windo ndow w
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12 13 14 15 16 17 18 19 20 21
ds=10^(-as/20); B = ( 0 .5 . 5 8 42 4 2 * ( as as - 21 2 1 ) ^ 0. 0 . 4) 4 ) + 0 . 07 0 7 8 86 8 6 * ( as as - 21 21 ) ;
N = ceil ( ( as a s - 8 ) / (2 ( 2 . 28 2 8 5 * Bw Bw ) ) ; r d er er o f t h e f i l t e r , N = ’ ) ; disp ( N , ’ O rd M=(N-1)*0.5; disp ( M , ’M = ’ ) ; w = window ( ’ kr ’ , N , 6 ) ; / / K a i s e r w in in do do w i=-M:1:M;
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Chapter 11 DSP Algorithm implementation
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19
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24 disp ( p ( 2 ) - a 1 * d ( 1 ) , ’ a2 a2 = ’ );
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20 x 1 = s i n ( % p i * n / L ) ;
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