DesignoftheWaveDigitalFilters 141 T h fo l pr o st r Fi g A 1 N G R 1 1 h e pro g ram for t h l lows, and the f r og ram for com p r ucture in the fi gu g . 4. Butterworth 1 =0.618146;B2=0. 2 N 2=0; N4=0; N6= 0                    G C . . G R R L G R G A G C G A G C G A G C G 0 1 1 2 1 2 2 2 0 1 0 1 2 3 2 3 4 51 4 5 1 618 1 0 618 1 2 h e anal y sis o f th e r equenc y respo n p utin g XN1-XN 4 u re 4. wave di g ital lo w 2 76524;A3=0.182 8 0 ; N8=0; N10=0; X     . . . . . A G 5 2 236 0 447 0 618 0 182 0 467 e wave di g ital fil t n se obtained is g 4 , BN4-BN1,N1- N w -pass filter of th e 8 58;B4=0.201756; A X N=1                   G G C R . G R R L G . R R B R L R B R L G A G C 3 2 3 3 3 4 3 4 4 4 1 2 1 2 3 4 3 4 5 52 4 2 1 0 4 2 1 0 4 0 0 2  G 5 5 t er of the 5 th ord g iven in fi g ure 4 N 9 and YN(i) w e 5 th order. A 51=0.467353;A 5 . . . . 2 447 4 08 026 4 93 276 0 201  .0 947 er written in M A 4 . The equation w as obtained fr o 5 2=0.947276; A TLAB in the o m the Fi g 5. A s fil t A 3 ch a hi g pr o for i=1:1:200 XN1=A1*X N XN3=N6- A BN4=XN 4 BN3=XN 3 BN2=XN 2 BN1=XN1 N1=XN* A N5=BN3- A N9=N10- A YN(i)=2* N N2=N1;N4 = end [h,w]=freqz(Y N Plot(w,20*lo g 10 g . 5. Frequency r e Design of the s a second exam p t er for n=5, A max =0.182, B 4 =0.19 2 a n g in g the valu e g h-pass filter in og ram we also h a N -A1*N2+N2; A 3*XN2-A3*N6; 4 -A51*XN4+2*N1 3 -B4*XN4-B4*BN 4 2 -A3*XN2+BN3+ N -BN2*XN2-B2*B N A 1-A1*N2+BN1; A 3*XN2-A3*N6; A 51*XN4-A51*N 1 N 10-A51*XN4-A5 = N3;N6=N5;N8= N N ,1,200) (abs(h))) e sponse in dB of t low-pass and p le we shall pr o =3 dB. From tab 2 , A 51 =0.383 an d e s of the coeffici e the fi g ure 6. A d a ve to chan g e N 2 XN2=XN1+N 4 XN4=XN3+ N 0-A51*N10-A52* N 4 ; N 6-A3*N6; N 2; N3=BN1+BN 2 N7=BN3+BN 4 1 0-A52*N10; 1*N10-A52*N10; N 7;N10=N9;XN = t he Butterworth l high-pass Ch e o pose low-pass a n le 7 we g et the v d A 52 =0.360. Us i e nts we g et the a t d ditionall y to g et 2 = -N1; N4= -N3; 4 ; N 8; N 10; 2 ; 4 ; = 0; l ow-pass filter . e bychev filter n d high-pass C h v alues of the W D i n g previous M t tenuation of C h the hi g h-pass f i N6= -N5; N8= - N h eb y chev wave D F A 1 =0.223, B 2 = M ATLAB pro g ra m h eb y chev low-pa i lter from the pr N 7; N10= -N9. digital = 0.226, m and ss and evious AUTOMATION&CONTROL-TheoryandPractice142 Fi g 6. In T h Ɵ = Fi g Pa ac c pr e g . 6. Attenuation Realization of fi g ure 7 the stru c h e values of the L = 30 o ), A max =1.249 g . 7. LC referenc e rallel resonant L c ordin g to Fi g ur e e sented in the Fi g of the Cheb y che v the low-pass C c ture of the 5 th o r L C filter was ob t 4 dB, A min =70.5 d e Cauer low-pass L C circuit in low - e 8. Structure of t g ure 9 . v low-pass and h i C auer WDF. r der ladder LC r e t ained from tabl e d B and Ω s= 2.0000 . filter - pass filter fi g ur e t he wave di g ital ig h-pass filter e ference Cauer l o e (Rudolf Saal, p . . (Saal 1979). e 7 wil be realiz e Cauer low-pass f o w-pass filter is s . 114, 1979) (C 0 5 e d b y di g ital st r f ilter of the 5th o s hown. 5 50 for r ucture rder is Fi g Fi g W i w a pr e in p C( 1 C( 2 N 1 R( 3 R( 5 K( 4 A ( A ( fo r g . 8. Discrete real i g . 9. Structure of t i th the assistanc e a ve di g ital filter e sented in Fi g ur e p ut data of the L 1 )=2.235878; C(3 ) 2 )=0.096443; C(4 ) 1 4=0; G(1)=1; G( 2 3 )=R(2)+1/(C(2) + 5 )=R(4)+1/(C(4) + 4 )=(L(4)*C(4)-1) / ( 3)=G(3)/(G(3)+ C ( 51)=2*G(5)/(G(5 ) r i=1:1:200 XN2=A(1)*XN1 + XN4=N8-A(3)* X BN5=XN5-A(51 ) BN4 =XN4-B(4) * BN3 =XN3-A(3) * BN2 =XN2-B(2) * N1 =A(1)*XN1- A N5=-K(2)*N3+K N9=BN4+BN5; i zation of parall e t he Cauer low-p a e of the followin g A 1 ,B 2 ,A 3 ,B 4 ,A 51 a e 10. The pro g ra m C filter was obta i ) =2.922148; C(5) = ) =0.257662; XN1 = 2 )=G(1)+C(1); R2 = + 1/L(2)); G + 1/L(4)); G / (L(4)*C(4)+1); A C (3)); B( 4 ) +C(5)+1); A ( + N2-A(1)*N(2); X N3-A(3)*N8; ) *XN5+2*N14-A( 5 * BN5-B(4)*XN5; * XN3+BN4+N8- A * XN3-B(2)*BN3; A (1)*N2+BN2; (2)*N6+N4: N N1 1 e l and serial LC c i a ss filter n=5. g MATLAB pro gr a nd A 52 . The att e m was obtained i ned from the cat = 2.092084; L(2)=0 . = 1; N2=0; N4=0; N = 1/G(2); G (3)=1/R(3); G G (5)=1/R(5); A (1)=G(1)/G(2); 4 )=R(4)/R(5); ( 52)=2/(G(5)+C( 5 XN3=XN2+ N XN5=XN4+N12 ; 5 1)*N14-A(52)*N A (3)*N8; N3=BN2*BN3; N 7=BN4-A(3)*X N 1 =N10-N9*K(4)+ N i rcuit. r am we can calc u e nuation of the l o from the struct u alo g ue of the Ca u 981174; L(4)=0.8 8 N 6=0; N8=0; N10 G (4)=G(3)+C(3); K (2)=(L(2)*C(2)- 1 B(2)=R(2)/R(3); 5 )+1); N 6; ; 14*; N 3-A(3)*N8; N 12*K(4); u late coefficients o w-pass Cauer f u re in the Fi g ure u er filter (Saal 19 7 8 9139; =0; N12=0; R(4)=1/G(4); 1 )/(L(2)*C(2)+1); of the ilter is 9. The 7 9.) DesignoftheWaveDigitalFilters 143 Fi g 6. In T h Ɵ = Fi g Pa ac c pr e g . 6. Attenuation Realization of fi g ure 7 the stru c h e values of the L = 30 o ), A max =1.249 g . 7. LC referenc e rallel resonant L c ordin g to Fi g ur e e sented in the Fi g of the Cheb y che v the low-pass C c ture of the 5 th o r L C filter was ob t 4 dB, A min =70.5 d e Cauer low-pass L C circuit in low - e 8. Structure of t g ure 9 . v low-pass and h i C auer WDF. r der ladder LC r e t ained from tabl e d B and Ω s= 2.0000 . filter - pass filter fi g ur e t he wave di g ital ig h-pass filter e ference Cauer l o e (Rudolf Saal, p . . (Saal 1979). e 7 wil be realiz e Cauer low-pass f o w-pass filter is s . 114, 1979) (C 0 5 e d b y di g ital st r f ilter of the 5th o s hown. 5 50 for r ucture rder is Fi g Fi g W i w a pr e in p C( 1 C( 2 N 1 R( 3 R( 5 K( 4 A ( A ( fo r g . 8. Discrete real i g . 9. Structure of t i th the assistanc e a ve di g ital filter e sented in Fi g ur e p ut data of the L 1 )=2.235878; C(3 ) 2 )=0.096443; C(4 ) 1 4=0; G(1)=1; G( 2 3 )=R(2)+1/(C(2) + 5 )=R(4)+1/(C(4) + 4 )=(L(4)*C(4)-1) / ( 3)=G(3)/(G(3)+ C ( 51)=2*G(5)/(G(5 ) r i=1:1:200 XN2=A(1)*XN1 + XN4=N8-A(3)*X BN5=XN5-A(51 ) BN4 =XN4-B(4) * BN3 =XN3-A(3) * BN2 =XN2-B(2) * N1 =A(1)*XN1- A N5=-K(2)*N3+K N9=BN4+BN5; i zation of parall e t he Cauer low-p a e of the followin g A 1 ,B 2 ,A 3 ,B 4 ,A 51 a e 10. The pro g ra m C filter was obta i ) =2.922148; C(5) = ) =0.257662; XN1 = 2 )=G(1)+C(1); R2 = + 1/L(2)); G + 1/L(4)); G / (L(4)*C(4)+1); A C (3)); B( 4 ) +C(5)+1); A ( + N2-A(1)*N(2); X N3-A(3)*N8; ) *XN5+2*N14-A( 5 * BN5-B(4)*XN5; * XN3+BN4+N8- A * XN3-B(2)*BN3; A (1)*N2+BN2; (2)*N6+N4: N N1 1 e l and serial LC c i a ss filter n=5. g MATLAB pro gr a nd A 52 . The att e m was obtained i ned from the cat = 2.092084; L(2)=0 . = 1; N2=0; N4=0; N = 1/G(2); G (3)=1/R(3); G G (5)=1/R(5); A (1)=G(1)/G(2); 4 )=R(4)/R(5); ( 52)=2/(G(5)+C( 5 XN3=XN2+ N XN5=XN4+N12 ; 5 1)*N14-A(52)*N A (3)*N8; N3=BN2*BN3; N 7=BN4-A(3)*X N 1 =N10-N9*K(4)+ N i rcuit. r am we can calc u e nuation of the l o from the struct u alo g ue of the Ca u 981174; L(4)=0.8 8 N 6=0; N8=0; N10 G (4)=G(3)+C(3); K (2)=(L(2)*C(2)- 1 B(2)=R(2)/R(3); 5 )+1); N 6; ; 14*; N 3-A(3)*N8; N 12*K(4); u late coefficients o w-pass Cauer f u re in the Fi g ure u er filter (Saal 19 7 8 9139; =0; N12=0; R(4)=1/G(4); 1 )/(L(2)*C(2)+1); of the ilter is 9. The 7 9.) AUTOMATION&CONTROL-TheoryandPractice144 en [h , H i N 1 fil t Fi g 7. W e T h of 12 A va Fi g N13=N14-A(51) * YN(i)=2*N14-A( N2=N1;N4=N3; N d , w]=freqz(YN,1, 2 ig h-pass can be o b 1 0=-N9; N12=-N 1 t er are presented . g . 10. Frequenc y r Design of the e shall realize b a h e values of the n Cauer filter (Saa A . With the tra n lues of the band- p g . 11.Tolerance s c * XN5-A(51)*N14 - 51)*XN5-A(52)* N N 6=N5;N8=N7; N 2 00); plot(w,20* l b tained b y chan g 1 1; N14=-N13. I n . r esponse of the C Cauer band-p a a nd-pass filter t h n ormalized low- p l 1979). The val u n sformation of th e p ass filter prese n c heme of the ban d - A(52)*N14; N 14-A(51)*N14; N 10=N9;N12=N1 1 l o g 10(abs(h))) g in g in the pro g r a n Figure 10 the C auer low-pass a n a ss filter h at accomplish t h p ass filter for C0 3 u es of the normal i e low-pass filter n ted in Fi g ure 12 B d -pass filter. 1 ;N14=N13;XN1 = a m N2=-N1; N4 = attenuation of l o n d hi g h-pass wa v h e tolerance sche m 3 50 θ=53 can be o i zed low-pass fil t to the band-pas s B . = 0; = -N3; N6=-N5; N 8 o w-pass and hi g v e digital filter m e g iven in Fi gu o btained from th e t er are shown in s filter can be o b 8 =-N7; g h-pass u re 11. e table fi g ure b tained Fi g af t Se r tr a Fi g pa      g . 12. A) LC lo w t er impedance tr a r ial-parallel res o a nsformed into p g . 13. Impedanc e rallel resonant ci r w -pass Cauer filt e a nsformation. o nant circuit in p arallel-parallel r e e transformation r cuit a e r, B) LC band-p the lon g itudin a e sonant circuit, fi of the serial-par a L A A L A A c a c          2 1 2 2 1 1 L a L u   4 1 C a C v   4 1 B A L L 1 2 1 B A L L 1 1 1 a ss Cauer filter, a l branch in th g ure 13, b y equa t a llel resonant L C A L A A L A c c  1 2 1 2 2 4 L A L A 2 1 C A C A 1 2 u 2 u 2 C) LC band-pas e fi g ure 12B) c t ions (5) up to (1 1 C circuit to the p a s filter c an be 1 ). a rallel- (5) (6) (7) (8) (9) DesignoftheWaveDigitalFilters 145 en [h , H i N 1 fil t Fi g 7. W e T h of 12 A va Fi g N13=N14-A(51) * YN(i)=2*N14-A( N2=N1;N4=N3; N d , w]=freqz(YN,1, 2 ig h-pass can be o b 1 0=-N9; N12=-N 1 t er are presented . g . 10. Frequenc y r Design of the e shall realize b a h e values of the n Cauer filter (Saa A . With the tra n lues of the band- p g . 11.Tolerance s c * XN5-A(51)*N14 - 51)*XN5-A(52)* N N 6=N5;N8=N7; N 2 00); plot(w,20* l b tained b y chan g 1 1; N14=-N13. I n . r esponse of the C Cauer band-p a a nd-pass filter t h n ormalized low- p l 1979). The val u n sformation of th e p ass filter prese n c heme of the ban d - A(52)*N14; N 14-A(51)*N14; N 10=N9;N12=N1 1 l o g 10(abs(h))) g in g in the pro g r a n Figure 10 the C auer low-pass a n a ss filter h at accomplish t h p ass filter for C0 3 u es of the normal i e low-pass filter n ted in Fi g ure 12 B d -pass filter. 1 ;N14=N13;XN1 = a m N2=-N1; N4 = attenuation of l o n d hi g h-pass wa v h e tolerance sche m 3 50 θ=53 can be o i zed low-pass fil t to the band-pas s B . = 0; = -N3; N6=-N5; N 8 o w-pass and hi g v e digital filter m e g iven in Fi gu o btained from th e t er are shown in s filter can be o b 8 =-N7; g h-pass u re 11. e table fi g ure b tained Fi g af t Se r tr a Fi g pa      g . 12. A) LC lo w t er impedance tr a r ial-parallel res o a nsformed into p g . 13. Impedanc e rallel resonant ci r w -pass Cauer filt e a nsformation. o nant circuit in p arallel-parallel r e e transformation r cuit a e r, B) LC band-p the longitudin a e sonant circuit, fi of the serial-par a L A A L A A c a c          2 1 2 2 1 1 L a L u   4 1 C a C v   4 1 B A L L 1 2 1 B A L L 1 1 1 a ss Cauer filter, a l branch in th g ure 13, b y equa t a llel resonant L C A L A A L A c c  1 2 1 2 2 4 L A L A 2 1 C A C A 1 2 u 2 u 2 C) LC band-pas e figure 12B) c t ions (5) up to (1 1 C circuit to the p a s filter c an be 1 ). a rallel- (5) (6) (7) (8) (9) AUTOMATION&CONTROL-TheoryandPractice146   Fr o pa ad fi g fi g Fi g Fr o an fi g in o m the circuit i n rallel resonant c aptors. Parallel r g ure 8. The coeff i g ure 8 can be obt a g . 14. Cauer LC b o m the port adm i d the coefficient s g ure 15. At the o u order to realize t h n the fi g ure 14 ca c ircuit and the r e r esonant circuit i cients K i i=(1, 2 , a ined by followi n K 1 = 0 K 3 = 0 and-pass filter a f i ttances Y i and i m G 1 =G 0- + Y R R 3 =R 2 +1 / s of the parallel a u tput of the ban d h e load resistor R G G G A R B R Z R B R Z G A G Y G A G Y               0 0 1 1 1 1 2 1 2 2 3 2 3 3 41 3 4 42 3 4 2 2 B A C aC 2 2 B A C aC 1 2 n be obtained t h e sistance R 1 , R 2 , can be realized , 3, 4) of the wa v ng equation    1 1 i i L C i i L C i K 0 .543105 K 2 = 0 .870278 K 4 = f ter the impedan c m pedances R i we Y 1 =4.39611 R 2 =R 1 +1/Y 2 =0.4 6 / Y 3 =0.581104 a nd serial adapt o d -pass filter the p a R 4 =1. . . . . . . . . . G . . G G . .              1 1 3 3969 3 4 4 4 0 2274 0 2274 0 2274 0 2364 0 4638 0 4638 0 1172 2 1 7 1 7208 3 2 1 7208 3 v1 2 v1 2 h e values Y i =C i + 1 R 3 and R 4 at t h b y the discrete v e di g ital band- p = -0.11659 = 0.543105 c e transformatio n g et the followin g R 1 =0.227432 6 3853 G 3 =1.720862 o rs of the wave d a rallel dependen t . . . . . .       0 4903 0 7982 7 208 0 5625 3969 1 0 3269 3969 1 1 /L i (i=1, 2, 3, 4) h e output port o structure prese n p ass filter prese n n g values, d i g ital band-pass t adaptor must b (10) (11) of the o f each n ted in n ted in in the e used Fi g 8. In st r te r pa th e 9. Fi g g . 15. Cauer wav e Tables of the W the fi g ure 16 th e r ucture of Butter w r minated at the p rallel dependent e end of structur e Tables of the g . 16. Frequenc y r e di g ital band-pa s W ave digital fi e re are frequenc y w orth WDF is c r p ort 3 with dela y adaptor to rea l e must be connec t Butterworth w r esponse and str u s s filter. lters. y response and s r eated b y conne c elements. At the l ize for n odd lo a t ed serial depen d ave digital filt e u cture of the But t s tructure of the w c tion of the para l end of the struc t a d resistance R L = d ent adaptor. e r t erworth wave di w ave di g ital filte r l lel and serial a d t ure must be con n = 1. In case of n e g ital filter. r . The d aptors n ected e ven at DesignoftheWaveDigitalFilters 147   Fr o pa ad fi g fi g Fi g Fr o an fi g in o m the circuit i n rallel resonant c aptors. Parallel r g ure 8. The coeff i g ure 8 can be obt a g . 14. Cauer LC b o m the port adm i d the coefficient s g ure 15. At the o u order to realize t h n the fi g ure 14 ca c ircuit and the r e r esonant circuit i cients K i i=(1, 2 , a ined b y followi n K 1 = 0 K 3 = 0 and-pass filter a f i ttances Y i and i m G 1 =G 0- + Y R R 3 =R 2 +1 / s of the parallel a u tput of the ban d h e load resistor R G G G A R B R Z R B R Z G A G Y G A G Y               0 0 1 1 1 1 2 1 2 2 3 2 3 3 41 3 4 42 3 4 2 2 B A C aC 2 2 B A C aC 1 2 n be obtained t h e sistance R 1 , R 2 , can be realized , 3, 4) of the wa v ng equation    1 1 i i L C i i L C i K 0 .543105 K 2 = 0 .870278 K 4 = f ter the impedan c m pedances R i we Y 1 =4.39611 R 2 =R 1 +1/Y 2 =0.4 6 / Y 3 =0.581104 a nd serial adapt o d -pass filter the p a R 4 =1. . . . . . . . . . G . . G G . .              1 1 3 3969 3 4 4 4 0 2274 0 2274 0 2274 0 2364 0 4638 0 4638 0 1172 2 1 7 1 7208 3 2 1 7208 3 v1 2 v1 2 h e values Y i =C i + 1 R 3 and R 4 at t h b y the discrete v e di g ital band- p = -0.11659 = 0.543105 c e transformatio n g et the followin g R 1 =0.227432 6 3853 G 3 =1.720862 o rs of the wave d a rallel dependen t . . . . . .       0 4903 0 7982 7 208 0 5625 3969 1 0 3269 3969 1 1 /L i (i=1, 2, 3, 4) h e output port o structure prese n p ass filter prese n n g values, d i g ital band-pass t adaptor must b (10) (11) of the o f each n ted in n ted in in the e used Fi g 8. In st r te r pa th e 9. Fi g g . 15. Cauer wav e Tables of the W the fi g ure 16 th e r ucture of Butter w r minated at the p rallel dependent e end of structur e Tables of the g . 16. Frequenc y r e di g ital band-pa s W ave digital fi e re are frequenc y w orth WDF is c r p ort 3 with dela y adaptor to rea l e must be connec t Butterworth w r esponse and str u s s filter. lters. y response and s r eated b y conne c elements. At the l ize for n odd lo a t ed serial depen d ave digital filt e u cture of the But t s tructure of the w c tion of the para l end of the struc t a d resistance R L = d ent adaptor. e r t erworth wave di w ave di g ital filte r l lel and serial a d t ure must be con n = 1. In case of n e g ital filter. r . The d aptors n ected e ven at AUTOMATION&CONTROL-TheoryandPractice148 In tables 1-5 are the elements of the Butterworth wave digital filters for various attenuation A max in the pass-band and n=3, 4, 5, 6 and 7. The tables was designed for sampling frequency f s =0.5. Table 1. Elements of the Butterworth WDF N=3, A max in dB. Table 2. Elements of the Butterworth WDF N=4, A max in dB. Table 3. Elements of the Butterworth WDF N=5, A max in dB. A max A 1 B 2 A 3 B 4 A 5 B 61 B 62 0.1 0.7256 0.4125 0.2871 0.2635 0.3355 0.6363 0.9895 0.2 0.7131 0.3944 0.2696 0.2458 0.3149 0.6170 0.9870 0.3 0.7066 0.3837 0.2595 0.2356 0.3029 0.6053 0.9854 0.4 0.7014 0.3761 0.2523 0.2284 0.2948 0.5968 0.9841 A max A 1 B 2 A 31 A 32 0.1 0.6517 0.3788 0.5494 0.9453 0.2 0.6246 0.3419 0.5096 0.9309 0.3 0.6082 0.3206 0.4856 0.9210 0.4 0.5962 0.3056 0.4682 0.9133 0.5 0.5867 0.2941 0.4545 0.9068 0.6 0.5789 0.2846 0.4331 0.9012 0.7 0.5721 0.2767 0.4334 0.8963 0.8 0.5662 0.2698 0.4249 0.8918 0.9 0.5609 0.2637 0.4174 0.8876 1.0 0.5561 0.2583 0.4105 0.8838 A max A 1 B 2 A 3 B 41 B 42 0.1 0.6764 0.3694 0.3210 0.5690 0.9679 0.2 0.6568 0.3424 0.2924 0.5385 0.9599 0.3 0.6449 0.3268 0.2760 0.5200 0.9545 0.4 0.6364 0.3157 0.2645 0.5067 0.9503 0.5 0.6295 0.3071 0.2556 0.4961 0.9468 0.6 0.6239 0.3000 0.2484 0.4874 0.9437 0.7 0.6190 0.2941 0.2423 0.4798 0.9409 0.8 0.6147 0.2889 0.2369 0.4732 0.9385 0.9 0.6109 0.2843 0.2326 0.4673 0.9362 1.0 0.6074 0.2802 0.2281 0.4620 0.9431 A max A 1 B 2 A 3 B 4 A 51 A 52 0.1 0.7022 0.3784 0.2876 0.3188 0.6021 0.9815 0.2 0.6872 0.3658 0.2655 0.2951 0.5781 0.9722 0.3 0.6782 0.3531 0.2532 0.2813 0.5636 0.9742 0.4 0.6716 0.3441 0.2446 0.2717 0.5530 0.9718 0.5 0.6664 0.3371 0.2380 0.2642 0.5446 0.9699 0.6 0.6621 0.3313 0.2325 0.2580 0.5376 0.9682 0.7 0.6584 0.3264 0.2280 0.2529 0.5317 0.9667 0.8 0.6551 0.3222 0.2240 0.2484 0.5264 0.9653 0.9 0.6522 0.3184 0.2205 0.2444 0.5217 0.9641 1.0 0.6495 0.3149 0.2173 0.2408 0.5174 0.9629 T a T a 1 0 In in Fi g T a 0.5 0.6 0.7 0.8 0.9 1.0 a ble 4. Elements o A max A 1 0.1 0.7 4 0.2 0.7 6 0.3 0.7 3 0.4 0.7 2 0.5 0.7 2 0.6 0.7 2 0.7 0.7 2 0.8 0.71 0.9 0.71 1.0 0.71 a ble 5. Elements o 0 . Tables of th e tables 6-9 are th e the pass-band a n g . 17. Frequenc y r a ble 6. Elements o 0.6072 0.3702 0.6938 0.3653 0.6909 0.3611 0.6883 0.3574 0.6859 0.3542 0.6838 0.3512 o f the Butterwort h B 2 A 3 4 62 0.4392 0.2 6 35 0.4236 0.2 3 07 0.4144 0.2 2 65 0.4078 0.2 2 31 0.4027 0.2 2 04 0.3984 0.2 2 80 0.3948 0.2 59 0.3916 0.2 40 0.3887 0.2 23 0.3862 0.2 o f the Butterwort h e Chebychev w e elements of C h n d n=3, 5, 7 and 9 r esponse and str u A max A 1 0.10 0.492 2 0.25 0.434 1 0.50 0.385 2 1.00 0.330 7 2.00 0.269 5 3.00 0.230 0 o f Cheb y chev W D 0.2468 0.2229 0.2422 0.2184 0.2384 0.2145 0.2350 0.2112 0.2321 0.2083 0.2294 0.2056 h WDF N=6, A ma x B 4 A 5 994 0.2497 0. 842 0.2352 0. 754 0.2268 0. 692 0.2208 0. 643 0.2162 0. 603 0.2125 0. 569 0.2093 0. 539 0.2065 0. 513 0.2041 0. 490 0.2019 0. h WDF N=7, A ma x w ave digital filt e h eb y chev wave d i . The tables was d u cture of Cheb y c h B 2 A 2 0.3022 0 1 0.2774 0 2 0.2599 0 7 0.2496 0 5 0.2445 0 0 0.2442 0 D F N=3, A max in d 0.2879 0.590 0.2824 0.584 0.2779 0.579 0.2739 0.575 0.2703 0.571 0.2671 0.568 x in dB. 5 B 6 A 2628 0.3597 0 2473 0.3415 0 2384 0.3308 0 2320 0.3232 0 2271 0.3173 0 2231 0.3125 0 2197 0.3083 0 2168 9.3047 0 2142 0.3015 0 2118 0.2986 0 x in dB. e r ig ital filters for v a d esi g ned for sa m h ev wave di g ital A 31 A 32 .4620 0.7574 .4310 0.6812 .4126 0.6114 .3995 0.5293 .3929 0.4331 .3925 0.3696 d B. 1 0.9830 5 0.9821 7 0.9831 4 0.9805 6 0.9798 1 0.9792 A 71 A 72 0 .6679 0.9940 0 .6522 0.9927 0 .6428 0.9917 0 .6358 0.9911 0 .6304 0.9904 0 .6258 0.9899 0 .6218 0.9895 0 .6184 0.9890 0 .6153 0.9887 0 .6124 0.9883 a rious attenuatio m plin g frequenc y f filter n A max f s =0.5. DesignoftheWaveDigitalFilters 149 In tables 1-5 are the elements of the Butterworth wave digital filters for various attenuation A max in the pass-band and n=3, 4, 5, 6 and 7. The tables was designed for sampling frequency f s =0.5. Table 1. Elements of the Butterworth WDF N=3, A max in dB. Table 2. Elements of the Butterworth WDF N=4, A max in dB. Table 3. Elements of the Butterworth WDF N=5, A max in dB. A max A 1 B 2 A 3 B 4 A 5 B 61 B 62 0.1 0.7256 0.4125 0.2871 0.2635 0.3355 0.6363 0.9895 0.2 0.7131 0.3944 0.2696 0.2458 0.3149 0.6170 0.9870 0.3 0.7066 0.3837 0.2595 0.2356 0.3029 0.6053 0.9854 0.4 0.7014 0.3761 0.2523 0.2284 0.2948 0.5968 0.9841 A max A 1 B 2 A 31 A 32 0.1 0.6517 0.3788 0.5494 0.9453 0.2 0.6246 0.3419 0.5096 0.9309 0.3 0.6082 0.3206 0.4856 0.9210 0.4 0.5962 0.3056 0.4682 0.9133 0.5 0.5867 0.2941 0.4545 0.9068 0.6 0.5789 0.2846 0.4331 0.9012 0.7 0.5721 0.2767 0.4334 0.8963 0.8 0.5662 0.2698 0.4249 0.8918 0.9 0.5609 0.2637 0.4174 0.8876 1.0 0.5561 0.2583 0.4105 0.8838 A max A 1 B 2 A 3 B 41 B 42 0.1 0.6764 0.3694 0.3210 0.5690 0.9679 0.2 0.6568 0.3424 0.2924 0.5385 0.9599 0.3 0.6449 0.3268 0.2760 0.5200 0.9545 0.4 0.6364 0.3157 0.2645 0.5067 0.9503 0.5 0.6295 0.3071 0.2556 0.4961 0.9468 0.6 0.6239 0.3000 0.2484 0.4874 0.9437 0.7 0.6190 0.2941 0.2423 0.4798 0.9409 0.8 0.6147 0.2889 0.2369 0.4732 0.9385 0.9 0.6109 0.2843 0.2326 0.4673 0.9362 1.0 0.6074 0.2802 0.2281 0.4620 0.9431 A max A 1 B 2 A 3 B 4 A 51 A 52 0.1 0.7022 0.3784 0.2876 0.3188 0.6021 0.9815 0.2 0.6872 0.3658 0.2655 0.2951 0.5781 0.9722 0.3 0.6782 0.3531 0.2532 0.2813 0.5636 0.9742 0.4 0.6716 0.3441 0.2446 0.2717 0.5530 0.9718 0.5 0.6664 0.3371 0.2380 0.2642 0.5446 0.9699 0.6 0.6621 0.3313 0.2325 0.2580 0.5376 0.9682 0.7 0.6584 0.3264 0.2280 0.2529 0.5317 0.9667 0.8 0.6551 0.3222 0.2240 0.2484 0.5264 0.9653 0.9 0.6522 0.3184 0.2205 0.2444 0.5217 0.9641 1.0 0.6495 0.3149 0.2173 0.2408 0.5174 0.9629 T a T a 1 0 In in Fi g Ta 0.5 0.6 0.7 0.8 0.9 1.0 a ble 4. Elements o A max A 1 0.1 0.7 4 0.2 0.7 6 0.3 0.7 3 0.4 0.7 2 0.5 0.7 2 0.6 0.7 2 0.7 0.7 2 0.8 0.71 0.9 0.71 1.0 0.71 a ble 5. Elements o 0 . Tables of th e tables 6-9 are th e the pass-band a n g . 17. Frequenc y r a ble 6. Elements o 0.6072 0.3702 0.6938 0.3653 0.6909 0.3611 0.6883 0.3574 0.6859 0.3542 0.6838 0.3512 o f the Butterwort h B 2 A 3 4 62 0.4392 0.2 6 35 0.4236 0.2 3 07 0.4144 0.2 2 65 0.4078 0.2 2 31 0.4027 0.2 2 04 0.3984 0.2 2 80 0.3948 0.2 59 0.3916 0.2 40 0.3887 0.2 23 0.3862 0.2 o f the Butterwort h e Chebychev w e elements of C h n d n=3, 5, 7 and 9 r esponse and str u A max A 1 0.10 0.492 2 0.25 0.434 1 0.50 0.385 2 1.00 0.330 7 2.00 0.269 5 3.00 0.230 0 o f Chebychev W D 0.2468 0.2229 0.2422 0.2184 0.2384 0.2145 0.2350 0.2112 0.2321 0.2083 0.2294 0.2056 h WDF N=6, A ma x B 4 A 5 994 0.2497 0. 842 0.2352 0. 754 0.2268 0. 692 0.2208 0. 643 0.2162 0. 603 0.2125 0. 569 0.2093 0. 539 0.2065 0. 513 0.2041 0. 490 0.2019 0. h WDF N=7, A ma x w ave digital filt e h eb y chev wave d i . The tables was d u cture of Cheb y c h B 2 A 2 0.3022 0 1 0.2774 0 2 0.2599 0 7 0.2496 0 5 0.2445 0 0 0.2442 0 D F N=3, A max in d 0.2879 0.590 0.2824 0.584 0.2779 0.579 0.2739 0.575 0.2703 0.571 0.2671 0.568 x in dB. 5 B 6 A 2628 0.3597 0 2473 0.3415 0 2384 0.3308 0 2320 0.3232 0 2271 0.3173 0 2231 0.3125 0 2197 0.3083 0 2168 9.3047 0 2142 0.3015 0 2118 0.2986 0 x in dB. e r ig ital filters for v a d esigned for sa m h ev wave di g ital A 31 A 32 .4620 0.7574 .4310 0.6812 .4126 0.6114 .3995 0.5293 .3929 0.4331 .3925 0.3696 d B. 1 0.9830 5 0.9821 7 0.9831 4 0.9805 6 0.9798 1 0.9792 A 71 A 72 0 .6679 0.9940 0 .6522 0.9927 0 .6428 0.9917 0 .6358 0.9911 0 .6304 0.9904 0 .6258 0.9899 0 .6218 0.9895 0 .6184 0.9890 0 .6153 0.9887 0 .6124 0.9883 a rious attenuatio m pling frequency f filter n A max f s =0.5. AUTOMATION&CONTROL-TheoryandPractice150 A max A 1 B 2 A 3 B 4 A 51 A 52 0.10 0.4658 0.2536 0.2161 0.2245 0.4179 0.7374 0.25 0.4197 0.2404 0.2059 0.2132 0.3987 0.6721 0.50 0.3696 0.2311 0.1975 0.2044 0.3869 0.5965 1.00 0.3190 0.2262 0.1911 0.1981 0.3798 0.5168 2.00 0.2610 0.2251 0.1857 0.1933 0.3797 0.4229 3.00 0.2231 0.2265 0.1828 0.1922 0.3830 0.3608 Table 7. Elements of Chebychev WDF N=5, A max in dB. A ma x A 1 B 2 A 3 B 4 A 5 B 6 A 71 A 72 0.10 0.4585 0.2437 0.2022 0.1947 0.1962 0.2123 0.4049 0.7313 0.25 0.4087 0.2316 0.1944 0.1884 0.1903 0.2028 0.3875 0.6550 0.50. 0.3653 0.2250 0.1893 0.1861 0.1867 0.1968 0.3872 0.5925 1.00 0.3158 0.2213 0.1847 0.1834 0.1836 0.1919 0.3735 0.4206 2.00 0.2587 0.2210 0.1805 0.1814 0.1812 0.1880 0.3734 0.3921 3.00 0.2213 0.2227 0.1783 0.1806 0.1802 0.1862 0.3782 0.3589 Table 8.Elements of Chebychev WDF N=7, A max in dB Table 9.Elements of Chebychev WDF N=9, A max in dB 11. Tables of Cauer wave digital filter In tables 10-12 are the elements of Cauer wave digital filters for various attenuation A max in the pass-band and N=3, 5 and 7. The tables was designed for sampling frequency f s =0.5. A max A 1 B 2 A 3 B 4 A 5 B 6 A 7 B 8 A 91 A 92 0.10 0.4554 0.2399 0.1970 0.1886 0.1854 0.1860 0.1908 0.2081 0.4000 0.7287 0.25 0.4064 0.2287 0.1912 0.1847 0.1821 0.1825 0.1863 0.1996 0.3836 0.6569 0.50 0.3636 0.2227 0.1867 0.1823 0.1800 0.1844 0.1834 0.1943 0.3751 0.5908 1.00 0.3145 0.2193 0.1826 0.1804 0.1783 9.1787 0.1811 0.1899 0.3708 0.5123 2.00 0.2578 0.2194 0.1789 0.1790 0.1769 0.1773 0.1793 0.1864 0.3273 0.4196 3.00 0.2205 0.2213 0.1769 0.1784 0.1763 0.1604 0.1647 0.1873 0.3753 0.3583 Fi g T a T a 0 0 0 0 g . 18. Frequenc y r C 0.000 4 0.001 7 0.003 9 0.010 9 0.027 9 0.043 6 0.098 8 0.177 3 0.280 3 1.249 4 a ble 10. Elements A max 0.0017 0.0039 0.0109 0.0279 0.0436 0.0988 0.1773 0.2803 1.2494 a ble 11. Elements A max A s 0 .0004 43.5 0. 6 0 .0017 49.5 0. 6 0 .0039 53.0 0. 6 0 .0109 57.5 0. 5 r esponse and str u A s 4 24.7 7 30.7 9 34.3 9 38.7 9 42.8 6 44.8 8 48.3 3 50.9 3 53.0 4 60.0 of Cauer WDF N A s A 1 41.3 0.657 44.8 0.627 49.2 0.585 53.3 0.543 55.3 0.522 58.8 0.479 61.4 0.446 63.5 0.418 70.5 0.309 of Cauer WDF N A 1 B 2 6 901 0.4211 6 543 0.3827 6 167 0.3611 5 776 0.3353 u cture of Cauer f i A 1 B 2 0.7504 0.57 6 0.7209 0.49 0 0.6673 0.45 6 0.6190 0.40 6 0.5706 0.36 1 0.5461 0.34 5 0.4983 0.31 6 0.4614 0.29 7 0.4306 0.28 5 0.3147 0.25 9 N =3, Ω s =4.8097, K 2 B 2 A 3 0.399 0.33 0 0.373 0.31 1 0.343 0.29 0 0.318 0.27 2 0.306 0.26 4 0.288 0.25 1 0.277 0.24 3 0.270 0.23 7 0.256 0.22 1 N =5, Ω s =2.000, K2 A 3 B 4 0.3521 0.4838 0.3240 0.4464 0.3093 0.4274 0.2924 0.4064 i lter A 31 6 6 0.7314 0 7 0.6664 6 9 0.6272 6 3 0.5778 1 2 0.5339 5 9 0.5140 6 1 0.4804 7 9 0.4590 5 5 0.4442 9 3 0.4118 2 =-0.93686 B 4 A 5 0 0.436 0.7 3 1 0.404 0.6 9 0 0.369 0.6 3 2 0.341 0.5 9 4 0.323 0.5 7 1 0.310 0.5 4 3 0.298 0.5 2 7 0.289 0.5 1 1 0.269 0.4 9 =-0.827, K 4 =-0.62 A 5 B 6 0.4446 0.49 7 0.4166 0.44 7 0.4015 0.42 0 0.3841 0.39 1 A 32 0.9629 0.9374 0.9160 0.8803 0.8365 0.8114 0.7573 0.7110 0.6699 0.4999 5 1 A 52 3 4 0.915 9 0 0.893 3 9 0.856 9 6 0.813 7 7 0.788 4 6 0.736 2 7 0.691 1 4 0.652 9 2 0.487 7 A 71 A 7 2 0.8781 0. 9 7 1 0.7973 0. 9 0 9 0.7502 0. 8 1 2 0.6972 0. 8 A 72 9 364 9 071 8 844 8 482 [...]... 0.4 274 0.4064 A5 0.4446 0.4166 0.4015 0.3841 A32 0.9629 0.9 374 0.9160 0.8803 0.8365 0.8114 0 .75 73 0 .71 10 0.6699 0.4999 A72 0.9 9364 0.9 9 071 0.8 8844 0.8 8482 152 0.0 279 0 0.0436 0 0.0988 0 0. 177 3 0 0.2803 0 1.2494 1 AUTOMATION & CONTROL - Theory and Practice 61.6 63.5 67. 1 69 .7 71 .7 78 .7 0.5 5 375 0.5 5168 0.4 475 8 0.4 4434 0.4 4158 0.3 3084 0.3134 0.3038 0.2880 0. 278 4 0. 272 0 0.2606 0. 278 5 0. 272 5 0.2624... 0.69 90 0.63 39 0.59 96 0. 57 77 0.54 46 0.52 27 0.51 14 0.49 92 A52 0.915 0.893 0.856 0.813 0 .78 8 0 .73 6 0.691 0.652 0.4 87 B6 0.4 97 72 0.4 47 71 0.420 09 0.391 12 A71 0. 878 1 0 .79 73 0 .75 02 0.6 972 able 11 Elements of Cauer WDF N N=5, Ωs=2.000, K2 =-0 .8 27, K4 =-0 .6 27 Ta Amax 0.0004 0 0.00 17 0 0.0039 0 0.0109 0 As 43.5 49.5 53.0 57. 5 A1 0.6 6901 0.6 6543 0.6 61 67 0.5 577 6 B2 0.4211 0.38 27 0.3611 0.3353 A3 0.3521... 0.2558 0.2511 0.2 378 0.38 97 0.3826 0. 371 2 0.3642 0.3593 0.3 475 0.3698 0.3636 0.3534 0.3 470 0.3425 0.3310 0.3 67 72 0.356 69 0.340 01 0.329 95 0.322 21 0.303 35 0.65 27 0.6293 0.6015 0.5823 0.56 97 0.5484 0.8 8056 0 .7 776 6 0 .7 7300 0.6 6862 0.6 6 472 0.4 4845 Ta able 12 Elements of Cauer WDF N N =7, Ωs=1.41421, K2 =-0 .78 15, K4 =-0 0.3491, K6 =-0 .48 67 7 12 Realization o wave digital filters with DS C 671 1 by SIMULINK... r g response and stru ucture of Cauer fi ilter C 0.0004 4 0.00 17 7 0.0039 9 0.0109 9 0.0 279 9 0.0436 6 0.0988 8 0. 177 3 3 0.2803 3 1.2494 4 As 24 .7 30 .7 34.3 38 .7 42.8 44.8 48.3 50.9 53.0 60.0 A1 0 .75 04 0 .72 09 0.6 673 0.6190 0. 570 6 0.5461 0.4983 0.4614 0.4306 0.31 47 B2 0. 576 66 0.490 07 0.456 69 0.406 63 0.361 12 0.345 59 0.316 61 0.2 97 79 0.285 55 0.259 93 A31 0 .73 14 0.6664 0.6 272 0. 577 8 0.5339 0.5140... Ωs=4.80 97, K2 =-0 .93686 Amax 0.00 17 0.0039 0.0109 0.0 279 0.0436 0.0988 0. 177 3 0.2803 1.2494 As 41.3 44.8 49.2 53.3 55.3 58.8 61.4 63.5 70 .5 A1 0.6 57 0.6 27 0.585 0.543 0.522 0. 479 0.446 0.418 0.309 B2 0.399 0. 373 0.343 0.318 0.306 0.288 0. 277 0. 270 0.256 A3 0.330 0 0.311 1 0.290 0 0. 272 2 0.264 4 0.251 1 0.243 3 0.2 37 7 0.221 1 B4 0.436 0.404 0.369 0.341 0.323 0.310 0.298 0.289 0.269 A5 51 0 .73 34 0.69... r the parallel and serial adaptor are 93 B2=0.249 96 A31=0 .70 3 32 A32=0.5293 K=0.1428 3 1=0.669 Fig 26 Low-pass wave digital filter Chebychev inverse with coefficients g s v B 2= =0.2496 A31= =0 .70 32 A32=0.5 5293 K=0.1428 =0.6693 1= 156 AUTOMATION & CONTROL - Theory and Practice No ormalized attenu uation of the low-pass and high-p pass filter is dem monstrated in the figure 27 Fig 27 Normalized attenuation... García-Ugalde and A Romero Mier y Teran.: Synthesis of the low-pass and highpass Wave digital filters.ICINCO 2008 International conference on informatics in control and Robotics May 1 1-1 5, 2008, Madeira-Portugal Rudolf Saal.: Handbuch zum Filterentwurf AEG Telefunken, 1 979 Sanjit K Mitra : Digital Signal Processing A computer-Based Approach McGraw-Hill Companies, INC New York, 1998 160 AUTOMATION & CONTROL - Theory. .. 21 Realization of fifth order wave digital filter by TMS320C 671 1 simulink This project created in code composer studio can be seen in figure 22 and can run on the DSPC 671 1 154 AUTOMATION & CONTROL - Theory and Practice Fig 22 Projects of t realization of Wave digital filte by TMS320C 67 g the er 71 1 13 Bank of filter with Chebyc 3 rs chev and inverse Chebychev wave digital filters v In this example... 1 972 ,25, pp .7 8-8 9 Design of the Wave Digital Filters 159 A.Sedlmayer and A Fettweis.: Digital filters with true ladder configurations Int J Circuit Theory and Appl., 1 973 , 1, pp 1-5 F García Ugalde, B Psenicka and M.A Rodríguez.: Implementation of Wave Digital Filters on a DSP Using Simulink IASTED, Circuits, Siognal and Systems, November 2 0-2 2, 2006, San Francisco, CA, USA B Psenicka, F García-Ugalde... is presented analy s ysis part of the filter bank This part is design s ned by Ch hebychev low-pass filter and inve erse Chebychev high-pass filter The coefficients of the che ebychev low-pas filter are A1= ss =0.33 07, B2=0.24 496, A31=0.3995 and A32=0.5293 The 3 coe efficients of inve erse Chebychev h high pass filter are A1=0.6692, B B2=0.2496, A31=0 0 .70 32, A3 32=0.5293 and K= =0.1428 In the ad dders . 0.813 7 7 0 .78 8 4 6 0 .73 6 2 7 0.691 1 4 0.652 9 2 0.4 87 7 A 71 A 7 2 0. 878 1 0. 9 7 1 0 .79 73 0. 9 0 9 0 .75 02 0. 8 1 2 0.6 972 0. 8 A 72 9 364 9 071 8 844 8 482 AUTOMATION & CONTROL - Theory and Practice1 52 0 0 0 0 0 1 T a . 0.1804 0. 178 3 9. 178 7 0.1811 0.1899 0. 370 8 0.5123 2.00 0.2 578 0.2194 0. 178 9 0. 179 0 0. 176 9 0. 177 3 0. 179 3 0.1864 0.3 273 0.4196 3.00 0.2205 0.2213 0. 176 9 0. 178 4 0. 176 3 0.1604 0.16 47 0.1 873 0. 375 3 0.3583. 3 9 0.856 9 6 0.813 7 7 0 .78 8 4 6 0 .73 6 2 7 0.691 1 4 0.652 9 2 0.4 87 7 A 71 A 7 2 0. 878 1 0. 9 7 1 0 .79 73 0. 9 0 9 0 .75 02 0. 8 1 2 0.6 972 0. 8 A 72 9 364 9 071 8 844 8 482 DesignoftheWaveDigitalFilters