T o d ete rm in e the p ro p e rties o f the lattice WDF using th e equations d eriv e d fo r the fr e q u e n c y and s e n s itiv ity re s p o n s e s , the tra n s fe r fu n c tio n s an d d e r iv a tiv e s fo r th e firs t and second o rd e r A P S 's are required. T h e tra n s fe r fu n ctio n o f th e first o rd e r A P S , illu strated by F ig .(4 .1 4 ), can be d eterm in ed from the s ca tte rin g m a trix f o r th e tw o -p o rt ad a p to r and th e relatio nsh ip b etw een the w ave p aram eters g iv en in E q.(4.3 3 ).
C o m b in in g th e eq u a tio n s o f E q .(4 .3 3 ), th e tra n s fe r fu n c tio n o f the firs t o rd e r s e c tio n can be derived and is g iv en by Eq.(4.34).
T h e a llp a s s n atu re o f th is f i r s t o rd e r sec tio n c a n b e seen from its tr a n s f e r fu n c tio n , w here if th e n u m e ra to r is G (z) then the d en o m in ato r h as th e fu n c tio n G ( z " ') and H i( z ) has a pole at a and a zero at 1 /a . T he s ta b ility o f th is tra n s f e r fu n c tio n is dependent upon th e po sition o f its pole w ith in the unit circ le in th e z
F ig u r e 4 .1 4 F ir s t order APS with w ave p aram eters.
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d o m ain . T o e n su re th a t the pole lies w ith in th e u n it c irc le , then th e section 's m u ltiplier m ust be lim ited to the range -1 < a < 1.
E v alu ation o f the g ro u p delay is based upon an ex p ressio n for the d eriv a tiv e o f the tra n s f e r fu n c tio n w ith re s p e c t to o>, d iv id e d by th at tra n s fe i fu n c tio n . T his p a ram eter for the first o rd e r section is giv en by E q.(4 .3 5).
T h e gain and p h ase c o e ffic ie n t se n s itiv itie s o f E q .(4 .2 6 ) and E q .(4 .2 7 ) are based u p o n an ex p re s s io n f o r the d iffe re n tia l o f e a c h sec tio n w ith re s p e c t to its m u ltip lie r( s ). T h is te rm fo r th e firs t o rd e r A P S o f F ig .(4 .1 4 ), w h ic h has a m u ltip lie r a , is g iv en by Eq.(4.36).
T h e Final ex p ressio n fo r the first order section is th e one required to ev a lu ate the g ro u p d e la y c o e f f ic ie n t s e n s itiv itie s . T h is p a r a m e te r can be d e te rm in e d from E q .(4 .3 7 ).
T h e tra n s fe r fu n c tio n . H2(z ). o f the second o rd e r A P S illustrated by F ig.(4 .1 3 ), can be d ete rm in e d from th e re la tio n s h ip betw een th e eq u a tio n s o f E q .(4 .3 8 ) and is g iven by Eq.(4.39). (4 .3 5 ) H i ( z ) d a ( - a + z / y • - a z ' l ) (4 .3 6 ) d / • I ( 4 a z -1 - ( 1 + a ^ ) ( l + t ^ ) ) ( - a + z ‘ * ) 2 ( 1 - a z* ' ) 2 (4 .3 7 ) d a A4 = z*1 . B4 , A3 - z ' 1 . B2 and A2 * B3 (4 .3 8 ) B | a t ( I • a ) P z ' 1 • z ' 2 **2 ^ A j - 1 + ( 1 - a ) P z * 1 + a z * 2 (4 .3 9 )
Chapter 4. Lattice W D F's page 4/20
A, B,
F ig u re 4 .1 5 S econd o rd e r A PS w ith wave param eters.
The stab ility o f this allp ass function is d e te rm in e d by the p o s itio n o f its com plex co n ju g a te p o les. The s tab ility crite ria o f th is sec o n d o rd e r APS c a n b e determ ined by c o m p a rin g its d e n o m in a to r to the d e n o m in a to r o f a s ta n d a rd sec o n d o rd er sec tio n , g iven by Eq.(4.40).
z 2 + 2 r cos( 0 ) z + r2 (4 .4 0 )
F o r th e stand ard second o rd e r section it is k n o w n ( ll ] stab ility re q u ire s Irl < 1. A p p ly in g th is lim it to the ap p ro p riate p a r a m e te r s o f E q .(3 .3 9 ) r e s u lts in the stability conditions -1 < a < 0 and -1 < p < 1 fo r the second order APS o f Fig.(4.15).
The e q u a tio n o f the seco nd o rd e r sectio n re q u ire d to d ete rm in e th e gro up delay response is show n by Eq.(4.41).
1 d H 2U ) _ Z -1 ( I ■ a » ) ( g - 2 z - l + 8 z - 2 )
H2(z) d o ( a + ( l - a ) P z ' 1 - z ' 2 ) ( - l ♦ ( l - a ) P z * 1 + a z ' 2 )
T h e te rm s re q u ire d fo r the c a lc u la tio n o f th e g ain an d p h a s e co e ffic ie n t s e n s itiv itie s , provided fo r the two m u ltip liers a and p , are g iv en by E q.(4 .42) and E q .(4 .4 3 ) resp ectiv ely .
1 d H2(z) a ________ ( z - 2 - I H 1 - 2 B z - U z ' 2 )_________ H2(x) d a ( a + ( l - a ) P z ' * - z ' 2 ) ( - l + ( l - a ) P z * 1 + a z * 2 ) (4 .4 2 ) 1 H 2(z) d H 2U) dp ____________ z ' 1 ( i • g 2 ) ( z :? _ - u _____________ ( a ♦ ( l - a ) P z ' 1 - z*2 ) ( -1 + ( l - a ) P z * 1 a z ' 2 ) (4 .4 3 )
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T h e fin a l ex p ressio n s fo r th is section are those requ ired to d e te rm in e the group dela y c o e ffic ie n t s e n sitiv itie s. T hese term s for the m u ltip liers a and p , are given by E q .(4 .4 4 ) and Eq.(4.45) respectively.
S o ftw are w ritte n to im p le m ent sim u ltan e o u s m agn itude and p h ase d e s ig n s on the lattice W D F structure fa lls into the two areas o f desig n and a n a ly sis . The design side o f th e softw are is prov ided through a m enu d riven program c a lle d "W DF”. This p ro gram is b ased upon th e optim iz atio n tec h n iq u e s and a lg o rith m s d iscu ssed for th e la d d e r W D F program . A m enu within this program allow s th e u s e r to en te r the o rd e r o f th e lattice, its in itial m u ltip lier v alu e s and frequency s p e c ific a tio n . The p o s itio n an d num ber o f first and second o rd e r A P S's are c a lc u la te d autom atically from th e filte r order. F requency sp ec ification s are en te red as a se t o f vectors that c o n ta in th e frequency ed g e and attenuatio n values. U n d e r th is v e c to r schem e any fi l t e r ty p e c a n be d e fin e d from a lo w p ass to a d u al b a n d p a s s s p ec ificatio n . F req u e n cy sp ec ificatio n s c a n also be d efin e d with d iffe re n t fre q u e n c y ed ges for th e g a in a n d group delay responses. The in fo rm ation ab o ut the la ttic e structu re, its p aram eters and frequency specification can then be recorded in to a data file.
A ll o p tim iz a tio n p aram eters o f th is lattice W D F prog ram are c o n ta in e d w ithin a s in g le m e n u . T h is m enu allo w s one o f the sing le, d ual o r id ea l lin e tem plate schem es to b e selected, along with the num ber and po sition o f th e erro r points at w h ic h th e te m p la te s arc d e fin e d . T h e w e ig h ts fo r th e gain a n d group delay te m p la te s m ay be set in d iv id u a lly o r c a lc u la te d au to m atica lly th ro u g h an o ption
: j z ' * ( p - 2 z * 1 + P z ' 2) ( a + ( l - a ) P z ' 1 - z ‘ 2) ' 2 (
2
z' 2 ( ( 1 - a ) 2 p 2 - 2 a ) - 2 p z * > ( l - a ) 2( l ♦ z * 2 ) + (1 + a 2 ) ( l + z ‘ 4 ) ) ( - l + ( l - a ) P z * 1 + a z ' 2) * 2 ( 4 . 44 ) dÎHÈ
Iz) • dp j z ' 1 ( a 2 - l ) ( a + ( l - a ) P z ' 1 - z * 2) ' 2 ( a ( l + z ' 6 ) + z *2 ( 1 + z *2 ) ( 1 + a ( a - 3 ) + P 2 ( 1 - a ) 2 ) - 4 p z ' 3 ( l - a ) 2 ) ( - l + ( l - a ) P z ' 1 + a z ' 2) 2 ( 4 . 4 5 )4 . 3
L a t t i c e W D F d e s ig n a n d a n a l y s i s s o f t w a r e
Chapter 4. Lattice W D F 's page 4/22
w ith in th e m enu th at ensures th a t an equal d ev ia tio n in each tem plate con trib u tes an eq u al erro r to the overall fu n c tio n . O ther o p tio n s in this m enu allow th e value o f the a n g le d line for tra n s itio n band d e fin itio n s to be a lte re d , th e o ptim iz atio n a lg o rith m to be ch an g ed and v a ria tio n o f the ra tio th a t d ete rm in e s th e re la tiv e c o n trib u tio n s o f the g ain and g ro u p delay erro rs to th e o v erall function. A menu w a lk -th ro u g h o f this program is prov ided in A p pen dix B l. alo ng with an exam ple to illu s tra te th e design p ro c e d u re and o ptim iz atio n op tion s.
A lim itatio n o f the ladder W D F program was im posed by the G H O ST routin es used fo r g ra p h ic a l ou tp ut. The G H O S T ro u tin es re q u ired an en v iro n m e n t w hich could su p p o rt a w indow system , ty p ic a lly a grap h ics w indow w ithin S un to ols. T h is m eant th a t th e la d d e r W D F program co u ld not be run o n d iffe ren t sy stem s ev e n when g ra p h ic s w ere not requ ired. F o r th is reason the an a ly tic al and g raph ical elem ents o f the lattice W D F softw are w ere not included w ith in the "W DF" design program . A m ore v ersatile graph ical system th an the G H O ST ap pro ach w as provided through a prog ram ca lle d M atL a b |9 |. W ith in M atLab a wide ra ng e o f an alytical and graphical p r o c e d u r e s c a n b e a c h ie v e d th ro u g h b u ilt- in fu n c tio n s . A p ro g ra m c a lle d " m lt w d f w as w ritten in the M alL a b pro cedural language to pro v id e an analysis o f any lattice W D F solutions g en e rate d from the "W DF" program .
T h e p ro g ram "m ltw d f" has th re e ele m e n ts. The firs t co n cerns th e en try o f data file s . T h e s e d a ta files arc s to re d in the M atL ab fo rm at and are create d by the d e s ig n p ro g ra m "W D F". T h e s e d a ta file s m ay b e lo aded in to "m ltw df" e ith e r in d iv id u a lly o r as a set. T h is a llo w s the p erform ance o f lattice W D F solutio n s under slig h tly d iffe re n t o p tim iz atio n p a ra m e te rs to be co m p a red d ire c tly . T h e o th e r two ele m en ts o f th is program relate d irec tly to the an a ly sis and d isplay o f a lattice WDF in the freq u en c y and tim e d o m a in s . T h e frequ en cy do m ain s id e o f th e program c a l c u l a t e s the m a g n itu d e , g a i n , p h a s e an d g ro u p dela y re s p o n s e s o v e r an a rb itra ry freq u en c y range. G a in , p h ase and g ro u p d ela y c o e ffic ie n t se n s itiv itie s c a n be ev a lu a te d for individ ual o r sets o f m u ltip lie rs w ithin the lattice stru ctu re , a g a in o v e r an a r b itr a ry f r e q u e n c y ra n g e . T h e fin a l e le m e n t w ith in the freq u en c y d o m ain part o f the p ro g ra m is co n c ern e d w ith the ca lcu latio n o f the p o le s and zeros o f th e stru c tu re . The program h ig h lig h ts the p o le s o f each lattice arm along w ith the zeros o f the overall structure. T he poles and zeros o f the lattice W D F s tru c tu re can be d e te rm in e d from th e o v e ra ll tran sfer fu n c tio n g iv en by E q .(4 .6 ).
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E x p re ssin g th e tra n s fe r fu n c tio n o f each b ra n c h o f the la ttic e in term s o f a n u m erator a n d d en o m in ato r p olynom ial. E q.(4.6 ) c a n be exp ressed as
The poles o f E q.(4.46) are the roots o f the tw o d en om in ator p olynom ials D '(z) and D"(z)- T h e z e ro s can be d eterm in ed from the ro o ts o f the num erator o f Eq.(4.46). T h is m eans th a t the zero s o f the structure cannot b e associated with a single lattice arm in the w a y the poles o f the lattice can.
Each o f th e respo n ses calcu lated is displayed to th e screen through M atLab and the u ser is g iv e n the option o f printing the graphs to a file or laser printer. W hile the fre q u e n c y d o m ain sid e o f the s o ftw a re program ca lc u la te s the filter