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Transfer functions with flat magnitude and flat delay Riml, Otfried Carl 1963

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TRANSFER FUNCTIONS WITH FLAT MAGNITUDE AND FLAT DELAY  by OTFRIED CARL RIML B.Sc.E.E., U n i v e r s i t y  o f M a n i t o b a , 1960  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS MASTER  FOR THE DEGREE OF  OF APPLIED SCIENCE  I n t h e D e p a r t m e n t of Electrical  We  accept  required  this  Engineering  thesis  as c o n f o r m i n g t o t h e  standard  Members o f t h e Department of E l e c t r i c a l The  University  Engineering  of B r i t i s h  .. .January, 1963  Columbia  In presenting  t h i s thesis i n p a r t i a l f u l f i l m e n t of  the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study.  I further agree that permission  for extensive copying of t h i s thesis f o r scholarly purposes may granted by the Head of my Department or by his  be  representatives.  It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of  Electrical  Engineering  The University of B r i t i s h Columbia, Vancouver 8, Canada. Date January 29. 1963.  ABSTRACT  The-coe-f f icien-t--and p o l e - z e r o - l o c a t i o n s o f a t r a n s f e r f u n c t i o n F(s) having imposing  a total  of  a t the  to  first  then  the  and  n poles  (m+n-l) c o n d i t i o n s  phase of F ( s ) adjust  m zeros  origin.  f u n c t i o n s may polynomial  these  be  The solutions,  are  special  m a g n i t u d e and  the flat  magnitude  conditions  relative are  by  and are  used  magnitude of F ( s ) , the  first  slope.  and  in this  w h i c h have n o t  delay  n,  determined  q,  a family  B u t t e r w o r t h and  of Bessel-  cases.  s t e p - f u n c t i o n response and  phase  i n w h i c h the  approach d e s c r i b e d  functions  the  used to a d j u s t  i n d i c e s m,  obtained  functions  A new transfer  be  be  q even d e r i v a t i v e s o f the  (m+n-l-q) e v e n d e r i v a t i v e s of t h e varying  on  I f q of these  (m+n-l-q) c o n d i t i o n s may  By  may  merits  t h e s i s y i e l d s some  been t r e a t e d  i n the  literature.  i s s t u d i e d f o r the r e a l i z a b l e of e m p h a s i z i n g  compared.  flat  ACKNOWLEDGEMENT  The  author would l i k e  t o thank the s u p e r v i s o r  p r o j e c t , D r . A. D. Moore, f o r h i s g u i d a n c e during  and a s s i s t a n c e  the course o f t h i s  research.  The  indebted to the National  Council  author i s a l s o  o f Canada f o r t h e a s s i s t a n c e  of this  received  Research  through a  S t u d e n t s h i p awarded i n 1961 and a d d i t i o n a l s u p p o r t f r o m a National  Research Council  Grant  (BT-68).  TABLE OF  CONTENTS Page  Abstract Table  i i  of Contents  i i i  List  of I l l u s t r a t i o n s  iv  List  of T a b l e s  v  Acknowledgement  vi  Chapter: 1.  Introduction  2.  Flat  3.  L i n e a r Phase A p p r o x i m a t i o n  7  3.1  The  7  3.2  General  All-pole  A New  5.  The. F a m i l y  Rational Functions  9  A p p r o a c h t o L i n e a r Phase A p p r o x i m a t i o n  Flat  of T r a n s f e r F u n c t i o n s w i t h F l a t  ......  13  Delay  Magnitude  21  E q u a l E m p h a s i s and  7.  F u n c t i o n s w i t h Almost  Quasi^-equal E m p h a s i s  26  E q u a l Emphasis  30  7.1  Introduction  30  7.2  Two  Excess  Magnitude C o n d i t i o n s  30  7.3  One  Excess  Phase C o n d i t i o n  32  7.4 7.5  Three E x c e s s M a g n i t u d e C o n d i t i o n s Two E x c e s s Phase C o n d i t i o n s  33 35  Results 8.1  8.2 9.  4  Function .  6.  8.  1  Magnitude A p p r o x i m a t i o n  4.  and  •  36  Coefficients o f the T r a n s f e r F u n c t i o n s R e a l i z a b i l i t y o f the F u n c t i o n  and 36  S t e p Response  39  Conclusion  References  •  54 •  55  L I S T OP  ILLUSTRATIONS  Figure I.  . 2.  3.  Page Linear-ghase Polynomials  Approximations D e r i v e d from B e s s e l 11  P y r a m i d C o n t a i n i n g the F a m i l y Functions  of " F l a t "  Low-pass 22  The F i r s t F o u r H o r i z o n t a l L a y e r s o f t h e T h r e e D i m e n s i o n a l R e p r e s e n t a t i o n o f t h e I n d i c e s m, n, and q i n F ( s , m, n, q)  23  4.  U n i t - s t e p R e s p o n s e s f o r F u n c t i o n s w i t h m=l  n=2  .  40  5.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=l,  n=3  .  41  6.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=l,  n=4  .  42  7.  U n i t - s t e p R e s p o n s e s f o r F u n c t i o n s w i t h m=l,  n=5  .  43  8.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=l,  n=6  .  44  9.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=2,  n=3  .  45  10.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=2,  n=4  .  46  II.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=2,  n=5  .  47  12.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=2,  n=6  .  48  13.  U n i t - s t e p R e s p o n s e s f o r F u n c t i o n s w i t h m=3,  n=4  .  49  14.  Unit-^step Responses  f o r F u n c t i o n s w i t h m=3,  n=5  .  50  15.  U n i t - s t e p Responses  f o r F u n c t i o n s w i t h m=3,  n=6  .  51  L I S T OF  TABLES  Table I.  II.  III.  Page P o l y n o m i a l s f o r some F l a t — m a g n i t u d e delay Transfer Functions .  Flat37-38  R i s e - t i m e and O v e r s h o o t f o r B e s s e l - d e r i v e d F u n c t i o n s w i t h m = 2 and n = 4  52  Comparison of F u n c t i o n s w i t h Magnitude C o n d i t i o n s  52  3 Phase and  3  1 TRANSFER FUNCTIONS WITH FLAT MAGNITUDE AND FLAT DELAY  1.  It  INTRODUCTION  i s a well-known f a c t t h a t  lumped l i n e a r t i m e - i n v a r i a n t forced  a network c o n s i s t i n g o f  elements g i v e s  r e s p o n s e t o an e x p o n e n t i a l  input.  an  exponential  The t r a n s f e r  function,  st or  ratio  type  of the output t o the input  excitation, e  o f network i s a r a t i o n a l f u n c t i o n  frequency,  s, o f t h e i n p u t .  , for this  of the g e n e r a l i z e d  That i s , m  P  ( ) a  N(s) _ D(s)  =  a _Q b  2  + a-, s + a~s 1 + b, s + b ~ s 0 1 2  1—  n  =  n  Z  m  +....+ a s SL_ +•.... + b s n  a. s i  i  /_  . ~  = 1 1  i=9 \  JL  , l b . s l  i=0  where m and n a r e i n t e g e r s , and t h e a's and b ' s a r e r e a l . transfer  function F(s)  (m+n+l) i n d e p e n d e n t approximation. normalization  The  has (m+n+2) c o e f f i c i e n t s and t h u s  c o e f f i c i e n t s or degrees of freedom i n  Two d e g r e e s o f f r e e d o m must be u s e d f o r magnitude and f r e q u e n c y s c a l i n g so t h a t  the remaining  (m+n-l) d e g r e e s o f f r e e d o m a r e a v a i l a b l e t o c o n t r o l t h e magnitude and then  phase c h a r a c t e r i s t i c s .  a r e q "magnitude  conditions",  (m+n-l-q) "phase c o n d i t i o n s " may be imposed. A  and  I f there  " d i s t o r t i o n l e s s " t r a n s f e r f u n c t i o n must have c o n s t a n t magnitude  l i n e a r phase  (or f l a t  delay) along  t h e w-axis i n t h e s - p l a n e .  One  method o f a p p r o x i m a t i n g  adjust derivatives band-centre are  function i sto  o f t h e m a g n i t u d e and phase f u n c t i o n s a t the  frequency.  F o r low-pass  adjusted a t the o r i g i n  w i t h the adjustment functions  such a t r a n s f e r  functions a l l derivatives  of the s-plane.  of the c o e f f i c i e n t s  This thesis  of r a t i o n a l  i n attempting to achieve f l a t  deals  low-pass.-  m a g n i t u d e and l i n e a r  phase. Some p a r t i c u l a r thoroughly function  investigated.  p o l y n o m i a l are used  are obtained.  s u c h an " a l l - p o l e "  linear-phase results  coefficients  to adjust f o r zero  derivatives  I f a l l a v a i l a b l e denominator  function  approximation,  are used  i n the  s=0, t h e B u t t e r w o r t h ^ ^  to achieve a  the B e s s e l - p o l y n o m i a l  t h e s e have b e e n s t u d i e d b y S t o r c h  Considering  have been  I f the numerator o f the t r a n s f e r  the magnitude f u n c t i o n a t the p o i n t  functions in  o f t h e above p r o b l e m  i s a c o n s t a n t and a l l a v a i l a b l e  denominator of  solutions  coefficient low—pass  solutions  (2) (3) ' and Thomson .  t h e B u t t e r w o r t h and B e s s e l t y p e s o f a p p r o x i m a t i o n (4)  as t h e extreme c a s e s obtained transfer two  requirements For  for all-pole  of f l a t  t h e case  magnitude and l i n e a r  obtained  solutions  for  certain variations  m >  Dennis  f o r the s p e c i a l  p l a c e d on m a g n i t u d e f l a t n e s s  other than those  phase  i s made.  i n which the numerator o f the t r a n s f e r (5)  i s not of zero degree,  for  has  f u n c t i o n s i n w h i c h a compromise between t h e  function  is  f u n c t i o n s , Golay  and L i n d e n  ' have  c o n d i t i o n where e q u a l  and phase l i n e a r i t y .  Except  on t h e B e s s e l - t y p e f u n c t i o n s , no  of Dennis  emphasis  solutions  and L i n d e n have p r e v i o u s l y b e e n  found  0. This thesis  i s then concerned w i t h  ( l ) t h e s t a t e m e n t of  the  g e n e r a l problem of f l a t - m a g n i t u d e  approximationj transfer  and (2) t h e t h e o r e t i c a l  linear-phase  characteristics  emphasis  o f m and n .  and d e n o m i n a t o r and t h e  flatness  o f f u n c t i o n s c a n be o b t a i n e d  and B e s s e l - p o l y n o m i a l  f l a t - m a g n i t u d e and  f o r a r b i t r a r y values  on magnitude  o f low—pass  n u m e r a t o r and  are used to achieve  By v a r y i n g t h e d e g r e e s o f n u m e r a t o r  family  development  f u n c t i o n s i n which a l l a v a i l a b l e  denominator c o e f f i c i e n t s  relative  and l i n e a r - p h a s e  and phase  linearity,  i n which the B u t t e r w o r t h  f u n c t i o n s are s p e c i a l  cases.  Certain  members o f t h i s f a m i l y o f f u n c t i o n s n o t p r e v i o u s l y known w i l l be  found.  a  2.  In  a l l available  at the o r i g i n  F(s) i s a constant*  semicircle In magnitude the  coefficients  i n the denominator o f the  f u n c t i o n are used to o b t a i n a f l a t  characteristic of  APPROXIMATION  the B u t t e r - w o r t h ^ ^ approximation f o r the a l l - p o l e  function, transfer  FLAT MAGNITUDE  magnitude  of the s-plane.  Since  t h e numerato  the p o l e s group around the o r i g i n i n a  i n the l e f t - h a l f  s-plane.  o r d e r t o o b t a i n a s u i t a b l e f u n c t i o n on w h i c h the conditions  c a n be imposed,  "squared-magnitude"  i t i s convenient to define  function m  r  A  2i  s  B  2i  s  2i  2i  i=0 ...(2-1)  It  i s also  c o n v e n i e n t t o d e f i n e G(0) = 1 .  The r e l a t i o n s h i p between t h e n u m e r a t o r  c o e f f i c i e n t s of  2 G(-s  ) and o f F ( s ) i s d e f i n e d b y t h e e q u a t i o n  W= [«][.]  ...(2-2)  where  A, 0  a0  A,  a.  A,  2m  L  [a]  =  a,  a  m  5  and  2a  0  \  2a,  -a^  2a,  -2a  \ a  3  2  x  m  The  matrix  [ o c ] i s lower  triangular  and  of order  (m+l)  by  (m+l).  Similarly,  •[BM  •..(2-3)  !»][•>]  where 0  '0  2n  n  [>] =  and  2b, 2b.  [(•]=  " l b  -2b.  b x 2  (-D b n  The  matrix  3  i s lower  triangular  and  of order  n (n+l) by  (n+l).  6 It  i s clear  that  °<- > = W0> 2  D(s)D(-s) - N ( s N(s)N(-s)  +  )N(-sT ...(2-4)  For  a q'th—order  magnitude a p p r o x i m a t i o n ,  |D(S)D(-S) - N(s)N(-s)J  function  band-centre  frequency*  t h e even  must have 2 ( q + l ) z e r o s  a t the  2 q even d e r i v a t i v e s o f Gr(—s ) have 2  Only  t h e r e f o r e been f o r c e d t o be z e r o , s i n c e G ( - s ) i s an even f u n c t i o n and a l l o f i t s o d d - o r d e r d e r i v a t i v e s a r e a u t o m a t i c a l l y zero. point  Since  o n l y low—pass f u n c t i o n s a r e c o n s i d e r e d h e r e * t h e  of approximation The  i s the o r i g i n  flat—magnitude  q+l  where and  is  q+l j^q+ij  a  r  e  s u  [ A ] and [ B ] r e s p e c t i v e l y . A  q+i  is The  of  B  q+i  c o n d i t i o n may be e x p r e s s e d A  q+l  a unit matrix bmatrices  /q+l  B  formed from the f i r s t The t y p i c a l  as ..•(2-5)  q+l  o f ( q + l ) ' t h o r d e r and  A  q+l  ( q + l ) rows o f  element ^ ( r + l )  ^  n  zero f o r m<^r^q<^n-l. degree  of the denominator o f F ( s ) l i m i t s  i n Equation  (2-5).  That  approximation,  q* must be l e s s t h a n  flat—magnitude  characteristic  equal  of the s-plane.  to ( n - l ) .  i s , the order n.  i s clearly  the order  o f t h e magnitude  Thus t h e b e s t p o s s i b l e o b t a i n e d when q i s  T h i s does n o t , however, mean t h a t magnitude  c o n d i t i o n s n e e d be r e s t r i c t e d  to c o n t r o l l i n g  d e n o m i n a t o r o f F ( s ) ; t h e y may a f f e c t  coefficients  coefficients  o f the  o f the numerator.  7  3.  3.1  LINEAR PHASE APPROXIMATION  The A l l - ^ p o l e F u n c t i o n  In the B e s s e l — p o l y n o m i a l a p p r o x i m a t i o n f o r the (2)  (3)  function  '  (7) a l l available  , x  coefficients  a t o r of the t r a n s f e r f u n c t i o n are used flat  i n the  to o b t a i n a  a p p r o x i m a t i o n t o t h e phase s l o p e a t the o r i g i n  s—plane.  all^pole.  denomin-  maximallyof the  The  phase o f t h e n o r m a l i z e d t r a n s f e r f u n c t i o n must —sTJ t h u s a p p r o x i m a t e t h e phase d e f i n e d by e 0 where XQ i s t h e -sT phase s l o p e o r d e l a y * W r i t i n g e 0 as e  0 =  c o s h SXQ  ...(3-1)  STQ  + sinh  -sT the a n g l e o f e  0 f o r s=j« i s e x p r e s s e d  Arg(e~ '^'0) s  = -tan ^ -  4  as  sinh  sT  0 c o s h sT, 0.  , -1 = -tan  1 — 3  tanh  si  ...(3-2) It  i s convenient at t h i s Q = The  1. numerator i n E q u a t i o n  denominator The  p o i n t t o n o r m a l i z e t h e phase s l o p e so  all-pole  ( 3 — l ) i s a c o n s t a n t and  i s w r i t t e n as the sum transfer  f u n c t i o n may  P  where the s u b s c r i p t s  < > S  "e" and  =  of an e v e n and s i m i l a r l y be  DTST"  =  "0" r e f e r  d  e  +  an odd  the function.  expressed  as  ..(3-3) 0  t o t h e even and  odd p a r t s  of  the  denominator p o l y n o m i a l  of F ( s )  of F ( s ) .  Therefore  the  Comparing E q u a t i o n s the  Since  angl  f o r s=g> i s  • •.(3-4)  Arg  that  phase  rational  only  function  are  (3-4),  should  apparent  approximate  tanh(s).  be  h e r e * the  origin  written  e x p a n s i o n a r o u n d the  tanh(s) =  i t i s now  considered  a p p r o x i m a t i o n i s the  f u n c t i o n t a n h ( s ) may  continued-fraction  and  6.^/6.^  low—pass f u n c t i o n s  o f emphasis i n t h e The  (3-2)  o f the  i n an  point  point s-plane.  infinite  s=0  as  1  j  1  s  1 1  Using  the  truncated  approximation to d /d n  e  must  continued-fraction  tanh(s)j  the  e x p a n s i o n as  continued-fraction  an  expansion  of  be  1 1 1 1  (2n-l) s  ...(3-5)  The  c o n t i n u e d - f r a c t i o n expansion  term  in anticipation  o f the  d e n o m i n a t o r o f F ( s ) must be i n the  the  realizable  as a t r a n s f e r  (3—3)  all—pole  the  been t e r m i n a t e d  that after  o f d e g r e e n.  Therefore  the  a l l coefficients  are  real  f u n c t i o n and  and  D(s)  From E q u a t i o n s  transfer  m a x i m a l l y - f l a t phase s l o p e may  Since  n'th  = d + g  &Q  F(s) = l/D(s) i s p h y s i c a l l y  function.  low—pass  a t the  . reassembly  of  f u n c t i o n i s a reactance  a Hurvitz polynomial.  T  fact  c o n t i n u e d — f r a c t i o n expansion  positive, is  has  (3-^5) and  f u n c t i o n which gives  a  i n g e n e r a l be w r i t t e n as  (2n)I *(s)=-^ Y~  ,..(3-6) (2n-i)l 2' N  The  denominator i n E q u a t i o n  the B e s s e l - p o l y n o m i a l . negative  an  Taylor series  c o n t a i n any  ii(n-i)i  X  (3-6)  i s the  the  s=0  transfer phase  terms between t h e  zeros  first  does  and  phasej not  the  of F ( s ) o f d e g r e e  (2n+l) th t  n.  Functions  f o l l o w i n g method may  be  used to  i f m and  n are n o t b o t h  odd,  the  i s o b t a i n e d when the B e s s e l - p o l y n o m i a l  f u n c t i o n of 2m  the  phase about  functions g i v i n g maximally-flat approximation  slopes  function  Rational  m>0>  HOT  polynomial.  expansion  of the  of  the  to l i n e a r  even o r odd  General  i s therefore  ( n — l ) ' t h - o r d e r approximation  power f o r a d e n o m i n a t o r p o l y n o m i a l  3.2  g e n e r a l i z e d form  phase o f F ( s )  o f the phase d e f i n e d by  In such the  The  i  g  (m+n)'th d e g r e e  i n quadrantal  i s m u l t i p l i e d by  symmetry  such t h a t m  find to  the  desired transfer all-pole  a polynomial zeros  canoel m  having poles  10 of  the a l l - p o l e f u n c t i o n .  m z e r o s and n p o l e s * Figure  The r e s u l t i n g t r a n s f e r f u n c t i o n has  Some i l l u s t r a t i v e  examples a r e shown i n  1. The  polynomial  m u l t i p l i c a t i o n o f t h e a l l - p o l e f u n c t i o n by a changes t h e m a g n i t u d e  change t h e phase f u n c t i o n . half-plane  poles  phase f u n c t i o n  c h a r a c t e r i s t i c b u t does n o t  That  i s , by r e p l a c i n g  by r i g h t - h a l f - p l a n e  a l o n g t h e (o-axis  zeros,  i s produced.  number o f d i f f e r e n t s o l u t i o n s  left-  no change i n t h e  more t h a n one s o l u t i o n f o r e a c h s e t o f v a l u e s the  some  Obviously there i s o f m and n«  If  i s N, t h e n f o r even m and  e v e n n,  and  f o r odd m and e v e n n,  \r  -  J  £^  In e v e r y one o f t h e s e N s o l u t i o n s  '.  f o r F ( s ) , a l l (m+n-l) a v a i l a b l e  degrees of freedom are used to approximate in  a maximally—flat  was imposed for  on F(s)«  F(s) will  sense; not a s i n g l e  constant  magnitude  I t i s however c o n c e i v a b l e  have a f l a t t e r  magnitude  time  delay  condition  that  one s o l u t i o n  c h a r a c t e r i s t i c than a l l  11 All-pole BesselPolynomial Approximatio^  x  Polynomial  o  Linear—phase Function with F i n i t e Zeros  o  O  y 5-  -y-  o a)  o  o  m = 2, n = 5  CO  O  o  O  o  <5~  (b)  m = 2, n =  (0  X  o  4.  A ©  o  T  A io  o  o  X 6-  -eX  o (c) Figure  1.  Linear-phase nomials.  o m = 3 , n = 4 .  Approximations  D e r i v e d from B e s s e l P o l y -  12 other  (N-l) s o l u t i o n s .  on a t r i a l - a n d — e r r o r  This  optimum  solution  c a n o n l y be  basis.  I f b o t h m and n a r e odd, t h e method d e s c r i b e d not l e a d  t o any  found  solution.  above does  13 4.  A NEW APPROACH TO LINEAR PHASE APPROXIMATION  Consider  the generalized  phase  slope  or delay  .••(4-1)  The f u n c t i o n P ( s ) / P ( ~ s )  i s an a l l - p a s s f u n c t i o n w h i c h , f o r  st=j<tt| has u n i t m a g n i t u d e and a phase of F ( s ) ,  angle which  i s twice  that  Writing  s  F  U  _ NU1 _ e - DTs) ~ d n  V( ) ;  +  0 + d n  ...(4-2) n  and  n  - n„ 0 d - d e 0 e  ...(4-3)  n  where n P(s)*  g  and n-Q a r e t h e even and odd p a r t s  and d  g  and  a r e t h e even and odd p a r t s  ln(n  1 2  of the numerator o f  d  ' + d », 0 d + d e 0 e  n  e  of the denominator*  + n ) - ln(d  e  + d ) - ln(n  + ln(d  e  - d )  Q  d  d » 0 d - d e 0 e  n  Q  e  - n ) Q  Q  n  '+ n ' e 0_ n + n„ e 0  n  n„ • 0 n - n„ e 0 e  ...(4-4) where t h e p r i m e d e n o t e s t h e d e r i v a t i v e w i t h  respect  A f t e r b r i n g i n g t o common d e n o m i n a t o r s , t h e d e l a y  to s.  i s found to  14  be  T(*\  0  d  d  e  - e d  V  d  0  n  0  n  -0 d  n  e  ' e  e  -  n  0  n  n  0  0  Since N(s)N(-a) = n  -  2  n  2 A  and  D(s)d(-s) = d ^ - d ' Q  e  and b o t h n u m e r a t o r s i n E q u a t i o n  (4-5) a r e even f u n c t i o n s  o f s,  t h e d e l a y becomes  m-rl  n—1 .2i  V  2i  c  2i  C  T  ( s )  =  i o_  ML-  f  2  ^  1  i=0  Equation  (2—3)•  The  __ (4  A A  2i  s  2  6)  i  i=0  where t h e d e n o m i n a t o r p o l y n o m i a l s (2-2) and  S  a r e as d e f i n e d  i n Equations  r e l a t i o n s h i p between t h e n u m e r a t o r s o f  (4-6) and t h e t r a n s f e r  function F(s) turns  [<=] = M M  o u t t o be  ..•(4-7)  15 with  a0  '0  a.  [0]  w =  =  C  a,  2m-2  0  a  m  and  a  ]  3a  c  5a  E  —a  2  3a,  a.  « •  m  0  The  matrix  is  0  lower  0  0  triangular  and  of order  (m+l)  by  (m+l)  Similarly  [B]  = [i]  [b]  .•.(4-8)  wher e D  0  '0  [D] = D  2n-2 n  l  b  3b  3  5b  5  \  ""2 -3b  *  \  4  •  *  • »  •  •  »  ( - l )  The  matrix The  £&J  i s lower  triangular  d e l a y f u n c t i o n may  T(s)  and  a l s o be  P(s)P(-s) _ ~ D(S)D(-S)  of order  n  ^  n"  (n+l) by  (n+l)  w r i t t e n as  Q ( s ) Q ( -s N ( S ) N ( -s  P(s)P(-s)N(s)N(-s) - Q(s)Q(-s)D(s)D(-s) D(s)D(-s)N(s)N(-sl  A s s u m i n g as b e f o r e j t h a t  TT(0)  =  = 1^ D  Q  -  C Q = b^ — a^ = 1  17  then  ...(4-9)  _ P(s)P(-s)N(s)N(-s)-Q(s)Q(-s)D(s)D(-s)-D(s)D(-s)N(s)N(-s) D(S)D(-S)N(S)N(-S)  must have m u l t i p l e z e r o s a t s=0 i f f l a t to be a c h i e v e d * of degree the  More p r e c i s e l y ,  d e l a y o r l i n e a r phase i s  s i n c e R ( s ) i s an e v e n p o l y n o m i a l  2(m+n)$ and R ( s ) / T ( s ) must have a t l e a s t  origin,  e a c h phase c o n d i t i o n must f o r c e an a d d i t i o n a l  of z e r o s t o l i e a t s=Oi  There c a n t h e n be a maximum o f  phase c o n d i t i o n s u s e d t o f l a t t e n  the d e l a y f u n c t i o n .  magnitude c o n d i t i o n s i m p o s e d , t h e r e r e m a i n conditions  two z e r o s a t  and t h e r e f o r e 2(m+n-q) z e r o s  Thus t h e f o l l o w i n g m a t r i x  [a]  equation  [»J  - [ffl]  (m+n-l)  With q  (m+n-l—q) phase  o f R ( s ) may  l i e a t s=0.  results?  [o  J - [a] W  (4-10)  •••  where  k  0  '0  '0  A, D.  N =  A 0  2m  '2m-2 0  pair  D 0  2n-2  18  with  A,  0  A. A.  A0  \  A,  A  0 \  [a] A  2m  O  2m  k  0  and  B  o  \  B  2  B  0  B  4  B  2  B  6  B  4  \  X  V B  •  [a] =  •  • •  •  * B  2  ft  \  X B  o  • • •  x  •  \ X  2n X  o The o r d e r written  of E q u a t i o n  X  •  B 2n  (4-10) i s (m+n-q).  B,  Equation  (4-10) may  be  as  ***(4-ll)  19 Since  the t r a n s f e r  (m+n+l) o f t h e s e  coefficients  to u n i t y .  Then a  Q  A  a single  are independent, the l e a d i n g  A l s o , t h e magnitude o f F ( s ) c a n be n o r m a l i z e d  so t h a t F ( 0 ) = 1.  The m a t r i x  and o n l y  &Q, i n t h e n u m e r a t o r p o l y n o m i a l o f P ( s ) c a n be made  coefficient, equal  f u n c t i o n has (m+n+2) c o e f f i c i e n t s  1t  =  D  = 1,  Q  p r o d u c t Q,]  Q B  = 1, = 1*  Q  i n Equation  matrix  (4-11) c a n be w r i t t e n as  Then  or  o  D  2  P  4  P  D  "l + c  "l  D  1  •  =  •  1  2  P  A  N  o.  C  2  y*- ->s.  4  P  6  P  1 ^  2  \  4  P  l  2  x  * • •  \ •  •  ••  •  •  ^n-2 •  0  2 +  0  •  \ •  *  A  2m-2  A  2m  +  C  2m-2  0  •  0  1  where  =  B  2  "  P  4  =  B  4  "  P  6  =  B  6  "  A  A  A  2 4  - P A 2 2  6  "  r  P  A  2 4 A  "  P  4 2 A  etc. The m a t r i x E q u a t i o n the q f l a t — m a g n i t u d e  (4-12) w i l l  be s i m p l i f i e d  c o n s i d e r a b l y when  c o n d i t i o n s on P ( s ) a r e t a k e n  into  account.  The  r e s u l t i n g m a t r i x [CP] w i l l  order  (m+n-q).  0 \k^(q+l), <  still  be l o w e r  t r i a n g u l a r and  The k ' t h magnitude c o n d i t i o n ,  will  make t h e t e r m  P  equal to zero  = B,^ whe i n [CP],  2k Therefore  \  [<p] =  1  2q+2 2q+4  -* x  P  2q+2  x  Q  2q+2  Although Equation  (4-12) does n o t y i e l d  f l a t - d e l a y approximation*  i t will  a simple  solution f  n e v e r t h e l e s s prove  useful  21 5.  THE FAMILY OF TRANSFER AND  From the p r e c e d i n g  FLAT MAGNITUDE  two c h a p t e r s ,  maximum number o f c o n d i t i o n s and  FUNCTIONS WITH FLAT DELAY  i t i s apparent  that the  on t h e phase o f F ( s ) i s (m+n—l)  t h a t t h e maximum number o f c o n d i t i o n s  on t h e m a g n i t u d e i s  (n-1). Where defined the  i t i s unique^ the t r a n s f e r f u n c t i o n i s  completely  i f t h e number o f z e r o s j m, t h e number o f p o l e s , n - and 9  number o f m a g n i t u d e c o n d i t i o n s , q, a r e s p e c i f i e d *  I t may  therefore  be w r i t t e n as P(s»m^n,q)«  remaining  (m+n-l-q) d e g r e e s o f f r e e d o m are u s e d t o a d j u s t t h e  phase s l o p e  in a flat  I t i s understood that the  sense*  I f m, n> and q a r e p l o t t e d a l o n g axes i n s p a c e , t h e l i m i t s  three  perpendicular  on t h e s e v a r i a b l e s m$ n  f  and q a r e  1 ^ i J t <^ oo m  <  O ^ q  These l i m i t s all is  define  n  <  n.  a rectangular  pyramid w i t h i n which  p o s s i b l e c o m b i n a t i o n s o f m, n, and q must lie„ a three-dimensional  sketch  of t h i s  pyramid,,  Figure  2  // / /  /  \  Figure  2.  The f i r s t  2  I  3  4  /  /  S  Pyramid C o n t a i n i n g the F a m i l y of " F l a t " Functions four horizontal  i n F i g u r e s 3 (a) t o  layers  (i.e.,  Low-Pass  m=0,l,2,3) a r e shown  (d)» (5)  For equal  e m p h a s i s ^ as d e f i n e d by D e n n i s and L i n d e n  ,  (m+n—1-q) = q  if  (m+n-l) i s even*  and /)  (m+n-l-q) = if  (q-1)  (m+n—l) i s odd. Only  h e r e be  the f i r s t  called  the t r u e sense  of these  conditions,  q = (m+n-l-q)^  will  e q u a l e m p h a s i s , s i n c e i t i s e q u a l emphasis i n (i.e.,  (m+n—l)/2  = q).  Number o f P o l e s , n 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8  17 8 9  1!,2 3 4 5 6 17 8 9  1 2 3 4 5 617 8 9 • V  B P P P P P I P P P \E x  R R C E R R C 0 E C R  1 2 3 4 5 6 7 8 9  Qk^TQ  Q Q Q Q Q Q Q Q \E \  \ c  R P E R| \C O E  7/ R  R  I  f  \0 E IR < C !0 E R. C R 0,  \y  x  r  X  C  [H>  x Ci m=0  m=l  m=2  m=3  (a)  (b)  '•(c)  (d)  B ~ Butterworth C ~ Butterworth—derived P ~ Bessel—polynomial Q ~Bessel—polynomial-derived (multiple solutions exist f o r m=l)  Figure  3.  The F i r s t F o u r  E~Equal—Emphasis 0 ~ Q u a s i — e q u a l emphasis G~ F u n c t i o n s o b t a i n e d b y G o l a y R ~ N e w l y added f u n c t i o n s  f o r m=0  except  H o r i z o n t a l Layers of the Three-dimensional I n d i c e s m,: n, and q i n F ( s , m j n , q ) .  Representation  of the  24 The  other  equal  (m+n-l-q) = ( q - l ) , w i l l be c a l l e d  quasi-  emphasis, because the d e r i v a t i o n of the t r a n s f e r f u n c t i o n  follows the  condition,  t h e same p a t t e r n  as t h a t f o r e q u a l  number o f magnitude c o n d i t i o n s  emphasis,  actually  although  e x c e e d s t h e number  o f phase c o n d i t i o n s b y o n e . 3,  In Figure identify  t h e equal-^emphasis c o n d i t i o n  quasi-equal-emphasis The  condition  "P"  letter  functions  derived  are used t o  (m+n--l even) and t h e  (m+n-1 o d d ) , r e s p e c t i v e l y .  i d e n t i f i e s the a l l - p o l e  approximations to f l a t  outlined  " E " and "0"  the l e t t e r s  delay*  while  Bessel-polynomial  "Q" i d e n t i f i e s  from t h e B e s s e l - p o l y n o m i a l s  i n C h a p t e r 3 ; t h e y w i l l be c a l l e d  rational,  by t h e p r o c e s s "Bessel-deriVed"...  F o r m 4 I, where s u c h s o l u t i o n s e x i s t j t h e y may be m u l t i p l e . 1  For  all-pole  functions  satisfying  magnitude c o n d i t i o n , t h e l e t t e r rational  functions  "B" i s u s e d ; f o r more  flat-  general  i n w h i c h t h e maximum p o s s i b l e number (n-d)  of magnitude c o n d i t i o n s identification.  the Butterworth  This  i s imposed, t h e l e t t e r  "C" i s u s e d f o r  t y p e o f f u n c t i o n may be o b t a i n e d  by t h e  (6) approximation technique devised will  be c a l l e d Functions  by  by Moore  v  1  ; such a f u n c t i o n  "Butterworth-deriyed". t h a t were f i r s t  found by G o l a y ^ ^  are i n d i c a t e d  t h e l e t t e r "G". I t w i l l be shown t h a t s o l u t i o n s c a n be e a s i l y  for  obtained  f u n c t i o n s which s a t i s f y the conditions  or  (m+n-d-q) = ( q - 2 ) ,  (m+n-1 e v e n ) ,  (m+n-l-q) = ( q + l ) ,  (m+n-1 o d d ) .  In the f i r s t  case, there  t h a n phase c o n d i t i o n s one  more phase  Relative  will  imposed;  condition  be two more m a g n i t u d e i n the second, there  than there  are magnitude  to the s o - c a l l e d equal-emphasis  o c c u r when one phase c o n d i t i o n condition  cases^  i s replaced  conditions vill  be  conditions.  these  solutions  by a magnitude  f o r (m+n-l) e v e n , and v i c e v e r s a  f o r (m+n-l) o d d .  These two c a s e s a r e r e f e r r e d t o as h a v i n g two e x c e s s m a g n i t u d e conditions Figure  and one e x c e s s phase c o n d i t i o n  3, t h e l e t t e r R W  The satisfying  letter the  M  respectively.  i s used t o i d e n t i f y  In  them.  "R" i s a l s o u s e d t o i n d i c a t e  functions  conditions  or  (m+n—1-q) = ( q - 3 ) ,  (m+n-l o d d ) ,  (m+n—1-q) = (q+2),  (m+n-l  even),  where t h e t r a n s f e r f u n c t i o n has a n u m e r a t o r p o l y n o m i a l o f first  degree It  several  (m=l).  i s apparent that  restrictions  Identification  some o f t h e f u n c t i o n s  s i m u l t a n e o u s l y ; where t h i s  i s made i n h i s t o r i c a l  sequence.  satisfy i s t r u e , the  26 6.  The general  EQUAL EMPHASIS AND QUASI-EQUAL EMPHASIS  derivation to follow  r e s u l t than that Writing  i s more d i r e c t and g i v e s  o f D e n n i s and L i n d e n  the t r a n s f e r f u n c t i o n P(s)  F(s)  = e0  0(s)  =  (  s  a more  '.  as  )  ...(6-1)  then  But  d_  and  d_ dco  If  flat  delay  will  .,.•(6-2)  a Ev  F(j<o)  In  and f l a t  equal-emphasis d0/ds -  Arg  ln[F(s)]  a  Od  d£ ds  d$ ds  m a g n i t u d e a r e combined  or the quasi—equal-emphasis have  (m+n—l) z e r o s  i n e i t h e r the  sense, the f u n c t i o n  a t the o r i g i n  of the s-plane.  Forming the d e r i v a t i v e of 0 ( s ) w i t h r e s p e c t  ds  d_ ds _ N-' ( s ) D ' (s ) ~ N(s) " D(s)  whe re  N  1  (s)  t o s,  ...(6-3)  (6-4)  27 and  4^ial  D-(.)  S i n c e T Q = 1, t h e e q u a t i o n  N'(s)P(s) - D((s)N(s) _ _ N(s)D(s) - l - U  has a r o o t Rewriting  of order  , ..,^-5; (  6  (m+n~l) a t s=0.  (6-5) as  N'(s)D(s)  - D'(s)N(s) = -N(s)D(s),  and s u b s t i t u t i n g f o r  N ( s ) = 1 + a, s + a s 1 2  2  0  N'(s)= a,+2a s +3a~s 1 2 3  2  0  D(s) = 1 + b,s + b ~ s 1 2 +2b s + 3 b s 1 2 3  D'U)^.,  the f o l l o w i n g  2  0  2a  4a^  2  l  -  l  b  - 2b  2  2a- b L  3  - 4b ^  etc.  m  + ... + a s , m +ma s m  +  + a s , n ' +na s " , n  ,  m - 1  n  + . . . . •.  1 1  1  i s obtained!  = - 1 = "  b  + &2^± — &-]j£>2 — 3b-j= - a  + 2a-jb-^ -  3  + . . . . .  s e t o f (m+n) e q u a t i o n s  a  3a^  a  2  + a~s 3  l" 2  -  a  i  a  i  b  a b 2  i 1  b  2  a  'l"2  3 ...(6~6)  28 F o r m=l, a ^ = 0 f o r i ^ > 1 , a n d t h e E q u a t i o n s expressed  i n terms o f t h e s i n g l e  variable,  (6-6) may be a-^.  Therefore  t>l = a ^ + 1  _ f l . 1_ II 21  •L D  2 -  +  a b  3  l  1  3T  2 l  =  +  ..,(6-7)  or a1  , b  r  ,1  (r-l).  =  +  rT  where 0<^r<C^ n + l j  Similarly,  and k  n + 1  =  Ov  f o r m=2 a ^ = 0 f o r i ^ > 2 , a n d f  b  l r  =  l  a  +  ^  (r-2)l  =  (r-l)  +  I  rT  +  •••( ~8) 6  where l < ^ r ^ n+2, and b  ,-j = b  n+1  n+2  = 0,  F o r m=3j t h e r e s u l t i s  = b  2  =  b  r  =  a  +' 1 2  +  a  l  IF  +  (r-3)!  +  U^2TT  +  (r-l)l  +  r t  ...(6-9)  29 where and  These r e s u l t s w i t h more t h a n t h r e e equal-emphasis  ,, = b , _ = b , „ = 0 . n+1 n+2 n+3  may be e x t r a p o l a t e d f o r l o w - p a s s f u n c t i o n s z e r o s where t h e e q u a l - e m p h a s i s  or quasi—  condition holds.  In g e n e r a l , with the  b  2<^r^n+3  coefficients  equal  emphasis  o r q u a s i - e q u a l -emphasis  f  o f F ( s ) may be w r i t t e n as  (m+n-r)j nl  v, b  r  (m+n)j ( n - r ) j vl  =  °< < r  n  ...(6-10)  =  / -| \ r (m+n-r) I ml (m+n) I ( m - r ) !  r i  ^/ / ° «  m  (6-11)  These  expressions r  compact and g e n e r a l for  f o r the c o e f f i c i e n t s  than those  the equal-emphasis  a  r  and b  r  a r e more , (5)  o b t a i n e d b y D e n n i s and L i n d e n  and q u a s i - e q u a l - e m p h a s i s  conditions.  30  7.  7.1  FUNCTIONS WITH ALMOST EQUAL EMPHASIS  Introduction  The leads  r e g u l a r i t y i n the equal-emphasis  to a search  for regularities  to the equal-emphasis focussed the  here  condition.  functions naturally  i n f u n c t i o n s which That  are c l o s e  i s , the i n t e r e s t i s  on t h e d e r i v a t i o n o f f u n c t i o n s w h i c h  satisfy -  requirements  or  These  (m+n-l-q) - ( q - 2 ) ,  (m+n-1 e v e n ) ,  (m+n-l-q) = ( q + l ) ,  (m+n-1 o d d ) .  two r e q u i r e m e n t s were d e f i n e d  i n Chapter  m a g n i t u d e c o n d i t i o n s " and "one e x c e s s phase  5 as "two e x c e s s  condition"  respectively.  7.2  Two E x c e s s M a g n i t u d e  Inspection that the  of Equations  (6-7),  (6-8),  and (6-9) w i l l  a l l c o e f f i c i e n t s , b ^ * may be e x p r e s s e d i n terms last  equation,  b . m +  = Oj i s o m i t t e d .  n  b  For  Conditions  r  rt  =  (  r  a  l  +  1 )  »  That  °< <^ r  of  i s , f o r m=l  n  b  r  =  n+l-r n rl,  , n+r ra, + —pr 1 n+1  ,  i f 9  •••(7-1)  m=2,  0<r<n  show  31 For  m=3,  (n+l-rj|n+2-r) ( n + l ) ( n ) ri  r  a  l  +  , n+2r "H+2 .(7-3)  n  Therefore,  the e x p r e s s i o n  f o r a denominator  coefficient  i n t h e t r a n s f e r f u n c t i o n w i t h m z e r o s and n p o l e s i s  b  =  ( n - l ) i (m+n-l-r): r i ( n - r ) . (.m+n-2)I  n + r(m-l) m+n-l  ra.  0<r<n  From E q u a t i o n nitude  (2—5)* the l a s t  conditions i s A ^  I n a l l i n s t a n c e s where A^  must be z e r o .  ...(7-4)  of the equations  expressing  = ^2q* (m+n-l-q) = ( q - 2 ) , t h e c o e f f i c i e n t  Thus t h e r e l a t i o n s h i p B,,^ = 0 may be  expressed  as  2b b b o ~ „ =0 n n—2 n—1 D  n  f o r m-m  = 3, *  or 2 b  n n-4 b  ~  2  W n - 3 ' n-2 = ° ° +  b  etc.  Using  Equation  f  rM  =  5  ' ...(7-5)  (7~4) i n E q u a t i o n  (7-5), the s o l u t i o n f o r  a.^ i s  a,  + pa, + q = 0  ...(7—6)  where P = 1 and q  _ ~  mn (m+n-1)(m+n+l)  2 Equation  (7-6) c a n n o t  have any complex r o o t s s i n c e 4q<^p «  That i s , 4mn (m+n)  2  <  - 1  i  or l<(n-m)  But  f o r l o w - p a s s f u n c t i o n s m<^n  complex r o o t s c a n e x i s t . roots root  of Equation  ( 7 - 6 ) must be r e a l  leads to a t r a n s f e r  d e f i n e d t o be t h e d e g r e e  7«3  function/which  Qne E x c e s s  unity,  and t h u s no  and n e g a t i v e .  t h e two  The l a r g e r  f u n c t i o n w i t h a denominator  f u n c t i o n which i s r e a l i z a b l e  transfer  by a t l e a s t  S i n c e p and q a r e p o s i t i v e j  which i s always non-Hurwitz*  is  2  polynomial  The s m a l l e r r o o t l e a d s t o a t r a n s f e r f o r n<^njj, where, f o r f i x e d  m,  of the denominator of the f i r s t  i s non-realizable.  Phase C o n d i t i o n  F o r t h e case  o f one e x c e s s  phase  c o n d i t i o n , the l a s t  e q u a t i o n i n the R e l a t i o n s h i p (4-12)* which expresses the conditions f o r f l a t in  d e l a y j must be u s e d  a^, u s i n g E q u a t i o n  (7—4).  t o determine  The r e s u l t may a g a i n be  the polynomial expressed  33 as  &J  2  where  n  ...(7-7)  p = 1  j a  + pa-,^ + q = 0  mn * = (m+n-l)(m+n+l) '  d  The  same argument as above i s u s e d  of Equation here  (7-7) a r e b o t h r e a l  t o prove  that the roots  and n e g a t i v e .  The s m a l l e r r o o t  again leads t o a t r a n s f e r f u n c t i o n which i s r e a l i z a b l e f o r  n<n .. R  7.4  Three  If  Excess  Magnitude C o n d i t i o n s  (m+n-l-q) = ( q — 3 ) ,  (6-6) may be u s e d according  The  the f i r s t  as b e f o r e *  to E q u a t i o n  The r e m a i n i n g  2, q < ^ n - l ,  two a r e e x p r e s s e d  (2-5) as  B  2(q-1) = °  B  2q  ...(7-8)  = °  right-hand side of Equation  Chapter  (m+n-2) e q u a t i o n s i n  (7-8) i s z e r o , b e c a u s e  from  so t h a t m<^n-4.  F o r m=l, E q u a t i o n s  b^ = K  ( 7 - 8 ) must be s o l v e d w i t h  (ra, + 1 ) ,  0<r<n-l  ...(7-9)  34  For  m=2,  r  b  rt  =  [ ( - ) 2 r  r  1  a  +  r  a  l  »  +  0<r<n  For  m=3,  ,  (n-r+l) r  /  ,\  2 2 n +nr+r -n-2r n-(n-l)  Since the transfer  equations  w h i c h must be  If Equations  and  3  Since  two  l  . +  a  r o o t s are  functions  complex  negative  which f a l l s  +  a^t  m=l.  or  i s the  (7-8)  3 the /  f o r m=lj  f o u r t h degree p o l y n o m i a l  a  l  +  (j>-6) i s  , 1 3(n+3)  of Equation  °  have a n e g a t i v e only r e a l  ...(7-12)  A =  i n a^  obtained:  (7-12) a r e real  r o o t , and  positive, root.  t h a t the  It other  conjugates.  real  r o o t f o r a^ p r o d u c e s a t r a n s f e r f u n c t i o n  i n t o the p a t t e r n e s t a b l i s h e d by  for  the  when m=2  substituted into  , n+4 3(n+3)  equation w i l l  shown t h a t t h i s  The  2 l  ...(7-11)  s o l v e d to d e t e r m i n e  analogous to  a l l coefficients  third-degree  can be  i n the  a polynomial  a  the  , 0<r<n  difficult.  (7-^9) a r e  the l e a d i n g c o e f f i c i e n t zero,  , r(n+r-2)  f u n c t i o n s a r e q u a d r a t i c i n a^ and  s o l u t i o n becomes much more  is  (7-10)  the  other  transfer  35 7.5  Two  Excess  If may  be  Phase C o n d i t i o n s  (m+n-l-q) =  used  (q+2), t h e f i r s t  as b e f o r e *  The  remaining  equations  i n R e l a t i o n s h i p (4-12).  equations  i n a^ and  equations w i l l  the  attempted  Equation  (7-8) the  s u b j e c t was  Here a l s o  The  solution  t h e two two  and  the  not pursued  last  ;"vv.''.:/rvV*.r:  to these  rr.U>-(i/>.  The  equivalent relationships  i s t o be  c o n d i t i o n s o f two  t o the nature any  (6-6)  argument  also.  solved i f a solution  Due  in  here.  cases w i t h t h r e e excess magnitude conditions.  are  t r e n d towards a s e t of e q u a t i o n s  d e g r e e w h i c h must be  phase  two  (7—12) i s a g a i n o b t a i n e d .  r o o t s h o l d s here  Equations delay e s t a b l i s h  result*  2  n o t be  F o r m=l, concerning  a  (m+n-2) e q u a t i o n s  further.  of these  for  of h i g h e r  found f o r excess  equations*  the  36 8,  8.1  C o e f f i c i e n t s o f t h e T r a n s f e r F u n c t i o n and R e a l i z a b i l i t y of t h e F u n c t i o n  The values be  RESULTS  coefficients  defining  o f m, n, arid q a r e g i v e n  compared w i t h F i g u r e 3*  transfer i n Table  Functions  functions f o r certain I; t h i s  i n the f i r s t  row o f F i g u r e s 3 (a) t o (d) a r e r e a l i z a b l e since  t h e B e s s e l p o l y n o m i a l s , from  have z e r o s o n l y i n t h e l e f t pole  half  f u n c t i o n (m=0), G o l a y ^ ^  are a l s o  obtained  also  should  horizontal  as t r a n s f e r f u n c t i o n s  which they  c a n be d e r i v e d ,  of the s-plane.  F o r the a l l -  has shown t h a t r e a l i z a b l e  i n the second  horizontal  With the e x c e p t i o n of the B u t t e r w o r t h has  table  functions  row o f F i g u r e 3 ( a ) .  functions (q=n-l),  shown t h a t n o n — r e a l i z a b l e f u n c t i o n s r e s u l t  Golay  f o r m=0,  n>4. If  t h e p r o c e s s by w h i c h t h e y a r e d e r i v e d i s c o n v e r g e n t ,  f u n c t i o n s which f a l l also  satisfy  under the heading  the Hurwitz  In those  of  "Butterworth-derived"  realizability condition.  f u n c t i o n s f o r which m ^ 0 r e a l i z a b i l i t y  was  assured f o r m = 1, n<^ 6, m = 2, n<^ 7, m = 3, n<^ 9.  Note t h a t t h e second of Dennis confirms n = 40.  and L i n d e n ^ \ their  estimate  who  of these  limits  contradicts that  estimated n<^6,  t h a t the l i m i t  lies  but t h a t the t h i r d  between n = 8 and  m  n  q.  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6  0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5  -4.30629 -3.33333 solution -2.76393 -2.50000 -2.74220 -2.39546 -2.00000 -2.11325 solution solution -1.72673 -1.66667 -1,86731 -2.01130 solution -1.54478 -1.42857 -1.46447 solution  not  2 2 2 2 2 2 2  3 3 3 4 4 4 4  0 1 2 0 1 2 3  multiple solution -4.00000 multiple solution -3.53615 -3.33333  solutions possible not f o u n d 5.00000 6.00000 solutions possible not f o u n d 3.84037 ° 6.46385 3.33333 6.66667  a xl0  2  2  a  Table  Q  = b  I.  not  not not  a xl0  3  3  found  found found ft  not  Q  found  b-jXlO  1  b xl0 2  b xl0'  b^lO*  3  b xl0-  b xl0  5  6  5.69371 6.66667  1.54812 1.66667  7.23607 7.50000 7.25790 7.60454 8.00000 7.88675  2.23607 2.50000 2.45420 2.60454 3.00000 2.88675  2.84701 4.16667 4.38120 4.68935 6.66667 6.10042  0.39091 0.83333 0.64459  8.27327 8.33333 8.13269 7.98980  3.27327 3.33333 3.13269 3.00890  8.03301 8.33333 7.33012 6.70000  1.28878 1.38889 1.05449 0.96960  1.13862 1.38889 0.75847 0.84734  0.36796  8.45552 8.57143 8.53553  3.45552 3.57143 3.53553  8.94275 9.52381 9.34434  1.59203 1.78571 1.72859  1.89674 2.38095 2.23139  0.66042 1.98413 1.68500  1.50000  1.66667  1.84789 2.00000  2.82629 3.33333  found  0.19327 0.27778  = 1  P o l y n o m i a l s f o r Some Flat-<  lay Flat-magnitude  Transfer Functions.  m  n  q  a^xlO  a xlO'  1  2 2 2 2 2 2 2 2 2 2 2  5 5 5 5 5 6 6 6 6 6 6  0 1 2 3 4 0 1 2 3 4 5  multiple solution solution -2.85714 -2.95876 multiple solution solution -2.56025 -2.50000 solution  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  4 4 4 4 5 5 5 5 5 6 6 6 6 6 6  0 1 2 3 0 1 2 3 4 0 1 2 3 4 5  multiple solution solution -4.28571 multiple solution solution -3.90891 -3.75000 multiple solution solution solution -3.33333 -3.41886  I.  bjXlO-  1  Q  = b  Q  b xlO  J  2  b xlO*  b-.xlO' 4  b5xlQ-  3  7.14285 7.04124  2.38095 2.29966  4.76190 4.45706  0.59524 0.52749  0.39683 0.31215  7.43975 7.50000  2.62836 2.67857  5.75155 5.95238  0.84265 0.89286  0.80918 0.89286  1.42857  1.90476  0.11905  1.67977 1.78571  2.65837 2.97619  0.24465 0.29762  0.10467 0.14881  2.08333 2.02224  3.96825 3.76461  0.49603 0.45530  0.39683 0.34592  solutions possible not found not found 7.14286 -4.76190 5.71429 solutions possible not found not found 5.88684 -3.50589 6.09109 5.35714 -2.97619 6.25000 solutions possible not found not found not found 4.16667 -1.98413 6.66667 4.41103 -2.18776 6.58114 a  Table  a^xlO"  2  b xlO 6  0.41235 0.49603  0.16534 0.13140  = 1  P o l y n o m i a l s f o r Some F l a t - d e l a y F l a t -magnitude T r a n s f e r  Functions(Cont' d).  ^ oo  Figure  The  resulting  realizability  3.  The e s t i m a t e d a r e a  patterns  a r e shown i n  o f n o n - r e a l i z a b i l i t y has been  shaded.  8.2  S t e p Response  Generally, transfer  for a unit-step  functions  shows s u s t a i n e d  which are  ringing.  input,  close  Figures  responses f o r t r a n s f e r functions  to the l i m i t  i s generally  Functions equal—emphasis realizability The extent  w i t h 0<^m<^3 and m<^n^6. rise-time with  the equal-emphasis  c o n d i t i o n a n d w h i c h are limit  d i s p l a y the highest  zeros,  Table  I t appears t h a t  lowest  f o r f u n c t i o n s w i t h q = 1.  i n f u n c t i o n s w h i c h were d e r i v e d  increases in  which s a t i s f y  of r e a l i z a b i l i t y  located  or quasi-  close  to the  overshoot.  o v e r s h o o t a n d r i s e — t i m e may be p r e d i c t e d  polynomials. the  obtained  with  4 t o 15 show t h e u n i t - s t e p  W i t h few e x c e p t i o n s * t h e b e s t overshoot  the output  t o some  from t h e B e s s e l  the l a r g e r the r a d i a l  distance to  t h e l o w e r i s t h e o v e r s h o o t ; however, t h e r i s e — t i m e appreciably  II.  f o r these f u n c t i o n s .  An example  i s given  Figure  5.  U n i t — s t e p Responses f o r F u n c t i o n s  w i t h m •==•!* n = 3  Figure  7.  U n i t - s t e p Responses f o r F u n c t i o n s  w i t h m = 1, n =  5.  Time  Figure  8.  Unit-step  Responses f o r F u n c t i o n s  with  m = 1,  n = 6,  ^  F i g u r e 10*  u n i t - s t e p Responses f o r F u n c t i o n s w i t h m = 2,  n -  4  Figure  14«,  U n i t — s t e p Responses f o r F u n c t i o n s  with  m = 3,  n =  5  Figure  15•  U n i t — s t e p Responses f o r F u n c t i o n s  with  m = 3  t  m =  6.  Zeros  Poles  Bandwidth (rad/sec.)  Rise-time (sec)  Overshoot (%)  2.516+J4.493  -4.248+J0.867 -3.735+J2.626  2.049  0.96  0.02  3.735+J2.626  -4.248+J0.867 -2.516+J4.493  5.750  0.50  2.80  4.428+J0.867  -3.735+J2.626 -2.516+J4.493  7.378  0.35  14.42  Table  I I . R i s e - t i m e and O v e r s h o o t f o r B e s s e l - d e r i v e d F u n c t i o n s w i t h m = 2 and n = 4.  If like  a c o m p a r i s o n i s made between f u n c t i o n s h a v i n g  number o f phase c o n d i t i o n s as w e l l as a l i k e  a  number o f  magnitude c o n d i t i o n s , t h o s e w h i c h have a n u m e r a t o r o f low d e g r e e have t h e h i g h e s t o v e r s h o o t of  while  m = n d i s p l a y the lowest  those  approaching  the c o n d i t i o n  Rise-time (sec.)  Overshoot  overshoot.  Bandwidth (rad/sec.)  (*)  1 zero  6 poles  5.62401  0.45  22.27  2 zeros  5 poles  5.62867  0.44  8.56  3 zeros  4 poles  6.98963  0.43  3.06  Table  III.  Comparison of F u n c t i o n s w i t h Conditions.  This behaviour  i s understandable,  3 Phase and 3 M a g n i t u d e  s i n c e the f u n c t i o n s  w h i c h have a n u m e r a t o r o f low d e g r e e a r e c l o s e r t o t h e l i m i t o f realizability. the  As t h i s  left-half-plane  limit  of r e a l i z a b i l i t y  p o l e s move c l o s e r  i s approached,  to the ©-axis.  The u n i t —  step response  therefore displays  m a g n i t u d e and t h e d e l a y r e s p o n s e peaks.  h i g h o v e r s h o o t and b o t h the show p r o n o u n c e d  resonance  54 9.  The  CONCLUSION  conditions f o r f l a t  been d e r i v e d .  can be d e t e r m i n e d .  f u n c t i o n s have b e e n c a l c u l a t e d  o t h e r s a method f o r s o l u t i o n has been  c h a n g i n g the emphasis from f l a t It  the  was  shown once more t h a t  Some of t h e s e  shown,  magnitude transfer  t o show t h e r e s u l t to f l a t  characteristic  as w e l l  as a good t r a n s i e n t  magnitude  response.  t h e same emphasis i s g i v e n t o phase and m a g n i t u d e  conditions*  transfer functions  a  response which i s improved c o n s i d e r a b l y  transient  transient the  delay.  f u n c t i o n s must be o f  non-minimum-phase t y p e i n o r d e r t o have a f l a t  If  have  completely while f o r  A number o f g r a p h s have been p r e s e n t e d of  delay  I t has b e e n shown t h a t by c o m b i n i n g them* a  number o f t r a n s f e r f u n c t i o n s transfer  m a g n i t u d e and f l a t  response of f u n c t i o n s  same emphasis on f l a t  f o r which  (n-m)  f o r which  (n-m)  d e l a y and on f l a t  i s small, e x h i b i t over the  i s large*  magnitude  with  i n both  cases. F u n c t i o n s w h i c h have equal  e m p h a s i s , e x h i b i t p r o n o u n c e d peaks i n t h e i r  characteristics close a  as w e l l  to the l i m i t  good compromise This  far  equal emphasis, or a p p r o x i m a t e l y  of n o n — r e a l i z a b i l i t y , . between f l a t  response i f they are Otherwise they  m a g n i t u d e and f l a t  study i s not complete because those  from equal  studied.  as p o o r t r a n s i e n t  emphasis i n t h e t a b l e  I t would appear t h a t  magnitude  i n Figure  such cases l y i n g  delay.  cases which l i e 3 have n o t b e e n some d i s t a n c e  f r o m t h e e q u a l - e m p h a s i s g r o u p m i g h t n o t have u n i q u e and  i t w o u l d be o f i n t e r e s t  give  solutions  t o d e t e r m i n e what t h e c r i t e r i o n f o r  s e l e c t i o n w i t h i n a s e t of s o l u t i o n s  should  be.  55 REFERENCES  1*  Butterworth,  S., "On t h e T h e o r y o f A m p l i f i e r s " , E x p . W i r e l e s s » 7*536-541 ( O c t o b e r , 1 9 3 0 ) .  2.  S t o r c h , L.,  3.  Thomson, W.E. , "Networks w i t h M a x i m a l l y F l a t D e l a y " , W i r e l e s s E n g r . . 29s256^263 ( O c t o b e r , 1 9 5 2 ) , and 298309 (November, 1 9 5 2 ) .  4.  G o l a y ^ M.J.E., " P o l y n o m i a l s o f T r a n s f e r F u n c t i o n s w i t h P o l e s O n l y S a t i s f y i n g C o n d i t i o n s a t the O r i g i n " , T r a n s . I R E . CT-78224-229 (September, I 9 6 0 ) ;  5*  D e n n i s j F.L.,  "Synthesis of Constant Time-delay Ladder Networks U s i n g Be.ssel P o l y n o m i a l s " * P r o c . IRE, 4281666-1675 (November, 1 9 5 4 ) .  and L i n d e n ^ D.A., "The D e r i v a t i o n o f P o l e — z e r o P a t t e r n s by D e r i v a t i v e A d j u s t m e n t " , J o u r n a l of t h e F r a n k l i n I n s t . , 2688283-293 ( O c t o b e r , 1959). 4  6.  M o o r e , A.D.,  "Synthesis of D i s t r i b u t e d A m p l i f i e r s f o r P r e s c r i b e d A m p l i t u d e R e s p o n s e " , Ph.D. Thesisj ( E l e c t r i c a l E n g i n e e r i n g ) $ S t a n f o r d U n i v e r s i t y j, S E L i Tech* R p t . 53 (September 1, 1 9 5 2 ) .  7.  Krall*  and F i n k j r Os* "A New C l a s s o f O r t h o g o n a l Polynomials! The B e s s e l P o l y n o m i a l s " , T r a n s . Am« Math* Soc., 65 ( J a n u a r y , 1 9 4 9 ) .  H.L.,  

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