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D e s ig n and Im p le m e n ta tio n
o f
Linear P h a se
W ave D ig ita l F ilters
B y
A. P. S. Jones
A thesis subm itted for the D egree of
Doctor o f Philosophy
D epartm ent o f E ngineering
U niversity o f W arwick
S y n o p s i s
A steady increase o f research w ithin the field o f digital sy stem s has resulted ip a w ide acceptan ce o f th e discrete approach to system design. R esearch has p roduced d iscre te tech n iq u es th a t com plem ent tho se alre ad y in use in th e analogu e dom ain. A rap id im p ro vem ent in the p erform ance and av a ila b ility o f d ig ita l hardw are h as prom pted a m ove fro m analogue to d ig ita l sy stem s, esp e cially w ithin the field o f s ig n a l p ro c e s s in g .
T h is th esis considers th e design o f W ave D ig ital F ilters (W D F 's) to satisfy arbitrary m a g n itu d e and p h a s e s p e c if ic a tio n s w ith fin ite w o rd le n g th c o e ffic ie n ts . It describ es the s tru c tu re s and properties o f lad d e r and lattice W D F's related to linear p h ase desig n th ro u g h co e ffic ie n t s e n s itiv ity and n o n m in im u m -p h a se.
T he in itial part o f t h i s thesis c o n c e n tra te s upon the d esig n and co m p ariso n o f o p tim iz a tio n te c h n iq u e s to s a tis fy m a g n itu d e -o n ly and s im u lta n e o u s lo w p ass frequ ency s p e c ific a tio n s upon lad d e r and la ttic e W D F's. E x p erim e n ts confirm the u n s u ita b ility o f th e la d d e r W D F fo r s im u lta n e o u s d e s ig n s b e c a u s e o f th e ir m in im u m -p h a s e c h a r a c te r is tic s . S u c c e s s fu l s im u lta n e o u s lo w p a s s d e s ig n s u p o n la ttic e W D F's w ere a c h ie v e d th rough q uasi-N ew to n alg o rith m s u sin g a dual line tem plate schem e and a w eighted L p -m e tric e r ro r fu n ctio n .
T he All P ass S e c tio n s (A P S 's ) u sed to c o n s tru c t the low p ass la ttic e W D F w ere in v estig ated and a ra n g e o f A P S's c o n sid ered th at w ou ld allo w the lattice W D F s tru c tu re to s a tis fy h ig h p a s s , s in g le b a n d p a s s and d u al b a n d p a s s freq u en c y s p e c ific a tio n s . S p e c ia l c a s e A P S 's for s in g le and dual b a n d p a s s d esig n s w ere generated by a p p ly in g frequency tran sfo rm atio n s to th e 1st and 2nd o rd e r low pass A P S 's. E q u atio n s an d c h a ra c te ris tic s fo r th ese A P S's are d e ta ile d along with a nu m ber o f exam ples o f filter deigns.
T he final area o f th is thesis co n cerns the desig n o f fin ite w o rdlcn gth solu tion s to m a g n itu d e -o n ly a n d s im u lta n e o u s fr e q u e n c y s p e c if i c a t i o n s , ra n g in g fro m lo w pass to dual b a n d p a s s type re sp o n ses. U sing the large w o rd lc n g th so lu tio n s g e n e ra te d th ro u g h t h e q u a s i-N e w to n o p tim iz a tio n t e c h n iq u e s as s ta r tin g co e ffic ie n ts, a H o o k c -J c e v e s direc t sea rch alg orithm w as im p le m en ted to g enerate fin ite w o rd le n g th s o lu tio n s .
T ec h n iq u es d eta ile d in th is thesis pro v id e a m ethod fo r the g en e ratio n o f finite w o rd le n g th c o e f f ic ie n ts th a t s a tis fy a r b itr a ry m a g n itu d e -o n ly and s im u lta n e o u s frequency s p e c ific a tio n s through o p tim iz atio n fo r the lattice W D F's.
C o n t e n t s
T itle P a g e ... i
D edication... ii
Synopsis... i i i C o n t e n t s ... iv
A c k n o w l e d g e m e n ts ... ix
D eclaratio n ... x
A bbreviations... xi
C hapter I In troduction... 1/1 1.1 D iscrete S ystem P ro p e rtie s... 1 / 1 I I I L i n e a r i t y ... 1 /2
I 1 2 S h i f t - I n v a r i a n c e ... 1 /3 1.1.3 S t a b i l i t y ... 1 /3
1.1.4 C a u s a lity ... 1 /3 1.2 Phase and G roup D e la y ... 1 /6
1.2.1 C h ara cteristics o f L in ea r P h a s e ... 1 /1 0
1.2.2 M in im um - and N o n m in im u m -P h a s e ... 1 /1 4 1.3 F in ite W ordlcngth E ffects... 1 /1 6 1.4 W ave Digital F ilter (W DF)... 1 /2 0
1.4.1 C ircu it D e s c rip tio n s ... 1 /2 0 1.4.2 S t r u c t u r e s ... 1 /2 5 1.5 R esea rch O b je c tiv e s ... 1 /2 8 1.6 S u m m a r y ... 1 /2 9
R e f e r e n c e s ... 1 /3 0
C hapter 2 Design A pp roaches... 2 /1 2.1 Existing M ethods... 2 /1
2.2 F ilte r S tr u c tu r e s ... 2 / 2 2.3 Dom ain O p tio n s ... 2 / 6
2.4 C o e ffic ie n t G e n e ra tio n ... 2 /1 1 2.4.1 O p tim iz atio n C o n s id e ra tio n s ... 2 /1 2
2.5 D esign C hoice - S um m ary... 2 /1 8 R e f e r e n c e s ... 2 /1 9
C hap ter 3 Ladder W D F s... 3 /1 3.1 D esign C h o ic e s ... 3 /1 3.1.1 R eference c irc u it o p tio n s... 3 / 2
3 .1 .2 O p tim iz a tio n c o n s id e r a tio n s ... 3 / 4 3.2 L adder W D F equations... 3 / 7
3 .2 .1 I n t e r c o n n e c t i o n ... 3 / 7 3 .2 .2 O v erall sy stem e q u a tio n s ... ... 3 /1 2
3 .2 .3 B uild in g B lo c k s ... 3 /1 7 3 .3 W A V E : two-port W D F design p ro g ram ... 3 /2 2
3.4 E x p e rim e n ta l R e s u lts ... 3 /2 7 3.4.1 M a g n itu d e -o n ly d e s i g n ... 3 /2 7
3 .4 2 S im u lta n e o u s d e s i g n s ... 3 /3 3 3.5 T w o -p o rt desig n c o n c lu s io n s ... 3 /4 1
R e f e r e n c e s ... 3 /4 3
C hap ter 4 Lattice W D F s... 4 /1 4.1 Design O ptions... 4 /1
4 .1 .1 L attice WDF structures... 4 /1
4 .1 .2 O p tim iz a tio n c o n s id e r a tio n s ... 4 / 8 4 .2 L attice WDF eq u a tio n s... 4 /1 1
4 .2 .1 O v erall system e q u a tio n s ... 4 /1 2
4 .2 .2 B uild in g B lo c k s ... 4 /1 8 4 .3 L attice W D F d esig n and analysis s o ftw a re ... 4 /2 1 4 .4 E x p e rim e n ta l R e s u lts ... 4 / 2 4 4 .4 .1 M a g n itu d e -o n ly d e s ig n ... 4 /2 4
4 .4 .2 S im u lta n e o u s d e s i g n s ... 4 / 2 6
4 .5 L attice W D F design conclusions... 4 / 4 3 R e f e r e n c e s ... 4 /4 5
C hap ter 5 WDF Frequency Transform ations... 5 /1 5.1 F requency T ra n sfo rm s... 5 /1
5 .2 F requen cy transform ed lattice W D F e le m e n ts ... 5 / 6 5.3 C h a ra c te r is tic s o f freq u en c y tra n s f o rm a tio n s ... 5 / 1 0 5 .4 D e sig n c o n s id e ra tio n s w ith freq uen cy t ra n s f o rm s ... 5 /2 2
5 .4.1 D e sig n a p p r o a c h e s ... 5 /2 2 5 .4 .2 O p tim iz a tio n c o n s id e r a tio n s ... 5 /2 4
5.5 D e sign e x a m p le s ... 5 /2 6 5 .5 .1 M a g n itu d e -o n ly d e s i g n ... 5 /2 6
5 .5 .2 S im u lta n e o u s d e s i g n s ... 5 /3 6 5 .6 C onclu sions... 5 /4 2
R e f e r e n c e s 5 /4 5
C h ap ter 6 F inite W ordlcn gth D esigns... 6 /1 6.1 Finite W ordlength E ffe c ts ... 6 /1
6.1.1 F req u e n cy d o m a in s im u la tio n ... 6 / 2 6 .1.2 T im e d o m ain s im u la tio n s ... 6 / 1 4 6 .1.3 L attice W D F im p le m e n ta tio n ... 6 / 2 0
6 .2 Design fo r finite w o rd lc n g th ... 6 /2 5 6.2.1 O p tim iz a tio n c o n s id e r a tio n s ... 6 /2 5
6 .2.2 Design te c h n iq u e s ... 6 / 2 7 6 .3 D esign e x a m p le s ... 6 / 2 9
6 .4 C onclusions... 6 /3 7 R e f e r e n c e s ... 6 /3 9
C hapter 7 L attice W D F D esign Exam ple... 7 /1 7.1 I n t r o d u c t i o n ... 7 /1
7 .2 F ilte r S p e c i f i c a t i o n ... 7 / 2 7.3 M ag nitud e-O nly D e s ig n ( I d e a l)... 7 / 6
7 .4 M agnitude-O nly D e sig n (F inite)... 7 / 1 0 7 .5 S im u ltan e o u s D e sig n ( I d e a l) ... 7 /1 2 7 .6 Sim ultaneous D esign (F in ite )... 7 / 1 6
7 .7 Design S um m ary... 7 / 1 9
C hap ter 8 Discussion and C onclusion s... 8 /1
8.1 P ro ject O u tlin e ... 8 /1 8 .2 Sum m ary o f W D F s tru ctu re s and p ro p e rties... 8 / 2 8.2.1 W D F s tru c tu re s ... 8 / 2
8 .2.2 F requency T ra n s fo rm s ... 8 / 3 8.2.3 F in ite W o rdlcn gth Effects... 8 / 3
8.3 Summ ary o f D esign O ptions... 8 / 4 8.3.1 O ptim ization T e c h n iq u es... 8 / 4
8.3.2 W D F Design M e th o d o lo g ies... 8 / 7 8 .4 C onclusions... 8 / 9
8.4.1 W D F's for L in ea r Phase D e sign ... 8 / 9 8.4.2 D esign T e c h n iq u e P erform an ce... 8 /1 2
8.5 Future W ork... 8 /1 3
A ppendix A Tw o-port B uilding B locks... A / l
A 1 S e rie s In d u c to r ... A /2 A2 S eries C a p a c ito r... A /7
A3 S eries T u n e d In d u c to r /C a p a c ito r... A /1 2
A 4 P a ra lle l I n d u c to r ... A /1 7 A5 P ara lle l C a p a c i to r ... A /2 2
A6 P ara lle l T u n e d In d u c to r/C a p a c ito r... A /2 7 A 7 Unit E le m e n t... A /3 2 A8 D esign E x a m p le s ... A /3 4
A8.1 S o urce D e sig n... A /3 4 A 8.2 Load D e sig n ... A /3 8
A8.3 M id d le D e s ig n ... A /4 1
Appendix B D e sig n Program D escriptio ns... B / l
B1 Design P ro g ra m ‘ellip*... B /2 B2 Design P ro g ra m * w d f... B /7
B3 A naly sis P ro g ra m ‘m l t w d f ... B /15
Appendix C Lattice W D F APS M odels (Frequency D om ain)... C /l Cl Highpass A P S M o d els... C /2 C l. l 1st o r d e r Highpass A PS... C /2
C1.2 2nd o r d e r Highpass A P S ... C /3 C2 Single B a n d p a ss APS M odels... C /4 C2.1 2nd o r d e r S ingle B andpass A P S ... C /4
C2.2 4th o r d e r S ingle B andpass A P S ... C /5
C3 Single B a n d sto p APS M odels... C/8 C3.1 2nd o r d e r S ingle B andstop APS... C/8 C3.2 4th o r d e r S ingle Bandstop A P S ... C /9
C4 Dual B an d p ass APS M o dels... C / l 2 C4.1 4th o r d e r Dual Bandpass APS... C / l 2 C4.2 8th o rd e r Dual Bandpass APS... C / l 5
C5 Dual B an dsto p APS M odels... C / l8 C5.1 4th o r d e r Dual B andstop A P S ... C / l8
C5.2 8th o rd e r Dual B andstop A P S ... C/21
Appendix D L attice W D F APS M odels (Time domain)... D / l
D1 Tw o-port A d a p to r M o del... D /2 D2 Lowpass A P S M odels... D/6
D2.1 1st o rd e r Lowpass APS M odel... D/6 D2.2 2nd o rd e r Lowpass APS M odel... D /7
D3 Highpass A P S M o d els... D/8 D3.1 1st o r d e r H ighpass APS M odel... D/8 D3.2 2nd o r d e r H ighpass APS M o d el... D /9
D4 Single Bandpass APS M odels... D /1 0 D4.1 2nd o rder Single B andpass APS M odel... D /1 0
D4.2 4th o rd er S ingle B andpass A PS M odel... D / l l
DS Single Bandstop APS M o dels... D /1 3 D5.1 2nd o rder Single B andstop APS M odel... D /13 D3.2 4th o rder Single B andstop A PS M odel... D /1 4
D6 Dual Bandpass APS M o d els... D /1 6 D6.1 4th o rd er Dual B andpass A PS M odel... D /1 6
D6.2 8th order Dual B andpass A PS M odel... D /18 D7 Dual Bandstop APS M odels... D /2 1 D7.1 4th o rder Dual B andstop A PS M odel... D/21
D7.2 8th order Dual B andstop A PS M odel... D /2 3
A c k n o w l e d g e m e n t s
T h e a u th o r w ould like to ac kn ow ledg e th e h e lp an d su p p o rt o f p e o p le ;
D r S tu art Law son for ideas, d iscussion an d en c o u rag e m en t as
on all aspects o f my thesis.
R ichard G reen for research ideas and CASE aw a rd with M arconi.
T o n y W icks for TM S32010 coding o f my lattice W D F structure and
T he a u th o r would also like to acknow ledge the fin a n c ia l support o f b o d ie s ;
S c ie n tific and E ngineering R esearch C o u n cil
M arconi U nderw ater System s Ltd
W a rw ic k U n iv ersity
the fo llow ing
m y sup erv isor
APS's.
the following
D e c l a r a t i o n
T h e work in this th esis h as been discussed in the follow ing c o n fe re n c e papers;
Law son. S. S. and Jones, A. P. S., "D esign o f W ave D igital F illers w ith Prescribed M agnitude and Phase R equirem ents". Proc. ECCTD '89, No. 308, F eb ru a ry 1989, p p l-
5.
Jones, A. P. S. and Law son, S. S., "An Approach to the design o f D ig ital Filters with
P re s c rib e d M ag n itud e and L in e a r P hase C h a ra c te ris tic s ”, 10th S a ra g a C olloquium 'D igital and Analogue F ilters and F ilters S ystem s', M ay 1990, p p 3 /l-4 .
Jo nes, A. P. S., Law son, S. S. and W icks, A., "Design o f Cascaded A llp a s s Structures w ith M ag n itu d e and D elay C o n s tra in ts usin g S im u late d A n n e a lin g and Q uasi-
New ton M ethods", Proc. ISC A S-91, S ingapore, IEEE.
Law son, S. S., W icks, A. and Jon es A. P. S., "Design and Im plem entation o f Cascaded A llp ass D igital F ilter S tru ctu re s with M agnitude and Delay C o n s tra in ts ", IEE Proc.
6th Int. C on f on DSP in C om m unications , Loughborough, 1991, p p31 -33.
W ith in th ese c o n fe re n c e p a p e rs th e au th o r w o u ld like to a c k n o w le d g e the c o lla b o ra tio n , ideas and d is c u s s io n s held with Dr. Law son c o n c e rn in g the nature
and c h a rac teristics o f the W ave D igital F ilter and A. W icks in v o lv in g Sim ulated A n n e a lin g o p tim iz a tio n te c h n iq u e s .
A b b r e v i a t i o n s
T h e fo llow ing th is ab b reviation s are used w ith in th is th esis
VVDF W ave Digital F ilter
D S P D igital S ignal P ro cessin g
L T I L in ea r T im e In v a ricn t
D FT D iscrete F o u rier T ansfo rm
F F T Fast F o u rier T ran sform
L B R L ossless Bounded Real
M A P M axim um A v ailable Pow er
A P S All Pass Section
D T L Doubly T erm inated L ossless
F I R F in ite Im pulse R esponse
I I R In fin ite Im p u lse R espon se
C h a p t e r 1
In trod u ction
D ig ital filters m ay be found in a large range o f d igital s y stem s, from dom estic co m p a ct disc p la y e rs to m issile g u idan ce sy stem s. A lthough the p rin cip les o f a d ig ita l filter are co m m o n across each ap p lica tio n , the p ro p e rties and perform ance
o f a specific d ig ita l fille r will depend upon the operation and requirem ent o f the o v erall system . A d ig ita l filter is designed to a lte r the frequency com ponents o f an
in p u t s ig n a l to a g iv en s p e c ific a tio n . F o r a n u m b er o f a p p lic a tio n s , th is s p e c ific a tio n is o n ly co n c ern e d w ith the m ag n itu d e c h a ra c te ris tic s o f a signal.
H o w e v er, a p p lic a tio n s th at also re q u ire th e p h ase re la tio n s h ip b etw e en the frequ ency c o m p o n e n ts o f a signal to rem ain u nd istorted , are c o n stra in ed to using
d ig ita l filters th at e x h ib it a lin ear p hase ch a rac teristic.
1 . 1
D i s c r e t e
S y s t e m
P r o p e r t i e s
A ny system m ay be d efined as an o p e ra to r o r tra n s fo rm a tio n , ac tin g upon an in p u t to p ro d u c e a co rresp o n d in g o u tp u t. T h e n atu re o f a tran sfo rm atio n is
d eterm in ed by these inputs and outputs. A d iscre te system uses in p uts and outputs th a t are a s e q u e n ce o f sam ples, re p rese n tin g a p a rtic u la r s ig n a l. Any discrete
tra n s fo rm w ould th e re fo re be con stra in ed to pro du ce a d is c re te ou tp u t from a d iscrete input. An in p ut sequence {..., x (i), x (i+ l), x (i+ 2 )... x(j),...} may be considered as a vector, x , o f w hich the "n,h sample" is x(n). T h is may be form ally written as
x » { x(n) } , -oo < n < oo
A d ig ital system w ou ld represent these sign als thro ugh a seq u e n ce built up from
sam p les o f the sig n al taken at a regular tim e interval. T his tim e interval is known as th e sam pling p e rio d , T , and is related to the sam pling freq u ency, Fs . by the
eq uatio n T = 1/FS. If a sequence represents a tim e varying signal then it is usual to
d e fin e the seq u en ces as having a fin ite num ber o f elem ents, N . tak en from when tim e equals zero. U n d e r these definitions, a sequence can be w ritten as,
X » (x(0), x (l), x(2)...x(n)... x(N -l)) . O s n S N -l
F o r every input seq u en ce, x . there will be a corresponding ou tpu t sequence, y . The o p eratio n o f a d is c re te system is therefore to use a set o f ru les or transform ations to c o n v e rt an in p u t s e q u e n c e to th e a p p r o p ria te o u tp u t s e q u e n c e . A
Chapter 1. Introduction page 1/2
ele m e n t o f a sequence in iso lation o r ab o u t previou s input an d /o r ou tpu t sam ples. E x am p les o f these types o f o p erations are g iven in E q .( l .l ) , w here E q .( l .l a ) show s
a s q u a rin g fu n c tio n , E q . ( l . l b ) g e n e ra te s an ou tp u t ele m e n t from a n u m b er o f in pu t ele m e n ts and E q . ( l .l c ) com b ines b o th input and o utp ut elem en ts to calculate th e n ex t o u tp u t elem ent.
y(n) = (x(n ))2 , -oo < n < oo (1 .1 a )
So if x * (..., x (i-l), x(i), x ( i + l ) ....) - > y a {..., (x (i-l))2, (x (i))2, (x (i+ l))2, ...) y(n) = x(n) + x (n -l) - x(n-2) , - o o < „ < o o (1.1b)
So if x = {..., x(i-3), x(i-2), x (i-l), x (i), ...) then
y ( i-l) = x (i-l) + x(i-2) - x(i-3) and y(i) = x(i) + x (i-l) - x(i-2)
y(n) = x (n + l) - 2 x(n) + 4 y(n-l) , -< » < n < o o ( 1 .1 c )
S o if x = {..., x (i-l), x(i), x (i+ l), ...J and y a {..., y (i-l), y(i), y (i+ l), ...} ,
t h e n y(i) = x (i+ l) - 2 x(i) + 4 y (i-l)
I f th e in p u t re p resents a sequence o f s a m p le s separated in tim e, then the present
o u tp u t sa m p le , y (i), m ust correspond in tim e to the p resent input sam ple, x (i). In th is w a y , a tran sfo rm is non -cau sal i f th e p resen t o u tp u t. y (i), re q u ires an input valu e , x ( i- f l) , that, as yet, does not ex ist. T herefore, the transform o f E q .( l .l c ) is
n o n - c a u s a l .
T h e b a s ic stru ctu re o f a discre te system is show n by F ig .(1 .1 ), w here th e o utput s eq u e n ce, y , Eq.(1.2), is related to the input sequence, x , and the transform ation, 01.
* —• —
* [ ]
—»— y
F ig u r e 1.1 D iscrete system w ith transfo rm atio n , 01.
y « * [ x ]
(
1
.
2
)
page 1/3
1 . 1 . 1 L i n e a r i t y
T h is p ro p e rty d e s c rib e s th e re la tio n s h ip b e tw e e n th e in p u t s ig n a l and the co rre s p o n d in g o u tp u t sig n al. L in e a rity m ay be d e fin e d u s in g the p rin c ip le s o f su p e rp o s itio n and sca lin g . A sy stem is lin ear, i f a lin e a r c o m b in a tio n o f input seq u e n ces m aps to a lin ear co m b in atio n o f o u tp u t seq u e n ces. T h erefo re, i f y i ( n ) and y 2 ( n ) are th e re s p o n s e s to in p u t s a m p le s X ) ( n ) and X 2 ( n ) , th ro u g h a
tra n s f o rm a tio n , 91. respectively, then a system w ill b e lin ear i f and only if
9l[ a x j(n ) + b X2(n)] = a 9t[ xi<n>] + b 9t[ X2(n)] ■ a y j(n ) + b y2(n)
fo r arb itrary co n stan ts a and b.
1 . 1 . 2 S h i f t * I n v a r i a n c e
T h is c h a ra c te ris tic d e s c rib e s h o w th e in p u t/o u tp u t re la tio n s h ip v a rie s a s the
input seq u e n ce is sh ifted. A sy stem is sh ift-in v aria n t i f th e resp on se to a shifted v ersio n o f th e in p u t s eq u e n ce, is iden tica l to a sh ifted v e rs io n o f the response
based upon the unshifted input. T his can be d escrib ed as, i f y (n ) = 9 l[ x ( n ) | then 91 is s h i f t - i n v a r ia n t w h e n y(n - nQ) = 9 t( x ( n - n o )] fo r all nD. W h e re the index n is
a s s o c ia te d w ith tim e , then s h ift-in v a ria n c e is d e sc rib e d a s tim e -in v aria n ce.
1 . 1 . 3 S t a b i l i t y
T h e sta b ility o f a tran sfo rm atio n ind icate s how a sy stem w ill behave to a given input. A tran sfo rm atio n is s ta b le i f it p ro d u c es a b o u n d ed ou tp u t sequence for
ev e ry b o u n d e d in p ut s e q u e n c e . T h is is re ferred to a s b o u n d ed in p ut bounded o u tp u t (B IB O ) stable.
1 . 1 . 4 C a u s a l i t y
C a u s a lity in d ic a te s w h e th e r a t r a n s f o rm a tio n c a n be re a lis e d . A c a u s a l
tra n s fo rm a tio n is o n e w ho se p re s e n t o u tp u t d e p e n d s o n ly o n p ast inpu ts and o u tp u ts an d the p resen t input. T h erefo re the tran sfo rm atio n o f E q .(1.3 ) is causal
y(m) ■ { a i x(n) + 82 x (n-l) ♦ 83 x(n-2) + ....
+ b i y(k) ♦ b2 y ( k -l) + b3 y (k -2) ♦ ....] (1.3)
Chapter 1. Introduction page 1/4
T r a n s f o r m a tio n s th a t m ee t th e lin e a r ity an d tim e -in v a r ia n c e re q u ir e m e n ts , satisfy a bro ad class o f D igital Signal P rocessing(D S P ) operations. A dig ita l filter is
an e x a m p le o f a L in ear T im e-Invariant(L T I) s tru ctu re and c a n be describ ed by the tr a n s f o rm a tio n , 91. o f F ig .(1 .1 ) and E q.(1.2). A tran sfo rm atio n can be com pletely
c h a rac terised by its response to the unit im pulse seq u e n ce. 5 . defined as
T h e un it im p u lse response, h , is the output seq u e n ce o f a system w hen the input sequence is the unit im pulse, 5 . T herefore for a transfo rm atio n . 91, its unit im pulse
re sp o n se is defined as
Any s e q u e n ce can be described as a sequence o f scaled unit im pulses delayed by
o ne s a m p le p erio d w ith re s p e c t to each other. A p p ly in g the p ro p e rtie s o f LTI s tru c tu re s , an o u tp u t seq u e n ce, y , can be c o n s tru c te d by sum m ing the system 's s ca le d u n it im p ulse resp o n ses fo r each elem ent o f th e in p ut seq u e n ce, x . T his process is described in E q .(l.S ).
E q .( l.S ) re p re s e n ts th e c o n v o lu tio n o f the in pu t sig n al w ith the s y s te m 's unit
im p u ls e re s p o n s e . U sing th e c o n v o lu tio n o p e r a to r. *, and th e u n it im pu lse re s p o n s e , h . then the output signal, y , o f a system to an input sequence, x . can be ex p re s s e d as
W ith th e d e s c rip tio n o f a L T I s tru c tu re g iv en b y E q .(1 .6 ), the b a s ic discrete s tru c tu re o f F ig.(1 .1 ), can be redraw n for a LTI s tru ctu re and is illu stra ted by
F ig .(1 .2 ) .
o t h e r w i s e n
h(n) » 91 [ 5(n)] -ex» < n < oo ( 1 .4 )
n
y(n) « X x <k > h < n-k). k-0
-oo < n < oo (1 .5 )
y(n) = x(n) * h(n) (1.6)
X
h
F ig u r e 1.2 D iscrete system in term s o f the
A c o n tin u o u s signal o r w aveform described in the tim e dom ain, m ay be redefined in th e freq u en c y d o m ain though the F o u rier tran sfo rm . A tim e d om ain w aveform
and th e c o rre s p o n d in g fre q u e n c y do m ain w a v efo rm , form a F o u rie r tra n s fo rm p air. T h e n atu re and p ro p e rtie s o f F o u rier tran sfo rm p airs are well k n o w n and c a n be ex te n d e d to in c lu d e d is c re te s ig n a ls [3 ]. U sin g th e D is c re te F o u rie r
T ra n sfo rm (D F T ), a tim e d om ain sequ ence, x . may be defined as a series. X , in the fr e q u e n c y d o m ain .
T he d is c re te frequency dom ain is com m only know n as the z do m ain, w here z is a
co m p lex v a ria b le . C o n v ersio n o f a tim e do m ain s eq u e n ce, x , into a z dom ain s e q u e n c e . X . is perfo rm ed th ro u g h the z tran sfo rm . T h e gen eral form s o f the z tra n s fo rm and th e in v e rs e z tra n s fo rm are g iv e n by E q .(1 .7 ) and E q .(1 .8 )
r e s p e c t i v e l y .
w here c re p resents a circ u lar c o n to u r centred at the o rig in o f the z do m ain , lying
in the re g io n o f convergence o f th e function, X (z).
If the com p lex variable, z. is d efined in its polar form as. z * r eJ®. then w hen r = 1
o r Izl = I , the z transform ation is equal to the DFT. U sing this idea, Eq.(1.8) can be m odified to define the inverse z transform when Izl a 1, as
The p ro p e rties o f the z transform can be used to desc rib e the function o f a discrete system in th e discrete frequency dom ain. F ig .(1 .3) sh ow s a basic discrete s y stem in term s o f th e z transform s o f an input sequence, x , the output sequence, y . an d the
unit im p u lse response,
h.
(1 .7 )
(1.8)
x
( 1 .9 )
X(z) H(z) - » --- Y(«>
[image:17.364.22.330.33.418.2]Chapter 1. Introduction page 1/6
The z tran sfo rm o f the un it im pulse response, h , is th e transfer function. H (z ). The re la tio n s h ip o f the tran sfer fu n c tio n to th e in p ut and ou tp u t sequ ences is given
by E q .(l.lO ).
Y(z) « X(z) H(z) (1 .1 0 )
T h e system eq uatio n o f E q .( l .l O ) is the freq uen cy d o m ain equ iv a le n t o f the time dom ain sy stem equation given by Eq.(1.6). From th ese equ ation s it can be seen that
m u ltip lic a tio n in the freq u en c y dom ain is e q u iv a le n t to c o n v o lu tio n in th e tim e d o m a in .
The sy stem eq uatio n s o f E q .(1 .6 ) and E q .(l.lO ) c a n b e re w ritten in term s o f the o p eration s th at occur within th e functions o f h and H (z ), as E q . ( l . l l ) and Eq.(1.12) r e s p e c t i v e l y .
n2
V1
y(n) * 2 * a i x(n- •) * X b i y ( n - i )
i- 0 i- 1
n 2 {U
Y(z) X bi z*i * X (z) 2 - * i ** *
i- 0 i- 0
n 1 num ber o f sam p les in X
n 2 num ber o f sam p les in
y
aj arbitrary co nstan ts, i =0. 1. 2... ni
b i arbitrary co nstan ts, i =1, 2, ti2 and bo
-E q u a tio n (l. 11) show s the g e n e ra l d ifferen ce eq u a tio n fo r a d iscre te sy stem , while E q .(1 .1 2 ) is th e equ iv alent g e n e ra l tran sfer fu n c tio n . E q .(l.lO ) and E q.( 1.12) can
be co m bined to express the tra n s fe r function, H (z), as.
page 1/7
1 . 2
P h a s e a n d G r o u p Delay
F u n c tio n s d e f in e d w ithin the z d om ain are c o m p le x in n atu re . T h e r e fo r e any fu n c tio n , G ( z ), m ay be represented as
G(z) = R e[ G(z) ] + j Im[ G(z) ]
o r in p o la r c o -o rd in ates given in E q .( l.lS ) .
G (z) = I G (z) I (cos (J) + j sin <J>)
G (z) = I G(z) I e J«** w h e r e
(1 .1 4 )
( 1 1 5 a )
( 1 .1 5 b )
«30)1 = V R e[G (z))2 + I m [G (,)] 2 » id 0 » u n > ( j
The action o f a digital filter is to accept o r reject th e frequency co m p o n e n ts o f an
in p u t s e q u e n c e by re ta in in g o r re d u cin g th e a m p litu d e o f each co m p o n e n t. A d ig ita l f i l t e r w ill also e f fe c t th e p hase re la tio n s h ip b etw e en th e freq u en c y
co m p o n e n ts o f th e input signal. A typical ph ase re sp o n se o f a lo w p ass filter is show n in F ig .(1 .4 ).
F ig u r e 1.4 T y p ic a l low pass p h a s e response.
E ach fre q u e n c y com p on ent o f a s te a d y state in p u t sequ ence p a s s e s th ro u g h a system in an eq u a l time period, tSys- T his system tim e delay, tsys> w ill c a u se each
[image:19.366.21.338.20.420.2]Chapter 1. Introduction
T h erefo re, if an input fu n c tio n o f the form
x(t) = C sin(o> t)
was applied to a LTI stru c tu re , th en the output w ould be
y(t) = D sin(o) (t - tsys)) = D sin(iot - <J>)
w here th e ratio o f D to C in d ic a te s the ch ange in am plitu de o f th e sine function and the p hase d iffe re n c e b etw e en th e input and o u tp u t v e rs io n s o f the sine
w aveform . F o r a LTI s tru c tu re to retain the p hase inform atio n o f an input signal,
th e p h ase re la tion sh ip b e tw e e n th e frequency co m p o n e n ts o f th at sig nal m ust be p re serv e d . C o n sid e r the in p u t fu n c tio n ,
x(t) = C i sin(coi t) + C2 sin(o>2 t) + C3 sin(o>3 t) (1 .1 6 )
and th e co rresp o n d in g o u tp u t fu n c tio n
y(t) = Di sin(o) 1 ( t - t s y s )) + D2 sin(o>2( t - t s y s )) + D3 sin((03( t - t sy s) ) (1 .1 7 )
U sing the p rin cip les o f s u p e r p o s itio n , the effec t on each frequ en cy co m p onen t o f the fun ction in E q .(1.1 6) c a n be con sid ered in is o la tio n and th en re com bined to prod u ce E q.(1.1 7 ). The in d iv id u a l input freq uency com p o n e n ts o f E q .(1 .1 6 ), along
page 1/9
F ig u r e 1.5 F requency com ponents (a ) © i , (b) © 2 and (c) © 3 o f the
input and output functions g iven in E q .(1 .1 6 ) and (1.17).
From F ig .(1 .5 ), it should be noted that all th e frequ en cy co m po n en ts o f the input fu n c tio n are in phase. F o r the system to p re s e rv e th is p h ase re la tio n s h ip , the freq u en c y co m po n en ts o f the ou tpu t fu n c tio n a r e a ls o re q u ired to b e in phase.
From E q.(1 .1 7 ), this w ill only occur when,
©1 t*y, - ©2 ttys “ “ 3 ttys ■ 0) ttys
T h erefo re, p hase linearity w ill be p reserved if a p h ase chang e, <)>j. at a frequency,
Chapter 1. Introduction page 1/10
A linear phase LTI structure will therefore have the characteristic
<J>(w) = to tsys
L in ea r p hase can be defined in term s o f the phase delay, a(o>), o r th e group delay,
t(<o) . Phase delay is defined as.
<{>(<■>)
a (a ) * - — — -7Ï < to < 71
A s tru ctu re will th ere fo re exh ibit exactly lin ear p hase if a is c o n s ta n t, illustrated in F ig .(1.6 ). G roup d ela y is d efin ed as the n egative d eriv a tiv e o f th e phase with re s p e c t to the frequ ency, so
T(d>) ■ d0(to)
dco ( 1 1 8 )
U sin g E q .(1.1 5 b) and E q.(1.18 ) the group delay can be ex p re s s e d in term s o f the tra n s f e r fu n c tio n , H (z).
In( H(z) ) = ln( I H(z) I ) + j <fr(o>)
dH(z) d u
___ Î___ ,dj.,H (z ) l d ^ )
I H (z ) I dto * J do>
T(«o) d H ( z )~l
du J
( 1 1 9 )page 1/11
1 . 2 . 1 C h a r a c t e r i s t i c s of L i n e a r Phase
F o r exactly lin e a r ph ase,
4>(w) a - a t ) - i c S t o S x
w here a is a co n stan t p hase delay. T o determ ine the natu re o f a transfer function th a t s atisfies th is co n d itio n , H(z) needs to be expressed in term s o f a . This can be ach ieved by co m b in in g Eq.(1 .1 5a) and Eq.(1.7).
T herefore, in o rd e r fo r a system described by h to possess a constant phase delay, o r exactly lin ear phase. Eq.(1.21) m ust be satisfied for all o f the sequence n = 1, N. A possible so lu tion to th is problem is.
F or the unit im pulse response to satisfy E q .(1.22), it m ust be sym m etrical about the sam ple (N + l)/2 o r a . The term , a , in Eq.(1.22) represents the constant angle o f the
phase response o r the p hase delay. C on sider a typical im pulse response, show n by F ig .(1 .7 ), w hich h as an odd num ber o f sam ples. N, and w hich satisfies E q.(1.22).
The phase d ela y , a , w ill be an in teg er and the sym m etry associated with lin ear phase, will o cc u r around a sam ple point equal to the value o f a .
N
H(ei“) = £ h ( " ) e*J“ n = I H (e*“ ) I (c o s(a o i) + j sin(aco) ) n *l
(1.2 0)
T aking the real and im aginary parts o f Eq.(1.20),
N
R e[ H( ei *) ] ■ I H(e->“ ) I cos(aco) = ^ h ( n ) c o s ( c o n )
N
Im [ H (ei“ ) ) = I H(ej“ ) I sin(aco) = £ h <n > s in (a > n )
n = l
t h e n
s in (a o ) ) c o s ( a u )
and where a * 0, then
N
y h (n ) s i n [ ( a - n ) to) ■ 0 (1 2 1)
Chapter 1. Introduction page 1/12
i c e n t r e o f s y m m e try
F ig u r e 1.7 S ym m etric im pulse resp on se w ith an o d d n u m b er o f sam ples.
I f th e n u m b er o f sam ples o f the unit im pulse response is even, then a is no longer an in teg er an d the sym m etry point fo r a lin e a r phase resp on se w ill ex ist between tw o sam ple p o in ts. T his is illustrated by F ig .(1.8).
F ig u r e 1 .8 S ym m etric im p ulse re sp o n se w ith an even num ber o f sam ples.
T he im pu lse re sp o n se sym m etry, in d icate d b y F ig .(1.7) and F ig .(1 .8 ), re la tes to a co n d itio n w h e n the fu n c tio n e x h ib its b o th c o n s ta n t p h ase dela y and co n stan t
g ro u p d elay. H ow ever, a full d efin itio n o f the transfer fu nction,
H(eJ-) - H*(ei°») ei*®> o r H(ei®) - ± I H(ei«*) I ©!♦<•>
sh o w s th at th e im pulse response w ill still p o sse ss lin ear p h ase i f it ex h ib its either
sy m m e try o r a n ti-s y m m e try . T h e a n ti-s y m m e try ca se re la te s to a 'p ie c e -w is e lin e a r' fu n c tio n , which has co n stan t gro u p d e la y but not co n stan t phase delay. In m o st p ra c tic a l d esig n ca s e s , p h ase d ela y is o f no in te re s t. W here th e filter's
im pu lse re s p o n s e cannot be d efin ed by a fin ite num ber o f sam ples, exactly linear p h ase is im p o ss ib le to obtain and the b est that can be achieved is approxim ately
page 1/t3
U sin g th e in fo rm atio n about th e u n it im pu lse respon se s y m m e try , the po sitio n o f
th e p o le s and zeros o f a fu n c tio n e x h ib itin g p h ase lin e a rity c a n be determ in ed. T h e p o s itio n and re la tio n s h ip o f th e z e ro s o f an e x a c tly lin e a r p h ase tran sfer fu n c tio n can be o b serv ed by co n sid erin g a FIR filter. In o rd e r to ex h ib it lin ear
p h ase a tra n s fe r fu nction. H ( z ), m ust p ossess a sym m etry o r an ti-sy m m etry o f its u n it im p ulse response, so
N
H(z) = X h (°) z ' n = h ( l ) + h(2) z"1 + h (3) z-2 + ... n= 1
± h(3)z-(N *2) ± h (2 )z* (N-») ± h ( l ) z - N
T h e plus sig n correspo n ds to a sym m etric response, w h ile the m in u s sign indicates
a n ti-s y m m e try . B eca u se o f th e sy m m e try o f th e u n it im p u ls e re sp o n se , the tra n s fe r fun ctio n. H (z) and its inverse, H ( z -') m ay be related by E q.(1.2 3 ).
H ( r ') - 1 xN H(D <1.23)
E q .(1.23 ) show s that the functions H (z) and H (z'*) are iden tical, ex c ep t for a delay o f N sam p les and ± 1 factor. U n d e r these con d itio n s th e tw o fu n c tio n s m ust posses
id en tica l zero s. T herefore to s atisfy E q.(1 .23 ), the ze ro s o f an ex a ctly lin ear phase s y stem m u st ex ist in s ets that c o m p rise a ze ro and its re cip ro c al about the unit circle, so H (z‘‘) will possess the sam e set o f zeros.
T h is p ro p e rty can be illu s tra te d i f H ( z ) has a facto r. H j( z ) , w h ich is a com plex
co n ju g a te zero pair at r e±J® w hen r * 1 and 0 * 0 o r n, sh ow n in F ig .(1.9 ) by points A an d C. T h e function H ( z - ') w ill have a co rresp o n d in g fu n c tio n H i( z -1). with a
co m p lex co n ju g ate zero p a ir at 1/r e*.!*, show n by points B and D in Fig.(1.9). To
sa tis fy E q .(1 .2 3 ), H (z) and H (z -') m ust possess the sam e zero s and so both functions m u st co n ta in factors to produce the zeros at A . B. C and O o f F ig .(1.9). If a factor H j(z ) p ro d u ces the zeros B and D , then H j ( z '') w ill g en e rate th e ze ro s A and C.
T h e re fo re E q.( 1.23) will on ly be satisfied if H (z) co n ta in s b o th fa c to rs H j(z ) and H j ( z ) , w here H j(z ) = 1 /H j(z). An ex a ctly lin e a r tra n s fe r fu n c tio n m u st th erefo re
Chapter 1. Introduction page 1/14
I m
/ / / / 1
B
' ( l / r , 9)
a. A / (r, ♦) \
c 1 ^
\ X . r . - w/ 1 -0
\ \
s C
1
X
( l / r , -9)
F igu re 1.9 Reciprocal complex conjugate zero positions for linear phase.
F ig.(l.lO ) show s the typical zero positions of linear phase FIR filters for the four possible cases o f linear phase design, odd or even filter order, N, with symmetrical o r anti-symmetrical unit impulse responses.
(c) (d)
F ig u re 1.10 Zero positions for the four possible exactly linear phase
[image:26.367.16.333.27.417.2]1 . 2 . 2 M in im u m -
a n d
N o n m i n i m u m - P h a s e
F ig .(1 .1 0 ) in d icate s the re la tio n s h ip b etw een z e ro s fo r exa ctly lin e a r p h ase FIR
s tru c tu re s . A ll lin e a r p h ase s y s te m s sh o u ld p o s s e s s z e ro s in th e s e ty p e s o f p o sitio n s, w h e th e r FIR o r IIR in n a tu re . IIR stru ctu re s a ls o p o ssess p o le s w ithin th e ir tra n s fe r fu n c tio n s that c o n s tra in the p o s s ib le p o s itio n s fo r its z e ro s . For
so m e IIR s tru c tu re s th e s e c o n s tra in ts m ake it im p o ss ib le to p la c e z e ro s in
re cip ro c al co m p lex co n ju g a te s e ts . T h e co n c ep t o f m in im u m - and n o nm in im um - p h ase can b e applied to a structu re to d ete rm in e i f its zeros can be arran g e d into re q u ir e d p o s itio n s . A fo rm al d e f in itio n o f m in im u m -p h a s e c a n be g e n e ra te d
th ro u g h the H ilbert T ra n sfo rm [2 9 ], o r fo r d is c re te sy s te m s , the D iscrete H ilbert T ra n s fo rm (D H T ).
The DHT provides a m ethod o f re la tin g the real part o f a frequency response in the d is c re te do m ain to its im aginary p art and vice versa. T h ese tw o re la tio n sh ip s form
a DHT pair. I f the z transform , X(z), o f a causal sequence x (n ), is described as
X(ei-) « XR(ei-) ♦ j Xi(ej-)
th en it has th e H ilbert transform p a ir n
x lW“> = ^ P J Xr(«»> CO. ( ^ d *
- I t
a n d
it
xr
(
w
-) - x.o)
-2
* rf
xi<«») cot
- it
w here P den o tes the C au chy p rin c ip le value o f th e in teg ra l! 18].
F or a system . H (ej“ ), to ex hib it m in im u m -p h ase then the com pon ents o f its tran sfer
fu n c tio n s , In[IH (ei“ )l] and arg[H (ei“ )] . m ust form a H ilbert transform pair. T h is may b e re-ex p ressed as
Jt
Chapter 1. Introduction page 1/16
a n d
it
argt H(eJ-) ] = £ P
J
ln [ I H ( e * ) l ] c o l-It
w here tf(z ) = In(H (z)) and li is the F o u rier tran sfo rm p a ir o f ft(z ). A ltern a tiv e ly a
sy stem , H (z ), w ill ex h ib it m inim um -phase if a c a u sal stable inverse sy stem , H **(z), ex is ts such th at
H (z)H 'U ) » 1.
S in c e H - '( z ) = 1/H (z), the transfer function, H ( z ), m ust have all its p o les and zeros in side the unit circ le in o rder for a stab le and ca u sal inverse system to exist.
T h e re q u irem en ts fo r m inim um -ph ase are c o n tra ry to those fo r lin e a r p h ase and th e re fo re , an ex a ctly lin e a r phase sy stem re q u ire s an o v erall n on m inim u m -p hase
s tru c tu re . T h is h o w ev er do es not elim in ate m in im u m -p h a se s tru c tu re s from linear p hase d esig n as any rational function, G (z ), m ay b e ex p ressed in the form
G(z) = Gniin(z) Gap(z)
w here G m in ( z ) <s a m inim um -phase function and G a p (z) is an all-pass function for w hich has IGap(ej“ )l - 1 for all <o.
T h e n atu re o f G a p (z ) is n on m in im u m -p h a se an d th e p o les and z e ro s o f this
fu n c tio n can b e used to produce an o v erall fu n c tio n that m eets the lin e a r phase req u irem en ts. A m inim um -ph ase function can th e re fo re be used in a lin e a r phase
d esig n p ro v id ed the o v e ra ll p hase re sp o n se is m o d ifie d by a p h a s e e q u a lise r. G a p (z ). L in ea r p h ase desig n s throu gh p hase eq u a lis a tio n are d iscu ssed in C hapter 2.
1 . 3
F i n i t e
W o r d l e n g t h
E ff e c t s
A larg e am ount o f research has been d irec ted at the effec ts o f fin ite w ordlength
on d ig ita l s y s te m s , es p e c ia lly fo r d ig ita l f ilte rs . I n itia l w ork by Jac k so n (1 4 ] o u tlin e d a sy stem atic ap pro ach to these fin ite w o rd le n g th effe c ts by d ete rm in in g th e re la tio n s h ip b etw e en ro u n d o ff n o ise and d y n a m ic range. T h is ap p ro a c h o f
F in ite w o rd le n g th effec ts m ay be collec ted u nd er fou r m ain headings;
( i ) C o n v ersio n o f an analogue signal to and from a d ig ita l eq u iv alent. T h is is u su ally know n as co n v e rsio n n oise and w ill d ep e n d upon the
q u a n t i z a t i o n s t e p , b e in g th e d i f f e r e n c e b e tw e e n c o n s e c u t i v e re p re s e n ta b le nu m b ers and th e ty p e o f q u a n tiz a tio n u s e d ; ro u n d in g , v a lu e tru n c a tio n o r m ag n itu d e tru n c a tio n .
( i i ) U n c o rre la tc d ro u n d o ff noise.
T h is is a g eneric term for the n oise introduced to a signal w ith in a Filter d ue to arithm etic operations. The m ain calcu lation to ca u se th is effect is
m u ltip licatio n . T h e bit length to accu rately re p rese n t th e p ro d u c t o f two b b it n um bers is 2b b its. T h is 2 b bit nu m b er ca n n o t be represented w ithin a system lim ited to b bits so the num ber has to be reduced either
th ro u g h ro u n d in g o r tru n c a tio n . T h is in tro d u c e s a ce rta in am o u n t o f u n co rrela te d n o is e into the o p eratio n o f th e filter. T h e v a ria n c e o f this u n co rrela tc d n o ise source w ill d ep en d upon the ty p e o f a rith m e tic used,
flo a tin g o r fix ed point, the sign al lim itatio n sch em e and the type o f n u m b er system used; l 's o r 2’s com p lem en t o r sig n ed -m a g n itu d e .
( i i i ) In a c c u ra c ie s in the filte r resp on se.
T h is n o ise so u rc e re su lts from an in ab ility to ac cu ra tely re p ro d u ce a filte r's freq uen cy response u sin g a fin ite n u m b er o f b its fo r th e filter c o e ffic ie n ts . T h is re su lts in a n o n -id ea l tra n s fe r fu n c tio n . T h is effec t
c a n b e offset i f filter co e ffic ie n ts are d esig n ed to a fin ite w ordlength, re s u ltin g in an a c ce p ta b le Finite w ord le n g th tra n s fe r fu n c tio n .
( i v ) C o rre la te d ro u n d o ff n oise (lim it cy c les).
T w o ty p es o f co rrela te d ro u n d o ff n o ise o r p a ra s itic o s c illa tio n ex ist, sm all scale (g ran u lar) and large scale (overflow ). T hese effe c ts are m ost
a p p a r e n t in fix e d p o in t re c u rs iv e d ig ita l f i l t e r s , w h e re in te rn a l
ro u n d i n g e r r o r s fo r a c o n s t a n t in p u t a r e h ig h ly c o r r e l a te d . Q u a n tiz atio n c a u s e s the no n -lin ea r m apping o f the lo w est o rd e r b its o f
an in tern a l sig n al u nder con stan t in put. T his g en e rate s lim it c y c les. For a re c u rs iv e f ilte r usin g ro u nd ing th is m ean s th a t th e re is n o u nique stead y state ou tp u t fo r a co n stan t in put. A so c a lle d d ea d b an d region
Chapter 1. Introduction page 1/18
L im it c y c le s are dependent upon a num ber o f factors, m ainly the filter realisatio n o r s tru c tu re a n d the q u a n tiz a tio n s tep . S ig n a l q u a n tiz a tio n th ro ug h ro u n d in g is
m o st s u s c e p tib le to lim it c y c le e ffe c ts. M ag n itu d e tru n c a tio n p ro v id es a b e tte r a l te r n a tiv e q u a n tiz a tio n p ro c e d u re , h o w e v e r, it d o e s n o t a lw a y s e lim in a te dead ban d lim it cycles.
F acto rs ( i i ) - ( i v ) are the o nly fin ite w ordlen gth effec ts th at re la te d irec tly to the d ig ita l f i l t e r 's o p eration . In tu rn , each o f th ese e ffe c ts d epend s on the filte r's
s tru ctu re a n d con figu ration. A gre at deal o f w o rk has been d irec ted at w ays to im plem en t a g iv en tran sfer fu n c tio n . H (z). E ac h digital filte r s tru c tu re pro p osed
co rresp o n d s to a d ifferent m ethod o f ex p ressin g the tra n s fe r fu n ctio n . A gen eral fu n c tio n , G ( z ) , may be d ivided into sm aller fu n ctio n s, G j(z ) and H j(z ), such the
[image:30.366.22.341.22.419.2]th e ir c o m b in a tio n eq uals G ( z ). T he general form for the co m b in atio n o f these functions, o r a Lagrange stru c tu re , is shown in F ig.(1 .1 1).
Figure 1.11 General Lagrange Structure.
The overall transfer function of the structure in Fig.( 1.11) is,
G(z) = G i(z) G 2(z) G3U) ( H i( z ) + H2U ) + H3U ) )
The G i(z ) fu n c tio n s o f F i g . ( l . l l ) arc connected in cascade, w hile the H i(z) fun ctio n s
are c o n n e c te d in p arallel. E ach m o d ific a tio n o f the L a g ran g e s tru c tu re w ill
p o ssess th e s am e p erform ance u n d er large ac curacy ca lcu latio n s. It is th e ir finite w o rd le n g th p e r fo rm a n c e , h o w e v e r, w h ich is o f in te re s t. T he form o f th e in d iv id u a l f u n c tio n s G j(z) and H i(z ) is arbitrary, and a w ide range o f com binations
ex ists fo r a g iv en transfer function. A d esire to analy se the ov erall stru ctu re for finite w o rd le n g th effects prom pts to a break dow n o f a response in to sm all regular
page 1/19
A ca scad e stru ctu re m ay be re p resen ted as
and H2i(z) = 1 + a i j z' 1 + a2i z ' 2 1 + b i J z * l + b2i z" ^
The cascade o f these s e c tio n s also allow s them to be d efined in term s o f functions w hich rep resent the n u m e ra to rs . N j(z) and den o m in ato rs. D i(z). o f each section, so
that H (z) could be expressed as
E q .(1.2 4 ) allow s a cascad ed s tru c tu re to be co nstru c ted from first and second o rder
s e c tio n s w ith arb itrary n u m e ra to r and d e n o m in a to r o rd e rin g s and pairin g s.
A stru cture w hich has a p a ra lle l form , may be ex pressed as,
w here H j(z) is eith er a first o r second order section o f the form.
T he n o ise p ro p e rties o f th e s e 1st and 2nd o rd e r se c tio n s are re la tiv e ly easy to
an a ly se [2 9 ] and the o v erall p e rfo rm an c e o f filte r s tru c tu re s u sin g these elem ents
can b e d eterm in ed . An im p o rta n t o bservation from th is a n a ly sis is th at the o rder and p airin g o f ca scaded s e c o n d o rd e r sectio ns can greatly e ffe c t the overall finite
w o rd le n g th p erfo rm an c e, b e c a u s e o f o v erflow w ith in the s tru c tu re .
A large n u m b er o f filter s tru c tu re s ex ist, each u sin g a d e riv a tiv e o f the g eneral
L a g ra n g e s tru c tu re , i n c l u d i n g th e D irect fo r m s th a t im p le m e n t a tra n s f e r fu n c tio n w ith o u t p a r titio n in g it in to sm a lle r fu n c tio n s . A la rg e am ount o f re search h as bee n d ire c te d at an a ly sin g and co m p a rin g th e s e v ario u s stru ctu re s
Chapter 1. Introduction page 1/20
im p ro v ed th e fin ite w ordlength p erfo rm an c e. A p ro p e rty su g g e s te d to m easure fin ite w o rd le n g th p e rfo rm a n c e co n c ern e d th e s e n s itiv ity o f th e s tru c tu re to
changes in its param eters. Bode defined a s e n s itiv ity fu nctio n, S, to determ ine this p ro p e rty by m ea su rin g how a function , F, c h a n g e s w ith re sp ect to o ne o f its
p a ra m e te rs , x. T h is p ro p e rty , d e fin e d in E q .(1 .2 5 ) , is c o n c e rn e d w ith sm all p aram eter ch a n g es and as a result, sm all s c a le sen sitiv ities.
F x dF
S(F.x) - S; = p - (1 .2 3 )
A n alog u e s tru c tu re s know n to po ssess low p a r a m e te r s e n s itiv ity in clu d e D oubly T erm inated L ossless(D T L ) netw orks. T hese s tru c tu re s s u ffer only a sm all am ount o f d isto rtio n o f th e ir m agnitude respp nses as th e c o m p o n e n ts' v alues are varied. T his
p roperty is related to the ability o f the DTL s tru c tu re to d eliv er m axim um pow er at poin ts ac ro ss its passband.
A t th e s e p o in ts o f M axim um A v a ila b le P o w e r(M A P ), th e d e r iv a tiv e s o f the a tte n u a tio n w ith re s p e c t to re activ e c o m p o n e n ts w ith in th e s tru c tu re are ze ro .
T h erefo re, at th ese M AP points the m ag n itu d e s e n s itiv ity to re activ e com po n ents is zero and b ecau se the sensitiv ity is a s m o o th co n tin u o u s fu nctio n, the sensitiv ity
in the region around these points is also likely to be low. T his effec t, togeth er with a m a t h e m a t i c a l e x p l a n a t i o n , h a s b e e n r e f e r r e d to a s O r c h a r d 's a r g u m e n t[2 6 ,2 7 ,3 7 .2 4 .2 5 ].
In an attem p t to reproduce the properties o f th e analog ue DTL netw o rk in a digital
c irc u it, F ettw eis in v estig ated a num ber o f m e th o d s o f co n v e rtin g a D TL structure in to th e d is c re te d o m ain . T he m etho d a d o p te d by F e ttw e is c o n c e n tra te d upon c re a tin g d ig ita l eq u iv a le n ts o f analogue co m p o n e n ts su ch as an in d u cto r, re sisto r,
v o ltag e so u rc e and tran sfo rm er. F irst by d e s c rib in g the an a lo g u e c o m p o n e n ts in term s o f w ave p aram eters and then c o n v e rtin g them in to th e d ig ita l d o m ain . A
d ig ita l e q u iv a le n t o f the DTL stru ctu re w as th e n co n stru c te d u sin g these dig ita l
c o m p o n e n ts .
T h e re s u ltin g W ave D igital F ilters(W D F ’s) h a s been w id ely re search ed and have
bee n sh o w n to p o s s e s s a s u p e rio r ro u n d o f f n o is e p e rfo rm a n c e c o m p a red to ex istin g d ig ita l filte r structu res[1 7,3 8,1 3,8 ,42 ]. T h e se n s itiv itie s o f W D F's and th eir re feren ce an alog u e D TL filters have also been co m p a red [4 3,28 ] and show n to bear
a c lo s e c o r re la tio n . F u rth e r w ork by F e tt w e i s [ 7 ,6 ,I 0 .2 ] and Jac k s o n [1 3 ) has ad v a n c e d a re la tio n s h ip b etw e en ro u n d o ff n o is e an d a tte n u a tio n c o e ffic ie n t
page 1/21
A n a l t e r n a t i v e ap p ro a c h s u g g e s te d by V a id y a n a th a n a n d M itr a [3 9 ,4 0 ,4 1 J
c o n c e rn e d d eriv in g d ig ita l s tru c tu re s in d ep e n d en t o f a n a lo g u e e q u iv a le n ts. The o b je c tiv e o f th is app roach w as to defin e a c la ss o f f u n c tio n based o n the r e q u ir e m e n ts fo r low c o e ffic ie n t s e n s itiv ity an d th en d e r iv e s tru c tu re s based upo n th e s e fu n ctio ns. T he re su lt co n sisted o f tw o -p o rt c h a in m atrice s w hich
d e s c rib in g L ossless Bounded R eal(LBR ) functions. A W D F s tru c tu re s atisfies a LBR fu nctio n an d the results from the two design m ethods are s im ila r in nature.
A co m p a riso n o f various filter structures by M atharu[21] u n d e r a num ber o f finite
w o rd le n g th effec ts, has also been ca rrie d out. The s tru ctu re s u n d er co n sid eratio n
w ere th e lad d er WDF, lattice WDF, unit elem ent W D F, G ray -M a rk el la ttice, d irect form I an d II. cascaded and parallel 2nd o rd e r sections. T h e re s u lts sugg est that c h o ic e o f filte r stru ctu re is not c le a r c u t and is d e p e n d a n t up o n th e filte r a rith m e tic and num bering sy stem . However, in all tests, th e p e rfo rm an c e o f W D F
s tru ctu re s placed them at o r nea r the top o f each com parison list.
1 . 4
W a v e Digital F i l t e r (WDF)
1 . 4 . 1 C i r c u i t
Descr i pt i ons
U sin g a D T L analogue filte r as a reference, F ettw eis b ro k e th e filte r in to its co n s titu e n t elem en ts and m odelled the circ u it as a co n n e ctio n o f o n e-p ort blocks.
A d ig ita l e q u iv a le n t o f e a ch analog ue co m p o n e n t w as th e n g en e rate d and a s tru c tu re co n s tru c te d usin g th ese dig ita l ele m e n ts. F e ttw e is trie d a n u m b er o f d iffe re n t tran sfo rm s to produce digital filters that retained th e p ro p e rties o f th eir
re feren ce s. A successful transfo rm adopted by F ettw eis was to replace th e vo ltag e and c u r re n t description o f an elem ent with an incident and re fle c te d voltag e w ave
Chapter 1. Introduction page 1/22
(1 .2 6 )
In the e q u a tio n s o f E q .(1 .2 6 ), the param eter. A , re p rese n ts th e in cid en t voltag e
w a v e, B th e re fle c te d v o lta g e w ave and R the p o rt re s is ta n c e o f the c irc u it. A p p lica tio n o f th is w ave n o tatio n allow s analog ue com p o nents to be describ ed in term s o f in c id e n t and re fle c te d w aves. A p p ly in g th e z tran sfo rm to analogue
c o m p o n e n t d e s c rib e d in te rm s o f w ave p ara m e te rs , g e n e ra te s a set o f dig ita l e le m e n ts th at c a n b e u s e d to c o n s tru c t d ig ita l s tru c tu re s th at p o s s e s s the p ro p e rties o f th e ir DTL re fe re n c e netw orks.
C o n sid e r the o n e -p o rt e le m e n t in F ig.(1.13). U sing E q .(1 .2 6 ) the reflected v oltage w ave, B, can be d esc rib ed in term s o f the incident voltage w ave. A, port resistance. R, and branch im p ed ance, Z . T his relationship is g iven in Eq.(1.27).
F ig u r e 1.13 O n e -p o rt circuit o f im pedance, Z , in term s o f v o ltag e and c u r re n t and incident and reflected voltage w aves.
A = I ( Z + R)
o r B = A R Z ^ _ R l lL(Z + R )J B = I ( Z - R)
I f the one-po rt b ran ch im p ed an c e. Z, represents a capacitor, C, then
‘ J C u,,u
T he b ilin e ar tran sfo rm is d e fin e d as
r o ' » c - * ) i ' L ( 1 / « C * R) J
(1 .2 7 )
w here T is th e sam plin g period.
2 ( ' ' ■ ' )
T ( « ♦ • ■ • )
( 1 2 9 )
C om binin g E q .(1 .2 8 ) and (1 .2 9 ) then
B = A ( T /2 C - R ) + z - » ( T / 2 C ♦ R )1 .( T /2 C + R ) + z" l ( T / 2 C - R )J
page 1/23
I f the ca p ac itan c e value is redefined as
t h e n
b = a [;
(l/c -
R ) ♦ z- ' n r
+ r i~.(l/c;
+ R ) + z - *( l / C ’ • R ).If the p o rt resistance, R, is set so R ■ I / C , then
B = A z*1
T h e r e fo r e th e dig ita l e q u iv a le n t o f a c a p a c ito r , C . u n d e r the w ave p a ram eter
m ethod sug g ested by F ettw eis, is a u n it d ela y , w ith a port resistance, T/2C . A list o f d ig ita l b u ild in g b lo ck s an d th e ir e q u a tio n s is g iv en in a re v ie w p a p e r by F e ttw e is [I I]. The port re sistan c e p lac es a co n s tra in t upon how the d igital elem ents
m ay b e c o n n e cted . T o use an e le m e n t w ith in a circ u it, the p o rt re sistan c e o f co n n e c te d o n e-p o rts m ust b e iden tica l. H o w e v e r, the port re sistan c e is pred efined
by the m o d elled co m ponent value. T o e lim in a te th is p roblem , F ettw eis also created ad a p to rs to eq u a lise the p o rt re sistan c e b e tw e e n tw o o r m ore d is s im ila r one-po rt e l e m e n t s .
C o n sid e r th e series ca p ac ito r o f the D TL n etw o rk show n in F ig .( l.I 4 ). To m odel this co m p o n e n t in a W DF, a sim p le delay is re q u ire d . H ow ever, to use th is elem ent it
needs to b e connected to the rest o f the netw o rk . T o this end a 3 -port series adapter is re q u ire d and is show n in F ig .( l.I S ) . T h e g en e ral eq u a tio n s d esc rib in g a series
c o n n e c te d c a p a c ito r, e x p re s s e d in its w a v e c h a in m atrix fo rm a t, is g iven by
E q .(l.S O ) .
F ig u r e 1.14 G eneral DTL n etw o rk w ith series ca p ac itor, C.
Vo
N N
[image:35.364.25.334.39.408.2]Chapter 1. Introduction page 1/24
’ A f r < 1 • I ' k i - r i ) ) ( 1 - Y l - Y ï - 2 - ' ( l - Y , ) ) l ‘A3"
( J - T i - T a ) O - i - 1) ( 2 - Y . - Yi) ( 1 - 2 - 1 )
O r , ■ a - ' d - Y i Y i ) ) U - Y . - 2*1)
Lb iJ L ( 2 -y, -t2 ) ( 1 - 2 - 1 ) < 2 - Y | - Y 2 ) ( l - 2 - 1) J Lb3J
It should be noted th at the port resistance R2. o f F ig .( l.l S ) , eq u a ls T/2C, w hile Ri and R3 w ill b e set by th e surro u n d in g circ u it. W hen t h e c irc u it is d esig n ed , h o w e v er, th e actual v alu e s o f R i o r R3 m ay not be p re -s e t and could be chosen
a rb itra rily . In th is ca s e , th e s e values m ay be used to e lim in a te y i o r 7 2- T h re e cases aris e for th is 3-port serie s adapter.
if Yl * 1. Y2 * 1 and R2 » 1/C . t h e n yv = R ^ + R ^ v+ 1 / c , . v - 1.2
o r
if Yi - 1 and R2 - 1/C. t h e n R1 ■ R3 + 1 /C . R i
n ■ R3 + l / C
0 r
if Y2 ■ 1 and R2 * 1/C . t h e n R3 - Rl ♦ 1/ C . Ri
T1 ■ R i ♦ 1 /C
-, ~ 2C
w h e r e C =
U sing th is tech niq ue, the overall com plexity o f a WDF circ u it m ay be reduced. ’ ch a in m a tric e s fo r the d e s ig n cases when Yi = 1 o r Y2 = 1 can be determ ined
su b s titu tio n into E q .(1 .3 0 ). A d eta ile d ex p la n atio n o f th e s e desig n procedures is g iv en in th e re v ie w p a p e r by F ettw eis. T h e fin al d e s c r ip tio n o f the o n e-p o rt c a p a c ito r ele m e n t and a 3 -p o rt series adap ter, g iven by E q .(1 .3 0 ) , was in the form
o f th e w a v e ch a in m a trix . T h e re fo re , th e o rig in a l o n e - p o r t ap p roach w as im p le m en ted w ith in th e circ u it as a tw o-port elem ent.
T h e n e c e s s ity o f usin g a sep a rate ad a p ter c irc u it can be av o id e d if a tw o -port
ap proach is used from the s tart. T his techn iqu e w as d e s c rib e d by Law son[19]. An
page 1/25
(■> ( b )
F igure 1.16 Two-port circuit o f series impedance, Z; (a) voltage and current parameters, (b) voltage wave parameters.
t h e r e f o r e
[;/
m
¿si •[;,’]
[
b
a
:
m
: : , ] • [ ; ; ]
—
(1 .3 1 )
( 1 .3 2 )
C o n s id e r a g a in , a s e r ie s c a p a c ito r . C . T h e c h a in m atrix f o r th is a n a lo g u e co m po n en t in term s o f s, is g iv en by E q.(1.33). It m ay be con verted into a d ig ita l w av e c h a in m atrix e q u iv a le n t, sh o w n by E q .(1 .3 4 ) , usin g th e v o lta g e w a v e
d e s c rip tio n s and the b ilin e ar tran sfo rm o f E q .(1 .2 9 ).
" A f r P i - 0 - P1 + P2) I - 1 - B l x ' 1 1 "A 2"
( 1 ♦ p 2 ) ( i - « - ' ) ( 1 ♦ P 2 ) ( l • • - ' )
01 - z * 1 ( 1 - 01 + 0 2 ) - 0 2 z - 1 Lb, J L ( l ♦ p 2 ) ( i • ( i + p 2 ) ( i - i - i ) J Lb2J
( 1 .3 3 )
( 1 3 4 )
A g a in , as w ith th e o n e-p o rt an d a d a p ter m eth o d , th e selec tio n o f R j or R2 o f
F ig .(1 .1 6 ), m ay not be p re-set by th e surrou nd in g c irc u it and eith er P i o r P2 m ay
[image:37.366.22.331.26.437.2]Chapter 1. Introduction page 1/26
p ro v id es three d esig n options.
i f R i a n d R2 a r e
P i R2 + R1 - 1/C* P2 R2 - R1- 1 / C ’
i n d e p e n d e n t t h e n ■ R2 + Rj + 1/C* * " R2 + R i + 1 /C ’
if R2 = R j + 1/C then Pi . " 1 + C 'R i *C O I P2= 0
if R i = R2 + 1 /C then Pi = l + p2. P2" 1 + c rc r22
w here C = 2C /T and T is the sam pling period. A fu ll d escription o f these design p rocedures and th eir e ffe c ts on realisation are discu ssed in C hapter 3.
B oth the o n e-p o rt and tw o -p o rt desig n tec h n iq u e s re ly up o n the u se o f v oltage w ave notatio n and the b ilin e a r tran sform . A lthough th is m ethod is w idely used, it
is not th e o n ly m e th o d to p ro v id e a v ia b le s o lu tio n . O th er m e th o d s w ere in v estig ated by L aw so n , w ho proposed a g en e ral W D F co n c ep t u s in g a chain
m atrix o f the form
[ : ; ] -
m
-
i
;
w h e re P and Q are 2 by 2 m a tric e s , th at re p re s e n t a nu m b er o f d iffe re n t t r a n s f o r m a t i o n s ^ , in clu d in g voltage, cu rrent and p o w e r w aves.
1 . 4 . 2 S t r u c t u r e s
DTL netw orks, w hich form the reference filters fo r W D F d esigns, m ay be defined
w ithin tw o groups; lad d e r and lattice stru ctures. T h e g en e ral DTL lad d e r netw ork, show n by F ig.(1.17), is w idely used in analogue c irc u its fo r radio and television as
no elem ent is m ore th an o ne node away from the g ro u nd line and is th erefo re less su sce p tib le to stray ca p ac itan c e.
F ig u r e 1.17 G eneral L ad der N e tw o rk .
T h e s in g le in p u t-o u tp u t path th ro u g h a la d d e r c ir c u it d e te rm in e s th a t the
s tru c tu re h as a m in im um -ph a se c h a ra c te ris tic .
A g e n e ra l la ttic e circ u it, show n by F ig .(1 .1 8 ), p o ssesses m o re than o n e input- o u tp u t p a th , an d m ay th e re fo re be c la s s e d as h a v in g a n o n m in im u m -p h a s e
c h a ra c te ris tic . T h e la ttic e s tru c tu re is m ore g e n e ra lly re d u ced to a b ala n c e d sym m etric form, w here Za ■ Z c and Zb = Z j.
F ig u re 1.18 G e n era l L attice N etw ork.
B oth la d d e r and lattice s tru ctu re s can be u sed as re fe re n c e s for W D F's. T hese d e s ig n s can be app roached through th e o n e o r tw o -p o rt tec h n iq u e s by redu cin g e a ch im p e d a n c e , Z \, into a sim ple ele m e n t, lik e a c a p a c ito r o r an in d u cto r, and
th e n g e n e ra tin g th e a p p r o p ria te W D F c o m p o n e n t. T h e s y m m e tric a l la ttic e stru c tu re , show n in Fig.(1 .1 9 ), bec au se o f its n o n m inim u m -p hase c h a ra c te ris tic , is ideal fo r im p lem en ting allpass functions and is w idely used a s phase e q u a lis e rs in
an a lo g u e d esig n s. L attice stru c tu re s p re sen t p ra c tic a l d e s ig n p ro b lem s, ho w e v er,
b e c a u s e th e p a irs o f b ranch im p e d a n c e s h a v e to be m atch ed w ith in a high to lera n ce. T h is is a difficu lt task as analo g ue com p on ents are hard to a d ju st, and ag e and c y c le w ith tem perature. T h ese effe c ts are not e v id e n t in d ig ita l desig n s
Chapter 1. Introduction page 1/28
T h e sym m etrical la ttic e o f F ig.(1 .19 ), is given in term s o f its canonic im pedances. Z a and Zb- I f th e co rrespo nd ing canonic reflectan ces fo r a sym m etric W D F lattice
are defined as S ' and S", then
‘ Z , ♦ R
Z h • R
w h ere R, bec au se th e stru ctu re is sy m m e trica l, re p rese n ts the p o rt re s is ta n c e o f
each end o f th e lattice. U sing these c a n o n ic re flec ta n ces, a general W D F lattice stru cture can be co n stru cted and is sho w n by Fig.(1.20).
F ig u r e 1.20 G eneral d is c re te sy m m etrical la ttic e with c a n o n ic re fle c ta n c e s .
I f the seco n d in p u t. A2. is set to zero and B i o r B2 ig n o re d , th en th is lattice
s tru c tu re can be s im p lifie d to p ro d u c e a s tru c tu re sh o w n by F ig .(1 .2 1 ). The tra n s fe r fu n c tio n o f th is s tru c tu re w ill then be the sum o r d iffe re n c e o f the
c a n o n ic re f le c ta n c e s .
F ig u r e 1.21 S im p lified sy m m etrical lattice
ca n o n ic re fle c ta n c e s .
B1
B2 S" + S'
2
with
T h e actual im plem entation o f S' and S" is a design param eter. Bartlett! 12] d evised a
m ethod o f g e n e ra tin g a lattice stru ctu re from a sy m m e tric lad d e r n etw ork. The
