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Opt ica l and Quantum E lec t ron ics 10 ( 1 9 7 8) 2 1 1 - 2 2 1

A s i m p l i f i e d a p p r o a c h t o d i g i t a l o p t i c a lr e c e i v e r d e s i g n

D . R . S M I T H , I . G A R R E T T

Post Off ice Research Ce ntre, Mart/esham Heath, Ipswich IP5 7RE, England

Received 10 Oc tobe r 197 7

A s i m p l i f i e d t h e o ry f o r t h e p e r f o rm a n c e o f a d i g i t a l o p t i c a l r e c e i v e r is d e v e l o p e d . Th e re c e i v e r s e n s i t i v i t y

is c a l c u l a t e d i n t e rm s o f c i r c u i t p a ra m e t e r s , r e c e i ve d a n d e q u a l i z e d p u l s e s h a pe s , p h o t o d i o d e p a ra m e t e r s

a n d b i t- r a te . A n e x c e l l e n t a g r e e m e n t b e t w e e n t h is t h e o r y a n d a m o r e c o m p l i c a te d a n aly sis b y

Pe rs o n i c k [ 4 ] is d e m o n s t ra t e d . I t is s h o wn t h a t t h e r e c e i v e r s e n s i t i v i t y m a y b e i m p ro v e d b y l a u n c h i n g

r e d u c e d - w i d t h p u ls es i n t o t h e f ib r e , p a r t i c u l a r l y i f f i b r e b a n d w i d t h is a s i g n i fi c a n t l i m i t a t i o n . R e d u c e d -w i d t h p u ls es b r in g b e n e f i t s in s o u rc e p o w e r c o n s u m p t i o n a n d l i fe t i m e , a n d i n t i m i n g r e c o v e r y .

1 . I n t r o d u c t i o n

Th e sen s i t i v i ty o f an o p t i ca l r eceiv e r u s in g an av a lan ch e p h o t o d io d e d ep en d s o n a b a l an ce b e tw een

s ig n a l - d ep en d en t an d s ig n a l - in d ep en d en t n o i se . Th i s p r o b lem h as b een d i scu ssed b y sev e r a l au th o r s

[ 1 - 4 ] , o f wh o m Pe r so n ick [ 4 ] , gives a d e t a il ed an d co m p r eh en s iv e an a ly s i s . Un f o r tu n a te ly , d e r iv in g

th e sh o t n o i se a t t h e d ec i s io n t im e l ead s to ex p r e ss io n s wh ich a r e n o t s im p le to ev a lu a t e . I n v i ew o f t h e

ap p r o x im a t io n s u sed in ex p r e ss in g th e ex cess m u l t ip l i ca t io n n o i se an d in ca l cu lat in g e r r o r r a t e s i t was

f e l t t h a t a s im p l i f i ca t io n o f th e t r e a tm en t o f sh o t n o i se wo u ld b e ju s t i f i ed an d co u ld l ead to a t h eo r y o f

o p t i ca l r ece iv e r d es ig n wh ich w o u ld b e ea s i e r t o a p p ly to p r ac t i ca l sy s t em s .W e h av e ach iev ed su ch a s im p l i f i ca t io n , w i th an in accu r a cy wh ich i s neg l ig ib le i n v i ew o f o th e r

a s s u m p t i o n s m a d e , b y r e l a ti n g t he s h o t n o i s e t o t h e a v e ra g e p h o t o c u r r e n t o v e r t h e b i t - ti m e . W e c o m p a r e

t h e r e s u lt s o f o u r t r e a t m e n t w i t h t h o s e o f P e r s o n i c k [ 4] a n d s h o w t h a t t h e m a x i m u m d i s a g re e m e n t is

o n l y 0 . 6 dB i n r e c ei v e d o p t i c a l p o w e r . W e th e n u s e o u r t h e o r y t o f i nd t h e o p t i m u m l a u n c h e d p u ls e

wid th f r o m a so u r ce su ch a s a sem ico n d u c to r l a se r , wh ich d eg r ad es a t a r a t e d ep en d in g o n th e em i t t ed

p o w er . W e sh o w th a t u s in g r ed u ced - w id th p u l se s p e r m i t s t h e n o i se - eq u iv a len t b an d w id th o f th e r ece iv e r

to b e r ed u ced an d c o n se q u en t ly th e r ece iv e r sen s i t i v i ty i s i n c r eased .

2 . T h e o r y

2 . 1 . T h e o p t i c a l r e c e i v e r

Th e b as i c r ece iv e r i s sh o w n a s a b lo ck d i ag r am in F ig . 1 , wh ich a l so sh o ws th e im p o r t an t n o i se so u r ces .

W e can r ep r e sen t t h e b i n a r y d ig i t al p u l se s t r eam in c id en t o n th e p h o to d io d e b y [ 5 ] :

p ( t ) = ~ b ~ h p ( t - - n T )

w h e r e p ( t ) i s t h e r ece iv ed o p t i ca l p o wer , T i s t h e b i t - t im e an d h p ( t ) i s t h e p u l se sh ap e . Th e am p l i tu d e

p a r a m e t e r , b n , c a n t a k e t w o v a lu e s b e n a n d b o e r c o r r e s p o n d i n g to O N a n d O F F p u l s e s, re s p e c ti v e ly .

W e t ak e

; _ ~ h p ( t ) d t = 1

s o t h a t b n i s t h e e n e r g y in th e n m p u l se . Th e m ea n o u tp u t cu r r en t f r o m th e p h o to d i o d e a t t im e t is :

9 1978 Chapman and Hall Ltd. Printed in Great Britain. 211

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D. R . S mi th , L Garret t

D e t e c t o r a n d b i a s A m p l i f i e r

Cd R b i b ( t ) R A CAh p ( t )

r

~e A ~ E q u a l i s e r

r - -~.- i v ~

i A ( t ) ~ >I " - - h a u l ( t )

Heq(f)

L H B I f ' ) ,

F i g u r e 1 A s c h e m a t i c d i a g r a m o f a n o p ti c a l r e c ei v er s h o w i n g c i rc u it c o m p o n e n t s a n d n o is e s o ur c e s.

T/eip ( t ) = ~-~#p( t )

w h e r e # i s t h e m e a n a v a l a n c h e g ai n , ~ is t h e q u a n t u m e f f i c ie n c y a n d h ~ i s t h e p h o t o n e n e r g y . T h i s

c u r r e n t c a u s e s a m e a n v o l t a g e a t t h e o u t p u t o f t h e e q u a l i z a t io n n e t w o r k g i v e n b y :

A T eVout(t) = ~ - gp ( t ) * h a ( t )* he q( t)

w h e r e

[ ' ]ha( t ) = g ( l / R ) +j27rfC

i .e . , h a ( t ) i s t he im pul se r e spo nse o f t he b i a s c i rc u i t a nd a mp l i f i e r, C be ing CA + Cb , ~ : i nd i c a t e s Fou r i e r

t r a n s f o r m a n d * d e n o t e s c o n v o l u t i o n . H e r e h eq ( t ) is th e i m p u l s e r e s p o n s e o f t h e e q u a l i z a t io n n e t w o r k .

C le a r ly Y our (t ) i s o f t he f o r m :

You = s b n h o u t ( t - - n T )

A ~ e gw h e r e h ~ = ~ [HP (f)HB(f)Heq(f)] ' h~2

H p ( f ) i s t h e F o u r i e r t r a n s f o r m o f t h e r e c e i v e d pu l s e s h a p e h p ( t ), H s ( f ) a n d H e q ( f ) a r e th e t r a n s f e r

f unc t io ns o f t he b i a s c i rc u i t a nd t he e qua l i z e r . T hus :

A ~ e ~

2 .2 . No i se i n t he op t i ca l r ece i ve r

I n c a l c u la t i n g t h e m e a n s q u a r e n o i s e v o l t a g e ( v ~ ) a t t h e d e c i si o n t i m e w e h a v e t o t a k e i n t o a c c o u n t s h o t -

no i se c on t r i bu t ions f r om a l l pu l se s i n t he pu l se t r a in i n so f a r a s t he y ove r l a p , a nd no t me r e ly f r om the

pu l se unde r de c i s ion . T he sh o t no i se a t the de c i s ion t ime thu s de pe nd s on t he sha pe o f t he r e c e ive d pu lse

a n d o n t h e s e q u e n c e o f O N a n d O F F p u l s es . W e c a lc u l a te t h e w o r s t c a s e o f s h o t n o i s e w h e n a l l n e i g h b o u r -

i n g p u l se s a r e O N . R a t h e r t h a n e v a l u a te t h e s h o t n o i s e a s a f u n c t i o n o f t i m e w i t h i n t h e b i t - ti m e , w e m a k e

t h e a p p r o x i m a t i o n t h a t t h e m e a n s q u a r e s h o t- n o i se v o l t a g e ( v ~ ) a t t h e d e c i s i o n t i m e i s re l a t ed t o t h e

m e a n u n i t y g a i n p h o t o c u r r e n t o v e r t h e b i t - ti m e (iO)T b y t h e n o r m a l s i m p l e s h o t n o is e e x p r e s s io n :

( v ~ ) = 2e (i o) T g2 B N R 2A 2 (1 )

w he r e g2 i s t he m e a n squa r e a va l a nc he ga in a nd BN i s t he no i se - e qu iva l e n t ba n dw id th o f t he b i a s c ir c u i t ,

a mpl i f i e r a nd e qua l i z e r :

_ 1

2B N R 2 L ~ i H e q( f ) H B ( f ) 12 d f

2 1 2

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A simpli f ied a pproach to digi tal opt ical receiver design

_ 1 2

. ,_ I d :

T he me a n un i ty -ga in pho to c ur re n t ove r the b i t - t ime i s, fo r a n ON pu lse :

~/e 1

~ T /2 h p ( t - - n T ) d t< i0 > ~ , o N : Y . . , , . . ~ b ~

r /e boN [ - h p ( t ) d t - n e b o N(2 )

hg2 T J = hfZ T

Fo r a n O FF pu lse , a s suming boF F i s z e ro , we ha ve :

r~e 1 ( r nUO)T, OFF =n~02 ~--~ bo N ~r- ",_T:2 h p ( t - - n T ) d t

_ r /e boN ( 1 - - 7 ) ( 3 )

h~2 T

[T /2 h p ( t ) d twh ere 3 ' = ~ - T n

i . e. , 3 ' i s the f r a c t ion o f the e ne rgy o f a s ing le pu l s e w hic h i s c on ta in e d w i th in i t s b i t - t ime . T he se va lue s o f

< i0>T c a n be su bs t i tu te d in E qua t io n 1 to c a lc u la te the w ors t - ca se me a n squa re sho t -no ise vo l ta ge s .

Pe r son ic k [4 ] c a lc u la te s the m e a n squa re sho t -no ise vo l ta ge as a func t ion o f t ime ins te a d o f a pprox i -

m a t i n g i t t o t h e s h o t n o i s e o n t h e m e a n u n i t y - g a in p h o t o c u r r e n t o v e r t h e b i t- t i m e . B y m a k i n g t h is

a pp rox i ma t ion we a c h ie ve subs ta n tia l s imp l i f i c a t ion o f the f ina l e xpre s s ions a nd c a lc u la t ions fo r the

re c e ive r s e ns i t iv i ty w i th ne g l ig ib le a dd i t iona l e r ro r .

F o r t h e t h e r m a l n o i se w e u s e t h e n o r m a l e x p r e s s i o n

< ~ > _ _ { 2 x o + S O A = f =H o u , ( f ) 2 d f + S E A 2 : ~ H e a ( f ) [ 2~ R b - = I H p 0 O

whe re 0 i s the a bso lu te t e m pe ra tu re , S I is the spe c t r a l de ns i ty o f the a mpl i f i e r no i s e c u r re n t sourc e a nd

SE i s the spe c t r a l he igh t o f the a mpl i f i e r no is e vo l ta ge sourc e . T he to ta l m e a n squa re no ise vo l ta ge a t the

de c i s ion t ime i s thus :

[ 2kO-t-Si] f : [H ~ 2 d f+ A 2 S E s (4 )< v ~> = A z e< i o > T g= + x + R b - I H p ( f ) I

We ha ve use d the a pp rox im a t ion g2 : ~ ~+x . A l te rna t ive ly , Mc In ty re ' s [6 ] m ore a c c ura te e xp re s s ion c ou ld

b e u s e d a t t h e e x p e n s e o f a l g eb r a ic c o m p l i c a t i o n i n f i n d in g t h e o p t i m u m a v a la n c h e g ai n.

We no w no rma l iz e , fo l lowing Pe r son ic k [4 ] , so tha t Oout( t = n T ) = b , , w h i c h m e a n s p u t t i n g

Arle~/hfZ = 1 a nd hou t ( t = 0 ) = 1 . I t is use fu l al so to m a ke the b a nd wid th in te g ra ls in E qu a t ion 4i n d e p e n d e n t o f b i t - ti m e T , s o t h a t t h e i r n u m e r i c al v al u e d e p e n d s o n l y o n t h e shape of the r e c e ive d a nd

e q u a l i z ed p u ls e s a n d n o t o n t h e i r scale. T o t h i s e n d w e i n t r o d u c e t h e d i m e n s i o nl e ss t im e a n d f r e q u e n c y

var iables z = t i T a nd r = f i r . We wa n t to f ind wh a t func t ion s H 'out(~b) a nd H~ ,(4) ) mu s t r e p la c e H ou t ( f

a n d H v ( J w h e n r i s s u b s t i t u t e d f o r f T a n d t h e in t e g r a t io n s p e r f o r m e d w i t h r e s p e c t t o r O n e ca n e a si ly

s h o w , b y c o n s i d er i n g t h e F o u r i e r tr a n s f o r m s o f h o u t ( z a n d h p ( r ) , t h a t

1H'out( r = ~ - H o u t ( j 0

a nd H p ( r = H p ( f )

: - I - o o , < : > = =

3 - - [ H I , ( : ) d : - T f _ d q L

213

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D. R. S m ith, L Garrett

Using Per son ick ' s no ta t io n [4 ] :

I H o ( 0 ) l :

1 3 = Y - ~ ~ lH ~ r d q~

z = + - - + / 2 + -a n d ~ I Rb Te 2 '3E13

i .e . Z i s a d im ens ion les s pa ra m e te r r ep resen t ing the s igna l - indepe nden t no i s e t e rm s , we ob ta in f rom

E q u a t i o n 4 :

a n dhf2 g . Z

<v~r> = <to>TTI2 9( s )

2 . 3 . C a l c u l a t i o n o f r e c e i v e r s e n s i t i v i t yW e w il l u s e t h e G a u s s ia n a p p r o x i m a t i o n t o c a l c u la t e t h e m i n i m u m e n e r g y p e r p u l se r e q u i r e d t o a c h ie v e

a p r e s c ri b e d m a x i m u m e r r o r r a t e . A l t h o u g h t h e s h o t n o i s e ha s i n f a c t a P o i ss o n d i s t r ib u t i o n , t h e

inacc uracy a r is ing f rom the Gaus s ian appro x im at ion i s no t l a rge [7 ] . W e def ine NON and NOFF as the

wors t - case va lues o f (v~r> fo r O N and O FF pu l s es , wh ich a re o b ta in ed b y subs t i tu t ing E qua t ion s 2 o r 3

f o r <io>Tn t o E q u a t i o n 5 . W e as s u m e t h a t t h e o u t p u t v o l t a g e is a G a u s s ia n r a n d o m v a r i a bl e w i t h m e a n

a n d v a r ia n c e a t t h e d e c i s i o n t i m e o f b o N a n d a ~ N = N O N f o r O N p u l se s , b O F F a n d a ~ F F = N O F F f o r

OF F pu l s es. W e a l so assum e tha t th e th re sho ld dec i s ion l eve l VD is s e t soa s to g ive equa l e r ro r p rob-

a b i l i ty P E f o r O N a n d O F F p u ls e s. T h e n

( b O N - - V D ) / a O N - - ( V D - - b o F F ) / a O F F =

a n d

1 89R E -

T h e n i f bO F F is z e r o , t h e r e q u i r e d e n e r g y p e r p u l se t o a c h ie v e th e m a x i m u m e r r o r r a t e c h a r a c t e r i z e d

by Q i s:

boN = g t V ] ( [ , ~ b o N I 2 + Z a / 2 + + x ~_ ~ o N ( l _ 7 ) i 2 + 1/2 ,

W ith a non-ava lanche p ho tod iod e , ~ = 1 and the sho t -no i s e t e rm s a re usua l ly negl ig ib le in com p ar i son

w i t h t h e t h e r m a l n o i s e . T h u s

bO N = 2 Q n a Z v Z , i f = 1rl

wh ic h is , o f cour se , iden t ica l to Per son ick ' s [4 ] r esu l t fo r a non-ava lanche pho tod iod e .

(6 )

2 . 4 . O p t i m u m a v a l a n ch e g ai n

Equating OboN/O t o z e r o , w e o b t a i n :

gov t u~ r~ 212 K

w h e r e

K = - - 1 + 1 + ~ - ~ - ]

F r o m E q u a t i o n s 6 a n d 7 w e o b t a i n o u r m a i n r e s u lt f o r t h e m i n i m u m r e q u i r e d p u ls e e n e r g y :

( 7 )

2 1 4

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A s i m p l i f i e d a p p r o a c h t o d i g i ta l o p t i c a l r e c e i v e r d e s i g n

b O N , m i n = Q(2+x) tQ+x)(~)ZX/(2+2x)I~/O+X)L ( 8 )

w h e r e

Ll+X= [ 2 ( 1 - - 7 ) ] ( [ K + I ] 1/2 1 1/2}

T h o u g h i t is a c u m b e r s o m e e x p r e s s i o n , L is a p a r a m e t e r w h i c h d e p e n d s o n l y o n t h e f r a c t io n 7 o f th e

p u l s e e n e r g y c o n t a i n e d w i t h i n t h e b i t - ti m e a n d o n t h e x - f a c t o r f o r t h e p h o t o d i o d e . F i g . 2 is a p l o t o f L

v e r su s 7 f o r t h r e e t y p i c a l v a l u e s o f x , a n d m a y b e u s e d t o f i n d L f o r a n y r e c e i v e d p u l s e s h a p e .

T h e r e q u i r e d e n e r g y p e r p u l s e a t o p t i m u m g ain i s g i ve n b y E q u a t i o n 8 , a n d c a n b e c a l c u l a t e d f o r a n y

r e c e iv e d a n d e q u a l i z e d p u l s e s h ap e s . T h e d e p e n d e n c e o n b i t - ra t e i s e n t i r e ly c o n t a i n e d i n Z , w h i c h i s

d o m i n a t e d b y t h e t e r m ( 2z rC ) 2 S E I 3 / T e 2 e x c e p t a t v e r y l o w b i t -r a t e s, a n d w h i c h e n t e r s t o t h e p o w e r

0 . 1 2 - 0 . 1 7 ( f o r x o f 0 . 3 - 0 . 5 ) . T h e r e q u i r e d p u l s e e n e r g y is t h e r e f o r e v e r y in s e n si ti v e t o b i t - ra t e , d e c r e a s -

i n g b y a f a c t o r o f 2 - 3 b e t w e e n 1 a n d 5 0 0 M b s - 1 , (a s s u m i n g S v. is c o n s t a n t w i t h b i t - r a te ) .

3 . Resul ts

3.1 . Depe ndence o f r eq u i red pu lse energy on pu l se w id thT h e o p t i m u m rece i v ed p u l s e s h ap e i s an im p u l s e [4 ] , s i n ce t h en [H ~(~)12 = 1 fo r a ll 4) an d t h e b an d -

w i d t h i n t eg ra l s I2 an d 1 3 a r e a s s m a l l a s t h e y can b e m ad e b y v a ry i n g H ~(q ~).

W e c a n c a l c u la t e t h e p o w e r p e n a l t y w h i c h h a s t o b e p a i d f o r h a v in g r e c e i v e d p u l s es w h i c h a r e n o t

i m p u l s e s. T h e p e n a l t y i n d B i s g iv e n b y :

I IX/(2+2x)I1/(l+x) L3 2 [

A b o N = 1 0 1 o g lo } l ~ [I : 3 , o p t 2 , o p t o p t )

4 .0

3 .5

3 .2

2 8

2 / .

2 0 1

0.1

~c=1

sc=0. :

I I I I I I I I0.3 0.5 0.7 0.9

> 7

Figure 2 A g r a p h o f t h e p a r a m e t e r L a s a

f u n c t i o n o f % t h e f r a c t i o n o f r e c ei v ed

o p t i c a l p u l s e e n e r g y w i t h i n i ts b i t * t i m e , f o r

v a lu e s o f t h e p h o t o d i o d e x - p a r a m e t e r o f0 .3 , 0 .5 a n d 1 .0 .

215

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18

16

14

m 1 2" o

S 10gQ. .

8

5

4

/ / /

// /

o11 o:2 o13 o[41 - y

Figure 3 T h e p e n a l t y in m i n i m u m

rece ived po we r f o r pu l se sp read i ng

ou t s i de t he b i t - t i m e , assum ing

Gauss ian rece ived pu lses. Th e do t tedl i ne i s rep rod uced f rom Person ick [ 4 ] .

f r o m E q u a t i o n 8 , i n w h i c h w e h a v e ig n o r e d t h e I 2 t e r m a g ai n i n Z i n c o m p a r i s o n w i t h t h e c a p a c i ta t i v e

t e r m . Th e p en a l ty th en d ep en d s o n th e sh ap e o f t h e r ece iv ed p u l se b u t n o t o n th e b i t - r a t e . F ig . 3 sh o ws

th e p en a l ty a s a f u n c t io n o f 7 f o r Gau ss i an r ece iv ed p u lse s , tak in g an x - f ac to r o f 0 .5 f o r t h e p h o to d io d e .

Th e r e su l ts o f Pe r so n ick [ 4 ] h av e b een r ep r o d u ce d f o r co m p ar i so n , an d th e ag r eem en t i s ex ce l l en t . On e

can see f r o m F ig . 3 t h a t t h e r e i s a sev e r e p en a l ty i f t h e p u l se en e r g y sp r ead s s ig n i f i can tly b ey o n d i t s

b i t - t im e .

3 .2 . L a u n c h e d p o w e r r e q u i r e m e n t s

Fr o m F ig . 3 , we d ed u ce th a t i n an o p t i ca l f i b r e sy s t em o n e wan t s t o u se a s n a r r o w a l au n ch ed p u l se a s

p o s s ib l e in o r d e r t o m i n i m i z e t h e p e n a l t y i n f l ic t e d b y d i s p er s io n a n d i n t e r s y m b o l i n t e r f e r e n c e o f t h e

r ece iv ed p u l se s , an d co n seq u en t ly r ed u ce th e n o i se eq u iv a len t b an d wid th o f t h e r ece iv e r. B y co n t r a s t , i n

a co ax ia l cab le sy s t em , th e r e i s an o p t im u m p u l se w id th ; a p en a l ty acc r u es i f t o o n a r r o w p u l se s ar e

l a u n c h e d b e c a u s e o f e x ce s s iv e a t t e n u a t i o n o f t h e h i g h - f re q u e n c y e n d o f t h e s p e c t r u m .

Ho w ev er , i n an o p t i ca l f i b r e sy s t em , th e p r ac t i ca l n a r r o win g o f t h e l au n c h ed o p t i ca l p u l se m ay b e

l i m i t e d b y t h e p e a k o u t p u t p o w e r w h i c h t h e o p t i c a l s o u r c e c a n e m i t w i t h o u t a n u n a c c e p t a b l e d e gr a-

d a t io n r a t e , f o r ex am p le , t h r o u g h f ace t d am ag e in a sem ico n d u c to r l a se r [ 8 ] . W e d o n o t h av e en o u g h

i n f o r m a t i o n a t r e l e v an t p u ls e r e p e t i t i o n r a t e s o n t h e e f f e c t o f p e a k p o w e r o n l a s e r d e g ra d a t io n , s o w e

a s s um e t h a t t h e d e g r a d a t i o n r a t e r D i s p r o p o r t i o n a l t o t h e e m i t t e d o p t i c a l p o w e r P t o s o m e p o w e r z . F o r

r ec t an g u la r l au n ch ed p u l se s , t h e d eg r ad a t io n r a t e d u r in g th e p u l se isr D ~ ( P , ~ ) ~

wh er e PL i s t h e p ea k em i t t ed p o wer . W e wil l i g n o r e th e lo ss in co u p l in g th e l a se r t o t h e f ib r e , a s t h i s d o es

n o t a f f e c t th e a r g u m e n t , a n d t a k e P L a s t h e p e a k l a u n c h e d p o w e r a l so . W e c an n o w c a l cu l a te t h e

n ecessa r y en e r g y p e r p u lse b o N r eq u i r ed to ach iev e a p r e sc r ib ed m ax i m u m e r r o r r a t e as a f u n c t io n o f

l au n ch ed p u l se w id th , w i th th e co n s t r a in t t h a t t h e to t a l so u r ce d eg r ad a t io n p e r u n i t t im e ( say p e r b i t -

t im e) i s h e ld co n s t an t a t it s m ax im u m a ccep tab le v a lu e R D . W e h av e :

1 [ T/2RD = -T" J-T~2 rD (t) dt"

Fo r r ec t an g u la r l au n ch ed p u l se s o f d u r a t io n - - a L T / 2 < t < a L T / 2 , a n d a s s u m i ng z e r o o u t p u t p o w e r

i n th e O F F s t a t e :R D c c p ~o ~L

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f u n c t i o n o f l a u n c h e d p u ls e w i d t h ~ L a n d

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t r a n s f e r f u n c t i o n , a n d n o r m a l iz e d t o f u l l -

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b a n d w i d t h .

b o N = PLa L T .R~/z ec bON (aLT)O/z) -1.

H e n c e , h o l d i n g R D c o n s t a n t , w e c a l c u l a t e t h e p e n a l t y i n re c e i v er s e n s i ti v i t y b o N a s a f u n c t i o n o f

l a u n c h e d p u l s e w i d t h a L ; b o N ~ a L - 1 /z .

W e h a v e p e r f o r m e d t h e c a l c u l a t i o n s a s s u m i n g a G a u s s i a n t r a n s f e r f u n c t i o n f o r t h e f i b re , a r e a s o n a b l e

a s s u m p t i o n g i v e n s u b s t a n t i a l m o d e - m i x i n g . T h u s t h e f i b r e t r a n s f e r f u n c t i o n i s :

a n d t h e l a u n c h e d p u l s e s h a p e i s :

T h e r e c e iv e d p u l s e s h a p e i s t h e n :

H ~ ( r = e - ( 2 ~ F r

h E ( t ) = 1/aLT l t l<O~LT/2

= 0 e l s e w h e r e .

o ( 0 - ' [ e , ' o2 (X L T t - - - - ~ J - - e r , c [ ~ T ~ - - - 2 - / / J "

W e h a v e a ls o c a l c u l a t e d t h e p o w e r p e n a l t y a s a f u n c t i o n o f p u l s e w i d t h f o r t h e c a s e o f a r ig i d u p p e r

l i m i t t o t h e p e a k p o w e r a v a il a b le f r o m t h e l a s e r, i n d e p e n d e n t o f p u ls e d u r a t i o n a n d r e p e t i t i o n r a t e, w h i c h

i s t h e m o s t p e s s i m i s t ic c a s e . T h e r e s u l ts f o r s e v e ra l f ib r e b a n d w i d t h s ( ~ v ) a r e s h o w n i n F i g . 4 , n o r m a l -

i z e d t o t h e c as e o f fu l l - w i d t h la u n c h e d p u l s es a n d i n f in i t e f i b r e b a n d w i d t h ( ~ L = 1 , a F = 0 ) . T h e r e is a

b r o a d m i n i m u m , c o r r e s p o n d i n g t o t h e o p t i m u m t r a d e - o f f b e t w e e n r e d u c in g th e p u l s e o v e r la p a n d r ed u c -

i n g t h e p u l s e e n e r g y b y n a r r o w i n g t h e p u l s e. T h e b e n e f i t o f u s i n g r e d u c e d - w i d t h p u l s e s is a f e w d e c i b e lsi f t h e f i b r e b a n d w i d t h i s a s i g n i f ic a n t l i m i t a t i o n .

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F i g u r e 5 T h e p e n a l t y in m i n i m u m l a u nc h e d

p o w e r a s a f u n c t i o n o f l a u n c h e d p u l s e w i d t h

f o r v a r i o u s va l u e s o f t h e s o u r c e d e g r a d a t i o n

p a r a m e t e r z , a s s u m i n g r e c t a n g u l a r l a u n c h e d

p u ls e s a n d i n f i n i t e f i b r e b a n d w i d t h , a n d

n o r m a l i z e d t o f u l l - w i d t h l a u n c h e d p uls es .

W e h a v e p e r f o r m e d t h e c a l c u l a t i o n s f o r s e v e ra l v a l u e s o f z w i t h a fi x e d f i b r e t r a n s f e r f u n c t i o n a r .

F i g . 5 s h o w s th e r e s u l t s f o r i n f i n i te b a n d w i d t h f i b re , a r = O . T h e c a s e o f z = 1 c o r r e s p o n d s t o c o n s t a n t

e n e r g y i n th e p u l se . F o r l a rg e r va l ue s o f z , t h e r e i s a m i n i m u m i n t h e p o w e r p e n a l t y . T h e c u r ve m a r k e d

A is f o r a s o u r c e w i t h a fi x e d u p p e r l i m i t t o t h e e m i t t e d o p t i c a l p o w e r a s i n F i g . 4 .

4 . D i s c u s s i o n

4 . 1 . C o m p a r i s o n w i t h P e r s o n i c k 's t h e o r yP e r s o n i c k g iv e s a n e x p r e s s i o n f o r th e m e a n s q u a r e s h o t - n o i s e v o lt a g e ( E q u a t i o n 1 7 o f [ 4 ] p . 8 5 2 ) :

( n~ (t ) ) = = e 2 ~ 2 ~ b . h p ( t - - n T ~ - ~ + X o h ~ ( t - - t ' ) d t

w h e r e X o is t h e ' d a r k c u r r e n t ' c o n t r i b u t i o n t o t h e p r i m a r y p h o t o c u r r e n t , a n d , w i t h r = 27r ,

1 Hout (Oa)~ : [ h i ( t - - t ') ] = g i ( L o ) = H e q (( .o ) ( I / R ) + j o a C H p ( c O )

I n t h e c a s e w h e r e a ll b n = b o N ( w o r s t - c a s e s h o t - n o i se f o r a n O N p u l s e ) , a n d i g n o r i n g X o , t h i s e x p r e s s i o n

b e c o m e s

( n ~ ( t = O ) ) = N O N = e 2 g - ~ b o N - - n T ) h ~ ( t - - t ') d t ' .

T h e c o r r e s p o n d i n g e x p r e s s io n i n t hi s t h e o r y is

N O N = e 2 ~2 fiY~ T

T h e a g r e e m e n t b e t w e e n t h e t w o t h e o r i e s t h u s d e p e n d s o n t h e s h a p e o f t h e r e c e iv e d p u ls e . D i sr e g ar d in g

t h e c o n s t a n t m u l t i p li e r , w h i c h i s th e s a m e i n b o t h e q u a t i o n s , t h e e x p r e s s io n d e r iv e d b y P e r s o n i c k c a nb e w r i t t e n :

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f ~ [ ~ n h p ( t ' - - n T ) ] h ~ ( t - - t ' ) d t '.

T h e q u a n t i t y i n t h e s q u a r e b r a c k e t s is a p e r i o d i c w a v e f o r m a n d s o c a n b e e x p r e s s e d a s a F o u r i e r s e r ie s :

, F ( t ) = ~ . h p ( t - - n T ) = • c m e - 2 = m j t / T

71 111w h e r e

1 ( T / 2 ~C m = -T J - T n n h e ( s - - n T ) e 2 r rm j s/ r d s

l f ~ h p ( s ) e 2 ~ m m /Tds- -

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1 H { 27 rm l = 1 H ; ( m ) .

I n pa r t i c u l a r , Co = 1 /T .

T hus P e r son i c k ' s e xpr e s s ion f o r t he w or s t - c a se sho t - no i se du r ing a n O N pu l se i s :

NON = ea g2 h~r/bONT ~ H ;(m ) 2 e -2 ~ rm i t/ T h ~ ( t - - t ' ) d t '

= eZ g 2 17 b oN Z1

hgZ T

T h u s t h e t h e o r y u s e d h e r e g iv es t h e s a m e r e s u lt a s P e r s o n i c k ' s w h e n 2 1 = / 2 . A s / 2 is t h e f i r st t e r m i n

2 ;1 , w e e x p e c t g o o d a g r e e m e n t w h e n H p ( m ) i s sma l l f o r [ m[ ~> 1 w hic h impl i e s b r o a d r e c e ive d pu l se s . I n

f a c t , i f H ~( m ) = 0 f o r im j ~> 1 t he r e c e ive d pu l se sha pe m us t s a t i s f y t he N y qu i s t c r i t e r i on , i .e . pos se s s

ske w sym m e t r y a bo u t t = T . U n le s s suc h a pu l se is r e c t a ngu l a r , c ons ide r a b l e e ne r gy is sp r e a d ou t s i de t he

b i t - t i m e .T h e c a s e o f a n O F F p u l s e i s p e r h a p s b e s t i n v e s t i g a te d b y d i r e c t c o m p a r i s o n o f t h e r e s u lt s o f t h e t w o

the or i e s .

I n F ig . 6 w e s h o w t h e e r r o r in b o N r e s u lt in g f r o m t h e a p p r o x i m a t i o n u s e d in t h is p a p e r c o m p a r e d

w i th P e r son i c k ' s mor e de t a i l e d a na lys i s , f o r G a uss i a n a nd r e c t a ngu l a r r e c e ive d pu l se s a nd a r a i s e d - c os ine

o u t p u t p u ls e . T h e m a x i m u m e r r o r o f 0 . 6 2 d B (t h e t h e o r y u s e d h e r e u n d e r es t im a t e s b o N ) o c c u rs w h e n

t h e r e c e i v e d p u ls e is a n im p u l s e . F o r w i d e r p u l se s t h e a g r e e m e n t is e x c e ll e n t. I n f a c t t h e m a x i m u m e r r o r

i n v o l v e d i s s m a l l e r t h a n t h e e r r o r w h i c h c o u l d r e s u lt f r o m u s in g t h e s i m p l e f ' ~ e x p r e s s i o n f o r t h e e x c e s s

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t h e o r y a n d t h a t d e r i v e d b y P e r s o n i c k [ 4 ]

a s a f u n c t i o n o f r e c e iv e d p u l s e w i d t hfo r r ec tang u la r and G auss ian pu lses.

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D. R. Smith, L Garrett

m u l t i p l i c a t io n n o i s e in s t e ad o f M c I n t y r e ' s [ 6 ] m o r e a c c u r a t e f o r m . W e e x p e c t o u r s i m p l e r t h e o r y t o

p r o v i d e r e a s o n a b le e s t i m a t e s o f t h e p e r f o r m a n c e o f r e a l s y s t e m s , a n d t o p r o v i d e a s o u n d b a s is f o r

c o m p a r i s o n o f s y s te m s .

4 . 2 . O p t i m u m l a u n c h e d p u l s e w i d t h

T he ca lcu la t ions desc r ibed in Se c t ion 3 . 2 and r ep rese n ted in F igs. 4 and 5 ind ica te tha t the r ece iver i sm o re s ens i t ive to r e duce d-w id th l aunch ed pu l s es , pa r t i cu la r ly i f the re i s s ign i f ican t pu l s e d i sper s ion in

t h e f i br e . T h e o p t i m u m p u l se w i d t h d e p e n d s o n t h e t r a n s f e r f u n c t i o n o f t h e f i b r e , t h e d e p e n d e n c e o f

o p t i c a l s o u r c e d e g r a d a t io n o n p u l s e d u r a t i o n a n d r e p e t i t i o n r a t e , a n d o t h e r f a c t o r s , b u t t h e r e i s a l w a y s

som e adva n tage in using r educed-w id th pu l ses .

A l t h o u g h t h e i m p r o v e m e n t s i n r e c e i v e r s e n s it i v it y t h a t c a n b e o b t a i n e d a r e n o t l a r ge ( a f e w d e c i b e l s

o n l y , e v e n w i t h s e v e re b a n d w i d t h l i m i t a t i o n i n t h e f i b r e ) a n i m p o r t a n t p o i n t i s t h a t t h e r e i s n o d is -

a d v a n ta g e i n u s in g r e d u c e d - w i d t h p u ls e s, d o w n t o a w i d t h d e p e n d i n g o n a r a n d z . T h u s r e t u r n - t o - z e ro -

l e ve l c o d in g m a y b e u s e d w i t h n o i n c re a s e ( in f a c t p o s s ib l y a d e c r e a s e) i n t h e f i b re b a n d w i d t h r e q u i r e d .

A m o r e i m p o r t a n t b e n e f i t b e c o m e s c l e a r w h e n o n e c o n s id e r s th e p o w e r r e q u i r e m e n t s o f a s e m i c o n d u c t o r

l a se r , u n d e r c i r c u m s t a n c e s w h e n i t m a y b e b i a s e d a t z e r o d r iv e c u r r e n t . B e c a u s e o f t h e n o n - l i n e a r

c h a r a c t e r is t i c o f o p t i c a l o u t p u t p o w e r a s a f u n c t i o n o f d r i ve c u r r e n t , t h e e l e c tr i c a l e n e r g y r e q u i r e d t op rod uce an op t ica l pu l s e o f g iven energy depends o n the pu l s e w id th , and i s l eas t fo r na r ro w pu ls es . So

t h e e l e c tr i c a l p o w e r r e q u i r e m e n t s o f t h e l a se r ( a n d h e n c e o f i ts c o n t r o l c i r c u i t, e t c . ) m a y b e r e d u c e d b y

us ing r educ ed-w id th pu l s es , w i th no d i s advan tage in r ece iver s ens i t iv i ty and f ib re bandw id th .

T h e o p t i m u m p u l se w i d t h d e p e n d s , t h e n , o n a b a la n c e o f s o u r c e l i f e ti m e a n d p o w e r r e q u i r e m e n t s ,

a n d o n f i b r e b a n d w i d t h . A p u l s e w i d t h o f a b o u t h a l f t h e b i t- t i m e w o u l d n o t b e f a r o f f th e o p t i m u m i n

m any cases .

One a r r ives a t a conc lus ion fo r the op t im um pu ls e w id th r a th er s im i la r fo r op t ica l f ib re sys tem s and

co-ax ia l cab le sys tem s , bu t fo r ve ry d i f f e ren t r easons . In the coax ia l cab le sys tem , the cab le ban dw id th

i s l i m i t e d b y f r e q u e n c y - d e p e n d e n t a t t e n u a t i o n , a p p r o x i m a t e l y a s X /f . A r e d u c e d - w i d t h p u l se e n a b l e s o n e

t o r e d u c e t h e h i g h - f r e q u e n c y l if t ( a n d h e n c e t h e n o i s e e q u i v a l e n t b a n d w i d t h ) o f t h e r e c e iv e r a m p l i f ie r .

T o o m u c h r e d u c t i o n o f p u l se w i d t h r e s u lt s in e x c e s si v e a t t e n u a t i o n i n t r a n sm i s s io n . I n a n o p t i c a l f i b re ,

t h e a t t e n u a t i o n i s i n d e p e n d e n t o f f r e q u e n c y ( o v e r a ra n g e o f th e o r d e r o f T H z ) ; t h e b a n d w i d t h i s l i m i t e d

b y d i s p e rs i o n w h i c h b r o a d e n s t h e p u l s e in t h e t i m e d o m a i n . A r e d u c e d - w i d t h p u l s e c o n f i n e s m o r e o f t h e

p u l s e e n e r g y w i t h i n i ts o w n b i t -t i m e a n d t h u s r e d u c e s i n t e r s y m b o l i n t e r f e r e n c e i n t h e r e c e iv e d p ul se s .

T h e n o i s e -e q u i v a l e n t b a n d w i d t h o f t h e r e c e iv e r c a n t h e r e f o r e b e r e d u c e d , i n c r e as i n g t h e r e c e i v er

sens i t iv i ty .

5 . Co n c l u s i o n s

W e h a v e d e s c r i b e d a s i m p li f ie d t h e o r y o f n o i s e i n a n o p t i c a l r e c e iv e r a n d h e n c e d e r i v e d t h e r e c e iv e r

s ens i t iv i ty . W e have shown tha t th i s theo ry g ives r esu l ts ve ry s im i la r to the r esu l t s o f a m o re de ta i l ed

t h e o r y b y P e r s o n i c k [ 4 ] f o r t h e t y p e s o f r e ce i v e d a n d e q u a l i z e d p u ls e s h ap e s o f p r a c t ic a l i n t e re s t . W eh a v e u s e d o u r s i m p l i fi e d t h e o r y t o c a l c u la t e t h e o p t i m u m l a u n c h e d p u l se w i d t h i n a s y s te m w i t h v a r i o u s

degrees o f f ib re d i sper s ion , and a l so fo r va r ious power - law depe nden ces o f l a s e r deg rad a t ion r a te on

op t ica l o u tp u t pow er . W e have show n tha t the re i s a sm al l inc rease in r ece iver s ens i t iv i ty to be ga ined by

us ing r educe d-w id th pu l s es in a ll cases . T hu s on e m ay use r e tu rn - to -ze ro - leve l cod ing , and pos s ib ly

r e d u c e t h e m e a n p o w e r r e q u i r e m e n t s o f t h e l a s er s i g n if i ca n t l y , w i t h n o p e n a l t y i n r e c e i v er s e n s it i v it y o r

s y s t e m b a n d w i d t h . T h e m e t h o d s u s e d i n t h is p a p e r m a y b e a p p l i e d t o t h e d e s i g n o f d i g it a l o p t i c a l

r ece iver s fo r a ny g iven r ece ived and equa l ized pu l s e shapes .

AcknowledgementsW e are g ra te fu l to Dr J . E . Midwin te r an d o th er co l l eagues fo r m an y s t im ula t ing d i s cuss ions, and to the

D i r e c t o r o f R e s e a r c h o f t h e P o s t O f f i c e f o r p e rm i s s i o n t o p u b l i s h t h is w o r k .

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A simplified approach to digital optical receiver design

References

1 . w . M . H U B B A R D , B e l l S y s t . Te c h. J . 52 (1973) 731-65.

2. R. DOGLIOTTI, A. GUARDIN CERRI and A. LUVISON, Op t . Qu a n t . E le c t . 8 (1976) 343-53.

3. J. E. MIDWINTER, i b i d 9 (1977) 299-304.

4 . S . D . P E R S O N I C K , B e l l S y s t . T e c h . J. 52 (1973) 843-86.

5. A.B . CARLSON, 'Com muni cati on Systems' , 2nd Edition , (McGraw-HiU, Kogakusha, 1975).

6. R. J. MCINTYRE, I EE E T r an s. E le c t r o n . De v ic e s ED-13 (1966) 164-8.7 . P . B A L A B A N , B e l l S y s t . T e ch . J. 55 (1976) 745-66.

8. N. CHINONE, R. ITO and O. NAKADA, J. A p p L P h y s. 47 (1976) 785-6.

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