F o r an arbitrary fix e d , but p ositive, fin ite real n u m b er A , let F A d en ote the corresp on din g class o f all distribution functions F ha vin g a ll th e mass points positions o n [ - A , + A ] . T h e mass points positions represent here the ch an n el input am plitude levels w h ich lie betw een - A and + A . A ls o , fo r certain fix e d v a lu e s o f SC , let th e output random variable Y take valu es between -S C , + S C , in steps o f 6q=2SC/(2b- l ) , assuming an b -b it quantiser. T h erefo re, fo r a particular SC value, the a v e ra g e mutual inform ation I ( X ; Y ) can be treated as a functional in the space FA, o f p ro b a b ility distributions F o f the input random variable X , and written as;
(3.25) M SC I ( F ; S C ) = Z Z p(Xj) p ( Y |
x)
lo g 2( p (Y | x J / p ^ Y )) i = l -SC w h ere M Pp( Y ) = Z p ( Y | Xj) p i x j (3.26) i = lH en ce th e capacity o f the IS A / O S C constrained channel [S m ith 1971] can be written as;
N o w th e ca p acity limits, fo r a fix e d A and S C , can be d e fin e d as the maxim um o f a function o f a fin ite dimensional vector, the com ponents o f w h ich are the mass point position s (input amplitude le v els ) and the mass points va lu es (th e probability o f o ccu rren ce o f each level).
S u p p o se the correct number o f mass points is know n (s a y n ) f o r particular values o f A and S C ; let ( x , ^ . . . . ^ ) denote the mass point position s o f an arbitrary input distribution F, and let ( q , . ^ ...q j denote the corresponding mass poin t values. Then the cu m u la tive distribution function F (x ) can be w ritten as;
C ( A ; S C ) = m ax I(F ;S C ) F e F A (3.27) n F ( x ) = Z
qj u(x-Xj)
(3 .28 ) i = lw h ere u (x -x ,) denotes the unit step function at x,.
Let Z = (Z ,....
Z
2J
be a vector comprising the components.Z ,= q, f o r a ll i= l,2 ,...,n (3.29a)
and
Z ^ = x, f o r a ll i= l,2 ,...,n (3.29b)
T h e n the output probability density function can be d efin ed as;
n
Pz<Y)=X Zi P (Y | Z ,J
(3.30)i= l
H en ce the a v e ra g e mutual inform ation can be treated as a function o f th e v e c t o r Z , and w ritten as;
L e t G , denote th e region o f n-dimensional Euclidean space in w h ich the v e cto r Z must lie ; let the f o llo w in g restrictions be im posed on the region G ;
( i ) a ll mass p o in t values are non-negative, ( i i ) a ll mass p o in t positions lie in [- A .+ A ], (ii i ) the sum o f a ll the mass point values is unity.
Thus, G is th e intersection o f all the restriction sets w ith in w h ich th e constraints are satisfied. T h en , th e ISA /O S C constrained capacity, C (A ;S C ), can b e d e fin e d as;
n S C
I (Z ; S C )= X X Z, p ( Y I lo g 2( p ( Y | Z ,J / Pz( Y ) ) i = l -S C
(3.31)
C (A ; S C )= m ax I (Z ; S C ) Z in G
(3.32)
A n optim isation a lg o rith m fro m the N A G com puter library routines [ N A G 1984] has been used to s o lv e th e pro blem o f m axim ising a know n function I (Z ;S C ) o v e r all vectors Z = ( Z , ...w h ic h lie in a w e ll d efin ed restriction region G . T h e optim isation theorem used guarantees the existence o f a unique m axim ising input distribution and provides necessary and su fficie n t conditions fo r achieving the m axim um [G a lla g e r 1968
PP82-97].
F o r a particular v a lu e o f SC, and an y arbitrary value o f am plitude lim it A , let n denote the number o f elem en ts in the v ecto r Z . I f n is known, then the determ ination o f the capacity C ( A ; S C ) is the w ell d efin ed optim isation problem as discu ssed above. In gen eral, i f n is not k n o w n , the fo llo w in g steps are necessary:
( i) Start w ith a v e ry s m a ll value o f A , assuming the optim um number o f m ass points M is tw o , and then fin d th e optimum capacity.
(ii) Increm ent A b y a s m a ll amount, check as A increases whether the n u m ber o f mass points M already used is sufficien t o r not. I f not, increment M b y on e and a p p ly the optim isation algorithm .
T h e program m in g p ro c ed u re used to test wh ether M is sufficien t o r not is based on whether the op tim isa tio n program output fo rc es the extraneous mass poin t valu es ( i f a larger valu e o f M is u s e d ) to zero o r n o t
S in ce th e G au ssian noise has a sym m etric probability density function, th e set o f mass points is also s ym m etric i.e. and xi=x_i. W ith this result, the optim isation problem s can b e fo rm u la te d as the determination o f the optim al pairs o f m ass points.
H ence, the op tim a l set o f mass point pairs is characterised by som e (q ,... q„, x ,,...,x j, w h ere n n o w d en o tes th e number o f mass poin t pairs (a mass point at the o r ig in is also treated as a pair) an d restricted by;
0 £ q, £ 1/2, (3.33a) n £ qi =
1
/2
, i = l (3.33b) and - A £ Xj £ 0 f o r all i=l,2 ,...,n (3.33c)S in ce £ qi= l/ 2 , th en th e number o f independent variables is further reduced as fo llo w s : ( i) i f n is od d (m ass p oin t pairs at the o rig in ), then the optim al set o f mass p o in t pairs is characterised as ( q , ...q ^ „ x , , . . . ^ , ) . w h ere q . = l- 2 £ qj fo r i= l,...,n - l and xB=0; ( i i ) i f n is e ven , th en the optim al set o f mass point pairs is characterised as (q ,...q » „ x „ . . . ^ 0 w h ere q ,= 0 .5 -£ q; fo r i= l,...,n -l.
T h e analysis p ro gra m em plo ys the a b o ve arrangement, which sim p lifies the problem further, and redu ces th e number o f variables o v e r w hich the function must b e optim ised. C onsequen tly, th e com putation tim e to fin d the optimum input distribution and the capacity fo r each f ix e d amplitude lim it is reduced. For a particular value o f am plitude
lim it. A , the op tim isa tio n program is tested w ith different values o f SC. T h e valu es o f the S C that g iv e s th e largest inform ation capacity value, and hence the op tim u m input distribution, is ch o se n to represent the optim um level o f signal clippin g.
T h e IS A P / O S C con strain ed capacity problem is sim ilar to the IS A /O S C shown above, w ith the added constraint o f the average signal pow er, o s 2, being chosen to g iv e the fix e d ratio A 2 /os 2 =2. F o r a n y A w ith a fix e d o s 2 and S C lim it, the IS A P / O S C capacity can b e defin ed as;
C ( A , o s 2;S C )= m ax I (F ; S C ) (3.34)