4.5.1 Noiseless Channel
G en era lly, the a ve ra ge mutual inform ation betw een the input and output alphabets can b e w ritten [G a lla g e r 1968 pp74], in bits per channel use, as fo llo w s ;
K L
I ( X ; Y ) = 2 2 p ( i ) p (j | i) log
2
(p (j | i)/ p (j)) (4.23) i = l j = lw h ere i and j are the i-th input and the j-th output, p (i) is the i-th input probability, p (j | i ) is the con ditional probability, and p (j) is the channel output distribution, which is g iv e n b y;
p ( j ) = 2 p (i) p (j | i ) (4.24)
F o r th e noiseless M A C , the average mutual inform ation o f M A C , reduces to th e channel output en tropy and can b e written as;
L
I ( X ; Y ) = - 2 p (j) lo g
2
( p (j)) (4.25)j = l
I f w e assume that the channel outputs are equiprobable, then the channel output p ro b a b ility can be w ritten as;
p ( j ) = 1/L; fo r j = l , 2 ... L (4.26)
Thus equ ation (4.25) is reduced to
I ( X ; Y ) = lo g
2
(L ); bits/channel use (4 .27 )T h is M A C cap acity w ith the assumption o f uniform distribution o f the output signals is called th e unconstrained capacity. H o w ever, the assumption o f uniform distribution o f output sign a ls is quite gen eral and dose not exh ibit the actual distribution o f output signals f o r th is channel m odels. C on sider the M A C capacity using the actual output probability distributions calculated previou sly fo r each channel m odel. I f it is assumed that during ea ch sym bol interval, each user transmit statistically independent sym bols w ith equal probability. T h en , substituting fo r the actual output distribution p (j), in to equation (4 .2 5 ), the m axim um avera ge mutual inform ation f o r the noiseless channel m odels ca n b e written as;
L - l
I ( X ; Y ) = -l/ M
7
2 d (j) l o g jf o g V M 1) (4 .28 ) j=0
w h ere d ( j ) is th e probability distribution fo r the j-th output s ym b ol g iven p reviou sly fo r each chan nel m odel. T h is is th e noiseless M A C capacity constrained b y the actual output distribution o f signals and w ill b e referred to, som e tim es, as the constrained capacity.
4.5.2 N o is y C h a n n d
C o n s id e r m ore practical situation where, a number o f transmitters attem pting to 88
communication w ith a sin gle receiver in th e presence o f A W G N [W y n e r 1974, El Gamal and C o v e r 1 9 8 0 , C hevillat 1981, G a lla g e r 1985, and H onary, A li and Darnell 1989]. T h e channel ou tput can be written as;
T
Y = L X j + N
(4.29)
i = l
where N is G aussian noise random va ria b le w ith zero m ean and variance o N2, independent o f the in p u ts X i. In the calculation o f this noisy channel capacity, w e use the cascaded noisy ch an n el model, in w h ich the noisy M A C can be characterised as, a noiseless T-user M - a r y adder M A C fo llo w e d b y an L-input, L-output noisy sin gle user channel. T h e n o isy s ta g e channel input sym b o ls are, Sit w h ere i= 0 ,l,...,L - l. Th erefo re, the average mutual in fo rm a tio n is, at m ost, the capacity o f sin gle input channel w ith it’ s input constrained t o th e average p o w er le v el. T h e capacity o f T -user adder M A C over A W G N channel is co m p u ted as the a vera ge mutual in form ation between the input and output o f the n o isy s in g le user channel w ith th e input sym bols, St. That is,
J ( S j r ) - g f p l s 1) p U \ s 1) l o g ^ p { r \ s i ) / p { Y ) ) d Y
(430)
where p ( Y | Sj) is th e channel transition probability, p(S,) is the i-th input sym bol probability, and p ( Y ) is the probability den sity function o f the output Y , w h ich can be written as;L - l
p ( Y ) = Z p (S t) p ( Y |
S)
(4.31)i
=0
F or practical system , th e signals w h ich can b e distinguished b y the receiver is lim ited b y the number o f quantisation levels at th e r ec e iv e r [H on ary, A l i and Darnell 1989, and H onary, A l i an d D arnell 1990]. T h erefo re, i f w e assume that the channel output Y takes values b e tw e e n -SC and + S C in steps o f
6
q g iv e n by;6
q = 2SC /(Q L-1) (4.32)w h ere SC and Q L are th e signal clip pin g and quantisation levels at the receiver, respectively. T h e quantisation levels can be w ritten in terms o f the number o f bits, b, in the quantiser as Q L = 2 b. T h erefo re, the channel output is quantised and equation (4.30) can b e m od ified in respon se to this as;
Lr\
SCI ( S ; Y ) =
2
2 p(S)
p ( Y | S.) lo g 2( Y | S ^ / p (Y )) (4.33) i= 0 -SCp ( Y | S,) is the con ditional pro b ab ility g ive n f o r G aussian distribution as;
p (Y | S,) = exp(-(y-Sj) 2/2aN2)/V(2x)oN
(4.34)
and since the channel ou tp u t is quantised, the p ( Y |
S)
can b e written as shown in Ap p en d ix A ;p ( Y | S,) = E rf((Y -S ,+ 6 q / 2 )/ o N) - E rf((Y-S ,-6q/2)/oN); fo r - S C < Y < S C , (4.35a) = E rf( ( Y-Si+6q/2)/oN); fo r Y = - S C , (4.35b) = Erfc((Y-S,-6q/2)/oN); for Y=+SC , (4.35c)
4.6 Simulation Results and Discussions
T h ree T-nser transmission system s are considered here and simulated fo r the calculation o f the inform ation c a p a c ity o f M A C models.
(i) B inary Signalling: In this transm ission scheme, the binary "0 " and "1 " are transmitted d irectly as signal le v e ls " 0 " and " A " , respectively, w h ere A is the signal amplitude. T h e signals fro m the T -u s e r are assumed to be superposed coh erently b y amplitude, g iv in g com posite sign al sym b o ls S, at the noiseless M A C o u tp u t T herefore, Sj fo r M = 2 can b e written as;
S, = W (2 E ); i = 0 , U . . i - l (4.36)
w h ere E = A 2/2 is the avera ge s ign a l energy per user. F or exam p le, fo r T = M = 2 , S ,e {0 ,A ,2 A } as shown in T a b le 4.1 b e lo w ;
X,
X 2
X , '
X 2'
S i
0
0
--- >0
1
0
> A
0
1
--- >0
1
1
> A
T able 4 1 C om po site S ig n a l S ym bols fo r B inary S ign a llin g (T = M = 2 )
w h ere X / is the i-th user transm itted signal.
(ii) Antipodal Signalling: In the b in a ry antipodal sign allin g schem e, the binary "0 " and "1 " are transmitted as signal le v e ls " - A " and " A " , respectively. T h e signals fro m the T -
users are assumed to be superposed by am plitude at the channel. T h e com posite signal sym bols
S,
fo r M = 2 can be written as;S, * (2 i-T W E ; i= 0 ,l,...,L - l (4.37)
w h ere