FT
Codes on
a
Fading Channel
Jiun Shiu and Ja-Ling WuTaipei, 10764, Taiwan, Republic of China Dept. of Comp. Sci. & Info. Eng., Nat'l Taiwan Univ.
Abstract --
T h e utilization of real-number DFT codes for a m u l t ~ ~ l i c a t ~ v e channel is introduced in y t h e proposed encoding proce- dundancies can be added into data. W i t h these redun- mes for t h e parameters of a a n be obtained from t h e re- T h e decoding algorithm for real- des can b e used to calculate t h e fading parameters with these syndromes.
I. INTRODUCTION
In 1951, Marshall first defined error control codes for real or complex data and suggested that real-number codes could have applications simi- lar to those of Reed-Solomon codes. Wolf, with a different view, took real-number codes as a new technique for solving signal processing problems such as impulsive noise cancellation in informa- tion transmission.
A common feature of previous studies is that the channel error model is assumed to be addi- tive. In this paper, the real-number decoding method for multiplicative channel error model (which corresponds to the situation of transmit- ting over a fading channel in practice) will be investigated.
11. ENCODING A N D DECODING
SCHEME
FOR A FADING CHANNELUsually the effect of a fading channel is mod- eled by a slowly varying component multiplying the transmitted signal, that is
ri = yi . e;
+
n; (1)where y; is the transmitted signal, ei the multi- plicative parameters of a fading channel, ni the background noise, and T ; the received signal. In
a block coding scheme, we can also assume that
A multiplication can be transformed into an addition by taking logarithm. However, since the signals under consideration are assumed to be complex, complex logarithm function are re- quired. It can be easily derived from eqn. (1) that
log, ~i = log, y;
+
logc e;+
6;
(2)
where fi; = logc(l
+
G).
It should noted that when n;<<
y; . e ; ,6;
will approach 0. Since e; is slowly varying, both e; and log, e; can be viewed as a lowpass signal. Therefore, it is reasonable to assume that log, e; can be obtained from the sum of some unknown low frequency componentsEk, that is
v
(3)
log, e; = Eki . e27r"'ii N 1=1
where
kl
is the location for a nonzero frequency components, and Ekl is the magnitude of thatcomponent. Now suppose that y; is encoded as
(4)
1
X; i i 0,1, ..
.,
N - Ir' - 1 N - I(, N - K+
1, . . .,
N - 1i
Yi
=The first N - K equations in eqn. (2) become the
desired syndromes
(5)
si
= log, T ; = log, e;+
f i ;These noisy syndromes can readily be input to some decoding algorithms for the DFT codes. to compute Eh, provided that the number of nonzero terms of Ek in eqn. (5). After Ek is com-
puted, an estimation of y; can then be derived. In this way, one can estimate out the channel parameters e; and, at the same time, the trans- mitted data x;.
the index z is in the range of 0 , 1 , 2 , . . .
