(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 1, January 2011
L(an )= ln
P(an = +1 ) P(an = -1 )
P(s',s, y) = P(yf s)P(s',s, yp , yn ) . Now apply Baye’s rule to the last term
P(s',s, yp , yn ) = P(s, yn s ', yp )P(s ', yp )
If an has two values +1 and -1, with equally likely, then this ratio is equal to zero. If the binary variable is not equally likely then P(an = -1)= 1- P(an = +1 ) .
L(an ) = ln
P(s',s,y)= P(y f s)P(s,yn s' ,yp )P(s',yp )
P(a n = +1) 1-P(a n = +1)
L1(an )= ln
P(an = -1 y1N )
(5)
,
(s')βn (s)γ n (s',s)
n−1
(s')βn (s)γ n (s',s)
, where
α n−1 ( s ') is
called
an =−1
the Forward metric,
β n ( s) is
called the Backward metric,
and γ n ( s ', s ) is called Transition metric. At the receiver, the received data sequence is combined according to the combining techniques described for STC-MC-CDMA system. The soft output of the combiner is applied directly to the deinterleaver, and then finally, it is applied to S-T Turbo decoder, such as the MAP algorithm, to decode the data.
L1(an )= L(a - priori) + L(channel) + L(extrinsic) The L(a - priori) is a-priori information about an ,
L(channel) is the channel value calculated from the knowledge of the SNR and received signal. The third term is called the a-posteriori term, also called the extrinsic L value.
= +1, y1N ) = ∑P(s ', s, y1N )
N
n−1
n
This LLR includes joint probabilities between the received bits and the information bit, the numerator of which can be written as n
γ n (s',s) = P(s, yn s ', yp ) P(s ', s, y) = αn−1 (s')βn (s)γ n (s',s)
∑α L(a ) = ∑α
P(y1N ,an =+1)/P(y1N ) P(y1N ,an =+1) =ln P(y1N ,an = -1)/P(y1N ) P(y1N ,an = -1)
∑P(a
and
,
an =+1
Here, first the S-T Turbo encoder encodes the source data. Next, the encoded data is applied to interleaver, and then mapped according to the desired signal constellation. Finally at each time interval, the signals are modulated and transmitted simultaneously over different transmit antennas [16]-[18]. Using Baye’s rule, the above equation can be reformulated as
L(an )=ln
αn-1(s') = P(s ', yp ) βn (s) = P( y f s)
Then The LLR equation for MAP algorithm can be written as
The LLR of N bit sequence is formulated as below. The lower indicator of y means it starts at time 1 and the upper indicator means the ending point at N.
P(an = +1 y1N )
Let
r
ur (s').β n (s).γne (s',s) + ur L(extrinsic)= ln a r α (s'). β n (s).γne (s',s) n-1 ∑
∑α
N
where s’ is starting state and s is ending state of trellis.
∑ P(s',s,y1N ) P(y1N ,an =+1) an=+1 L(an )= ln = P(y1N ,an = -1) ∑ P(s',s,y1N )
n-1
(6)
a-
During each iteration the decoder produces the extrinsic value, this extrinsic value becomes the input to the next decoder. The decision is made about the bit by looking at
an =-1
N y1N = y1k-1, yn , yn+1 = yp , yk , y f
uur
We take the N bit data sequence and separate this into three
the sign of the L value. an = sign {L1 (an ) , The process can continue until the extrinsic values change becomes insignificant or the algorithm can allow for a fixed number of iterations.
nth point, and then from n+1 N to N. P(s ', s, y1 ) = P(s ', s, y p , yk , y f ) where yp is past
pieces, from 1 to n-1, then the
, sequence, which is the part that came before the current data point. yn current data point and y f is the part that come
VI. STTUC CONCATENATED WITH STBC MC-CDMA SYSTEM MODEL In order to further improvement in the performance of STTuC-MC-CDMA system, we can use S-T turbo code in concatenation with S-T block code MC-CDMA system. The STTuC-STBC-MC-CDMA system provides both diversity and coding gain with a reasonable increase in complexity. Figure 3 show the general block diagram of concatenated system. First, the STTuC encoder encodes the source data. Next, the encoded data is applied to S-T block encoder &
after the current point. Using Baye’s rule
P(s ', s, y) = P(s ', s, yp , yn , y f ) = P(y f s' ,s, y p , yn )P(s',s, y p , yn ) The term y f is the future sequence and we consider that it is independent of the past and only depend on the present state s. We can remove these dependencies to simplify the above equation.
112
http://sites.google.com/site/ijcsis/ ISSN 1947-5500