TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 185

Now, since 6 t determines Pr( t i jd i ) and thus I ( d i , t i ), and similarly 6 Y determines
Pr( Y i jd i ) and thus I ( d i , Y i ), the transfer function 6 t ¼ f inner ( 6 Y ) can be rephrased
as I ( d i , t i ) ¼ g inner [ I ( d i , Y i )]. Similarly, the transfer function 6 Y ¼ f outer ( 6 t ) can be
rephrased as I ( d i , Y i ) ¼ g outer [ I ( d i , t i )], and successive iterations are then described as

I ( d i , t ( m ) ) ¼ g inner [ I ( d i , Y ( m )

)]

i i

I ( d i , Y ( mþ 1)
i

) ¼ g outer [ I ( d i , t ( m )
i

)]

B EXAMPLE 3.12
EXIT Chart Construction. Figure 3.18 shows the extrinsic information transfer
function

I ( d i , Y i ) ¼ g outer [ I ( d i , t i )]

for the decoder corresponding to the rate 1 / 2 encoder from Figure 3.4. The curve is
obtained by randomly generating pseudo priors U i ( d i ) such that their log ratios
Y i ¼ log [ U i (1) =U i (0)] follow the conditional Gaussian distribution from (3.11),
for a given value of 6 Y . These values, when fed to the decoder, give extrinsic infor-
mation values T i . The histogram of their log ratios t i ¼ log [ T i (1) =T i (0)] was
empirically verified to fit the conditional distribution (3.10), and the value 6 t can
be estimated from the mean of the values t i . By repeating this experiment for a
range of values for 6 Y , and transforming to mutual information values I ( d i , Y i )
and I ( d i , t i ), the plot of Figure 3.18 results.

Figure 3.18 Extrinsic information transfer function for the outer decoder corresponding
to the rate 1 / 2 encoder of Figure 3.4.