TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 180

Figure 3.14 (a) Differential encoder; (b) One section of the trellis diagram, denoting
transition branches as d i / e i .

in which the latter equality holds because each e i is either þ 1or 2 1. We observe that a
sign change applied to the sequence ( e i ) cancels in each term of the product e i e i 2 1 , and
so does not affect ( d i ). This is the desired behavior.
In the presence of channel noise, the decoding formula d i ¼ e i e i 2 1 is less than
reliable. Instead, the operation of the differential encoder may be viewed as a rate-
one convolutional encoder [57], as sketched in Figure 3.14( a ). A single section of
the trellis diagram for the encoder is shown in Figure 3.14( b ), to which the forward-
backward algorithm may be applied for the purposes of decoding. The forward
recursion takes the form

a i ( þ 1) / a i 1 ( þ 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
þa i 1 ( 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
a i ( 1) / a i 1 ( þ 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
þa i 1 ( 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)

(3 : 8)

while the backward recursion becomes

b i 1 ( þ 1) / b i ( þ 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
þb i ( 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
b i 1 ( 1) / b i ( þ 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)
þb i ( 1) Pr( e i ¼ 1 jd i ¼ 1) Pr( d i ¼ 1)

(3 : 9)

using the boundary values

a 0 ( þ 1) ¼ Pr( e 0 ¼þ 1) b n ( þ 1) ¼ Pr( e n ¼þ 1)
a 0 ( 1) ¼ Pr( e 0 ¼ 1) b n ( 1) ¼ Pr( e n ¼ 1) :

These probabilities may be estimated from the first and last received symbols. Note in
particular that this requires transmitting the seed value e 0 of the differential encoder.
Integrating this element into the overall communication chain then appears as in
Figure 3.15 at the transmitter, and Figure 3.16 at the receiver. Observe that we now