Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 195

s ( n ) x ( n )

z −1

q ( n )

0.5

s ( n )

y ( n )

est

r ( n )

(a)

+

detector

1.5

1 s ( n ) = 1

thick arrows represent
s ( n ) =

− 1

thin arrows represent

the two states

−0.5

0.5

numbers on the edges represent
output values

−1

(b)

−1.5

Figure 5.16 . (a) A first-order FIR channel, and (b) the trellis module to be used in
Viterbi's algorithm. The input is assumed to be from a 1-bit PAM constellation.

cost 0.25

1.5

1

cost 0.25

−0.5

0.5

cost 2.25

−1

−1.5

cost 6.25

Figure 5.17 . The first stage of the Viterbi algorithm. With output r (0) = 1, the
costs of all possible state transitions are indicated.

Starting from this output sequence, and without knowledge of the noise variance
σ q , we show how to estimate the closest path in the trellis which would have
given rise to this output. This process will automatically yield estimates of the
state sequence x ( n ) , the noiseless output sequence y ( n ) , and, most importantly,
the input sequence s ( n ) .

To initialize the procedure, note that if the initial state x (0) is 1 and the
input s (0) is also 1, then the noise-free output would be y (0) = 1 . 5, as shown
by the top horizontal path in Fig. 5.17. We define the cost associated with this