Introduction - Codes and Turbo Codes - page 8

Figure 1.2 - Binary symmetric channel with error probability p .

When the signal received by the decoder comes from a device capable of
producing estimations of an analogue nature on the binary data transmitted,
the error correction capability of the decoder can be greatly improved. To show
this using the example of the extended Hamming code, we must first change
alphabet and adopt an antipodal (or symmetric) binary alphabet. We will make
the transmitted values x =-1and x = +1 correspond to the systematic binary
data d =0and d = 1, respectively. Similarly, we will make the transmitted
values y =-1and y = +1 correspond to the redundant binary data r =0and
r = 1, respectively. We then have:

1) d x =2 d − 1= − ( −

(1.7)

1) r

y =2 r − 1= − ( −

Figure 1.3 gives an example of a transmission during which the transmitted
values -1 and +1 are altered by additive noise of an analogue type. The values
at the output of the transmission channel are then real variables, which must in
practice be clipped then quantified when the decoder is a digital processor. The
number of quantization bits, denoted N q , does not need to be high: 3, 4 or 5
bits are sucient to closely represent analogue samples. One time out of two on
average, the noise is favourable as it has the same sign as the transmitted value.
In the other case, the amplitude of the signal is attenuated and, when this
unfavourable noise is large, the sign can be inverted. An immediate decision
taken per threshold (that is, is it larger or smaller than the analogue zero?)
would then lead to an erroneous binary value being given.

Figure 1.3 - Transmission channel with additive noise of an analogue type.

Since the decoder has information about the degree of reliability of the values
received, called soft or weighted values in what follows, the decoding of the