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24 Introduction Chap. 1
Constraint length = K = 3
Three stage shift register
Data = x
Input
S 1 S 2 S 3
Modulo 2 adders
Commutator
Convolutional code
Output
Figure 1–6
1
Convolutional encoder for a rate
2, constraint length 3 code.
the input sequence x 11 = 1010 is fed into the input (with the most recent input bit on the
right), the corresponding encoded output sequence is y 11 = 11010001, as shown by path
A in Fig. 1–7.
A convolutionally encoded signal is decoded by “matching” the encoded received data
to the corresponding bit pattern in the code tree. In sequential decoding (a suboptimal
technique), the path is found like that of a driver who occasionally makes a wrong turn at a
fork in a road but discovers the mistake, goes back, and tries another path. For example, if
y 11 = 11010001 was received, path A would be the closest match, and the decoded data
would be x 11 = 1010. If noise was present in the channel, some of the received encoded bits
might be in error, and then the paths would not match exactly. In this case, the match is found
by choosing a path that will minimize the Hamming distance between the selected path
sequence and the received encoded sequence.
An optimum decoding algorithm, called Viterbi decoding, uses a similar procedure.
It examines the possible paths and selects the best ones, based on some conditional
probabilities [Forney, 1973]. The Viterbi procedure can use either soft or hard decisions.
A soft-decision algorithm first decides the result on the basis of the test statistic †
being above or below a decision threshold and then gives a “confidence” number that
specifies how close the test statistic was to the threshold value. In hard decisions, only the
† The test statistic is a value that is computed at the receiver, based on the receiver input during some
specified time interval. [See Eq. (7–4).]