In fact, the following metric is a suggestion
for soft decision decoding
(5.53)
where yk is the multilevel-quantized soft symbol output and xk is the locally generated code word bit
stream that takes on the values of 1 or 1. By using this metric we have essentially de-emphasized the
small amplitude symbols and emphasized the large amplitude symbols in the decision process [17, 18].
Let us take a moment to recall the aim in using FEC codes was to improve system performance
[19, 20]. This will be accomplished in the following discussion of coding gain. Very simply, coding
gain is defined as follows (for a particular value of BER).
(5.54)
Recall the following formula:
(5.55)
For uncoded Quaternary Phase Shift Keying (QPSK) modulation with square-root raised cosine filtering
at both the transmitter and the receiver we have the following:
(5.56)
Consider a coded QPSK system with a rate R 1/2 convolutional code and the coded bit energy
(defined as energy of the encoded bits), we now have
(5.57)
Then our uncoded equation becomes
(5.58)
In the above discussion, uncoded refers to the bit energy at the output of the FEC decoder [21, 22].
Below we plot the Bit Error Rate (BER) for R 1/2 convolutional code for various constraint values,
K, using a hard decision decoder in an AWGN channel.
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