Page 353 - Satellite Communications, Fourth Edition

Error Control Coding 333

1.544 8/7 1.765 Mb/s. From Eq. (10.16), the required bandwidth is
B 1.765 (1.2)/2 1.06 MHz.
IF

11.8 Coding Gain
As shown by Eqs. (11.12) and (11.13), the probability of bit error for
a coded message is higher (therefore, worse) than that for an uncoded
message, and therefore, to be of advantage, the coding itself must
more than offset this reduction in performance. In order to illustrate
this, the messages will be assumed to be BPSK (or QPSK) so that
the expressions for error probabilities as given by Eqs. (11.12) and
(11.13) can be used. Denoting by BER the bit error rate after demod-
U
ulation for the uncoded message and by BER the bit error rate for
C
the coded message after demodulation and decoding, then for the
uncoded message
BER P eU (11.14)
U
Certain codes known as perfect codes can correct errors up to some
number t. The BER for such codes is given by (see Roddy and Coolen,
1995)
(n 1)! t 1
BER P eC (11.15)
C
t!(n 1 t)!
where x! x(x 1)(x 2) . . . 3.2.1 (and n is the number of bits in a
codeword). The Hamming codes are perfect codes that can correct one
error. For this class of codes and with t 1, Eq. (11.15) simplifies to

2
(n 1)P (11.16)
BER C eC
A plot of BER and BER against [E /N ] is shown in Fig. 11.8 for
b
U
0
C
the Hamming (7, 4) code. The crossover point occurs at about 4 dB,
so for the coding to be effective, [E /N ] must be higher than this.
b
0
5
Also, from the graph, for a BER of 10 , the [E /N ] is 9.6 dB for the
b
0
uncoded message and 9 dB for the coded message. Therefore, at
this BER value the Hamming code is said to provide a coding gain
of 0.6 dB.
Some values for coding gains given in Taub and Schilling (1986) are
block codes, 3 to 5 dB; convolutional coding with Viterbi decoding, 4 to
5.5 dB; concatenated codes using R-S block codes and convolutional
decoding with Viterbi decoding, 6.5 to 7.5 dB. These values are for a P e
5
value of 10 and using hard decision decoding as described in the fol-
lowing section.