Page 352 - Satellite Communications, Fourth Edition

332 Chapter Eleven

Since r c is always less than unity, then R c R b always. For constant
carrier power, the bit energy is inversely proportional to bit rate (see Eq.
10.22), and therefore,

E c
r c (11.11)
E b
where E b is the average bit energy in the uncoded bit stream (as intro-
duced in Chap. 10), and E is the average bit energy in the coded bit
c
stream.
Equation (10.18) gives the probability of bit error for binary phase-shift
keying (BPSK) and quadrature phase-shift keying (QPSK) modulation.
is just the E of Eq. (10.18), and the proba-
With no coding applied, E b b
bility of bit error in the uncoded bit stream is

1 E b
erfc° ¢
P eU (11.12)
2 Å N 0

For the coded bit stream, the bit energy is E r E , and therefore,
c
b
c
Eq. (10.18) becomes
1 r E b
c
erfc° ¢ (11.13)
P eC
2 Å N 0
This means that P eC P , or the probability of bit error with coding
eU
is worse than that without coding. It is important to note, however, that
the probability of bit error applies at the input to the decoder. For the
error control coding to be effective, the output BER should be better
than that obtained without coding. More will be discussed about this
later.
The limitation imposed by bandwidth also must be considered. If the
time for transmission is to be the same for the coded message as for the
uncoded message, the bandwidth has to be increased to accommodate
the higher bit rate. The required bandwidth is directly proportional to
bit rate (see Eq. 10.16), and hence it has to be increased by a factor 1/r .
c
If, however, the bandwidth is fixed (the system is band limited), then
the only recourse is to increase the transmission time by the factor 1/r .
c
For a fixed number of bits in the original message, the bit rate R enter-
b
ing into the encoder is reduced by a factor r compared with what it could
c
have been without coding.
As an example, it is shown in Sec. 10.4 that the TI message rate is
1.544 Mb/s. When 7/8 FEC is applied, the transmission rate becomes