Page 357 - Satellite Communications, Fourth Edition

Error Control Coding 337

The bit rate is R 1/T . Substituting this in Eq. (11.20) then in the
b
b
inequality (11.19) gives
R b R b E b
log a1 b (11.21)
2
W W N 0
/W in the
As noted in Sec. 10.6.3, the ratio of bit rate to bandwidth (R b
inequality (11.21) is an important characteristic of any digital system.
The greater this ratio, the more efficient the system. The limiting case
is when the inequality sign is replaced by the equal sign.

R E
R b b b
log a1 b (11.22)
2
W W N 0
Keep in mind that this relationship is for the condition of arbitrarily small
probability of bit error. A plot of R /W as a function of E /N is shown in
b
0
b
6
5.75

5

4
Bit Rate/Bandwidth 3

2 QPSK

1 BPSK

−2 −1 0 1 2 3 4 5 6 7 8 9 10
E b
(dB)
N 0
Figure 11.10 Graph showing the Shannon limit, Eq. (11.22). The units for
the y-axis are (bits/s)/Hz (a dimensionless ratio in fact). The points for
5
BPSK and QPSK are evaluated for a P e 10 .