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Sec. 1–11 Coding 27
10 –1
P e = Probability of bit error
10 –2
Shannon’s ideal system
with coding (1–14)
Turbo code
[Sklar, 1997]
10 –3
Coding gain = 1.33 dB
BPSK with
(23, 12) Golay coding
10 –4
10 –5
Coding gain = 8.8 dB
Ideal coding gain = 11.2 dB
Coding gain = 2.15 dB
Polar baseband or
BPSK without
coding (7–38)
10 –6
–2
–1
–1.59
Figure 1–8
0 1 2
3
4 5
E b /N 0 (dB)
Performance of digital systems—with and without coding.
6
7
8
9
10
there is some noise in the channel. We will now find the E b /N 0 required so that P e : 0 with
the optimum (unknown) code. Assume that the optimum encoded signal is not restricted in
bandwidth. Then, from Eq. (1–10),
C = lim bB log 2 a1 + S
B:q
N br = E b T b
lim b B log 2 a1 +
N 0 B b r
B: q
lim b log 2 31 + (E b N 0 T b )x4
r
x
=
x:0
where T b is the time that it takes to send one bit and N is the noise power that occurs within
the bandwidth of the signal. The power spectral density (PSD) is n (f) = N 0 /2, and, as
shown in Chapter 2, the noise power is
B
B
N = n (f)df = a N 0
3 3 2 b df = N 0B
-B
-B
(1–12)