Page 356 - Satellite Communications, Fourth Edition

336 Chapter Eleven

Comparing it with the stored version of 000 results in

2 2 2
[0.5 ( 1)] [0.7 ( 1)] [ 2 ( 1)] 6.14
The distance determined in this manner is often referred to as the
Euclidean distance in acknowledgment of its geometric origins, and the
distance squared is known as the Euclidean distance metric. On this
basis, the received codeword is closest to the 000 codeword, and the
decoder would produce a binary 0 output.
Soft decision decoding results in about a 2-dB reduction in the
[E /N ] required for a given BER (Taub and Schilling, 1986). This ref-
0
b
erence also gives a table of comparative values for soft and hard deci-
sion coding for various block and convolutional codes. Clearly, soft
decision decoding is more complex to implement than hard decision
decoding and is only used where the improvement it provides must
be had.

11.10 Shannon Capacity

In a paper on the mathematical theory of communication (Shannon,
1948) Shannon showed that the probability of bit error could be made
arbitrarily small by limiting the bit rate R to less than (and at most
b
equal to) the channel capacity, denoted by C. Thus
R C (11.17)
b
For random noise where the spectrum density is flat (this is the N 0
spectral density previously introduced) the channel capacity is given by

S
a1 b (11.18)
C W log 2
N
Here, W is the baseband bandwidth, and S/N is the baseband signal to
noise power ratio (not decibels). Shannon’s theorem can be written as

S
R W log a1 b (11.19)
2
b
N
Letting P represent the average signal power, and T the bit period
b
R
then as shown by Eq. (10.17) the bit energy is E P T . The noise
b
R
b
power is N WN and the signal to noise ratio is
0
S P R (11.20)

N N
E b

T WN 0
b