B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

13.4 Channel Capacity of a Discrete Memoryless Channel 753
Fi g ure 13.4
Binary symmetric
channel capacity
as a function of
error
probability Pe
.
(..)
·51
c
]
-
"
(.)
"
1 .0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0
Error probability
P e ---+-
I
I
I
From Fig. 13.4, which shows C s vs. P e , it follows that the maximum value of C s is
unity. This means we can transmit at most l bit of information per binary digit. This is the
expected result, because one binary digit can convey one of the two equiprobable messages.
The information content of one of the two equiprobable messages is log 2 2 = 1 bit. Second,
we observe that C s is maximum when the error probability P e = 0 or P e = I. When the
error probability P e = 0, the channel is noiseless, and we expect C s to be maximum. But
surprisingly, C s is also maximum when P e = l. This is easy to explain, because a channel
that consistently and with certainty makes errors is as good as a noiseless channel. All we
have to do to have error-free reception is reverse the decision that is made; that is, if O is
received, we decide that 1 was actually sent, and vice versa. The channel capacity C s is
zero (minimum) when P e = ½ - If the error probability is ½ , then the transmitted symbols
and the received symbols are statistically independent. If we received 0, for example,
either 1 or O is equally likely to have been transmitted, and the information received
is zero.
Channel Capacity per Second
The channel capacity C s in Eq. (13.22) gives the maximum possible information transmitted
when one symbol (digit) is transmitted. If K symbols are being transmitted per second, then the
maximum rate of transmission of information per second is KC., . This is the channel capacity
in information units per seconds and will be denoted by C (in bits per second):
C=KCs
A Comment on Channel Capacity: Channel capacity is the property of a particular
physical channel over which the information is transmitted. This is true provided
the term channel is correctly interpreted. A channel means not only the transmission
medium, it also includes the specifications of the kind of signals (binary, r-ary, etc., or
orthogonal, simplex, etc.) and the kind of receiver used (the receiver determines the error
probability). All these specifications are included in the channel matrix. A channel matrix
completely specifies a channel. If we decide to use, for example, 4-ary digits instead of
binary digits over the same physical channel, the channel matrix changes (it becomes a
4 x 4 matrix), as does the channel capacity. Similarly, a change in the receiver or the
signal power or noise power will change the channel matrix and, hence, the channel
capacity.