Digital Communications 1. Safwan El Assad
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Special cases.
– Noiseless channel: X and Y symbols are linked, so:I(X, Y)=H(X)=H(Y)
– Channel with maximum power noise: X and Y symbols are independent, therefore:I(X, Y) = 0
2.7. Capacity, redundancy and efficiency of a discrete channel
Claude Shannon introduced the concept of channel capacity, to measure the efficiency with which information is transmitted, and to find its upper limit.
The capacity C of a channel: (information bit/symbol) is the maximum value of the mutual information I(X, Y) over the set of input symbols probabilities
[2.54]
The maximization of I(X, Y) is performed under the constraints that:
The maximum value of I(X, Y)occurs for some well-defined values of these probabilities, which thus define a certain so-called secondary source.
The capacity of the channel can also be related to the unit of time (bitrate Ct of the channel), in this case, one has:
[2.55]
The channel redundancy Rc and the relative channel redundancy pc are defined by:
[2.56]
[2.57]
The efficiency of the use of the channel
[2.58]
2.7.1. Shannon’s theorem: capacity of a communication system
Shannon also formulated the capacity of a communication system by the following relation:
[2.59]
where:
– B: is the channel bandwidth, in hertz;
– Ps: is the signal power, in watts;
is the power spectral density of the (supposed) Gaussian and white noise in its frequency band B;
is the noise power, in watts.
EXAMPLE.– Binary symmetric channel (BSC).
Any binary channel will be characterized by the noise matrix:
If the binary channel is symmetric, then one has:
p(y1/x2) = p(y2/x1) = p
p(y1/x1) = p(y2/x2) = 1 − p
Figure 2.5. Binary symmetric channel
The channel capacity is:
The conditional entropy H(Y/X) is:
Hence:
But max H(Y) = 1 for p(y1) = p(y2). It follows from the symmetry of the channel that if p(y1) = p(y2), then p(x1) = p(x2) = 1/2, and C will be given by:
Figure 2.6. Variation of the capacity of a BSC according to p
2.8. Entropies with k random variables
The joined entropy of k random variables is written:
[2.60]
Furthermore, by setting down in the relationship:
One has:
[2.61]
Equality occurs when the variables are independent.
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