.65) calculated using the fact that the total noise within the channel bandwidth is given by Equation 4.59 where N/W is the two-sided A decoder scheme may be defined as a function that assigns to power spectral density. The Shannon-Hartley theorem may now be each output sequence (B1,...,Bn) an input sequence (a1,...,an) and written as in Equations 4.60 and 4.61. the ideal observer is the scheme which minimises the overall prob- ability of error for the given input distribution, and this is found by N = nB (4.59) maximising the conditional probability as given by Equation 4.66. C - B10B (1+ 5) (4.66) (4.60) p(B1,....,Bn) (4.61) When all the inputs are equally likely, then Equation 4.67 may be obtained and for a fixed y, maximising p(x,ly) is equivalent to maximising the inverse probability p(y/x). This is referred to as Using Equation 4.62 gives Equation 4.63. the maximum likelihood decision scheme. (4.62) (4.67) (4.63) 4.6 Discrete channels with memory The Shannon-Hartley theorem allows for a trade off between Previously we have described channels that have no memory, that bandwidth and signal to noise ratio, in the following sense. If, for is channels in which the occurrence of errors during a particular example, the signal to noise ratio S/N = 7, and B = 4kHz, we obtain symbol interval does not influence the occurrence of errors during C = 12x10³ bits/s. If the signal to noise ratio is increased to 15, and succeeding symbol intervals. However in many realistic channels, the bandwidth increased to 3kHz, the channel capacity remains the errors do not occur as independent random events, but tend to occur same. in bursts. These channels are said to have memory. Examples of We have seen that in the case of a noiseless channel, the capacity channels with memory would be telephone channels that suffer is infinite, and no matter how restricted the bandwidth, it is always from switching transients, and microwave radio links that suffer possible to receive a signal without error, as predicted by the from fading, etc. Sha