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Estimates of \epsilon capacity for certain linear communication channels

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The following simple abstract model for a class of communications problems is adopted: the set of possible transmitted signalsxis taken to be the unit ball in theL_{2}space of functions defined on[-T, T](bounded energy); the transmitted signal is assumed to be operated on by a convolution operatorH; and the final observed received signalzisz = Hx + n, wherenis an unknown error, caused either by additive noise, lack of complete knowledge ofH, or other causes, of norm less than some specifiedepsilon(not necessarily small). The problem is to determine how many "distinguishable" signals can be sent, i.e., how manyx_{i}there are such that they_{i} = Hx_{i}are separated in norm by at leastepsilon. The chief results are asymptotic upper and lower bounds on the rate of error-free transmission possible, i.e., the ratio of the logarithm of the number of distinguishable signals to the time interval2TasT rightarrow infty. These estimates are in terms of the Fourier transform of the kernel of the convolution operatorH. The suitability of the model and the nature of the results are discussed.

Published in:

Information Theory, IEEE Transactions on  (Volume:14 ,  Issue: 3 )

Date of Publication:

May 1968

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