Estimates of capacity for certain linear communication channels

The following simple abstract model for a class of communications problems is adopted: the set of possible transmitted signals is taken to be the unit ball in the space of functions defined on (bounded energy); the transmitted signal is assumed to be operated on by a convolution operator ; and the final observed received signal is , where is an unknown error, caused either by additive noise, lack of complete knowledge of , or other causes, of norm less than some specified (not necessarily small). The problem is to determine how many "distinguishable" signals can be sent, i.e., how many there are such that the are separated in norm by at least . 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 interval as . These estimates are in terms of the Fourier transform of the kernel of the convolution operator . The suitability of the model and the nature of the results are discussed.