On the estimation in the presence of noise of the impulse response of a random, linear filter

A sounding signal is transmitted through a transmission medium which may be characterized as a linear filter whose impulse response is random, i.e., drawn according to some probability law from an ensemble of possible impulse responses. To the output of the medium is added random noise; the resultant waveform is the received signal. A receiver is required to operate on this received signal so as to make a linear, minimum-mean-square-error estimate of the impulse response of the transmission medium. Two problems concerned with the design of such a sounding system are considered in this paper. The first is the determination of the transfer function of the optimum linear estimating filter in the receiver. The second is the optimization of the spectrum of the transmitted sounding signal.