Xiv. Processing and Transmission of Information': A. Estimating Filters for Linear Time-variant Channels

In other work (1, 2) it has been shown that channel estimating filters play an important role in optimum receivers for linear time-variant channels. In this report we discuss these filters in greater detail, chiefly to supplement later discussions of receivers. The relation between minimum variance and maximum likelihood estimates is shown, even in the case of a "singular" channel. Some observations on the general estimation problem are also made. 1. Definition of the Problem The situation that we shall consider is diagrammed in Fig. XIV-1. A known signal, x(t), of limited duration is transmitted through a random linear time-variant channel, A, of finite memory. The result is a waveform, z(t), which is further corrupted by additive x'(t) A z(kt) y(t noise, say n(t), before it is available to the TRANSMITTED n RECEIVED SIGNAL n(t) SIGNAL receiver. Let y(t) denote the final received CHANNEL ADDITIVE signal-that is, y(t) = n(t) +-z(t)-and let T NOISE denote the duration of y(t). Fig. XIV-1. The channel. Our problem is: Given y(t), and the knowledge of the statistical parameters of y(t) and n(t), we wish to derive estimates of z(t) on a minimum-variance and maximum-likelihood basis. For the minimum variance estimate we need only assume knowledge of the auto-correlation functions of y(t) and n(t); for the maximum likelihood estimate we have to assume that y(t) and n(t) are Gaussian. If we make these assumptions, we shall find that