The optimal discrete filter corresponding to a given analog filter

In this paper, the discrete filter to be used in place of a given analog filter so as to result in minimum mean-square error at the sampling instants is derived. The spectra of the signals to be filtered are assumed to be either 1) band limited, or 2) of known rational form. The condition which the optimal discrete filter must satisfy is obtained by minimizing the expected value of the square of the error, formed by the difference between a digital and sampled analog channel. For rational spectra, the optimum discrete filter is a function of both the frequency characteristics of the analog filter and the spectrum of the process. The optimal discrete filter for this case can be placed in rational form. For band-limited spectra, it is demonstrated that the corresponding optimal discrete filter is dependent only upon the frequency characteristics of the analog filter. The optimal discrete filter for this case, however, cannot, in general, be placed in rational form. Two methods are described for placing this optimal discrete filter in rational form. One of these methods, which involves obtaining the best first-order rational approximation to the optimum transformation, reduces to the well-known Tustin transformation.