A Technique for Finding Approximate Inverse Systems and Its Applications to Equalization

A technique for designing a sampled-data version of the approximate inverse of a class of linear systems is described. Coefficients of the inverse are computed from a polynomial formed by taking equispaced samples of the impulse response of the given linear system. Long division has previously been used [2], [11] to invert polynomials. The quotient, however, does not converge when some roots of the polynomial lie outside the unit circle in the complex plane. In a general case, therefore, the method [3], [2], [11] used depended on the determination of roots of high-degree polynomials, a very time consuming computation. The method described here involves continued long division with proper treatment of the remainder. It is applicable in all cases except when the polynomial to be inverted has a root on the unit circle. The technique lends itself to machine computation and requires only a time-domain description of the system whose inverse is to be found. It is, therefore, suitable for the evaluation of tap gains of transversal filters used for automatic equalization of data transmission systems. It may also be used to design the inverse in the form of a recursive digital filter. The performance of nonrecursive and recursive equalizers designed by this method is compared with the zero forcing (ZF) and the least mean square (LMS) equalizers. The technique described here tends to minimize peak distortion as does the ZF method, however it is shown to work even when the initial peak distortion is greater than 100 percent and the ZF method may fail. Moreover, the method selects a favorable position for the reference tap thus sometimes resulting in less mean-square distortion than the LMS equalizer with the reference tap in the center.