Transmission line discriminators for FM noise measurements

Abstract

A number of different structures have been proposed in recent years to realize a digital transfer function H(z). Of these, the well-known parallel realization has been found to exhibit low roundoff noise properties in many cases [ 11 . Here, a partial fraction expansion of H(z)/z is first derived and then each term in the expansion is multiplied by z to obtain a sum-type expansion of H(z). These terms are then realized individually and connected in parallel to yield the final structure.Jackson has labeled the final structure as 1P and its transpose as 2P [ 1 ] .In this note we propose an alternate parallel structure. This new structure is obtained by making a partial-fraction expansion of H ( z ) directly, realizing each term of the expansion individually, and then connecting them in parallel. This new type of parallel configuration will be designated as 3P and its transpose as 4P.Without any loss of generality we make the following assumptions:i) The degree of the numerator of H ( z ) is less than or equal to that A direct partial-fraction expansion of H ( z ) will then take the form:of the denominator; and ii) All poles of H ( z ) are simple.In writing (1) we have combined a pair of real poles into one second order factor. If necessary, the real poles could be left as first order factors in the expansion. An implementation of (1) incorporating scaling multipliers is shown in Fig. l(a). Note that the hardware requirements of the new structure is identical to that of the old parallel realization. Another new structure is obtained by taking the transpose of Fig. l(a) and is shown in Fig. l(b).Assuming the noise to be uncorrelated from sample to sample, and from one error source to another, we can make a roundoff noise analysis of the new structure and its transpose following Jackson's approach [ 11 . The scaled versions of the 3P structure and its transpose 4P structure are shown in Fig. 1. The detailed noise analysis of these structures have been omitted for brevity.For uniformity we use Jackson's notations [ 11. Let the variance of the rounding error from the output of each multiplier (or other rounding points) be denoted as og, and ws = 2n/T be the radian sampling frequency with T as the sampling interval. Furthermore, we let and let [ .] denote the "integer part of." the 3P parallel structure is then given asThe variance or total average power of the output roundoff noise for where and m is the order of the filter given. lel structure is given by The corresponding expression for the noise variance for the 4P paral-As seen from expressions (2) and (3), there is really little advantage to be gained by using form 3P and 4P (or vice versa) as in the case with forms 1P and 2P, But there might be some advantage to using form 3P in place of form l P when we seek low noise realization, as illustrated by the following example. ExampleConsider the realization of the elliptic high-pass transfer function [2] : (Z -1 ) * (~* -0.7072 + 1) H(z) = (z2 +0.7772 + 0.3434) (zz + 0.018772 + 0.801)' (4 1 UIII-(a) (b) Fig. 1 . Propo...