Digital Programming for the Inversion of Z Transforms

N E a(1)iMod 3 from of G(z) be represented by the sequence Rothstein1 has described a method for i=o determining the least non-negative residue g*(t)=g,I(t)+g,a(t-T)+g23(t-2T) modulo three of a number expressed in -[aoe aD3 a2 0 ... f (-l)nan]Mod3+3* +gka(t-kT)+... ( binary notationi. This note describes a method for determiining this residue which This congruence is perhaps most easily where gk denotes the value of g*(t) at the is somewhat less intricate in its application, applied as follows. kth sampling instant and T is the sampling and which appears more tractable to mech1) Obtain N,, the modulo-three ring period. aniization. As in Rothstein's method, actual count of the ones in the even orders of Cross-multiplying (2) yields division by three is Inot necessary. N. (Even orders are those in which This method makes use of the conigruence the radix 2 is raised to an even power.) (1 + b,z-' + b2z-2 + + bz-P)G(z) 2) Obtain N, the modulo-three ring n counit of the ones in the odd orders. = ao + a1z1 + 02z2 + + azP.(4) N -= ai(-p)iMod (r + p), 3) Then N= (Ne-No)MOd 3. If i=° (N,,-N,) is non-negative, it is the Taking the inverse transform of the above least non-negative residue. If (N. -No) equation term-by-term leads to a difference where N iS an integer expressed in the numis negative, 3-f(Ne-No) is the least equation ber system of radix r, ai is the digit in the ri order of N, and p is an integer. The following non-negative residue. gn + blgn_l+ b2gn_9 + + bpgn_p proof of this congruence is an easy extension JOEL H. GERMEROTH of the proof of the validity of the method of McDonnell Aircraft Corp. = ao5o + a,a, + a252 + * + a.ap (5) casting out nines in the decinmal number St. Louis, Mo. system:2 xvhere a,, I,, a2 * * represent the sequence of a unit sampling fuinction and they equal N-arn + + air' + aor airi one at the respective sampling instanits and i=0 zero otherwise.