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When digital signal processing operations are implemented on a computer or with specialpurpose hardware, errors due to finite word length are unavoidable. Most of the literatures dealing with finite word length effects in recursive digital filters have studied limit cycles and roundoff errors separately. This paper studies a systematic approach to synthesize limit cycle-free digital filters with minimum noise gain. A sufficient condition for the absence of limit cycles including overflow oscillations is given for the class of fixed-point digital filters described by the state equations, and the constraint for the transformation matrix to synthesize limit cycle-free digital filters is obtained. The problem of noise minimization subject to the constraint that guarantees the absence of limit cycles is easily formulated, since asymptotic stability of the actual digital filter is invariant under diagonal and orthogonal transformations. A method of realizing limit cycle-free digital filters with the local minimum of the noise gain is given, and a numerical example is shown to illustrate the computational procedure. The noise gain of the limit cycle-free digital filter synthesized here is nearly equal to that of the minimum unit noise digital filter by Hwang that can not guarantee the absence of limit cycles.

Synthesis of Digital Filters Based on the State Equations