A necessary and sufficient condition for the BIBO stability of general-order bode-type variable-amplitude wave-digital equalizers

Recently, the authors developed a new synthesis technique for the design of higher-order Bode-type variable-amplitude (VA) wave-digital (WD) equalizers. The salient feature of the resulting VA WD equalizers is that they permit the continuous variation of the WD equalizer transfer function from a shaping transfer function to its inverse by changing the value of a single variable digital multiplier only. The proposed design technique was based on the WD realization of the corresponding positive-real analog prototype shaping impedance function, and on the realization of the equalizer transfer function as the reflectance of the shaping impedance function with respect to the constituent variable digital multiplier. This paper is concerned with an investigation of the bounded-input bounded-output (BIBO) stability of general-order VA WD equalizers. It is shown that the resulting conditions are both necessary and sufficient for the BIBO stability of the VA WD equalizers for the entire range of values for the variable digital multiplier. These conditions can be checked in a straightforward fashion in terms of the characteristics of the shaping transfer function alone. An application example is given to illustrate the main results.