We use a result due to Craven — the “Integer Noise Shaper Theorem” — to show that the internal system dynamics of the class of sigma-delta modulators (or equivalently noise shapers) with integer-coefficient error-feedback filters can be completely understood from the action of simple, linear pre- and de-emphasis filters surrounding a (possibly dithered) quantizer. In these mathematically equivalent models, there is no longer any feedback around the quantizer. The major stumbling block, which has previously prevented a complete dynamical analysis of such systems of order higher than one, is thus removed. The class of integer noise shapers includes, but is not restricted to, the important family of “Pascal” shapers, having all their zeros at dc. Before examining the “integer” shaper case, we discuss and extend Gerzon’s generic “Generalized Noise Shaper Theorem.
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