A multiplier-free digital sinusoid generator based on sigma-delta modulation

In this paper sigma-delta (Σ-Δ) sequences derived from a single-loop numeric Σ-Δ modulator with sinusoidal input are investigated, A class of limit cycles shows a special kind of symmetry, designated as “half-symmetry”. It is shown that half-symmetric Σ-Δ sequences are especially suited for automatic generation by means of an autonomous oscillator. The design of such an oscillator is presented. The central part is a single-loop numeric Σ-Δ modulator. The two-level output sequence of the Σ-Δ modulator is accumulated in lossless discrete integrators and fed back to the input. If the initial conditions of the oscillator registers are chosen properly in relation to a system constant F, the oscillator generates a two-level Σ-Δ sequence identical to the sequence of a Σ-Δ modulator which is externally driven by a pure sinusoid. Conditions for system constant F and initialization for periodic operation are derived, If constant F is an integer, the operation of the oscillator is exclusively based on integers. Thus, although the network is highly recursive, truncation- or roundoff errors are avoided and problems with respect to stability do not occur