A robust O(N log n) algorithm for optimal decoding of first-order Σ-Δ sequences

An exact recursive formula is derived to describe the structure of an ideal first-order Σ-Δ output sequence as a function of its input. Specifically, it is shown that every Σ-Δ sequence generated by the constant input x∈[0, 1] can be decomposed into a shorter E-A subsequence whose input x´∈[0, 1) may be used to recover that of the original Σ-Δ sequence. This formula is applied to develop an O(N log N) algorithm for decoding an N-length sequence. Without knowledge of the modulator´s initial state, it exhibits an average improvement, over all initial states, of 4.2 dB in output signal-to-noise ratio (SNR) compared with a near-optimal linear finite impulse response (FIR) filter. The regularity of the ideal first-order Σ-Δ structure with constant inputs permits the algorithm to be extended to bandlimited and noise-corrupted data. A simple error correction procedure is demonstrated, and it is shown that the recursive algorithm can outperform FIR filters on sequences of length N<64 having input SNRs as low as 30 dB