Fast and efficient algorithm to design noise-shaping FIR filters for high-order overload-free stable sigma-delta modulators

This paper presents a fast and efficient algorithm to design FIR noise transfer functions (NTFs) for overload-free stable high-order ΣΔ modulators. In the design, three parameters, filter order, oversampling ratio (OSR) and l₁-norm of the impulse response are specified, and the in-band peak of the amplitude response is minimized under the specification. The design problem is formulated as a linear minimax problem, in which the stability condition limits the l₁-norm of the impulse response of the NTF. The proposed algorithm can fast and efficiently solve the problem by iteratively utilizing the simplex method. In numerical examples, it is confirmed that the proposed algorithm can design near optimum ΣΔ modulators and besides is far faster and more efficient than the conventional design methods. Besides, by the proposed algorithm simulation results reveal the existence and achievable performance of rigorously stable ΣΔ modulators up to the order 199. Then it is revealed that a single-bit ΣΔ modulator for OSR 64 can have an in-band peak amplitude -96.2[dB], while rigorously ensuring the stability, and this modulator achieves a high SNDR 107.6[dB] for a sinusoidal input signal. In addition, it is demonstrated that a rigorously stable nine-level ΣΔ modulator has an in-band peak amplitude -105.3[dB] for OSR 16 only by optimizing the high order NTF.