Design of optimal and narrow-band Laguerre filters for sigma-delta demodulators

Conventional sigma-delta (Σ-Δ) decimation methods are not optimal because the downsampling operation following the noise-shaping filters enables some quantization noise to alias back into the baseband. This paper proposes a novel decimation method that simplifies the implementation of the decimation process. This method makes use of an optimal Laguerre filter as the noise-shaping filter followed by another Laguerre filter that acts as a narrow-band filter. Conventional finite-impulse response (FIR) low-pass filters such as the optimal FIR filter and Sinc^k filter are commonly used as noise-shaping filters in Σ-Δ demodulators. As an alternative, optimal infinite-impulse response (IIR) filter architecture based on orthonormal Laguerre functions is used in the proposed decimation scheme. An optimal Laguerre IIR filter design methodology is presented via the optimization of a quadratic function subject to a linear and quadratic constraint. An efficient procedure to perform the optimization is introduced. A Laguerre IIR filter can also be used as the narrow-band filter before the decimation process. In this paper, a narrow-band Laguerre filter design methodology is presented via a digital frequency transformation. The narrow-band Laguerre filter can then be efficiently designed using a min-max criterion via a modified Parks-McClellan FIR filter design algorithm.