Optimal design of both multi‐input multi‐output loop filter and multi‐input single‐output reconstruction filter of sigma delta modulator

Abstract

This paper considers a sigma delta modulator (SDM) with the outputs of the quantizers directly fedback to the inputs of the loop filter without being subtracted from the input sequence. The design of the multi-input multi-output (MIMO) loop filter is formulated as an optimization problem. In particular, the sum of the inner products between the elements in the signal transfer function (STF) and the corresponding elements in the noise transfer function (NTF) of the SDM is minimized. The constraints of the optimization problem are defined based on the specifications on the maximum absolute difference between the magnitude response of the designed STF and that of the desirable STF as well as that between the magnitude response of the designed NTF and that of the desirable NTF of the SDM. Similarly, the design of the multi-input single-output (MISO) reconstruction filter is also formulated as an optimization problem. In particular, the sum of the total absolute difference between the designed magnitude response of the reconstruction filter and the desirable magnitude response of the STF of the SDM is minimized. The constraint of the problem is defined based on the specification on maximum absolute difference between the designed magnitude response of the reconstruction filter and the desirable magnitude response of the STF of the SDM. In this paper, the genetic algorithm is employed to find a near global solution of the formulated optimization problems. In order to reduce the required computational power, the matrix inversion lemma is applied. The computer numerical simulation results show that the SDM designed by our proposed method outperforms the SDMs designed by the Matlab function in terms of the signal to noise ratio (SNR) performance.