Characterization of sigma-delta modulators using ARMAX identification methods

Sigma-delta modulators are becoming increasingly more popular in electronic circuits since they make optimal use of the high-speed capabilities of novel technologies. Such ΣΔ-modulators can be characterized via their signal and noise transfer function. These transfer functions are primarily used during the design of the modulators and hence, their pole/zero locations give valuable information about the system under test. The equivalent model with a signal and a noise transfer function is basically an ARMAX model. The identification of ARMAX systems is well studied under relatively general conditions. However, ΣΔ-modulators are designed such that they violate various of these general assumptions. This paper gives an overview of the properties of ΣΔ-modulators and compares the pros and contras of both time and frequency domain identification techniques for ARMAX systems. Finally, it is shown that only a modified frequency domain approach is capable to identify the ΣΔ-modulator. This technique is illustrated on a realistic low-pass ΣΔ-modulator, showing an excellent agreement between the estimated noise transfer function and the noise transfer function predicted using linear system theory.