An ARMAX Identification Method for Sigma–Delta Modulators Using Only Input-Output Data

Sigma-delta modulators are becoming increasingly more popular in electronic circuits. They are characterized via their signal and noise transfer functions (STF and NTF). This linearized model is basically an autoregressive moving average model with exogenous inputs (ARMAX) model, which is used during the design of the modulator. Hence, the pole/zero locations of its transfer functions give valuable information about the system. The identification of the poles and zeros from input/output data enables the verification of the complete ???? modulator and can be used on actual measurements where the internal signals are not accessible. The identification of ARMAX models from input-output data is well studied in the literature under conditions that are generally met in control applications. However, ???? modulators are designed such that these general assumptions are violated. This paper gives an overview of the properties of ???? modulators and compares the pros and cons of both time- and frequency-domain identification techniques for ARMAX systems. Then, it discusses a modified frequency-domain approach to identify the ???? modulator starting from its input-output data. The technique is illustrated on both low-pass and bandpass ???? modulators, showing an excellent agreement between the theoretical and estimated transfer functions.