Background Calibration of Gain Errors in $\Pi\Delta \Sigma$ A/D Converters

This paper describes a method of calibrating gain errors in each channel of Hadamard modulated parallel delta-sigma (ΠΔΣ) converters. The ΠΔΣ converters extend traditionally bandwidth-limited ΔΣ modulators to applications such as wireless communications that require high-resolution wide-bandwidth converters. The focus of this paper is on the development and convergence of an adaptive calibration algorithm. A hardware implementation that verifies this paper can be found in an earlier work. The calibration is done in real time and is accomplished by adding an additional channel that is linearly dependent on the ΠΔΣ channels that are used. By introducing this redundancy, an adaptive recursive least squares (RLS) algorithm is used to correct for gain errors within the channels. To demonstrate the calibration scheme, an eight-channel ΠΔΣ converter with second-order ΔΣ modulators and an oversampling ratio of 4 was simulated in MATLAB. Simulation results show a 40-dB reduction in unwanted distortion tones caused by 1% gain mismatches.