mismatch and offset of the delta-sigma ADC

All said the requiement of mismatch and offset of the delta-sigma ADC is relaxed. However, relaxed to what kind of level is not expressed. Is there anyone can explain it?

you can simulate with mismatch and offset in matlab.
and the file you can download from matlab website

A so profound question. I can not understand

noiseless,

All said the requiement of mismatch and offset of the delta-sigma ADC is relaxed. However, relaxed to what kind of level is not expressed. Is there anyone can explain it?

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Why oversampling?

It's about the property of relaxation that oversampling offers. This is achieved by the use of sigma-delta modulator. Such relaxation basically relaxes cutoff frequency or -3dB bandwidth.

We musn't forget that quantisation noise is caused by quantisation error always inherent in ADC because quantisation can only linearly approximate a digital equivalent to represent an analog value over an analog dynamic range.

In a layman terms, oversampling is to sample an analog value many times, to minimise deviation error or quantisation error. The higher sampling frequency, the more accurate.

Sigma-delta suppresses quantisation noise by spreading the quantisation noise energy over the wider sampling bandwidth defined by the oversampling frequency, therefore the higher the sampling frequency, the wider the sampling bandwidth, the greater the noise suppression.

In addition, with the use of noise-shaping such as a HPF, this noise reduced to the minimum or almost removed.

Thus SNR of a sigma-delta modulator is increased to 6.02N + 1.76 + 10log(OSR), where the oversampling ratio, OSR is fs/2fo, where 2fo = nyquist-rate or minimum sampling frequency.

SNR of sigma-delta with 1st order noise-shaping = 6.02N + 1.76 - 5.17 + 30log(OSR)

SNR of sigma-delta with 2nd order noise-shaping = 6.02N + 1.76 - 12.9 + 50log(OSR)

These properties of bandwidth relaxation and removal quantisation noise offer relaxation or greater tolerance in mismatches and offsets.

Still no one can explain why oversampling can relax the mismatch and offset. No mathmatics expression. Is that mean every designer use simulation to decide the extent of relaxation?

I think you don't understand Quantisation error and Oversampling at all. Oversampling properties which I have explained above aim to minimise quantisation error.

Where do you think this quantisation error comes from?

Don't forget that quantisation error is directly related to the input signal in rms.

Beside linear approximation bounded by resolution, offsets in circuits due to variation in bias, and mismatches in transistor aspect ratios also contribute to the variation of the input signal, hence quantisation error too.

SDM = Oversampling + Noise Shaping
Opamp offset and other mismatch is not seriously effect the SDM if ur design parameter and architecture choose well. U have to watch some papers about this topic.

offset can be cancelled using auto-zeroing technique, that is usually applied in SDM design.