Over-sampling analog-to-digital converter (ADC) A practical difficulty with analog to digital conversion is the requirement for a low pass analog anti-aliasing prefilter to band limit the signal to less than half of the sampling rate. High-order analog filters are very expensive, and they are also difficult to take in calibration. The combination of over- sampling given by down-sampling may be used to transfer some of the anti-aliasing burden from the analog into the digital domain, and thereby need a simpler low order analog filter. As an example, a typical compact disk encoding system can employ an over-sampling sigma-delta A/D converter which over-samples at 3175.2 kHz  which is then brought back to the CD sampling rate of 44.1 kHz. That amounts to over-sampling by a factor of 72 (= 3175.2/44.1). 

 Consider the range of frequencies of interest in the signal x(t) is 0 ≤ |F| ≤ FM. Usually we could band-limit x(t) to the maximum frequency FM with a sharp cut-off analog low pass filter and sample it at a rate of Fs = 2FM at least (strictly, Fs ≥ 2FM). Consider that we instead over-sample x(t) by an integer factor M, at Fs = M (2FM). That significantly replaces the needs for the anti-aliasing filter which can be specified more leniently as

Though that is still an ideal low pass filter in the pass band and stop band, its transition band width is no longer zero and it can be approximated with an inexpensive first- or second-order Butterworth filter as given below.

We are operating the price of a higher sampling rate for the profit of a cheaper analog anti-aliasing filter. The result is a discrete-time signal that is sampled at a much higher rate than 2FM. Following the sampling operation, we may reduce that sampling rate to the lower value using a decimator. The resulting structure of the over-sampling ADC is given in the block diagram below. There are two anti-aliasing filters, a low-order analog filter H_a(s) with cut-off frequency FM rad/sec., and a high-order digital filter H_d(z), with a cut-off frequency (π/M) rad/sample.

A second benefit of using the over-sampling ADC is the reduction in quantization noise. If q is the quantization step size (precision), then the quantization noise in x(n) is σ_x² = q/12 ,and the noise appearing in the output y(n) in the above scheme is σ_y² = σ_y² /M q²/12= a reduction by a factor of M.

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