Noise shaping is a technique used in analog-to-digital conversion and digital-to-analog conversion in conjunction with oversampling, which permits the trade-off between bits (i.e. the number of quantization levels) and samples (i.e. the bandwidth used).

In an ordinary ADC, a (band-limited) signal must be sampled above its Nyquist frequency in order to be able to reconstruct it without errors. Since there is no reason to go above it, most systems use a sampling frequency of about 1.2 times the Nyquist frequency in order to keep the aliasing low.

In such a system, the quantization noise can be seen as a white noise covering the entire spectrum from minus half the Nyquist frequency to plus half the Nyquist frequency. In a noise-shaping system such as a sigma-delta ADC, the signal is sampled well above the Nyquist frequency (often up to a factor of 128). In such a system, the useful signal only occupies a fraction of the bandwidth of the sampled signal, whereas the quantization noise is spread uniformly over all the bandwidth. Noise-shaping systems try to push quantization noise out of the band of interest (which lies in the low frequencies region) out to high frequencies using clever electronic designs. The quantization noise is thus separated from the useful signal in such a way that the appropriate low-pass filter eliminates much of the noise without affecting the signal.

The performance of noise-shaping systems depends mainly on the oversampling factor used and the order of the transfer function used to separate the quantization noise from the useful signal. For example, using an oversampling factor of 16 with a first-order noise-shaping filtering yields the same signal-to-noise ratio than a signal sampled at the Nyquist frequency but using 5 supplementary bits. This is quite surprising considering that a 1-bit ADC with noise-shaping and 16-times oversampling (that is, a superfast comparator whose output is simply a square wave with a varying duty cycle) has the same performance as a 6-bits ordinary ADC.

The example in the last paragraph is from Introduction to Signal Processing, S.J. Orfanidis, Prentice-Hall Editor (1996)

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