How Noise Shaping Works
Noise shaping works by putting the quantization error in a feedback loop. Any feedback loop functions as a filter, so by creating a feedback loop for the error itself, the error can be filtered as desired. The simplest example would be
where y is the output sample value that is to be quantized, x is the input sample value, n is the sample number, and e is the quantization error made at sample n (error when quantizing y). This formula can also be read: The output sample is equal to the input sample plus the quantization error on previous sample.
Essentially, when any sample's bit depth is reduced, the quantization error between the rounded (truncated) value and the original value is measured and stored. That "error value" is then added to the next sample prior to its quantization. The effect here is that the quantization error itself (and not the valid signal) is put into a feedback loop. This simple example gives a single-pole filter (a first-order Butterworth filter), or a filter that rolls off 6 dB per octave. The cutoff frequency of the filter can be controlled by the amount of the error from the previous sample that is fed back. For example, changing the value for A1 in the formula
will change the frequency at which the feedback loop is centered.
More complex algorithms can be used which use more samples' errors' worth of feedback in order to create more complex curves. The formula
is that of a ninth order noise shaper, and can allow very complex noise shaping.
Noise shaping must also always involve an appropriate amount of dither within the process itself so as to prevent determinable and correlated errors to the signal itself. If dither is not used then noise shaping effectively functions merely as distortion shaping — pushing the distortion energy around to different frequency bands, but it is still distortion. If dither is added to the process as
then the quantization error truly becomes noise, and the process indeed yields noise shaping.
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