Low-pass boxcar filter example

For example, consider the feedback system:

${\displaystyle y[n]=x[n]+b\cdot e[n-1],}$

where b is a constant, n is the cycle number, x[n] is the input sample value, y[n] is the value being quantized, and e[n] is its quantization error:

${\displaystyle \ e[n]=y_{\text{quantized}}[n]-y[n].}$

In this model, when any sample's bit depth is reduced, the quantization error is measured and on the next cycle added with the next sample prior to quantization. The effect is that the quantization error is low-pass filtered by a 2-sample boxcar filter (also known as a simple moving average filter). As a result, compared to before, the quantization error has lower power at higher frequencies and higher power at lower frequencies. The filter's cutoff frequency can be adjusted by modifying b, the proportion of error from the previous sample that is fed back.

Impulse response filters in general

More generally, any FIR filter or IIR filter can be used to create a more complex frequency response curve. Such filters can be designed using the weighted least squares method.^[1] In the case of digital audio, typically the weighting function used is one divided by the absolute threshold of hearing curve, i.e.

${\displaystyle \ W(f)={\frac {1}{A(f)}}.}$

Dithering

Adding an appropriate amount of dither during quantization prevents determinable errors correlated to the signal. 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

${\displaystyle \ y[n]=x[n]+b\cdot e[n-1]+\mathrm {dither} ,}$

then the quantization error truly becomes noise, and the process indeed yields noise shaping.

Noise shaping and 1-bit converters

Since around 1989, 1-bit delta-sigma modulators have been used in analog-to-digital converters. This involves sampling the audio at a very high rate (2.8224 million samples per second, for example) but only using a single bit. Because only 1 bit is used, this converter only has 6.02 dB of dynamic range. The noise floor, however, is spread throughout the entire non-aliased frequency range below the Nyquist frequency of 1.4112 MHz. Noise shaping is used to lower the noise present in the audible range (20 Hz to 20 kHz) and increase the noise above the audible range. This results in a broadband dynamic range of only 7.78 dB, but it is not consistent among frequency bands, and in the lowest frequencies (the audible range) the dynamic range is much greater — over 100 dB. Noise shaping is inherently built into the delta-sigma modulators.

The 1-bit converter is the basis of the DSD format by Sony. One criticism of the 1-bit converter (and thus the DSD system) is that because only 1 bit is used in both the signal and the feedback loop, adequate amounts of dither cannot be used in the feedback loop and distortion can be heard under some conditions (more discussion at Direct Stream Digital § DSD vs. PCM).^[4][5]

Most A/D converters made since 2000 use multi-bit or multi-level delta-sigma modulators that yield more than 1 bit output so that proper dither can be added in the feedback loop. For traditional PCM sampling the signal is then decimated to 44.1 kHz or other appropriate sample rates.

In modern ADCs

Analog Devices uses what they refer to as "Noise Shaping Requantizer",^[6] and Texas Instruments uses what they refer to as "SNRBoost"^[7][8] to lower the noise floor approximately 30db compared to the surrounding frequencies. This comes at a cost of non-continuous operation but produces a nice bathtub shape to the spectrum floor. This can be combined with other techniques to further enhance the resolution of the spectrum.