Quantization error

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The difference between the actual analog value and quantized digital value due is called quantization error. This error is due either to rounding or truncation.

Many physical quantities are actually quantized by physical entities. Examples of fields where this limitation applies include electronics (due to electrons), optics (due to photons), biology (due to DNA), and chemistry (due to molecules). This is sometimes known as the "quantum noise limit" of systems in those fields. This is a different manifestation of "quantization error," in which theoretical models may be analog but physics occurs digitally. Around the quantum limit, the distinction between analog and digital quantities vanishes.

1 Quantization noise model of quantization error
2 References
3 See also
4 External links

[edit] Quantization noise model of quantization error

Quantization noise. The difference between the blue and red signals in the upper graph is the quantization error, which is "added" to the original signal and is the source of noise.

Quantization noise is a model of quantization error introduced by quantization in the analog-to-digital conversion (ADC) process in telecommunication systems and signal processing. It is a rounding error between the analogue input voltage to the ADC and the output digitized value. The noise is non-linear and signal-dependent. It can be modelled in several different ways.

In an ideal analog-to-digital converter, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the signal-to-noise ratio (SNR) can be calculated from

$\mathrm{SNR_{ADC}} = 20 \log_{10}(2^Q) \approx 6.0206 \cdot Q\ \mathrm{dB} \,\!$

The most common test signals that fulfil this are full amplitude triangle waves and sawtooth waves.

In this case a 16-bit ADC has a maximum signal-to-noise ratio of 6.0206 · 16=96.33 dB.

When the input signal is a full-amplitude sine wave the distribution of the signal is no longer uniform, and the corresponding equation is instead

$\mathrm{SNR_{ADC}} = \left ( 1.761 + 6.0206 \cdot Q \right )\ \mathrm{dB} \,\!$

Here, the quantization noise is once again assumed to be uniformly distributed. When the input signal has a high amplitude and a wide frequency spectrum this is the case.^[1]

In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB.

For complex signals in high-resolution ADCs this is an accurate model. For low-resolution ADCs, low-level signals in high-resolution ADCs, and for simple waveforms the quantization noise is not uniformly distributed, making this model inaccurate.^[2] In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.

Quantization Noise

An example of audio with progressively worsening quantization noise.

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[edit] References

^ Pohlman, Ken C. (1989). Principles of Digital Audio 2nd Edition. SAMS, 60.
^ okelloto, tom (2001). The Art of Digital Audio 3rd Edition. Focal Press. ISBN 0240515870.