Quantization error

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.

Quantization noise model of quantization error

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 analog 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

:$mathrm\{SNR\_\{ADC\; =\; left\; (\; 1.761\; +\; 6.0206\; cdot\; Q\; ight\; )\; mathrm\{dB\}\; ,!$

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. [cite book
last = Pohlman
first =Ken C.
title = Principles of Digital Audio 2nd Edition
publisher = SAMS
date = 1989
pages = 60 ]

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. [cite book
last = okelloto
first = tom
title = The Art of Digital Audio 3rd Edition
publisher = Focal Press
date = 2001
isbn = 0240515870 ] In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.

References

See also

*Round-off error
*Dither
*Analog to digital converter
*Quantization
*Quantization noise
*Discretization error
*Signal-to-noise ratio
*Bit resolution
*SQNR

External links

* [http://www.mit.bme.hu/books/quantization/ Quantization noise in Digital Computation, Signal Processing, and Control] , Bernard Widrow and István Kollár, 2007.
* [http://www.techonline.com/community/related_content/20771 The Relationship of Dynamic Range to Data Word Size in Digital Audio Processing]
* [http://ccrma.stanford.edu/~jos/mdft/Round_Off_Error_Variance.html Round-Off Error Variance] — derivation of noise power of q²/12 for round-off error
* [http://ieee.li/pdf/dynamic_evaluation_dac.pdf Dynamic Evaluation of High-Speed, High Resolution D/A Converters] Outlines HD, IMD and NPR measurements, also includes a derivation of quantization noise
* [http://www.dsplog.com/2007/03/19/signal-to-quantization-noise-in-quantized-sinusoidal/ Signal to quantization noise in quantized sinusoidal]