Quantization noise

Quantization noise. Waveform of sound, amplitude quantisized. Upper diagram: one bit = 2 values. Lower diagram: 4 bits = 16 values.

Quantization noise is a noise error introduced by quantization in the analogue 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 modeled in several different ways.

It is expressed as a root-mean-square error as

$N_{Q}={\frac {\left({\frac {V_{\mathrm {AD} }}{2^{Q}}}\right)^{2}}{6\cdot T_{\mathrm {S} }\cdot R_{\mathrm {L} }^{2}}}$

where $V_{\mathrm {AD} }$ is the analogue voltage range of the converter (volts) $Q$ is the number of bits of the converter, that is, bit resolution of the converter

$T_{\mathrm {S} }$ is the sample interval of the converter (seconds), 

and $R_{\mathrm {L} }$ is the load resistance of the converter (ohms).

In an ideal analogue-to-digital converter, the signal-to-noise ratio (SNR) can be calculated from

$\mathrm {SNR_{ADC}} =20\log _{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm {dB}$

For instance, 16-bit audio has a quoted dynamic range of 6.02 · 16 = 96.3 dB.

This comes from a model of quantization noise in an ideal ADC where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB. The signal is also assumed to have a uniform distribution covering all quantization levels, and the most common test signals that fulfill this are full amplitude triangle waves and sawtooth waves.

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

$\mathrm {SNR_{ADC}} =\left(1.761+6.02\cdot Q\right)\ \mathrm {dB}$

Here, the quantization noise is once again assumed to be uniformly distributed. This is very close to the truth for high resolution ADCs, but does not accurately model the noise in low resolution ADCs (e.g. ~4 bits) where the quantization noise distribution is strongly affected by the exact amplitude of the signal.

See also

• Signal-to-noise ratio
• Bit resolution
• Quantization error
• SQNR

External links

• The Relationship of Dynamic Range to Data Word Size in Digital Audio Processing
• Round-Off Error Variance — derivation of noise power of q²/12 for round-off error
• Dynamic Evaluation of High-Speed, High Resolution D/A Converters Outlines HD, IMD and NPR measurements, also includes a derivation of quantization noise
• Signal to quantization noise in quantized sinusoidal