Quantization error
Encyclopedia
In analog-to-digital conversion, the difference between the actual analog value and quantized digital value is called quantization error or quantization distortion. This error is either due to rounding or truncation. The error signal is sometimes considered as an additional random signal called quantization noise because of its stochastic
behaviour.
. When this is the case, the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution
. In the rounding case, the quantization error has a mean of zero and the RMS
value is the standard deviation
of this distribution, given by
. In the truncation case the error has a non-zero mean of 
and the RMS value is 
. In the eight-bit ADC example, the RMS rounding error represents 0.113% of the full signal range.
At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the audio band. In order to make the quantization error independent of the input signal, noise with an amplitude of 2 least significant bits is added to the signal. This slightly reduces signal to noise ratio, but, ideally, completely eliminates the distortion. It is known as dither
.
in the analog-to-digital conversion
(ADC) 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-quantization-noise ratio (SQNR) can be calculated from
Where Q is the number of quantization bits.
The most common test signals that fulfill this are full amplitude triangle wave
s and sawtooth wave
s.
For example, a 16-bit
ADC has a maximum signal-to-noise ratio of 6.02 × 16 = 96.3 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
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. In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle/sawtooth.
Quantization noise power can be derived from
where
is the voltage of the level.
(Typical real-life values are worse than this theoretical minimum, due to the addition of dither
to reduce the objectionable effects of quantization, and to imperfections of the ADC circuitry. On the other hand, specifications often use A-weighted measurements to hide the inaudible effects of noise shaping
, which improves the measurement.)
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. In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.
The calculations above, however, assume a completely filled input channel. If this is not the case - if the input signal is small - the relative quantization distortion can be very large. To circumvent this issue, analog compressors and expanders can be used, but these introduce large amounts of distortion as well, especially if the compressor does not match the expander.
(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 physically occurs digitally. Around the quantum limit
, the distinction between analog and digital quantities vanishes.
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
behaviour.
Quantization error models
In the typical case, the original signal is much larger than one LSBLeast significant bit
In computing, the least significant bit is the bit position in a binary integer giving the units value, that is, determining whether the number is even or odd. The lsb is sometimes referred to as the right-most bit, due to the convention in positional notation of writing less significant digits...
. When this is the case, the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
. In the rounding case, the quantization error has a mean of zero and the RMS
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...
value is the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of this distribution, given by
. In the truncation case the error has a non-zero mean of and the RMS value is . In the eight-bit ADC example, the RMS rounding error represents 0.113% of the full signal range.At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the audio band. In order to make the quantization error independent of the input signal, noise with an amplitude of 2 least significant bits is added to the signal. This slightly reduces signal to noise ratio, but, ideally, completely eliminates the distortion. It is known as dither
Dither
Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images...
.
Quantization noise model
Quantization noise is a model of quantization error introduced by quantizationQuantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...
in the analog-to-digital conversion
Analog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...
(ADC) in
telecommunication systems
Communications system
In telecommunication, a communications system is a collection of individual communications networks, transmission systems, relay stations, tributary stations, and data terminal equipment usually capable of interconnection and interoperation to form an integrated whole...
and signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of 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-quantization-noise ratio (SQNR) can be calculated from
Where Q is the number of quantization bits.
The most common test signals that fulfill this are full amplitude triangle wave
Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape.Like a square wave, the triangle wave contains only odd harmonics...
s and sawtooth wave
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....
s.
For example, a 16-bit
16-bit
-16-bit architecture:The HP BPC, introduced in 1975, was the world's first 16-bit microprocessor. Prominent 16-bit processors include the PDP-11, Intel 8086, Intel 80286 and the WDC 65C816. The Intel 8088 was program-compatible with the Intel 8086, and was 16-bit in that its registers were 16...
ADC has a maximum signal-to-noise ratio of 6.02 × 16 = 96.3 dB.
When the input signal is a full-amplitude sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
the distribution of the signal is no longer uniform, and the corresponding equation is instead
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. In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle/sawtooth.
Quantization noise power can be derived from
where
is the voltage of the level.(Typical real-life values are worse than this theoretical minimum, due to the addition of dither
Dither
Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images...
to reduce the objectionable effects of quantization, and to imperfections of the ADC circuitry. On the other hand, specifications often use A-weighted measurements to hide the inaudible effects of noise shaping
Noise shaping
Noise shaping is a technique typically used in digital audio, image, and video processing, usually in combination with dithering, as part of the process of quantization or bit-depth reduction of a digital signal...
, which improves the measurement.)
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. In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.
The calculations above, however, assume a completely filled input channel. If this is not the case - if the input signal is small - the relative quantization distortion can be very large. To circumvent this issue, analog compressors and expanders can be used, but these introduce large amounts of distortion as well, especially if the compressor does not match the expander.
Other fields
Many physical quantities are actually quantized by physical entities. Examples of fields where this limitation applies include electronicsElectronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
(due to electrons), optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
(due to photons), biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
(due to DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...
), and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
(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 physically occurs digitally. Around the quantum limit
Quantum limit
Quantum limit in physics refers to a limit on measurement accuracy at quantum scales.It may refer to the minimum level of quantum noise which is obtainable without squeezed states. For instance, the sensitivity of optical measurements has a standard quantum limit...
, the distinction between analog and digital quantities vanishes.
External links
- Quantization noise in Digital Computation, Signal Processing, and Control, Bernard Widrow and István Kollár, 2007.
- 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