Signal-to-noise ratio (SNR or S/N) is a measure used in
science and engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
that compares the level of a desired
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
to the level of background
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
. SNR is defined as the ratio of signal power to the
noise power, often expressed in
decibels. A ratio higher than 1:1 (greater than 0 dB) indicates more signal than noise.
SNR,
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
, and
channel capacity of a
communication channel
A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for inform ...
are connected by the
Shannon–Hartley theorem
In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding ...
.
Definition
Signal-to-noise ratio is defined as the ratio of the
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
of a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
(meaningful input) to the power of background
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
(meaningless or unwanted input):
:
where is average power. Both signal and noise power must be measured at the same or equivalent points in a system, and within the same system
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
.
Depending on whether the signal is a constant () or a random variable (), the signal-to-noise ratio for random noise becomes:
:
where E refers to the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
, i.e. in this case the
mean square of ,
or
:
If the noise has
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of zero, as is common, the denominator is its
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
, the square of its
standard deviation .
The signal and the noise must be measured the same way, for example as voltages across the same
impedance. The
root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of th ...
s can alternatively be used in the ratio:
:
where is
root mean square (RMS) amplitude (for example, RMS voltage).
Decibels
Because many signals have a very wide
dynamic range
Dynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base- ...
, signals are often expressed using the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
ic
decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as
:
and
:
In a similar manner, SNR may be expressed in decibels as
:
Using the definition of SNR
:
Using the quotient rule for logarithms
:
Substituting the definitions of SNR, signal, and noise in decibels into the above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels:
:
In the above formula, P is measured in units of power, such as watts (W) or milliwatts (mW), and the signal-to-noise ratio is a pure number.
However, when the signal and noise are measured in volts (V) or amperes (A), which are measures of amplitude, they must first be squared to obtain a quantity proportional to power, as shown below:
:
Dynamic range
The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un-
distorted signal on a
channel and the minimum discernible signal, which for most purposes is the noise level. SNR measures the ratio between an arbitrary signal level (not necessarily the most powerful signal possible) and noise. Measuring signal-to-noise ratios requires the selection of a representative or ''reference'' signal. In
audio engineering, the reference signal is usually a
sine wave
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
at a standardized
nominal
Nominal may refer to:
Linguistics and grammar
* Nominal (linguistics), one of the parts of speech
* Nominal, the adjectival form of "noun", as in "nominal agreement" (= "noun agreement")
* Nominal sentence, a sentence without a finite verb
* Nou ...
or
alignment level
The alignment level in an audio signal chain or on an audio recording is a defined anchor point that represents a reasonable or typical level. It does not represent a particular sound level or signal level or digital representation, but it can ...
, such as 1 kHz at +4
dBu
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whose ...
(1.228 V
RMS).
SNR is usually taken to indicate an ''average'' signal-to-noise ratio, as it is possible that instantaneous signal-to-noise ratios will be considerably different. The concept can be understood as normalizing the noise level to 1 (0 dB) and measuring how far the signal 'stands out'.
Difference from conventional power
In physics, the average
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
of an AC signal is defined as the average value of voltage times current; for
resistive
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
(non-
reactive) circuits, where voltage and current are in phase, this is equivalent to the product of the
rms voltage and current:
:
:
But in signal processing and communication, one usually assumes that
so that factor is usually not included while measuring power or energy of a signal. This may cause some confusion among readers, but the resistance factor is not significant for typical operations performed in signal processing, or for computing power ratios. For most cases, the power of a signal would be considered to be simply
:
Alternative definition
An alternative definition of SNR is as the reciprocal of the
coefficient of variation
In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...
, i.e., the ratio of
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
to
standard deviation of a signal or measurement:
[Bushberg, J. T., et al., ]
The Essential Physics of Medical Imaging
'' (2e). Philadelphia: Lippincott Williams & Wilkins, 2006, p. 280.
:
where
is the signal mean or
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
and
is the standard deviation of the noise, or an estimate thereof.
[The exact methods may vary between fields. For example, if the signal data are known to be constant, then can be calculated using the standard deviation of the signal. If the signal data are not constant, then can be calculated from data where the signal is zero or relatively constant.] Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and
luminance
Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls withi ...
), and it is only an approximation since
. It is commonly used in
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
, where the SNR of an
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
is usually calculated as the ratio of the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
pixel value to the
standard deviation of the pixel values over a given neighborhood.
Sometimes SNR is defined as the square of the alternative definition above, in which case it is equivalent to the
more common definition:
:
This definition is closely related to the
sensitivity index
The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected.
Definition
The discriminability index is ...
or ''d'', when assuming that the signal has two states separated by signal amplitude
, and the noise standard deviation
does not change between the two states.
The ''Rose criterion'' (named after
Albert Rose) states that an SNR of at least 5 is needed to be able to distinguish image features with certainty. An SNR less than 5 means less than 100% certainty in identifying image details.
[
Yet another alternative, very specific, and distinct definition of SNR is employed to characterize ]sensitivity
Sensitivity may refer to:
Science and technology Natural sciences
* Sensitivity (physiology), the ability of an organism or organ to respond to external stimuli
** Sensory processing sensitivity in humans
* Sensitivity and specificity, statisti ...
of imaging systems; see Signal-to-noise ratio (imaging)
Signal-to-noise ratio (SNR) is used in imaging to characterize image quality. The sensitivity of a (digital or film) imaging system is typically described in the terms of the signal level that yields a threshold level of SNR.
Industry standa ...
.
Related measures are the "contrast ratio
The contrast ratio (CR) is a property of a display system, defined as the ratio of the luminance of the brightest shade (white) to that of the darkest shade (black) that the system is capable of producing. A high contrast ratio is a desired aspe ...
" and the "contrast-to-noise ratio
Contrast-to-noise ratio (CNR) is a measure used to determine image quality. CNR is similar to the metric signal-to-noise ratio (SNR), but subtracts a term before taking the ratio. This is important when there is a significant bias in an image, su ...
".
Modulation system measurements
Amplitude modulation
Channel signal-to-noise ratio is given by
:
where W is the bandwidth and is modulation index
Output signal-to-noise ratio (of AM receiver) is given by
:
Frequency modulation
Channel signal-to-noise ratio is given by
:
Output signal-to-noise ratio is given by
:
Noise reduction
All real measurements are disturbed by noise. This includes electronic noise
In electronics, noise is an unwanted disturbance in an electrical signal.
Noise generated by electronic devices varies greatly as it is produced by several different effects.
In particular, noise is inherent in physics, and central to the ...
, but can also include external events that affect the measured phenomenon — wind, vibrations, the gravitational attraction of the moon, variations of temperature, variations of humidity, etc., depending on what is measured and of the sensitivity of the device. It is often possible to reduce the noise by controlling the environment.
Internal electronic noise of measurement systems can be reduced through the use of low-noise amplifier
A low-noise amplifier (LNA) is an electronic amplifier that amplifies a very low-power signal without significantly degrading its signal-to-noise ratio. An amplifier will increase the power of both the signal and the noise present at its inp ...
s.
When the characteristics of the noise are known and are different from the signal, it is possible to use a filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
to reduce the noise. For example, a lock-in amplifier
A lock-in amplifier is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment. Depending on the dynamic reserve of the instrument, signals up to a million times smaller than noise components, p ...
can extract a narrow bandwidth signal from broadband noise a million times stronger.
When the signal is constant or periodic and the noise is random, it is possible to enhance the SNR by averaging
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
the measurements. In this case the noise goes down as the square root of the number of averaged samples.
Digital signals
When a measurement is digitized, the number of bits used to represent the measurement determines the maximum possible signal-to-noise ratio. This is because the minimum possible noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
level is the error
An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'.
In statistic ...
caused by the quantization of the signal, sometimes called quantization noise
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and t ...
. This noise level is non-linear and signal-dependent; different calculations exist for different signal models. Quantization noise is modeled as an analog error signal summed with the signal before quantization ("additive noise").
This theoretical maximum SNR assumes a perfect input signal. If the input signal is already noisy (as is usually the case), the signal's noise may be larger than the quantization noise. Real analog-to-digital converter
In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide ...
s also have other sources of noise that further decrease the SNR compared to the theoretical maximum from the idealized quantization noise, including the intentional addition of dither
Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither is routinely used in processing of both digital audio and video data, and is often ...
.
Although noise levels in a digital system can be expressed using SNR, it is more common to use Eb/No, the energy per bit per noise power spectral density.
The modulation error ratio (MER) is a measure of the SNR in a digitally modulated signal.
Fixed point
For ''n''-bit integers with equal distance between quantization levels ( uniform quantization) the dynamic range
Dynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base- ...
(DR) is also determined.
Assuming a uniform distribution of input signal values, the quantization noise is a uniformly distributed random signal with a peak-to-peak amplitude of one quantization level, making the amplitude ratio 2''n''/1. The formula is then:
:
This relationship is the origin of statements like " 16-bit audio has a dynamic range of 96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB.
Assuming a full-scale sine wave
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
signal (that is, the quantizer is designed such that it has the same minimum and maximum values as the input signal), the quantization noise approximates a sawtooth wave
The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...
with peak-to-peak amplitude of one quantization level[Defining and Testing Dynamic Parameters in High-Speed ADCs](_blank)
— Maxim Integrated Products
Maxim Integrated, a subsidiary of Analog Devices, designs, manufactures, and sells analog and mixed-signal integrated circuits for the automotive, industrial, communications, consumer, and computing markets. Maxim's product portfolio includes ...
Application note 728 and uniform distribution. In this case, the SNR is approximately
:
Floating point
Floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
numbers provide a way to trade off signal-to-noise ratio for an increase in dynamic range. For n bit floating-point numbers, with n-m bits in the mantissa and m bits in the exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
:
:
:
Note that the dynamic range is much larger than fixed-point, but at a cost of a worse signal-to-noise ratio. This makes floating-point preferable in situations where the dynamic range is large or unpredictable. Fixed-point's simpler implementations can be used with no signal quality disadvantage in systems where dynamic range is less than 6.02m. The very large dynamic range of floating-point can be a disadvantage, since it requires more forethought in designing algorithms.[Fixed-Point vs. Floating-Point DSP for Superior Audio](_blank)
— Rane Corporation
Rane Corporation is an American pro audio equipment manufacturer. Located in Mukilteo, Washington, it was made up of former employees of Phase Linear Corporation, and started out with products aimed at small live bands. They now carry many prod ...
technical library[Often special filters are used to weight the noise: DIN-A, DIN-B, DIN-C, DIN-D, CCIR-601; for video, special filters such as ]comb filter
In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly space ...
s may be used.[Maximum possible full scale signal can be charged as peak-to-peak or as RMS. Audio uses RMS, Video P-P, which gave +9 dB more SNR for video.]
Optical signals
Optical signals have a carrier frequency
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
(about and more) that is much higher than the modulation frequency. This way the noise covers a bandwidth that is much wider than the signal itself. The resulting signal influence relies mainly on the filtering of the noise. To describe the signal quality without taking the receiver into account, the optical SNR (OSNR) is used. The OSNR is the ratio between the signal power and the noise power in a given bandwidth. Most commonly a reference bandwidth of 0.1 nm is used. This bandwidth is independent of the modulation format, the frequency and the receiver. For instance an OSNR of 20 dB/0.1 nm could be given, even the signal of 40 GBit DPSK would not fit in this bandwidth. OSNR is measured with an optical spectrum analyzer.
Types and abbreviations
Signal to noise ratio may be abbreviated as SNR and less commonly as S/N. PSNR stands for peak signal-to-noise ratio
Peak signal-to-noise ratio (PSNR) is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic ...
. GSNR stands for geometric signal-to-noise ratio. SINR is the signal-to-interference-plus-noise ratio In information theory and telecommunication engineering, the signal-to-interference-plus-noise ratio (SINR) (also known as the signal-to-noise-plus-interference ratio (SNIR)) is a quantity used to give theoretical upper bounds on channel capacity ...
.
Other uses
While SNR is commonly quoted for electrical signals, it can be applied to any form of signal, for example isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers ( mass number ...
levels in an ice core
An ice core is a core sample that is typically removed from an ice sheet or a high mountain glacier. Since the ice forms from the incremental buildup of annual layers of snow, lower layers are older than upper ones, and an ice core contains ...
, biochemical signaling
Signal transduction is the process by which a chemical or physical signal is transmitted through a cell as a series of molecular events, most commonly protein phosphorylation catalyzed by protein kinases, which ultimately results in a cellula ...
between cells, or financial trading signals. The term is sometimes used metaphorically to refer to the ratio of useful information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
to false or irrelevant data in a conversation or exchange. For example, in online discussion forums and other online communities, off-topic posts and spam are regarded as that interferes with the of appropriate discussion.
See also
* Audio system measurements
Audio system measurements are a means of quantifying system performance. These measurements are made for several purposes. Designers take measurements so that they can specify the performance of a piece of equipment. Maintenance engineers mak ...
* Generation loss
Generation loss is the loss of quality between subsequent copies or transcodes of data. Anything that reduces the quality of the representation when copying, and would cause further reduction in quality on making a copy of the copy, can be cons ...
* Matched filter
In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal w ...
* Near–far problem
The near–far problem or hearability problem is the effect of a strong signal from a near signal source in making it hard for a receiver to hear a weaker signal from a further source due to adjacent-channel interference, co-channel interference, d ...
* Noise margin
In electrical engineering, noise margin is the maximum voltage amplitude of extraneous signal that can be algebraically added to the noise-free worst-case input level without causing the output voltage to deviate from the allowable logic voltage le ...
* Omega ratio The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some thres ...
* Peak signal-to-noise ratio
Peak signal-to-noise ratio (PSNR) is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic ...
* Signal-to-noise statistic
* Signal-to-interference-plus-noise ratio In information theory and telecommunication engineering, the signal-to-interference-plus-noise ratio (SINR) (also known as the signal-to-noise-plus-interference ratio (SNIR)) is a quantity used to give theoretical upper bounds on channel capacity ...
* Signal-to-noise ratio (imaging)
Signal-to-noise ratio (SNR) is used in imaging to characterize image quality. The sensitivity of a (digital or film) imaging system is typically described in the terms of the signal level that yields a threshold level of SNR.
Industry standa ...
* SINAD
Signal-to-noise and distortion ratio (SINAD) is a measure of the quality of a signal from a communications device, often defined as
:
\mathrm = \frac,
where P is the average power of the signal, noise and distortion components. SINAD is usually ex ...
* Subjective video quality
Subjective video quality is video quality as experienced by humans. It is concerned with how video is perceived by a viewer (also called "observer" or "subject") and designates their opinion on a particular video sequence. It is related to the fiel ...
* Total harmonic distortion
The total harmonic distortion (THD or THDi) is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. Distortion fact ...
* Video quality
Video quality is a characteristic of a video passed through a video transmission or processing system that describes perceived video degradation (typically, compared to the original video). Video processing systems may introduce some amount of dist ...
Notes
References
External links
*
ADC and DAC Glossary
– Maxim Integrated Products
Maxim Integrated, a subsidiary of Analog Devices, designs, manufactures, and sells analog and mixed-signal integrated circuits for the automotive, industrial, communications, consumer, and computing markets. Maxim's product portfolio includes ...
Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise floor
– Analog Devices
Analog Devices, Inc. (ADI), also known simply as Analog, is an American multinational semiconductor company specializing in data conversion, signal processing and power management technology, headquartered in Wilmington, Massachusetts.
The c ...
The Relationship of dynamic range to data word size in digital audio processing
* ttp://www.vias.org/simulations/simusoft_spectaccu.html Learning by simulations – a simulation showing the improvement of the SNR by time averaging
Dynamic Performance Testing of Digital Audio D/A Converters
Fundamental theorem of analog circuits: a minimum level of power must be dissipated to maintain a level of SNR
Interactive webdemo of visualization of SNR in a QAM constellation diagram
Institute of Telecommunicatons, University of Stuttgart
*
Quantization Noise
Widrow & Kollár Quantization book page with sample chapters and additional material
Signal-to-noise ratio online audio demonstrator - Virtual Communications Lab
{{Noise
Engineering ratios
Error measures
Measurement
Electrical parameters
Audio amplifier specifications
Noise (electronics)
Statistical ratios
Acoustics
Sound