Signal-to-noise ratio

Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. 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, and channel capacity of a communication channel are connected by the Shannon–Hartley theorem.

Definition

Signal-to-noise ratio is defined as the ratio of the power of a signal (meaningful input) to the power of background noise (meaningless or unwanted input):
:$\mathrm = \frac,$
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.

Depending on whether the signal is a constant () or a random variable (), the signal-to-noise ratio for random noise becomes:
:$\mathrm = \frac$
where E refers to the expected value, i.e. in this case the mean square of ,
or
:$\mathrm = \frac$

If the noise has expected value of zero, as is common, the denominator is its variance, 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 squares can alternatively be used in the ratio:
:$\mathrm = \frac = \left ( \frac \right )^2,$
where is root mean square (RMS) amplitude (for example, RMS voltage).

Decibels

Because many signals have a very wide dynamic range, signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as

:$P_\mathrm = 10 \log_ \left ( P_\mathrm \right )$

and

:$P_\mathrm = 10 \log_ \left ( P_\mathrm \right ).$

In a similar manner, SNR may be expressed in decibels as

:$\mathrm = 10 \log_ \left ( \mathrm \right ).$

Using the definition of SNR

:$\mathrm = 10 \log_ \left ( \frac \right ).$

Using the quotient rule for logarithms

:$10 \log_ \left ( \frac \right ) = 10 \log_ \left ( P_\mathrm \right ) - 10 \log_ \left ( P_\mathrm \right ).$

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:
:$\mathrm = .$

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:

:
\mathrm = 10 \log_ \left \left ( \frac \right )^2 \right = 20 \log_ \left ( \frac \right ) = 2 \left ( \right ).

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 at a standardized nominal or alignment level, such as 1 kHz at +4 dBu (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 of an AC signal is defined as the average value of voltage times current; for resistive (non- reactive) circuits, where voltage and current are in phase, this is equivalent to the product of the rms voltage and current:
:$\mathrm = V_\mathrmI_\mathrm$

:$\mathrm= \frac = I_\mathrm^ R$
But in signal processing and communication, one usually assumes that $R=1 \Omega$ 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
:$\mathrm= V_\mathrm^$

Alternative definition

An alternative definition of SNR is as the reciprocal of the coefficient of variation, i.e., the ratio of mean 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.
:$\mathrm = \frac$
where $\mu$ is the signal mean or expected value and $\sigma$ 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 $\sigma$ can be calculated using the standard deviation of the signal. If the signal data are not constant, then $\sigma$ 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), and it is only an approximation since \operatorname\left ^2 \right= \sigma^2 + \mu^2 . It is commonly used in image processing, where the SNR of an image is usually calculated as the ratio of the mean 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:
:$\mathrm = \frac$

This definition is closely related to the sensitivity index or ''d'', when assuming that the signal has two states separated by signal amplitude $\mu$, and the noise standard deviation $\sigma$ 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 of imaging systems; see Signal-to-noise ratio (imaging).

Related measures are the "contrast ratio" and the " contrast-to-noise ratio".

Modulation system measurements

Amplitude modulation

Channel signal-to-noise ratio is given by
:$\mathrm = \frac$
where W is the bandwidth and $k_a$ is modulation index

Output signal-to-noise ratio (of AM receiver) is given by
:$\mathrm = \frac$

Frequency modulation

Channel signal-to-noise ratio is given by
:$\mathrm = \frac$

Output signal-to-noise ratio is given by
:$\mathrm = \frac$

Noise reduction

All real measurements are disturbed by noise. This includes electronic noise, 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 amplifiers.

When the characteristics of the noise are known and are different from the signal, it is possible to use a filter to reduce the noise. For example, a lock-in amplifier 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 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 level is the