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 noise power, often expressed in decibels. A ratio higher than 1:1 (greater than 0 dB) indicates more signal than noise. SNR is an important parameter that affects the performance and quality of systems that process or transmit signals, such as communication systems, audio equipment, radar systems, imaging systems, and data acquisition systems. A high SNR means that the signal is clear and easy to detect or interpret, while a low SNR means that the signal is corrupted or obscured by noise and may be difficult to distinguish or recover. SNR can be improved by various methods, such as increasing the signal strength, reducing the noise level, filtering out unwanted noise, or using error correction techniques. SNR also determines the maximum possible amount of data that can be transmitted reliably over a given channel, which depends on its bandwidth and SNR. This relationship is described by the Shannon–Hartley theorem, which is a fundamental law of information theory. SNR can be calculated using different formulas depending on how the signal and noise are measured and defined. The most common way to express SNR is in decibels, which is a logarithmic scale that makes it easier to compare large or small values. Other definitions of SNR may use different factors or bases for the logarithm, depending on the context and application.

== Definition ==

One definition of signal-to-noise ratio is the ratio of the power of a signal (meaningful input) to the power of background noise (meaningless or unwanted input): $\mathrm {SNR} ={\frac {P_{\mathrm {signal} }}{P_{\mathrm {noise} }}},$ The signal-to-noise ratio of a random variable (S) to random noise N is: $\mathrm {SNR} ={\frac {\mathrm {E} [S^{2}]}{\mathrm {E} [N^{2}]}}\,,$ If the signal is simply a constant value of s, this equation simplifies to: $\mathrm {SNR} ={\frac {s^{2}}{\mathrm {E} [N^{2}]}}\,.$ The signal and the noise must be measured the same way, for example as voltages across the same impedance. Their root mean squares can alternatively be used according to: $\mathrm {SNR} ={\frac {P_{\mathrm {signal} }}{P_{\mathrm {noise} }}}=\left({\frac {A_{\mathrm {signal} }}{A_{\mathrm {noise} }}}\right)^{2},$

=== 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 $\mathrm {SNR_{dB}} ={P_{\mathrm {signal,dB} }-P_{\mathrm {noise,dB} }}.$ 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: $\mathrm {SNR_{dB}} =10\log _{10}\left[\left({\frac {A_{\mathrm {signal} }}{A_{\mathrm {noise} }}}\right)^{2}\right]=20\log _{10}\left({\frac {A_{\mathrm {signal} }}{A_{\mathrm {noise} }}}\right)={A_{\mathrm {signal,dB} }-A_{\mathrm {noise,dB} }}.$ 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 VRMS). 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: $\mathrm {P} =V_{\mathrm {rms} }^{2}$ 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: $\operatorname {E} \left[X^{2}\right]=\sigma ^{2}+\mu ^{2}$
. 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: $\mu$
, and the noise standard deviation $\sigma$ 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 spe...

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Signal Processing - Engineering