Signaltonoise ratio (often abbreviated SNR or S/N) is an electrical engineering measurement, also used in other fields (such as scientific measurement or biological cell signaling), defined as the ratio of a signal power to the noise power corrupting the signal. A ratio higher than 1:1 indicates more signal than noise.
In less technical terms, signaltonoise ratio compares the level of a desired signal (such as music) to the level of background noise. The higher the ratio, the less obtrusive the background noise is.
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In engineering, signaltonoise ratio is a term for the power ratio between a signal (meaningful information) and the background noise:
where P 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. If the signal and the noise are measured across the same impedance, then the SNR can be obtained by calculating the square of the amplitude ratio:
where A is root mean square (RMS) amplitude (for example, typically, RMS voltage). Because many signals have a very wide dynamic range, SNRs are usually expressed in terms of the logarithmic decibel scale. In decibels, the SNR is, by definition, 10 times the logarithm of the power ratio:
A common alternative definition of SNR is the ratio of mean to standard deviation of a signal or measurement:^{[1]}^{[2]}
where μ is the signal, or the mean or expected value of the signal, or some measure of signal strength, and σ 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 σ can be calculated using the standard deviation of the signal. If the signal data are not constant, then σ can be calculated from data where the signal is zero or relatively constant.
Often the signals being compared are electromagnetic in nature, though it is also possible to apply the term to sound stimuli. Due to the definition of decibel, the SNR gives the same result independent of the type of signal which is evaluated (such as power, current, or voltage).
Signaltonoise ratio is closely related to the concept of dynamic range, where dynamic range measures the ratio between noise and the greatest undistorted signal on a channel. SNR measures the ratio between noise and an arbitrary signal on the channel, not necessarily the most powerful signal possible. Because of this, measuring signaltonoise ratios requires the selection of a representative or reference signal. In audio engineering, this reference signal is usually a sine wave, sounding a tone, at a recognized and standardized nominal level or alignment level, such as 1 kHz at +4 dBu (1.228 V_{RMS}).
SNR is usually taken to indicate an average signaltonoise ratio, as it is possible that (near) instantaneous signaltonoise 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'. In general, higher signal to noise is better; the signal is 'cleaner'.
In image processing, the SNR of an image is usually defined as the ratio of the mean pixel value to the standard deviation of the pixel values, that is, SNR = μ / σ (the inverse of the coefficient of variation). Related measures are the "contrast ratio" and the "contrasttonoise ratio".
The connection between optical power and voltage in an imaging system is linear. This usually means that the SNR of the electrical signal is calculated by the 10 log rule. With an interferometric system, however, where interest lies in the signal from one arm only, the field of the electromagnetic wave is proportional to the voltage (assuming that the intensity in the second, the reference arm is constant). Therefore the optical power of the measurement arm is directly proportional to the electrical power and electrical signals from optical interferometry are following the 20 log rule.^{[3]}
The Rose criterion (named after Albert Rose) states that an SNR of at least 5 is needed to be able to distinguish image features at 100% certainty. An SNR less than 5 means less than 100% certainty in identifying image details.^{[4]}
Any measurement device is disturbed by parasitic phenomena. This includes the electronic noise as described above, but also any external event that affects the measured phenomenon — wind, vibrations, 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. Otherwise, when the characteristics of the noise are known and are different from the signal's, it is possible to filter it or to process the signal.
When the noise is a random perturbation and the signal is a constant value, it is possible to enhance the SNR by increasing the measurement time. Similarly, nonconstant signals may be enhanced by calculating the average of several repeated measurements.
When using digital storage the number of bits of each value determines the maximum signaltonoise ratio. In this case the noise is the error signal caused by the quantization of the signal, taking place in the analogtodigital conversion. The noise level is nonlinear and signaldependent; different calculations exist for different signal models. The noise is modeled as an analog error signal being summed with the signal before quantization ("additive noise").
The modulation error ratio (MER) is a measure of the SNR in a digitally modulated signal. Like SNR, MER can be expressed in dB.
For nbit integers with equal distance between quantization levels (uniform quantization) the dynamic range (DR) is also determined.
Assuming a uniform distribution of input signal values, the quantization noise is a uniformlydistributed random signal with a peaktopeak amplitude of one quantization level, making the amplitude ratio 2^{n}/1. The formula is then:
This relationship is the origin of statements like "16bit audio has a dynamic range of 96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB.
Assuming a fullscale sine wave 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 with peaktopeak amplitude of one quantization level^{[5]} and uniform distribution. In this case, the SNR is approximately
Floatingpoint numbers provide a way to trade off signaltonoise ratio for an increase in dynamic range. For n bit floatingpoint numbers, with nm bits in the mantissa and m bits in the exponent:
Note that the dynamic range is much larger than fixedpoint, but at a cost of a worse signaltonoise ratio. This makes floatingpoint preferable in situations where the dynamic range is large or unpredictable. Fixedpoint'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 floatingpoint can be a disadvantage, since it requires more forethought in designing algorithms.^{[6]}
Informally, "signaltonoise ratio" refers to the ratio of useful information to false or irrelevant data.
In online discussion forums and other online communities, offtopic posts and spam are regarded as "noise" that interferes with the "signal" of appropriate discussion.

