Signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in
SNR is an important parameter that affects the performance and quality of systems that process or transmit signals, such as
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
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.
The signal-to-noise ratio of a random variable (S) to random noise N is:[1]
where E refers to the expected value, which in this case is the mean square of N.
If the signal is simply a constant value of s, this equation simplifies to:
If the noise has expected value of zero, as is common, the denominator is its variance, the square of its standard deviation σN.
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:
where A is
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
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,[note 1] 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-
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
But in signal processing and communication, one usually assumes that [3] 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, i.e., the ratio of mean to standard deviation of a signal or measurement:[4][5]
where is the signal mean or expected value and is the standard deviation of the noise, or an estimate thereof.[note 2] 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 . It is commonly used in
Sometimes[further explanation needed] 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 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.[5][10]
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
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
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
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 converters 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.
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 (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 2n/1. The formula is then:
This relationship is the origin of statements like "
Assuming a
Floating point
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.[12][note 3][note 4]
Optical signals
Optical signals have a
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. GSNR stands for geometric signal-to-noise ratio.[13] SINR is the signal-to-interference-plus-noise ratio.
Other uses
While SNR is commonly quoted for electrical signals, it can be applied to any form of signal, for example
SNR can also be applied in marketing and how business professionals manage information overload. Managing a healthy signal to noise ratio can help business executives improve their KPIs (Key Performance Indicators).[15]
Similar Concepts
The signal-to-noise ratio is similar to
See also
Notes
- 20 log rule.[2]
- ^ 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.
- ^ 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 filters 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.
References
- ISBN 9780387331393.
- ^ Michael A. Choma, Marinko V. Sarunic, Changhuei Yang, Joseph A. Izatt. Sensitivity advantage of swept source and Fourier domain optical coherence tomography. Optics Express, 11(18). Sept 2003.
- S2CID 125414987. Retrieved 5 May 2021.
- ^
D. J. Schroeder (1999). Astronomical optics (2nd ed.). Academic Press. p. 278. ISBN 978-0-12-629810-9., p.278
- ^ a b Bushberg, J. T., et al., The Essential Physics of Medical Imaging, (2e). Philadelphia: Lippincott Williams & Wilkins, 2006, p. 280.
- ISBN 978-0-13-168728-8.
- ISBN 978-0-12-372529-5.
- ISBN 978-1-4398-0003-4.
- ISBN 978-0-8493-7254-4.
- ^
Rose, Albert (1973). Vision – Human and Electronic. Plenum Press. p. 10. ISBN 9780306307324.
[...] to reduce the number of false alarms to below unity, we will need [...] a signal whose amplitude is 4–5 times larger than the rms noise.
- Maxim Integrated ProductsApplication note 728
- ^ Fixed-Point vs. Floating-Point DSP for Superior Audio — Rane Corporation technical library
- ISBN 9783319023090.
- ISBN 9781932340020.
- ^ "What Is Signal To Noise Ratio?". www.thruways.co. Retrieved 2023-11-09.
- ^ https://blog.minitab.com/en/adventures-in-statistics-2/understanding-t-tests-1-sample-2-sample-and-paired-t-tests
External links
- Walt Kester, Taking the Mystery out of the Infamous Formula,"SNR = 6.02N + 1.76dB," and Why You Should Care (PDF), Analog Devices, archived (PDF) from the original on 2022-10-09, retrieved 2019-04-10
- ADC and DAC Glossary – Maxim Integrated Products
- Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise floor – Analog Devices
- The Relationship of dynamic range to data word size in digital audio processing
- Calculation of signal-to-noise ratio, noise voltage, and noise level
- 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
- Bernard Widrow, István Kollár (2008-07-03), Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications, Cambridge University Press, Cambridge, UK, 2008. 778 p., ISBN 9780521886710
- Quantization Noise Widrow & Kollár Quantization book page with sample chapters and additional material
- Signal-to-noise ratio online audio demonstrator - Virtual Communications Lab