Signal-to-noise ratio

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Signal-to-noise ratio (often abbreviated SNR or S/N) is meaningful both in the context of Electrical engineering and, informally, for Usenet or other newsgroup-like services.

Contents

1 Technical sense

1.1 Electrical SNR and acoustics
1.2 Image processing and interferometry

2 Digital signals

2.1 Fixed point
2.2 Floating point
2.3 Notes

3 Informal use
4 See also
5 References
6 External links

Technical sense

Signal-to-noise ratio is an engineering term for the power ratio between a signal (meaningful information) and the background noise:

[\mathrm = \over P_\mathrm} = \left ( \over A_\mathrm } \right )^2]

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 20 times the base-10 logarithm of the amplitude ratio, or 10 times the logarithm of the power ratio:

[\mathrm = 10 \log_ \left ( \over P_\mathrm} \right ) = 20 \log_ \left ( \over A_\mathrm} \right )]

where P is average power and A is RMS amplitude. Both signal and noise power are measured within the system bandwidth.

Electrical SNR and acoustics

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).

Signal-to-noise ratio is closely related to the concept of dynamic range, where dynamic range measures the ratio between noise and the greatest un-distorted 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 signal-to-noise 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 magnitude, 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 (near) 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'. In general, higher signal to noise is better; the signal is 'cleaner'.

Image processing and interferometry

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

Digital signals

When using digital storage the number of bits of each value determines the maximum signal-to-noise ratio. In this case the noise is the error signal caused by the quantization of the signal, taking place in the analog-to-digital conversion. The noise level is non-linear and signal-dependent; different calculations exist for different signal models.[Introduction to DSP: Quantisation] — Bores Signal Processing The noise is modeled as an analog error signal being summed with the signal before quantization ("additive noise").

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:

[\mathrm = \mathrm = 20 \log_(2^n) \approx 6.02 \cdot n]

This is the origin of statements like "16-bit audio has a dynamic range of 96 dB". Each extra quantisation bit reduces the level of the quantisation noise by roughly 6 dB.

Assuming a full-scale sine wave signal, the quantization noise approximates a sawtooth wave with peak-to-peak amplitude of one quantization level.[Defining and Testing Dynamic Parameters in High-Speed ADCs] — Maxim IC Application note 728 In this case, the SNR is:

[\mathrm = 10(2n-1) \cdot \log_(2)+10\log_(3) \approx 6.02 \cdot n + 1.761]

(With this signal model, 16-bit audio has an SNR of 98.1 dB.)

Floating point

Floating-point 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:

[\mathrm = 6.02 \cdot 2^m]

[\mathrm = 6.02 \cdot (n-m)]

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] — Rane Corporation technical library

Notes

Analog-to-digital converters have other sources of noise that decrease the SNR compared to the theoretical maximum from the idealized quantization noise.
Often special filters are used to weight the noise: DIN-A, DIN-B, DIN-C, DIN-D, CCIR-601, and special filters in video. (Kammfilter)
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.
It is more common to express SNR in digital systems using Eb/No (the Energy per bit per noise power spectral density).

Further information: Quantization noise, Bit resolution

Informal use

Informally, "signal-to-noise ratio" is often used to describe the ratio of useful information to false or irrelevant information, for example in an online discussion forum.

The term has been used on Usenet, for instance, where off-topic posts and spam are regarded as "noise" that interferes with the "signal" of interesting discussion, or Bugzilla, where "please fix this" comments clutter up the discussion without helping to solve the bug.

Many Internet users prefer moderated forums, for instance, because moderation can improve the SNR of a forum. The wiki collaboration model addresses the same question in a different way, by granting every user the power to "moderate" content, adding signal while removing noise. The assumption is that a majority of users are motivated by belief in the project goals, which leads to improved SNR.

References

External links

[Learning by Simulations] A simulation showing the improvement of the SNR by time averaging
[ADC and DAC Glossary] — Maxim IC
[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]

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