# Signal to noise ratio

The signal-to-noise ratio , and signal to noise ratio or (signal) noise ratio , abbreviated SRV or S / N or SNR or S / N of English signal-to-noise ratio is a measure of the technical quality of a useful signal z (. B Voice or video) on which a noise signal is superimposed. It is defined as the ratio of the mean power of the useful signal to the mean noise power of the interfering signal . ${\ displaystyle a}$ ${\ displaystyle a_ {R}}$

The SNR is a term from high frequency , measurement and communication technology as well as acoustics , which is also used in many other areas such as automation technology or signal and image processing . Related quantities are the peak signal-to-noise ratio (PSNR), the carrier-to-noise ratio (C / N) and the carrier-to-interference ratio (C / (I + N) or C / I).

## Applications

• The signal-to-noise ratio serves as a rating number for assessing the quality of an ( analog ) communication path. In order to be able to extract the information reliably from the signal, the useful signal must stand out clearly from the background noise, so the SNR must be sufficiently large. If the SNR falls, the error rate increases in digital transmissions .
• As a parameter of a receiver , the SNR characterizes when the receiver can differentiate between noise and signal. For a human being, a noisy signal requires at least an SNR of approx. 6 dB in order to be able to hear the speech contained therein.
• The SNR is also used to evaluate analog-to-digital converters . The quantization error is seen as noise and can be related to the signal. If the system is linear, this value can also be used to determine the effective number of bits .
• In terms of electromagnetic compatibility , the signal-to-noise ratio is a quality criterion for signal transmission.

## definition

The signal-to-noise ratio is defined as the ratio of the existing mean signal power P signal to the existing mean noise power P noise (the integral of the spectral noise power density over the bandwidth ), whereby the origin of the noise power is not taken into account.

The signal-to-noise ratio is dimensionless as the ratio of quantities with the same unit of measurement . So it is:

${\ displaystyle \ mathrm {SNR} = {\ frac {\ text {Useful signal power}} {\ text {Noise power}}} = {\ frac {P _ {\ text {Signal}}} {P _ {\ text {Noise}} }}}$

Since the signal power in many technical applications is several orders of magnitude greater than the noise power, the signal-to-noise ratio is often represented on a logarithmic scale . The pseudo unit decibel (dB) is used for this :

${\ displaystyle \ mathrm {SNR} = 10 \; \ lg \ left ({\ frac {\ text {useful signal power}} {\ text {noise power}}} \ right) {\ text {dB}} = 10 \; \ lg \ left ({\ frac {P _ {\ text {Signal}}} {P _ {\ text {Noise}}}} \ right) {\ text {dB}}}$

### To noise ratio

At low frequencies and narrow-band electromagnetic useful signal and noise power, signal-to-noise ratios can be expressed in terms of effective voltage or current amplitudes (→  noise voltage ). This is e.g. B. common in audio technology . Since the available power in this case is proportional to the square of the rms value of the voltages ( u eff, signal , u eff, noise ), the following applies:

${\ displaystyle \ mathrm {SNR} = {\ frac {P _ {\ text {signal}}} {P _ {\ text {noise}}}} = {\ frac {u _ {\ mathrm {eff, signal}} ^ { 2}} {u _ {\ mathrm {eff, noise}} ^ {2}}}}$

From which follows:

${\ displaystyle \ mathrm {SNR} = 10 \; \ lg \ left ({\ frac {P _ {\ text {Signal}}} {P _ {\ text {noise}}}} \ right) {\ text {dB} } = 10 \; \ lg \ left ({\ frac {u _ {\ mathrm {eff, signal}} ^ {2}} {u _ {\ mathrm {eff, noise}} ^ {2}}} \ right) { \ text {dB}} = 20 \; \ lg \ left ({\ frac {u _ {\ mathrm {eff, signal}}} {u _ {\ mathrm {eff, noise}}}} \ right) {\ text { dB}}}$

### Alternative definition

An alternative definition of the signal-to-noise ratio is mainly used, for example, in spectroscopy or image processing (in particular in medical imaging ). Here the SNR is defined as the ratio of the mean signal amplitude A signal (instead of the power) and the standard deviation  σ noise of the noise:

${\ displaystyle {\ text {SNR}} = {\ frac {\ text {Useful signal amplitude}} {\ text {Noise standard deviation}}} = {\ frac {A _ {\ text {Signal}}} {\ sigma _ { \ text {Noise}}}}}$

This must be differentiated from the previous definition based on the voltage amplitudes, since there the power is first calculated using the squared amplitudes, while here the non-squared amplitude ratio is the basis. When using this definition, the conversion to decibels is also less common; The SNR is usually given as a unit-less size of dimension 1.

## Carrier-to-noise and carrier-to-interference ratios

With modulation methods such as phase modulation or frequency modulation , signal and carrier power cannot be separated from one another. Therefore, it does not apply where the noise on the signal S , but the carrier C (engl. Carrier ). The ratio is called the carrier-to-noise ratio (C / N).

In addition to the noise, interference I can also superimpose the signal. The signal can interfere with itself through multipath reception caused by reflections, as well as with similar signals, for example from neighboring radio cells in mobile communications. Depending on whether the noise power is also taken into account, the carrier-to-interference ratio is abbreviated as C / I or C / (I + N).

The carrier-to-noise ratio C / N of a radio link improves with the transmission power P t and the antenna gains G t and G r of the transmitter and receiver. This is reduced by the noise power, the product of the Boltzmann constant k , the noise temperature T and bandwidth B . In addition, it increases with the free space attenuation F = (4π · R / λ) 2 from ( R is the distance, λ is the wavelength):

${\ displaystyle C / N = {\ frac {G _ {\ text {r}} \ cdot G _ {\ text {t}} \ cdot P _ {\ text {t}}} {k \ cdot T \ cdot B \ cdot F}}}$

Changing the sizes provides the relationship between carrier-to-noise ratio and reception quality (G / T).

## Peak Signal-to-Noise Ratio (PSNR)

If an image or video is transmitted in compressed form , it must be decompressed and displayed at the receiving end. The peak-signal-to-noise ratio (PSNR ) is used as a parameter for the quality of this transmission . Typical values ​​are 30 dB to 40 dB with a bit depth of 8  bits . With a bit depth of 16 bits, values ​​between 60 dB and 80 dB are common.

As a fault value which is usually mean square deviation (English mean squared error , the two MSE) used m × n -Black and white images I and K , one of which is the original, the other approach for the disturbed (., By (lossy) compression and decompression), is specified as follows:

${\ displaystyle {\ text {MSE}} = {\ frac {1} {mn}} \ sum _ {i = 0} ^ {m-1} \ sum _ {j = 0} ^ {n-1} ( I (i, j) -K (i, j)) ^ {2}}$

The PSNR is thus defined as:

${\ displaystyle {\ text {PSNR}} = 10 \ cdot \ lg {\ frac {I _ {\ max} ^ {2}} {\ text {MSE}}} \, {\ text {dB}} = 20 \ cdot \ lg {\ frac {I _ {\ max}} {\ sqrt {\ text {MSE}}}} \, {\ text {dB}} = (2 \ cdot \ lg I _ {\ max} - \ lg { \ text {MSE}}) \ cdot 10 \, {\ text {dB}}}$

I max is the maximum possible signal intensity (in the case of an image, the maximum possible pixel value). If 8  bits are used to represent a sampled value, this is 255. If linear pulse code modulation (PCM) is used, these are generally B bits for a sampled value; the maximum value of I max is then 2 B −1.

For color images with three RGB values ​​per pixel, the definition of the PSNR is the same; the MSE is then the sum of all difference values ​​divided by the image size and divided by 3.

However, this metric ignores many effects in the human visual system ; other metrics are structural similarity ( SSIM , English for "structural similarity") and DVQ.

According to a study by the developers in 2007, the PSNR-HVS-M metric, which has been expanded to include contrast perception and masking criteria, offers the best approximation to date of the subjective assessments of human observers, with a large lead over PSNR, UQI and M SSIM but also a clear distance to DCTune and PSNR-HVS.

## Improvement of the SNR

The more is known about the useful signal, the more the SNR can be increased. Some techniques for improving SNR are listed in the following sections.

### Increase the signal strength

With a constant noise component, the SNR increases if the useful signal is increased. Whispering can hardly be heard in a noisy crowd, while loud shouting can be heard clearly.

### Compressor / Expander Systems

With constant noise (e.g. from a magnetic tape), the SNR for small signals is very small. Compressor / expander systems, the so-called companders , therefore reduce the dynamic range. For example, with the Dolby system, quiet sections are recorded with excessive volume. The procedure ensures that the system remembers the correct volume during playback.

### Filter

Noise occurs in the entire frequency spectrum. To limit it, you filter it out of the system's bandwidth . For example, a low-pass filter on a telephone ensures that frequencies above approx. 3 kHz are suppressed.

In digital transmission methods (for. Example, telephone modem , any type of digital wireless data transmission), a is in the receiver for optimization of the SNRs matched filter (engl. Matched filter ) are used. Put simply, the same filter characteristics are used in the receiver as in the transmitter. A root raised cosine filter is often used here .

### Autocorrelation

If you are not interested in the entire signal, but only in its frequency, for example, you can amplify the signal using autocorrelation .

Although the noise is significantly reduced, the useful signal is also weakened. With this method one can not go below the Cramer-Rao limit . The Cramer-Rao limit specifies the minimum size for the frequency uncertainty as a function of the sampling frequency, the number of signal periods present and the SNR.

### Averaging

The noise can be reduced by sending information several times. Since noise occurs stochastically, the standard deviation of the noise signal only increases by the factor when transfers are added , while the signal increases by the factor . The SNR related to the signal amplitudes (a common convention in image processing) increases by . This results from the central limit theorem . ${\ displaystyle n}$${\ displaystyle {\ sqrt {n}}}$${\ displaystyle n}$${\ displaystyle {\ frac {n} {\ sqrt {n}}} = {\ sqrt {n}}}$

Noisy image (left), averaged 2 times and 8 times.

The partial image on the left is one of 8 images that were noisy with a Gaussian blurring of approx. 80 gray value differences. The result of averaging two images shows the middle partial image. The SNR has increased from approx. 6 dB to 7 dB. After the summation of 8 images, right part of the image, it increases to approx. 10 dB. The SNR of the images was determined from the ratio of the contrast range of the image and the scatter of a low-contrast sub-area. ${\ displaystyle {\ sqrt {2}}}$${\ displaystyle {\ sqrt {8}}}$

The averaging of image data is often used in astronomy, for example, in lucky imaging technology . In principle, very sharp images are possible through the earth's atmosphere , but long exposures suffer from the unrest in the air - the stars appear blurry. If you take several thousand short-term recordings, it is by pure chance (hence the name of the method) that several hundred of them are quite sharp. These images are then averaged in order to improve the signal-to-noise ratio and to reconstruct a long-term recording.

## literature

• Jürgen Detlefsen, Uwe Siart: Basics of high frequency technology . 2nd, expanded edition. Oldenbourg, Munich a. a. 2006, ISBN 3-486-57866-9 ( limited preview in Google Book Search).
• Hubert Henle: The recording studio manual. Practical introduction to professional recording technology . 5th, completely revised edition. Carstensen, Munich 2001, ISBN 3-910098-19-3 .
• Thomas Görne: Microphones in theory and practice . 8th new, revised and expanded edition. Elektor-Verlag, Aachen 2007, ISBN 978-3-89576-189-8 .
• Thomas Görne: Sound engineering . Fachbuchverlag Leipzig in the Hanser-Verlag, Munich a. a. 2006, ISBN 3-446-40198-9 ( limited preview in Google Book Search).
• Curt Rint (ed.): Handbook for high frequency and electrical technicians . 13th revised edition. tape 2 . Hüthig and Pflaum, Heidelberg a. a. 1981, ISBN 3-7785-0699-4 .

## Individual evidence

1. Joachim Franz: EMC. Fail-safe construction of electronic circuits. Teubner, Wiesbaden / Stuttgart a. a. 2002, ISBN 3-519-00397-X , Chapter 2.3: The signal-to-noise ratio as a quality criterion , pp. 9-10 ( limited preview in the Google book search).
2. ^ Daniel J. Schroeder: Astronomical optics . 2nd Edition. Academic Press, San Diego CA et al. a. 2000, ISBN 0-12-629810-6 , pp. 433 ( limited preview in Google Book search).
3. Jerrold T. Bushberg, J. Anthony Seibert, Edwin M. Leidholdt Jr., John M. Boone: The Essential Physics of Medical Imaging . 2nd Edition. Lippincott Williams & Wilkins, Philadelphia PA et al. a. 2002, ISBN 0-683-30118-7 , pp. 278 ( limited preview in Google Book search).
4. ^ DVQ: A digital video quality metric based on human vision . ( Memento of the original from March 9, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF)
5. Nikolay Ponomarenko, Flavia Silvestri, Karen Egiazarian, Marco Carli, Jaakko Astola, Vladimir Lukin: On between-coefficient contrast masking of DCT basis functions . In: CD-ROM Proceedings of the Third International Workshop on Video Processing and Quality Metrics for Consumer Electronics VPQM-07, 25. – 26. January 2007 . Scottsdale AZ 2007 ( ponomarenko.info [PDF]).