# Signal-to-Noise (S/N) Ratio: Definition & Formula

Lesson Transcript
Instructor
Marc Chiacchio

Marc has taught Bachelor level students climate science and has a PhD in climate science.

Expert Contributor
Christianlly Cena

Christianlly has taught college physics and facilitated laboratory courses. He has a master's degree in Physics and is pursuing his doctorate study.

Signal-to-noise ratio (SNR) is the measure of the power of desired signal relative to the background noise level. Learn the definition and formula of SNR, and explore related concepts such as surface temperature change signal and time of emergence. Updated: 01/05/2022

## Signal and Noise

The terms signal and noise are used in many different contexts, but in this lesson we'll explore what they mean in a physical engineering or statistical sense. The terms actually come from radio engineering, in which a signal is the noise-free signal, and noise is the white noise you hear when you can't tune a radio to a particular station. Signal processing is the statistical technique used to extract information from the raw signal.

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## Signal-to-Noise Ratio

In order to determine the strength of a signal it's necessary to calculate what is called the signal-to-noise-ratio (or SNR). The higher the ratio, the easier it becomes to detect a true signal or extract useful information from the raw signal. Thus, signal-to-noise-ratio is defined as the ratio as the power (P) of a signal to the power (P) of the background noise.

The knowledge of this ratio has many important applications in applied mathematics, analytical chemistry, electronics, and the geosciences. In electronics, signal and noise are measured in decibels, which is a measure of volume. In other disciplines, the SNR is also known as the effect size.

Imagine you're having a conversation with a friend on a quiet street. Now imagine you two are talking in a crowded pub or restaurant. The noise (in the background) and the signal (your voices) will both be a lot louder, but the SNR may be about the same—just strong enough for you to understand each other. If you were to make a video of your conversation, you could easily tell how loud the video is, but you'd have to do some processing to determine the power of each element.

Now imagine you and your friend are talking about the acorns you see on the sidewalk. You notice that you tend to see bigger acorns under bigger oaks. This could be your imagination, though. To figure out whether bigger oaks really make bigger acorns, you'd have to measure the natural size variability among acorns (the noise) and the correlation with oak size (the signal) to find out how much of the variability is due to oak size as opposed to some other reason (the signal-to-noise ratio or effect size). If you really wanted to, you could measure it in other towns to see if the SNR was any different there.

Since you're talking about oaks, the conversation may naturally turn to climate change, Then you and your friend can really dig deep into signal-to-noise ratios. Climate scientists use a similar (although more sophisticated) approach to determine the signal of surface temperature change against the very noisy backdrop of natural climate variability.

## Surface Temperature Change Signal

Since 1880, the global surface temperature has increased by 0.8° Celsius (1.4° Fahrenheit). This does not sound like much, but such a change is large considering the consequences that a rise in temperature could have on the melting of the icecaps in the poles as well as on extreme weather in the tropics.

This is the long-term global surface temperature change, but we look at temperature on a year-to-year or by decade-to-decade basis, we see large amplitudes of variability in temperature through time. A lot of this variability is seasonal. Some of it comes from less predictable sources, including volcanic ash or air pollution, both of which can block sunlight and cause local surface temperatures to drop. Ocean currents and weather patterns also influence surface temperature. Against this noise, climate scientists need to find the signal—the rising temperature that we know is the result of growing levels of greenhouse gases in the atmosphere. These types of changes are both superimposed on the long-term climate signal, and it is a challenge to determine whether the signal of human-made climate change has emerged from the noise of natural variability.

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## Signal-to-Noise Ratio: Multiple Choice Exercise

This activity will help you assess your knowledge regarding the properties and applications of the signal-to-noise ratio.

### Directions

For this activity, carefully read and select the best answer that completes each of the given statements. To do this, print or copy this page on blank paper and circle the letter of your answer.

### Multiple Choice

1) __________ is the common measure of sound intensity, otherwise known as volume.

A. Hertz

B. Min, Max

C. Decibels

D. Pitch

2) __________ is defined as the ratio between the power of the signal to the power of the background noise.

A. Effect size

B. Signal processing

C. Surface temperature

D. Time of emergence

3) The signal of surface temperature change against the __________ can be determined in a similar approach as signal-to-noise ratios.

A. global warming

B. natural climate variability

C. surface temperature correlation

D. None of the above

4) Which of the following is TRUE about the data for global surface temperature?

A. The global surface temperature has increased periodically by 0.1 degrees Celsius since the 1880s.

B. The data is not linked to the rise in temperature leading to the melting of the icecaps in the poles.

C. The tropics have the smallest signal relative to the noise of climate variability.

D. There is a large amplitude of variability in temperature over the years.

5) __________ is a parameter used to map other values of SNR, providing a better understanding of how fast the signal is increasing in a particular region.

A. Effect size

B. Signal processing

C. Surface temperature

D. Time of emergence

1. C
2. A
3. B
4. D
5. A

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