1. Introduction

In telecommunications, radar, audio and image processing, or sensor networks, we repeatedly encounter the term Signal-To-Noise Ratio (SNR).

In this tutorial, we’ll go over the definition of SNR. Following that, we’ll present examples of how to calculate and interpret the SNR in different scenarios.

2. How to Compute the SNR?

2.1. Definition

The SNR represents the ratio between the signal and the background noise. If we have a high SNR, it means that the strength of our signal is significantly greater than the noise.

When we’re discussing the SNR in data communication, a high SNR will indicate that we’re successfully transmitting information. Once we have a low SNR, we’ll start losing information. In critical scenarios, the noise can even become larger than the signal, compromising the whole communication.

2.2. Equation for SNR

Now, let’s see how to compute the SNR. As an illustration we consider a signal with power P_{signal} and noise power P_{noise}, we obtain the SNR as:

    \[SNR = 10 \cdot \log_{10} \left( \frac{P_{signal}}{P_{noise}} \right)\]

We often express the SNR in decibels (dB), given its relative nature.

But in some cases, we might not have the information about the noise power. When this happens, and we have at least access to the total power (P_{total} = P_{signal} + P_{noise}), we compute the SNR with a slightly different formula:

    \[SNR = 10 \cdot \log_{10} \left( \frac{P_{signal}}{P_{total} - P_{signal}} \right)\]

We should be careful when we have SNR = 0 \textrm{ dB}. It doesn’t mean that we have a noiseless scenario. In fact, an SNR equal to 0 \textrm{ dB} characterizes a very challenging setup since \frac{P_{signal}}{ P_{noise}} = 1.

It means that our signal and noise have the same amplitude. In this case, it can be hard to separate the signal from the noise. If we return to the SNR formula, we see that:

    \[SNR = 10 \cdot \log_{10} \left( \frac{P_{signal}}{P_{noise}} \right) = 10 \cdot \log_{10} (1) = 0 \textrm{ dB}\]

Now that we know what SNR is and how to compute it, let’s see some examples.

3. Examples

We’ll now see examples of calculating the SNR for discrete and continuous signals in different fields.

3.1. Discrete Signal

In the first and simplest example, we consider dealing with digital data transmission. That is, we have a signal made only of 0s and 1s with P_{signal} = 20 \textrm{ mW}. In this case, we know that the noise power is P_{noise} = 5 \textrm{mW}, so we obtain the SNR for this signal:

    \[SNR = 10 \cdot \log_{10} \left( \frac{20 \textrm{ mW}}{5 \textrm{ mW}} \right) = 10 \cdot \log_{10} (4) \approx 6.02 \textrm{ dB} \right)\]

This indicates that we have a signal power roughly 6 decibels higher than the noise power. Moving forward, let’s see what changes if we have a continuous signal.

3.2. Continuous Signal

Now, let’s suppose we’re evaluating with a communication system. More specifically, this system is responsible for dealing with the transmission of audio signals captured by a microphone. Usually, we represent this type of signal as a sinusoidal wave.

If our signal has a frequency of 1000 \textrm{ Hz} and amplitude A=3 \textrm{V} we can compute P_{signal} as:

    \[P_{signal} = \frac{A}{2} = \frac{(3 \textrm{ V})^2}{2} = 4.5 \textrm{ V}^2\]

We need to consider that under the transmission several factors can influence the signal. For instance, we might have interference in the network, distortion, or even electronic issues in the equipment.

In this hypothetical scenario, we assume P_{noise} = 0.2 \textrm{ W} to account for all these factors, and the SNR is given by:

    \[SNR = 10 \cdot \log_{10} \left( \frac{4.5 \textrm{ V}^2}{0.2 \textrm{ W}} \right) = 10 \cdot \log_{10} (22.5) \approx 13.52 \textrm{ dB}\]

Usually, if we have an SNR of 13.49 \textrm{ dB}, we have a reliable transmission over the communication channel. But other applications might require even greater values.

We can think of magnetic resonance imaging (MRI), in which medical diagnosis relies on acquired data. In that case, we need a considerably higher SNR.

But what if we don’t have direct access to the noise power of a signal? Depending on the context, we might need to use different techniques.

3.3. Unknown Noise

Let’s suppose we’re working for a mobile communication company. For this purpose, we need to evaluate the quality of the signal between a transmission antenna and the clients. We can’t measure the noise in this transmission due to the complexity and dynamic nature of the channel.

But we can measure the total power of the received signal. This power will include both the power of the signal and the noise. In fact, our measurement yielded P_{total} = 19 \textrm{ mW}.

The power of the signal can be either predefined or determined with power control algorithms. With that in mind, we know that P_{signal} = 10 \textrm{ mW}. Now, we can compute the SNR as:

    \[SNR = 10 \cdot \log_{10} \left( \frac{P_{signal}}{P_{total} - P_{signal}} \right) = 10 \cdot \log_{10} \left( \frac{10 \textrm{ mW}}{19 \textrm{ mW} - 10 \textrm{ mW}} \right) = 0.41 \textrm{ dB}\]

We observe a very low SNR, close to the threshold of 0 \textrm{ dB}. In this case, we have limited reliability in the transmission of this antenna for this area. Even without directly measuring the noise, we can conclude that the ratio between the signal and the noise is close to 1.

4. Conclusion

In this article, we discuss how we can estimate the ratio between a signal and its associated noise. This is especially relevant in wireless communication, audio, and image processing. In summary, the SNR value assesses the quality of a transmitted signal. However, different scenarios require specific considerations and equations based on the fact that we might not have direct access to noise measurements.

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