Table of Contents
Introduction
This blog performs a simple math derivation for SNR and then verifies the results with a Python simulation.
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Calculating SNR Mathematically
An example received signal x[n] is the addition of the signal s[n] and the noise w[n],
(1) ![\begin{equation*}x[n] = s[n] + w[n].\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-98532eecdd7ae16a2cd8fc8e41a7384d_l3.png "Rendered by QuickLaTeX.com")
The SNR of the received signal x[n] is the ratio of the power for the signal 
to the power of the noise 
,
(2) 
where
(3) ![\begin{equation*}P_{s} = \mathbb{E} \{ s[n] s^*[n] \},\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-76448b42aa86c67a7ee8086ddfbf9e5f_l3.png "Rendered by QuickLaTeX.com")
(4) ![\begin{equation*}P_{w} = \mathbb{E} \{ w[n] w^*[n] \}.\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-279a9c8a3a7168f53834d0b8a8823196_l3.png "Rendered by QuickLaTeX.com")
The expectation operator 
operates as an average or mean. It is assumed that x[n] is N samples long beginning at time n=0, therefore the average signal power (3) can be written according to
(5) ![\begin{equation*}\begin{split}P_{s} & = \mathbb{E} \{ s[n] s^*[n] \} \\& = \frac{1}{N} \sum_{n=0}^{N-1} s[n] s^*[n] \\& = \frac{1}{N} \sum_{n=0}^{N-1} |s[n]|^2.\end{split}\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-1a30cac064e0ddcbc5da150fcba0ae88_l3.png "Rendered by QuickLaTeX.com")
Similarly, the average power of the noise (4) is
(6) ![\begin{equation*}P_{w} = \frac{1}{N} \sum_{n=0}^{N-1} |w[n]|^2.\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-aaae1bd12f611ee286ba206755469a00_l3.png "Rendered by QuickLaTeX.com")
The SNR of x[n] (2) is therefore [Lyons2011, p.875]
(7) ![\begin{equation*}SNR_{x} = \frac{\frac{1}{N} \sum_{n=0}^{N-1} |s[n]|^2}{\frac{1}{N} \sum_{n=0}^{N-1} |w[n]|^2}\end{equation*}](https://www.wavewalkerdsp.com/wp-content/ql-cache/quicklatex.com-86dba2066b701592358d12da3974bd94_l3.png "Rendered by QuickLaTeX.com")
which can be written in decibels as
(8) 
Calculating SNR with Python
Python is used to generate two signals, a BPSK signal and real Gaussian noise. The BPSK signal is generated by the following Python code:
import numpy as np
numSamples = 4096
symbolMap = np.array([-1, 1])
mapIndex = np.random.randint(0,len(symbolMap),numSamples)
symbols = symbolMap[mapIndex]
The average power of the symbols (5) is calculated and printed by:
Ps = np.mean(np.abs(symbolMap)**2)
print('signal power = ' + str(np.round(Ps,2)))
which results in:
signal power = 1.0
The real Gaussian noise is generated by:
noise = np.sqrt(0.01)*np.random.normal(0,1,numSamples)
The average noise power (6) is calculated and printed by:
Pn = np.mean(np.abs(noise)**2)
print('noise power = ' + str(np.round(Pn,2)))
which results in:
noise power = 0.01
The SNR is the ratio of the signal power to the noise power, which is calculated and printed by:
SNRdB = 10*np.log10(Ps/Pn)
print('SNR (dB) = ' + str(np.round(SNRdB)))
which results in:
SNR (dB) = 20.0
Figure 1 is an example of s[n], w[n] and x[n] in the time domain as generated by the example Python code.
![A BPSK signal s[n], real Gaussian noise w[n], and the received signal x[n] = s[n] + w[n] for SNR = 20 dB](https://www.wavewalkerdsp.com/wp-content/uploads/2023/12/HowToComputeSignalToNoiseRatioSNRInSimulation_signalNoiseTimeDomain-768x614.png)
Conclusion
This blog provided a simple mathematical backing for how to compute SNR mathematically and then provided Python code for how to generate an example BPSK and then compute it’s SNR.
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