How to verify the obtained SNR of the noisy signal after adding noise?

Question

I defined the SNR as follows:

$$\rm SNR = \frac{\text{Signal power density}}{\text{Noise power density}}$$

Given noise-free signal, I added the noise to this signal by using the formula above. Now, I am looking for a way to verify if the obtained SNR of the noisy signal is equal to the SNR value that I set when I generated the noise.

Thanks in advance!

Answer 1

Here is a common/straight-forward way using the discrete time domain samples. If you have the noise free signal, $x[n]$, and you created a noise signal, $w[n]$, then you can calculate the SNR by using the formula: $$\hat{\text{SNR}}=\frac{\sum_n |x[n]|^2}{\sum_n |w[n]|^2}$$

Edit

An unstated assumption above is that $\mathbb{E}\big[x[n]\big]=\mathbb{E}\big[w[n] \big]=0$. In general, you'd use the variance not the sum of squared values, as to take care the possible non-zero means:

$$\hat{\text{SNR}}=\frac{\text{var}(x[n])}{\text{var}(w[n])}$$

Answer 2

Assuming you are simulating everything and the expected values of both the signal $x[n]$ and the noise $w[n]$are both $0$, you can calculate the SNR by dividing their mean squared sum and extracting $20$ times the logarithm of the result, as in:

$$SNR=10 log\left(\frac{\sum_{k=0}^N|x[k]|^2}{\sum_{k=0}^N|w[k]|^2}\right)$$

The $log$ is there in order to convert the units into %db$ which is the convention on measuring SNR.

After calculating the existing SNR, you can set the required SNR. This is usually done by setting the gain on the noise $w[n]$, because when we simulate we assume the signal is given. Let the required SNR be denoted by $\hat{SNR}$ and a gain $A$ on $w[n]$:

$$\hat{SNR}=10 log\left(\frac{\sum_{k=0}^N|x[k]|^2}{\sum_{k=0}^N|Aw[k]|^2}\right)=10 log\left(\frac{\sum_{k=0}^N|x[k]|^2}{A^2\sum_{k=0}^N|w[k]|^2}\right)$$ $$\hat{SNR}=10 log\left(\frac{\sum_{k=0}^N|x[k]|^2}{A^2\sum_{k=0}^N|w[k]|^2}\right)-10log(A^2)=SNR-20log(A)$$ $$A=10^{\frac{SNR-\hat{SNR}}{20}}$$

After calculating $A$, you can check for the requested SNR by:

$$\hat{SNR}=10 log\left(\frac{\sum_{k=0}^N|x[k]|^2}{\sum_{k=0}^N|Aw[k]|^2}\right)$$

which is quite redundant. You can now create your results by adding them together using $y[n]=x[n]+Aw[n]$. There is no way to test the SNR of $y[n]$ without using either $x[n]$ or $w[n]$.