Let us say say we have a noised constant signal and want to evaluate the standard deviation (std) of the noise. We calculate the std of the said noised signal and call it σ1. Now we process the signal, by interpolating it (linear) and stretching it on a larger area and obtain a signal with more samples, which contains the same amount of information with noise because we did the interpolation. We calculate the standard deviation σ2. How do they relate? Can we just compare them or do we need to make some adjustments based on the sample sizes. I get that σ2 is less than σ1.

But it is wrong that the noise was reduced by just stretching the signal along with the noise. So which metric tells us that the noise is the same.

Here it is for clarification in images

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We assume the signal to be 0.5 and the noise in the first and the forth quadrant to be -0.5 and +0.5. \begin{bmatrix}0&0.5\\0.5&1\end{bmatrix} So σ1 = 0.3535. We interpolate the signal by stretching it in both directions.

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The std of the 2nd signal σ2 = 0.2795.

So what gives? How can we compare the signal with the noise in both cases?


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