Can we normalize data to unit variance without distorting the phase? Consider we have one dimension time samples. The noise in the samples made some portions of the time samples amplitude very high. Obviously the short variance for those time samples will have high variance. I want to do running variance (of some window size)  and normalize data by dividing by the variance. Is this procedure would destroy the phase of the phase of the signal? Is there a way to normalize data with unit variance without distorting the phase?
"Normalizing the variance": Since variance is a real number, normalizing it means multiplying the signal with a real number.

That doesn't change the phase, ever.