I have read that for non-stationary signal we break the signal into smaller segments by applying a window function . My question is how this can help to make the signal has a fixed features or to became stationary signal even it's not?
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\$\begingroup\$ I know what a stationary process is, but I've never heard of a stationary signal... \$\endgroup\$– Vladimir CraveroAug 9, 2014 at 9:02
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\$\begingroup\$ @VladimirCravero A stationary signal is one whose frequency does not changes over time. \$\endgroup\$– LearnerAug 9, 2014 at 9:05
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\$\begingroup\$ That's quite a new definition for me. Have you got some reference for me? I think you are mixing things up either with periodic vs non periodic signals or with the power spectral density of a process, that is the F-transform of its autocorrelation iff the process is stationary, but I may be wrong of course :) \$\endgroup\$– Vladimir CraveroAug 9, 2014 at 9:09
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\$\begingroup\$ maybe a stationary signal is a signal which FFT does not change "too much" over time? \$\endgroup\$– Vladimir CraveroAug 9, 2014 at 9:12
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\$\begingroup\$ @VladimirCravero from wiki answers wiki.answers.com/Q/What_is_a_non-stationary_signal , is says that a nonstationary signal is one whose frequency changes over time.So the sationary signal is the signal is one whose frequency does not changes over time ! :) \$\endgroup\$– LearnerAug 9, 2014 at 9:21
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2 Answers
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The discrete Fourier transform works on the assumption that your signal is periodic.
So, say we start with this arbitrary time-domain signal:
If it's periodic, we should be able to repeat it:
Well I guess that works, but we've introduced a discontinuity. This is like adding a square wave to your signal: you are going to see a new frequency component emerge equal to the DFT period, plus all of its odd harmonics.
In other words, the DFT sees any discontinuities even if they are at the ends of the signal. In fact, since the signal is periodic, it doesn't matter if we rotate all the inputs. If we do that with our original, we end up with:
This is exactly the same input as the first, as far as the DFT is concerned.
A window function works by tapering the ends to some similar value (usually 0) gradually, thereby making them equal. But it does so gradually, so that a minimum of extra frequency components are introduced. If we apply a window function to our original signal, you get something like this:
Which when duplicated, gives you:
or rotated:
No discontinuities! Now our non-periodic signal looks like a periodic signal, and we made it so while introducing a minimum of frequency-domain distortion. Of course, different window functions define "minimum distortion" in different ways, according to what you are trying to accomplish with the transform.
answered Aug 9, 2014 at 10:41
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\$\begingroup\$ But do you really call non periodic signals also "stationary" signals? \$\endgroup\$ Aug 9, 2014 at 12:59
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\$\begingroup\$ @VladimirCravero I think what the OP had in mind was stationary process, which is certainly related but not the same. The approach, so far as I can infer, is to take a non-stationary process and slice it into short time segments, such that each segment is approximately stationary, then analyze that segment. That's rather orthogonal to the periodic requirement of the Fourier transform. \$\endgroup\$ Aug 9, 2014 at 13:04
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\$\begingroup\$ Well thanks, that's what I thought. A "stationary signal" is a misuse of words. \$\endgroup\$ Aug 9, 2014 at 13:05
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\$\begingroup\$ @VladimirCravero Maybe so, but I understood the question to be "why do we need a window function?", and as it turns out, the reason has nothing to do with stationary anything, though there seems to be some assumption by the OP to the contrary. \$\endgroup\$ Aug 9, 2014 at 13:07
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\$\begingroup\$ You've got a point. Maybe OP cofusion is just confusing me. I find your answer more than adequate for this question though. \$\endgroup\$ Aug 9, 2014 at 13:09
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Your post is a little unclear on what you want to do with the signal but going by the Fourier and Signal Processing tags on the post I guess you want to process your non-stationary signal in some way like an FFT or something.
The FFT considers the signal to be periodic in the window in which it is applied. So if you apply 2048 samples to the FFT it's assumed that the next 2048 are exactly the same, as are the 2048 preceeding values etc..
The purpose of the window is to reduce the weighting of the signal at the ends of the window so that the assumed periodicity in the window size has a reduced affect on the FFT. In reality there is an effect; causing side lobes on the signal of interest. Different window types produce different magnitudes of side lobes (e.g. Hamming, Blackman etc..).
Specifically in your case of a non-stationary signal the window would serve to "narrow down" the signal to a small segment that could be considered to be stationary for the purposes of the FFT.
The windows themselves have an FFT spectum that cause the lobes and leakage. See how a rectangular window (same as no window) has significant leakage and lobes:
These are reduced significantly in a simple triangular window:
This is significant for you because it "separates" your time variant signal periods from one another, allowing processing such as an FFT or Auto-correlation function.
