B Time-evolving Fourier transform

[entropy1]

I am a little familiar with Fourier Analysis, but I don't know where to get tools to get the answer to this question:

Consider a discrete signal A_[0..N-1], consisting of N samples. Suppose we Fourier transform it and get a series of harmonics.

Now, consider the discrete signal A_[1..N], that is equal to signal A_[0..N-1] on the corresponding indices [1..N-1], and has the next sample A_N added to it, to obtain N indices.

We subject the second signal to Fourier transform too.

Will the set of harmonics of the first signal and the set of harmonics of the second be quite similar (having similar amplitudes), or could they differ considerably?

I relate this to a spectral analyser display of audio signals. If the Fourier transform is done on a fixed interval, each next transform done from one sample further on, the (virtual) frequencies in the audio signal shouldn't jump around too much on that instance, should they? I am not sure about white noise though.

I hope the question is clear. Answers are very welcome!

 

[FactChecker]

You are right. The idea of the Fourier transform is to decompose a signal from the time domain into its frequency content. That should not change due to a one-sample shift. You would not expect a guitar string to sound different just because you plucked it a fraction of a second later.

EDIT: If the first point of the first series shows a very sharp transient from it to the second point, then there might be more high frequency content in the first series. It takes a lot of high frequency to make a step function. But if there is nothing special about that first point, it should not make much difference.

 

[chiro]

Hey entropy1.

It might make sense to do the actual integral over a number of harmonics for a number of functions to see the effects itself for a class of functions.

As FactChecker pointed out above - it will depend on the function quite a lot along with the size of the window you have that is different between the two functions.