Short Time Fourier Transform - invertible?

[jiapei100]

Short Time Fourier Transform -- invertible?

On Wikipedia,
http://en.wikipedia.org/wiki/Short-time_Fourier_transform"

The STFT is invertible, that is, the original signal can be recovered from the transform by the Inverse STFT.

However, it's also said

It can be seen, comparing to above that windowed "grain" or "wavelet" of x(t) is

http://www.visionopen.com/iGabor.png

the inverse Fourier transform of X(τ,ω) for τ fixed.

That is to say, Gabor is invertible, it's able to obtain the original signal, but modulated.

original signal is obviously x(t),
w(t-τ) is the window function used to extract a local signal within this window,
which can also be looked on as a kind of modulation.

Therefore, in the above function (attached picture),
x(t)w(t-τ) can be computed, from the Gabor transformed data,
But, I'm dropping questions to ask, whether the true original data x(t) can be finally recovered?
as it's declared by Wiki itself

The STFT is invertible, that is, the original signal can be recovered from the transform by the Inverse STFT.

Can anybody help to make me clarified?


Best Regards
JIA Pei

 

[IttyBittyBit]

I'll answer your question by asking you another question.

Let's say I have a time-varying signal. I multiply it by a gaussian then send it to you. Will you be able to recover the original signal?

 

[jiapei100]

Sorry for my stupidity.

Your word "multiply" hints me. !

This is a "element-wise" multiplication! Right? Yes, it should be.

Thanks for your answering to clarify my doubts.

Best Regards
JIA Pei

I'll answer your question by asking you another question.

Let's say I have a time-varying signal. I multiply it by a gaussian then send it to you. Will you be able to recover the original signal?

 

[IttyBittyBit]

Yes, it's element-wise.

 

[CellCoree]

I can confirm that the Short Time Fourier Transform (STFT) is indeed invertible. This means that the original signal can be recovered from the transformed signal using the Inverse STFT. However, it is important to note that the recovered signal may be modulated due to the use of a window function in the STFT.

The window function is used to extract a local signal within a certain time window, and this can be seen as a form of modulation. Therefore, the recovered signal may not be exactly the same as the original signal, but it will still contain the same information. It is important to carefully consider the window function used in the STFT to ensure that the recovered signal is as close to the original signal as possible.

In summary, the STFT is invertible and can be used to recover the original signal, but the use of a window function may result in a modulated signal. Further research and experimentation may be needed to optimize the window function and improve the accuracy of the recovered signal.