# Short Time Fourier Transform - invertible?

• jiapei100
And yes, you should be able to recover the original signal using the Inverse STFT. The STFT is invertible, meaning the original signal can be recovered from the transform. This is done by multiplying the Gabor transformed data with the window function, which can also be seen as a form of modulation. Therefore, the original signal can be obtained from the Gabor transformed data.

#### jiapei100

Short Time Fourier Transform -- invertible?

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

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 [Broken]

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

Last edited by a moderator:

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?

Sorry for my stupidity.

Your word "multiply" hints me. !

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

Best Regards
JIA Pei

IttyBittyBit said: