Windowing in Frequency Domain: Exploring Effects on Time Domain Data

[DSPly]

Hey @ all,

when windowing in DSP in time domain one multiplies all recorded time samples with a weighting factor (hanning, hamming, etc.), followed by a Fourier transform (FFT) to reduce sidelobes in the spectral domain.

But now when thinking about starting up in frequency domain where I have multiplied my frequency data with a rectangular window, i.e. I have only non-zero frequencies from fstart to fend ( ... 0 0 0 0 0 0 0 fstart f1 f2 f3 f4 f5 fend 0 0 0 0 ... ). (or alternatively I only have frequency datas recorded at finite points.) What happens to my time domain data after performing the inverse FFT due to the rectangular window?


But in general: How do one has to perform windowing in frequency domain? Really by multiplying the "origianal" (i.e. time domain) window-coefficients with the spectral components? Or performing convolution (with the origignal window, or the Fourier transformed coefficients?) since this is the fourier-pair to multiplication?

Thanks for any ideas.

 

[rbj]

there is a theorem of Duality with the Fourier Transform (even with the DFT). whatever effect in the time domain there is from a frequency-domain operation is just like the effect in the frequency domain from the same operation done in the time domain (except for a reversal of time or frequency in one or the other and also with scaling).

 

[MisterX]

With the DFT, a (sampled) rectangle function in the time domain corresponds to a (sampled) Dirichlet kernel in the frequency domain. In general the Fourier transform or inverse Fourier transform of a rectangle function will be a "sinc-like" function.

For the DFT, multiplying in one domain (applying the window function to the samples in the time domain) corresponds to convolution in the other domain. So to perform the time domain windowing in the frequency domain, you could convolve the DFT of the time domain signal with a particular (sampled) Dirichlet kernel function. For the data that one would use with the DFT/iDFT, circular convolution is appropriate.Using the frequency domain rectangular window can remove certain frequencies from the signal.

 

[DSPly]

thank you for the answers.

Okay, so as I want to window my sampled time domain signal I can decide weather I want to do this directly in time domain (multiply samples with window) or in frequency domain (convolve Fourier transformed samples with the transformed window).

And if I got it right I perform windowing on my recorded frequency domain data just the same way, interchanging only "time" with "frequency" due to the theorem of duality: Either multiplying my frequency samples with the window coefficients or convolvong the iDFT of my frequency data with the inverse transformed window coefficients.

Right?

 

[MisterX]

Yes
_

 

[rbj]

you are perfectly correct in this specific case.

one thing to remember is that, when applying duality, you swap time with the negative of frequency. when the time functions are even symmetry (like the rectangular window), so also are the frequency functions so this negation of time or frequency does not matter in the case of even symmetry.

also, i just remembered, that the scaling of $1/N$ in front of one of the DFT summations makes a difference. if the DFT is defined with $1/\sqrt{N}$ in front of both summations, then you have a nice symmetry and you need not worry about the difference of scaling.