DFT - Removing window effect in spectral domain with convolution I was thinking about the DFT windowing subject and a thought came to my mind. A DFT will yield the spectrum of a signal convoluted with spectrum of the window used, therefore having a main lobes and side lobes.

I figured it would be possible to remove the window effect on the spectrum of the signal by convoluting again both the signal and the window  spectrum magnitude, and it did indeed works as you can see on the following image.



Left is the original spectrum generated with a hanning window. Right is the spectrum convoluted by the DFT of a hanning window. Top is the Spectrum itself, bottom is MATLAB findpeaks result.

I never read anything regarding this technique, but I am pretty sure I haven't invented anything there. So I am wondering if there is a benefit of doing this processing on the spectrum or if there is a downside to it that I am not seeing.

From what I see, this could help peak detection as we can see on the previous image. Also, it looks like the spectrum is a little bit distorted as we can see on the 2 following images. :




Where the blue graph is the spectrum and the red graph the post-convoluted spectrum.


Any thought about this? 
Is there a problem that could arise from this post-FFT convolution? 
Any paper that treats the subject?


EDIT

You can find a script here that will generate the following graph:
There is, in fact, a downside to your suggestion: the signals you showed are all very clearly divided into their frequency components, but in general, real-life signals tend to be more noisy. 

Depending on the application, you'd want as much leakage dampening (higher main lobes/smaller side lobes of a signal frequency in the transformed spectrum) or, as another example, the narrowest possible main lobe.

In your plots, it is visible that smoothing the magnitude spectra with a window does the opposite of these: the main lobes get wider and smaller, while leakage products from transforming a finite time signal gain in power. Applied to noisy signals, this would yield a significant disadvantage.

Your suggestion, however, is still pretty useful for peak identification!