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Why window functions can't have both narrow main lobe and low-level side lobes?

i got a hint from my mentor that it's quite often expained in books about wavelets...

I thought it could be associated with uncertainty heisenberg principle but as understand this principle more about time-frequency uncertainty i.e. size of the window but not a lobes...

Narrow main lobe => high frequency resolution, low-level side lobes => low spectral leakage. Attempting to made main lobe narrower leads to rise of side lobes level and vice versa... That's exactly what we usually observe but what's mathematical reason for it and how it can be explained more precisely? Why there doesn't exist window functions with both narrow main lobe and low-level side lobes?

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The idea is that windows make a tradeoff between resolution (the ability to resolve signals of similar amplitude when frequencies are close) and dynamic range (the ability to distinguish between far away frequencies with different amplitudes). The Heisenberg uncertainty principle applies to this pair of properties of the signal, ie you cannot resolve close frequencies with similar amplitudes and far apart frequencies with dissimilar amplitudes simultaneously.

Thinking about this analytically, the Fourier transform of a rectangular window is a sinc. This means that if you use a rectangular window, you convolve the signal spectrum with the window spectrum, which is a sinc. The narrower the time domain window, the broader the sinc mainlobe and lower sidelobes, and vice versa. Convolving with a broad sinc leads to low resolution, but convolving with a narrow sinc leads to low dynamic range.

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  • $\begingroup$ I don't really understand why convolving with a narrow sinc leads to low dynamic range? $\endgroup$
    – Alexander
    Dec 30, 2023 at 20:01
  • $\begingroup$ A narrow sinc has higher sidelobes because the period is shorter. The first null occurs quicker, which means the denominator is still smaller, leading to a higher sidelobe. $\endgroup$
    – Baddioes
    Dec 30, 2023 at 21:49

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