# What is the physics behind the width of a main lobe?

We know that the square window gives the lowest main lobe width possible, and that other windows after that trade main lobe width for side lobe height. I also understand that the main lobe width is inversely proportional to the length of the DFT aperature.

Given this, is there then a physical limit to how narrow a main-lobe can be, for a fixed aperture length? For example, can we ever hope to have main lobes that are narrower than that given by a square window, (for a fixed aperture length again), via windowing or any other method?

A second related question I have here has to do with energy conservation - is the interpretation that windows trade side lobe height for main lobe width, because energy must be conserved? Thus any reduction in side lobe height means the energy "has to go somewhere" and ends up in the main lobe? Would this be a correct interpretation?

• I would use the term "mathematics" instead of "physics". There isn't necessarily any physical relationship implied in the Fourier transform process. – Jason R Sep 24 '13 at 16:54
• @JasonR Yes, but in terms of the physics involves, (conservation of energy), there might be. That is what I would like to ascertain. (Second paragraph). – TheGrapeBeyond Sep 24 '13 at 17:42
• @TheGrapeBeyond: You seem to be referring to Parseval's theorem, which specifies that the unitary Fourier transform is energy-preserving. The properties of the Fourier transform are independent of any physical phenomena, so there will be no required tie-in to any conservation laws in physics. – Jason R Sep 24 '13 at 18:14
• @JasonR Thus I take it that by extension, the trade where sidelobes are minimized at the expense of an expansion of the main lobe has absolutely nothing to do with preserving energy? – TheGrapeBeyond Sep 24 '13 at 18:23
• The time-bandwidth product of signals (product of the rms time duration and the rms or Gabor bandwidth) has a lower bound which all signals must satisfy. Only Gaussian pulses e.g. $x(t) =e^{-\pi t^2}, -\infty < t < \infty$ meet this bound exactly; all others have larger time-bandwidth products. This is akin to the Heisenberg uncertainty principle in quantum mechanics (physics again, JasonR!) and in fact, the calculations are mathematically identical in the two cases. – Dilip Sarwate Sep 24 '13 at 19:29