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Uncertainty Principle without quantum woo-woo - as a pulse becomes shorter in time domain, its spectrum gets fuzzier in the frequency domain, and vice versa. It's impossible to perfectly localize a signal in both time and frequency domain simultaneously simply due to how Fourier transform works. #electronics
Figure 6.4: The “spread” of a signal and its Fourier transform are inversely proportional. (a) A constant function in the time domain corresponds to a unit impulse (Dirac delta function) in the frequency domain. (b, c) As the width of a pulse in the time domain decreases, its spectrum in the frequency domain spreads (spectrum shown is schematic). (d) A unit impulse in the time domain has a spectrum which is a constant function.
From these equations we get the general Gabor Uncertainty Principle: ΔfΔt >= 1 It should be noted that such an uncertainty principle applies whenever we make simultaneous measurements of a function and its Fourier transform. The implications of Gabor’s principle are easier to understand by looking at it in “Fourier space,” where the abscissa reflects the time domain and the ordinate the frequency domain (Fig. 6.8). Then Gabor’s principle says that the spreads or uncertainties in the time and frequency measurements must define a rectangle in Fourier space whose area is at least 1. Thus we can decrease Δt and so localize it better in the frequency domain, or decrease Δf, and so localize it better in the frequency domain, but we cannot localize it arbitrarily well in both domains simultaneously. The most we can localize signals in the Fourier domain is into rectangles of size ΔfΔt = 1
Figure 6.8: Minimum possible localization of signal in Fourier space. The product of the nominal duration
#electronics

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