Fourier series power spectra question Watch

trm90
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Report Thread starter 7 years ago
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Hey guys

I've derived the following equation for the power spectrum (given as |F(w)|2, where F(w) is the f.t. of some function f(t)):

\dfrac{\omega_0^{2} \tau^4}{[1 + (\omega - \omega_0)^2 \tau^2][1 + (\omega + \omega_0)^2 \tau^2]}

I'm asked to:

i) Sketch the function as a function of \omega

ii) Using the approximation that \omega + \omega_0 is constant in the region of positive frequency peak, to show that the half width at half maximum \Delta \omega satisfies \tau \Delta \omega = 1.

For i), I've had to do this using the help of wolfram. The shape of the graph is basically two peaks symmetric about the y-axis, and the function decreaes rapidly and tends to zero as the function tends to +/- infinite. But what are some tricks I can use to sketch this graph without using help from graphical help from programs?

For ii) I have absolutely no idea how to show this. Is there an actual expression for \Delta \omega?
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