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There's a question in my signal processing textbook that says:

Verify that $\int_{-\infty}^{\infty} sinc^2 (kx)dx = \frac{\pi}{k}$ by signal energy method.

I'm unsure what "signal energy method" means here. I've tried finding the energy ($E_g = \int_{-\infty}^{\infty} (g(t))^2dt$) of the signal, but I'm at a loss as to how to make sense of the integral. Also looked at using Parseval's Identity, which doesn't seem to apply here. It seems like there should be a relatively simple solution, but I'm not sure what 'signal energy method' means.

Rome_Leader
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1 Answers1

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You're calculating the square of the $L^2$ norm of a function. Physically this can be understood as the energy of the signal. So you can use that the $L^2$ norm is preserved under the Fourier transform and inverse Fourier transform, provided both are normalized to be unitary. Physically this statement can be understood as saying that the energy has the same representation in real and in Fourier space.

Anyway, as for actually doing the problem, it turns out that sinc is the Fourier transform of the indicator function of an appropriately chosen interval centered at zero. See http://en.wikipedia.org/wiki/Sinc_filter

Ian
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