# How to prove energy is preserved in sampling using Parseval's Relation?

I have a continuous signal,xc(t) which is bandlimited to =pi/T , and x[n] = xc(nT) are samples of this signal. I need to show there is a Parseval's relation between the original signal and the sampled version.That is I need to prove that energy will be conserved! Can anyone please help?

• Parseval's theorem specifies that energy is preserved by the Fourier transform across the time and frequency domains. It doesn't relate to the conversion between continuous-time and discrete-time signals. – Jason R Feb 27 '14 at 12:54
• @JasonR: Which is true in general, but wrong for bandlimited signals. See the sampling theorem. – Lutz Lehmann Feb 28 '14 at 15:43
• @LutzL: As you pointed out in your answer, that's Plancherel's theorem, a generalization of Parseval's theorem. I explained before that Parseval's theorem specifically describes energy conservation across the time and frequency domains. – Jason R Feb 28 '14 at 15:48
• But for band limited functions it really is Parsevals theorem for a countable orthonormal basis of the subspace of band-limited functions and their coefficient functions which just happen to be the function values at the sample points. Plancherel is only needed if one wants to check the orthonormality of the basis. – Lutz Lehmann Feb 28 '14 at 15:51

$$x_c(t)=\sum_{n\in\Bbb Z} x_c(nT)\operatorname{sinc}\left(\tfrac tT-n\right)$$
where $\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}=\int_{-1/2}^{1/2}e^{i2\pi fx}\,df$.
By the Plancherel identity of the continuous Fourier transform this leads to $$\langle \operatorname{sinc}(x-m),\,\operatorname{sinc}(x-n)\rangle = \int_{-1/2}^{1/2}e^{i2\pi f(n-m)}\,df = \delta_{m,n},$$ i.e., orthogonality of the unscaled functions. In the scaled variant above one gets, first by orthogonality and then by rescaling of the time parameter, the Parseval identity for the orthonormal basis $\tfrac1{\sqrt{T}}\operatorname{sinc}(\tfrac tT-n)$, $n\in\Bbb Z$, of the space of band-limited functions,
$$\|x_c\|_2^2=\sum_{n\in\Bbb Z} |x_c(nT)|^2 \langle \operatorname{sinc}(\tfrac tT-n),\,\operatorname{sinc}(\tfrac tT-n)\rangle=T\,\sum_{n\in\Bbb Z} |x_c(nT)|^2$$