Question 7.6: Signal energy of a sinc signal Find the signal energy of x[n......

The signal energy of a signal is defined as

E_x=\sum\limits_{n=-\infty }^{\infty }{\left|x[n]\right|^2 } .

But we can avoid doing a complicated infinite summation by using Parseval’s theorem. The DTFT of x[n] can be found by starting with the Fourier pair

sinc(n/w)\xleftrightarrow[]{\mathcal{F} }wrect(wF)*\delta _1(F).

and applying the linearity property to form

(1/5)sinc(n/100)\xleftrightarrow[]{\mathcal{F} }20rect(100F)*\delta _1(F).

Parseval’s theorem is

\sum\limits_{n=-\infty }^{\infty }{\left|x[n]\right|^2 }=\int_{1}^{}{\left|X(F)\right| } ^2 dF.

So the signal energy is

E_x=\int_{1}^{}{\left|20rect(100  F)*\delta _1(F)\right| }^2 dF =\int_{-\infty }^{\infty }{{\left|20rect(100  F)\right| }^2 dF} .

or

E_x=400\int_{-1/200}^{1/200}{dF} =4.