- #1

mathmari

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A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$.

I want to calculate $\int_0^{T_0}x^2(t)\, dt$.

I have done the following:

It holds that $$\frac{1}{T_0}\int_{T_0}|x(t)|^2\, dt=\sum_{k=-\infty}^{+\infty}|X_k|^2$$ right? (Wondering)

Then do we get $$\int_{T_0}|x(t)|^2\, dt=2\sum_{k=-\infty}^{+\infty}|X_k|^2=2\left [\left(\frac{2}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{4}\right )^2+1\right ]$$ But the result that I get is not one of the choices. So have I done something wrong? (Wondering)