I was asked to show that this convolution integral results in the answers also given in the image. Not quite sure how to approach this integral, everything seems to be coupled together.

Does anyone know how this kind of integrals is solved?

enter image description here

  • $\begingroup$ Multiply the integrand by a rectangular window whose support is $[-T,T]$ and integrate over $\mathbb R$. Recall that multiplication in one domain corresponds to convolution in the other domain. $\endgroup$ – Rodrigo de Azevedo Oct 16 '19 at 20:01

As $T\rightarrow \infty$, the integral becomes the convolution integral. You can use the fact that convolution in the time domain is multiplication in the frequency domain and then you have a single carrier being passed through a low pass filter. The term where $\omega=\omega_c$ just has to do with how you define the ideal low pass filter and in this case you'd say that the value at exactly $\omega=\omega_c$ is equal to $\frac{1}{2}$.

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  • $\begingroup$ Good answer! Welcome... $\endgroup$ – Royi Oct 16 '19 at 18:50

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