Calculate $\int_{-\pi}^\pi\delta(\omega-\omega_0) e^{j\omega t}d\omega$.

Question

Let us consider the Dirac delta function $\delta(\omega)$ Calculate: $$\int_{-\pi}^\pi\delta(\omega-\omega_0) e^{j\omega t}d\omega$$ The presence of Delta function gives me some problems.

I would appreciate some help with this problem.

Answer 1

The Dirac Delta is a linear functional that operates on a test function and returns the value of the function evaluated at the origin. Here, the functional spans $-\pi$ to $\pi$, while the Dirac Delta is "active" at $\omega_0$. So, if $|\omega_0|<\pi$, then we have

$$\int_{-\pi}^{\pi}\delta(\omega-\omega_0)e^{j\omega t}\,d\omega=e^{j\omega_0t}$$

If $|\omega_0|>\pi$, then

$$\int_{-\pi}^{\pi}\delta(\omega-\omega_0)e^{j\omega t}\,d\omega=0$$

Heuristically, we can think of the Dirac Delta as "sifting" the test function at the point for which its "argument" is zero. Here, that is when $\omega=\omega_0$. But, we need to ensure that the "integral limits" (this is not actually an integral, but rather a functional) include the point for which the argument is zero.