Analytic signal of the Dirac delta function

Question

Does anyone know of any derivations of the analytic signal of the Dirac delta function? I suppose that it can be found by first working from the definition of the Hilbert transform, since the analytic representation of a real-valued function is the analytic signal, comprising the original function and its Hilbert transform? Appreciate any insight.

Answer 1

The Hilbert transform is defined as $$ H[u](t) = \lim_{\epsilon\rightarrow 0} \int_\epsilon^\infty \frac{ u(t+\tau) - u(t-\tau)}{\pi \tau} d\tau. $$ Inserting the Dirac delta for $u$, we have $$ H[\delta(t)](t) = \lim_{\epsilon\rightarrow 0} \int_\epsilon^\infty \frac{\delta(\tau+t)-\delta(\tau-t)}{\pi \tau} d\tau.$$ Now using the sifting property of the Delta function, we obtain $$ H[\delta(t)](t) = \lim_{\epsilon\rightarrow 0} \Big[\frac{\Theta(\tau+t)}{\pi t} +\frac{\Theta(\tau-t)}{\pi t}\Big]\Bigg|_\infty^\epsilon \\ = \lim_{\epsilon\rightarrow 0} \Big[\frac{\Theta(\epsilon+t)-1}{\pi t} + \frac{\Theta(\epsilon-t)-1}{\pi t}\Big]\\ = \lim_{\epsilon\rightarrow 0} \frac{\Theta(-\epsilon-t)+\Theta(t-\epsilon)}{\pi t}\\ = \frac{1}{\pi t},$$ thereby arriving at the solution on wikipedia