Question 8.4: Determine the FT of the unit-impulse signal v(t) = δ(t).

As it encloses unit area at t = 0, we get
X(jw)=\int\limits_{-\infty}^{\infty }{δ(t)e^{-jwt} }dt = e^{-jw0}\int\limits_{-\infty}^{\infty }{δ(t) }dt=1 and δ(t) ↔ 1.
Since the spectrum is 1, the impulse is composed of components of all frequencies from ω = −∞ to ω = ∞ in equal proportion. That is,
δ(t) =\frac{1}{2\pi }\int\limits_{-\infty}^{\infty } {e^{jwt}}dw=\frac{1}{\pi }\int\limits_{0}^{\infty }cos(wt)dw.
The real exponential signal is also important in that the natural response of systems is of that form and it occurs in problems involving exponential growth and decay of signals, such as capacitor discharge and computation of compound interest.