The derivative of the unit step function \(u\left(t\right)\) is the unit impulse function \(\delta\left(t\right)\).
The unit impulse function \(\delta\left(t\right)\) (also known as the Dirac delta function) is zero everywhere except at \(t = 0\), where it is undefined.
\(\boxed{\delta\left(t\right) = \frac{d}{dt} u\left(t\right) = \begin{cases}0\,, & t \neq 0 \\ \infty\,, & t = 0\end{cases}}\)
Impulsive currents and voltages occur in electric circuits as a result of switching operations or impulsive sources.
The unit impulse may be regarded as an applied or resulting shock. It may be visualized as a very short duration pulse of unit area. This may be expressed mathematically as
\(\boxed{\int_{-\infty}^{\infty} \delta\left(t\right)\,dt = \int_{0^-}^{0^+} \delta\left(t\right)\,dt = 1}\)
where \(t = 0^-\) denotes the time just before \(t = 0\) and \(t = 0^+\) is the time just after \(t = 0\).
The area of the impulse \(A\,\delta(t)\) equals \(A\) and is also called its strength.
Consider the rectangular pulse \((1 / \tau)\,\mathrm{rect}(t / \tau)\) of width \(\tau\) and height \(1 / \tau\). As \(\tau\) decreases, the width shrinks and the height increases proportionately to maintain unit area. As \(\tau \to 0\), the rectangular pulse becomes a narrow spike with unit area, exhibiting properties consistent with an impulse \(\delta(t)\).
Properties of Impulse function
Scaling Property
Since \(\delta(t)\) has unit area, the time-compressed impulse \(\delta(\alpha t)\) should have an area of \(1 / |\alpha|\). Since the impulse \(\delta(\alpha t)\) still occurs at \(t = 0\), it may be regarded as an unscaled impulse \(\delta(t) / |\alpha|\). Since a time shift does not affect areas, the general result is
\(\boxed{\delta\left[\alpha(t - \beta)\right] = \frac{1}{|\alpha|}\delta(t - \beta)}\)
Product Property
The product of a signal \(x(t)\) with an impulse \(\delta(t)\) at \(t = \alpha\) is also an impulse \(\delta(t)\) at \(t = \alpha\) whose height is scaled by the value of \(x(t)\) at \(t = \alpha\).
\(\boxed{x(t)\,\delta(t - \alpha) = x(\alpha)\,\delta(t - \alpha)}\)
Sifting Property
The product property immediately suggests that the area of the product \(x(t)\,\delta(t - \alpha) = x(\alpha)\,\delta(t - \alpha)\) equals \(x(\alpha)\). In other words, \(\delta(t - \alpha)\) sifts out the value of \(x(t)\) at the impulse location \(t = \alpha\), and thus,
\(\boxed{\int_{-\infty}^{\infty} x(t)\,\delta(t - \alpha)\,dt = x(\alpha)}\)