The derivative of an impulse \(\delta(t)\) is called a doublet and denoted by \(\delta^{\prime}(t)\).
\(\boxed{\delta^{\prime}(t) = \frac{d}{dt} \delta(t) = \begin{cases}0\,, & t \neq 0 \\ \textrm{undefined}\,, & t = 0\end{cases}}\)
Consider the triangular pulse \(x(t) = (1 / \tau)\,\mathrm{tri}(1 / \tau)\). As \(\tau \to 0\), \(x(t)\) approaches \(\delta(t)\). Its derivative \(x^{\prime}(t)\) should then correspond to \(\delta^{\prime}(t)\). Now, \(x^{\prime}(t)\) is odd and shows two pulses of height \(1 / \tau^2\) and \(-1 / \tau^2\) with zero area. As \(\tau \to 0\), \(x^{\prime}(t)\) approaches \(+\infty\) and \(-\infty\) from below and above, respectively.
Thus, \(\delta^{\prime}(t)\) is an odd function characterized by zero width, zero area, and amplitudes of \(+\infty\) and \(-\infty\) at \(t = 0\).
\(\boxed{\int_{-\infty}^{\infty} \delta^{\prime}(t)\,dt = 0}\)
\(\boxed{\delta^{\prime}(-t) = -\delta^{\prime}(t)}\)
The two infinite spikes in \(\delta^{\prime}(t)\) are not impulses (their area is not constant), nor do they cancel. In fact, \(\delta^{\prime}(t)\) is indeterminate at \(t = 0\). Even though its area is zero, its absolute area is infinite.
Properties of Doublet Function
Scaling Property
The scaling property of the doublet can be obtained by taking the derivative of scaled impulse.
\(\boxed{\delta^{\prime}\left[\alpha (t - \beta)\right] = \frac{1}{\alpha |\alpha|}\delta^{\prime}(t - \beta)}\)
Product Property
The product property of the doublet can be obtained by comparing the derivative of \(x(t)\,\delta(t - \alpha)\) to the derivative of the product property of impulses.
\(\boxed{x(t)\,\delta^{\prime}(t - \alpha) = x(\alpha)\,\delta^{\prime}(t - \alpha) - x^{\prime}(\alpha)\,\delta(t - \alpha)}\)
Sifting Property
The doublet \(\delta^{\prime}(t - \alpha)\) sifts out the negative derivative of \(x(t)\) at \(t = \alpha\).
\(\boxed{\int_{-\infty}^{\infty} x(t)\,\delta^{\prime}(t - \alpha)\,dt = -x^{\prime}(\alpha)}\)