Pulse Signals

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 \(\delta(t)\).

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.