91Ó°ÊÓ

Express the following signals in terms of singularity functions. (a) \(v(t)=\left\\{\begin{aligned} 0, & t<0 \\\\-5, & t>0 \end{aligned}\right.\) (b) \(i(t)=\left\\{\begin{aligned} 0, & t<1 \\\\-10, & 15 \end{aligned}\right.\) (c) \(x(t)=\left\\{\begin{array}{ll}t-1, & 11 \end{aligned}\right.\)

Sketch the waveform that is represented by $$v(t)=u(t)+u(t-1)-3 u(t-2)+2 u(t-3)$$

Evaluate the following integrals involving the impulse functions: (a) \(\int_{-\infty}^{\infty} 4 t^{2} \delta(t-1) d t\) (b) \(\int_{-\infty}^{\infty} 4 t^{2} \cos 2 \pi t \delta(t-0.5) d t\)

Evaluate the following integrals: (a) \(\int_{-\infty}^{\infty} e^{-4 t^{2}} \delta(t-2) d t\) (b) \(\int_{-\infty}^{\infty}\left[5 \delta(t)+e^{-t} \delta(t)+\cos 2 \pi t \delta(t)\right] d t\)

The voltage across a 10 -mH inductor is \(20 \delta(t-2) \mathrm{mV} .\) Find the inductor current, assuming that the inductor is initially uncharged.

Find the solution of the following first-order differential equations subject to the specified initial conditions. (a) \(5 d v / d t+3 v=0, \quad v(0)=-2\) (b) \(4 d v / d t-6 v=0, \quad v(0)=5\)

In designing a signal-switching circuit, it was found that a \(100-\mu \mathrm{F}\) capacitor was needed for a time constant of \(3 \mathrm{ms}\). What value resistor is necessary for the circuit?

An \(R L\) circuit may be used as a differentiator if the output is taken across the inductor and \(\tau \ll T\) (say \(\tau<0.1 T\) ), where \(T\) is the width of the input pulse. If \(R\) is fixed at \(200 \mathrm{k} \Omega\), determine the maximum value of \(L\) required to differentiate a pulse with \(T=10 \mu \mathrm{s}\).