110 C H A P T E R 1: Continuous-Time Signals
(c)Issinh(t)even or odd?
(d)Obtain an expression forx(t)=e−tu(t)in terms of the hyperbolic functions. Use symbolic MATLAB to
plotx(t)=e−tu(t)and to plot your expression in terms of the hyperbolic functions. Compare them.
1.7. Impulse signal generation—MATLAB
When defining the impulse orδ(t)signal, the shape of the signal used to do so is not important. Whether
we use the rectangular pulse we considered in this chapter or another pulse, or even a signal that is not a
pulse, in the limit we obtain the same impulse signal. Consider the following cases:
(a) The triangular pulse,31 (t)=1
1(
1 −∣∣
∣∣t
1∣∣
∣∣)
(u(t+1)−u(t−1))Carefully plot it, compute its area, and find its limit as 1 → 0. What do you obtain in the limit? Explain.
(b)Consider the signalS 1 (t)=
sin(πt/1)
πt
Use the properties of the sinc signalS(t)=sin(πt)/(πt)to expressS 1 (t)in terms ofS(t). Then find its
area, and the limit as 1 → 0. Use symbolic MATLAB to show that for decreasing values of 1 theS 1 (t)
becomes like the impulse signal.
1.8. Impulse and unit-step signals
By introducing the impulseδ(t)and the unit-stepu(t)signals, we expand the conventional calculus. One
of the advantages of having theδ(t)function is that we are now able to find the derivative of discontinuous
signals. Let us illustrate this advantage. Consider a periodic sinusoid defined for all times,x(t)=cos( 0 t) −∞<t<∞and a causal sinusoid defined as
x 1 (t)=cos( 0 t)u(t)
where the unit-step function indicates that the function has a discontinuity at zero, since fort= 0 +the
function is close to 1 , and fort= 0 −the function is zero.
(a) Find the derivativey(t)=dx(t)/dtand plot it.
(b)Find the derivativez(t)=dx 1 (t)/dt(treatx 1 (t)as the product of two functionscos( 0 t)andu(t)) and
plot it. Expressz(t)in terms ofy(t).
(c)Verify that the integral ofz(t)gives you backx 1 (t).
1.9. Series RC circuit response to a unit-step signal
A unit-step functionu(t)can be considered a causal constant source (e.g., a battery in a circuit if the units
ofu(t)is volts).
(a) From basic principles consider the response of an RC circuit tou(t)—that is, a battery connected in
series with the resistor and the capacitor. Remember that the voltage across the capacitor results
from an accumulation of charge, and that the presence of the resistor simply means that the charge is
slowly accumulated. Therefore, plot what would be the voltage across the capacitor fort> 0 (assume
the capacitor has no initial voltage att= 0 ).
(b)What would be the voltage across the capacitor in the steady state? Explain.
(c)Finally, suppose that the capacitor is disconnected from the circuit at some timet 0 >> 0. Ideally, what
would be the voltage across the capacitor from then on?
(d)If you disconnect the capacitor, again att 0 >> 0 , but somehow it is left connected to the resistor, so
they are in parallel, what would happen to the voltage across the capacitor? Plot approximately the
voltage across the capacitor for all times and explain the reason for your plot.