Signals and Systems - Electrical Engineering

160 C H A P T E R 2: Continuous-Time Systems

FIGURE 2.23

Problem 2.6.

1F

+ −

+

−

is(t) vs(t)

1H

iL(t)

2.7. Time-varying capacitor

A time-varying capacitor is characterized by the charge–voltage equation

q(t)=C(t)v(t)

That is, the capacitance is not a constant but a function of time.

(a) Given thati(t)=dq(t)/dt, find the voltage–current relation for this time-varying capacitor.

(b)LetC(t)= 1 +cos( 2 πt)andv(t)=cos( 2 πt). Determine the currenti 1 (t)in the capacitor for allt.

(c)LetC(t)be as above, but delayv(t)by 0.25 sec. Determinei 2 (t)for all time. Is the system TI?

2.8. Sinusoidal Test for LTI

A fundamental property of linear time-invariant systems is that whenever the input of the system is a sinu-

soid of a certain frequency, the output will also be a sinusoid of the same frequency but with an amplitude

and phase determined by the system. For the following systems let the input bex(t)=cos(t),−∞<t<

∞, and find the outputy(t)and determine if the system is LTI.

(a)y(t)=|x(t)|^2

(b)y(t)=0.5[x(t)+x(t− 1 )]

(c)y(t)=x(t)u(t)

(d)y(t)=

1

2

∫t

t− 2

x(τ)dτ

2.9. Testing the time invariance of systems

Consider the following systems and find the response tox 1 (t)=u(t)andx 2 (t)=u(t− 1 ). Determine from

the corresponding outputs whether the system is time-varying or not.

(a)y(t)=x(t)cos(πt)

(b)y(t)=x(t)[u(t)−u(t− 2 )]

(c)y(t)=0.5[x(t)+x(t− 1 )]

Ploty 1 (t)andy 2 (t)for each case.

2.10. Window/modulator

Consider the system where for an inputx(t)the output isy(t)=x(t)f(t)for some functionf(t).

(a) Letf(t)=u(t)−u(t− 10 ). Determine whether the system with inputx(t)and outputy(t)is linear, time

invariant, causal, and BIBO stable.

(b)Supposex(t)=4 cos(πt/ 2 ), andf(t)=cos( 6 πt/ 7 )and both are periodic. Is the outputy(t)also

periodic? What frequencies are present in the output? Is this system linear? Is it time invariant?

Explain.