Signals and Systems - Electrical Engineering

Problems 161

(c)Letf(t)=u(t)−u(t− 2 )and the inputx(t)=u(t). Find the corresponding outputy(t). Suppose you

shift the input so that it isx 1 (t)=x(t− 3 ). What is the corresponding outputy 1 (t). Is the system time

invariant? Explain.

2.11. Initial conditions, LTI, steady state, and stability
The input–output characterization of a system is

y(t)=e−^2 ty( 0 )+ 2

∫t

0

e−^2 (t−τ)x(τ)dτ t≥ 0

and zero otherwise. In the above equationx(t)is the input andy(t)is the output.
(a) Is this system LTI? Is it possible to determine a value fory( 0 )that would make this an LTI system?
Explain.
(b)Find the differential equation that also characterizes this system.
(c)Suppose forx(t)=u(t)and any value ofy( 0 ), we wish to determine the steady-state response of the
system. Is the value ofy( 0 )of any significance—that is, do we get the same steady-state response if
y( 0 )= 0 ory( 0 )= 1? Explain.
(d)Compute the steady-state response wheny( 0 )= 0 andx(t)=u(t)using the convolution integral. To
do so, first find the impulse response of the systemh(t). Then relate the integral in the equation given
above with the convolution integral and graphically compute it.
(e) Suppose the input is zero. Is the system depending on the initial condition BIBO stable? Find the
zero-input responsey(t)wheny( 0 )= 1. Is it bounded?
2.12. Amplifier with saturation
The input–output equation characterizing an amplifier that saturates once the input reaches certain values
is

y(t)=







100 x(t) − 10 ≤x(t)≤ 10

1000 x(t) > 10

− 1000 x(t) < 10

wherex(t)is the input andy(t)is the output.

(a) Plot the relation between the inputx(t)and the outputy(t). Is this a linear system? Explain.

(b)For what range of input values is the system linear, if any?

(c)Suppose the input is a sinusoidx(t)=20 cos( 2 πt)u(t). Carefully plotx(t)andy(t)fort=− 2 to 4.

(d)Let the input be delayed by two units of time (i.e., the input isx 1 (t)=x(t− 2 )). Find the corresponding

outputy 1 (t)and indicate how it relates to the outputy(t)due tox(t)found above. Is the system time

invariant?

2.13. QAM system
A quadrature amplitude modulation (QAM) system is a communication system capable of transmitting two
messagesm 1 (t),m 2 (t)at the same time. The transmitted signals(t)is

s(t)=m 1 (t)cos(ct)+m 2 (t)sin(ct)

Carefully draw a block diagram for the QAM system.

(a) Determine if the system is time invariant or not.

(b)Assumem 1 (t)=m 2 (t)=m(t)—that is, we are sending the same message using two different modu-

lators. Express the modulated signal in terms of a cosine with carrier frequencyc, amplitudeA, and

phaseθ. ObtainAandθ. Is the system linear? Explain.