Signals and Systems - Electrical Engineering

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

FIGURE 2.20

Problem 2.2.

− 4 − 3 − 2 − 1 0 1 2 3

− 1

−0.5

0

0.5

1

t(sec)

x

(t

)

2.3. Analog averaging system

Consider the analog averager

y(t)=

1

T

t+∫T/ 2

t−T/ 2

x(τ)dτ

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

(a) Find the impulse responseh(t)of the averager. Is this system causal?

(b)Letx(t)=u(t). Find the output of the averager.

2.4. LTI determination from input–output relation

An analog system has the input–output relation

y(t)=

∫t

0

e−(t−τ)x(τ)dτ t≥ 0

and zero otherwise. The input isx(t)andy(t)is the output.

(a) Is this a linear time-invariant system? If so, can you determine without any computation the impulse

response of the system? Explain.

(b)Is this system causal? Explain.

(c)Find the unit-step responses(t)and from it find the impulse responseh(t). Is this a BIBO-stable

system? Explain.

(d)Find the response due to a pulsex(t)=u(t)−u(t− 1 ).

2.5. p-n diode—MATLAB

The voltage–current characterization of a p-n diode is given by (see Figure 2.21)

i(t)=Is(eqv(t)/kT− 1 )

wherei(t)andv(t)are the current and the voltage in the diode (in the direction indicated in the diode),Is

is the reversed saturation current, andkT/qis a constant.