Signals and Systems - Electrical Engineering

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

The relation between the impulse response and the unit-step and the ramp responses can be gen-

eralized for any system as the impulse responseh(t), the unit-step responses(t), and the ramp

responseρ(t)are related by

h(t)=

{

ds(t)/dt

d^2 ρ(t)/dt^2

(2.19)

This can be shown by computing firsts(t)(the output due to a unit-step input):

s(t)=

∫∞

−∞

u(t−τ)h(τ)dτ=

∫t

−∞

h(τ)dτ

since

u(t−τ)=

{

1 τ≤t

0 τ >t

The derivative ofs(t)ish(t).

Similarly, the ramp responseρ(t)of a LTI system, represented by the impulse responseh(t), is

given by

ρ(t)=

∫∞

−∞

h(τ)(t−τ)u(t−τ)dτ=

∫t

−∞

h(τ)(t−τ)dτ=t

∫t

−∞

h(τ)dτ−

∫t

−∞

h(τ)τdτ

and its derivative is

dρ(t)

dt

=

∫t

−∞

h(τ)dτ+th(t)

︸ ︷︷ ︸

d(t

∫t

−∞

h(τ)dτ)/dt

− th(t)

︸︷︷︸

d(

∫t

−∞

h(τ)τdτ)/dt

=

∫t

−∞

h(τ)dτ

so that the second derivative ofρ(t)ish(t)—that is,

d^2 ρ(t)

dt^2

=

d

dt





∫t

−∞

h(τ)dτ



=h(t)

Using the Laplace transform, one is able to obtain the above relations in a much simpler way. n

nExample 2.10

The outputy(t)of an analog averager is given by

y(t)=

1

T

∫t

t−T

x(τ)dτ