Examples of Convolution 3.2

Graphical Examples Impulse Response of an Integrator circuit due to the unit impulse function Response of an integrator due to a unit step function Response of an Integrator circuit due to the unit ramp input Response of an integrator due to a rectangular pulse

Impulse Response of an Integrator =δ(t) 0 (τ)(τ) 0 (τ)(τ) interval of integration

Graphical Examples Impulse Response of an Integrator circuit due to the unit impulse function Response of an integrator due to a unit step function Response of an Integrator circuit due to the unit ramp input Response of an integrator due to a rectangular pulse

Response of an Integrator Due to a Unit Ramp Function =tu(t) Why is h(t)=u(t) ?

Response of an Integrator Due to a Unit Step Function =tu(t) Constant!

Graphical Illustration (t=-1) No area of intersection

Graphical Illustration (t=1)

Response of an Integrator Due to a Unit Step Function =tu(t) Unit step function is only 1 when the argument is greater than 0. It does not make sense to integrate all the way to infinity.

A system with rectangular impulse response δ(t) u(t) u(t-2)

Example (why not take advatage of linearity ? )

Focus on x 1 (t) δ(t)→h(t) δ(t+3)→h(t+3)

Understanding h(t-τ)

h(t-τ) when t=-1 h(τ) h(-1+τ) h(-1-τ)

Integration when t<0 h(t-τ) for t<0

h(t-τ) when t=0.5 h(τ) h(0.5+τ) h(0.5-τ)

Integration when 0<t<2 h(t-τ) for 0<t<2 (move to the right as t increases )

h(t-τ) when t=3.5 h(τ) h(3.5+τ)h(3.5-τ)

Integration when t>2 t t-2