# Convolution with a Rectangular Pulse

Convolution with a Rectangular Pulse

The output signal of an LTI (linear time-invariant) system with the impulse response is given by the convolution of the input signal with the impulse response of the system. Convolution is defined as . In this example, the input is a rectangular pulse of width and , which is the impulse response of an RC low-pass filter.

y(t)

h(t)

x(t)

y(t)=x(t)⋆h(t)=x(τ)h(t-τ)dτ

∞

∫

-∞

x(t)

T

0

h(t)=exp(-t/RC)

1

RC

The upper figure shows (red) and (blue). The gray area in the figure is the area under the product of the two functions. It equals the value of the convolution integral and the value of the output signal at time . This value is marked as a blue point in the lower figure.

x(t)

h(t-τ)

t