Convolution with a Rectangular Pulse

time

t

-0.99

pulse width

T

0

1

The output signal

y(t)

of an LTI (linear time-invariant) system with the impulse response

h(t)

is given by the convolution of the input signal

x(t)

with the impulse response of the system. Convolution is defined as

y(t)=x(t)⋆h(t)=

∞

∫

-∞

x(τ)h(t-τ)dτ

. In this example, the input

x(t)

is a rectangular pulse of width

T

0

and

h(t)=

1

RC

exp(-t/RC)

, which is the impulse response of an RC low-pass filter.

The upper figure shows

x(t)

(red) and

h(t-τ)

(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

t

. This value is marked as a blue point in the lower figure.