impulse response

The unit impulse response of a system is its output when presented with a unit impulse (Dirac delta () in the continuous case and unit sample sequence in the discrete case).

For linear time-invariant (LTI) systems, the impulse response completely characterizes the system’s behavior. If we know , we can determine the system’s output for any arbitrary input as a convolution:

Intuition of Continuous Impulse Response

We can conceptualize any input signal as a sum of infinitesimally narrow rectangular pulses. As the width of these pulses approaches zero, they effectively become Dirac delta function (impulses). ¹

If a system is linear, we can get each individual response from those impulses, and sum those responses to get the overall response. This is exactly what the convolution operation accomplishes.

Related

• transfer function - Laplace transform of the impulse response (continuous), Z-transform (discrete)
• frequency response - Fourier transform of the impulse response

Footnotes

1. Linear Systems and Signals, 3rd Edition, 2.3 ↩