# Why is impulse response of LTI system important

Why is the impulse response for a LTI system also called its system response? What is special about the output when the input is impulse signal? I understand that impulse signal contains all the frequencies from zero to infinity and also the response for any input is convolution with the impulse response but how is the system nature or system parameters concluded by observing the response of the system due to impulse input.

• The Laplace transform of the unit impulse is 1. Therefore a LTI system's response to a unit impulse is its transfer function.
– Chu
Jun 13, 2016 at 12:58

What is special about the output when the input is impulse signal?

How is the system nature or system parameters concluded by observing the response of the system due to impulse input?

The Fourier transform of the impulse response is the transfer function of the system, i.e. a function that outlines the behavior of the system at all frequencies. Since you can, in practice, build any signal in time as a sum of sinusoidal functions (also called informally frequencies, although it is not rigorous), if you know the transfer function then you can predict the behavior of the system for any input signal.

The response of a system to ANY input can be calculated by the time convolution or the frequency product of the response of the system to the impulse signal and the input signal.

https://en.wikipedia.org/wiki/LTI_system_theory

In reality an impulse is rarely used (in reality is impossible to make an impulse, buy you can approximate). One type of commonly used method is the analysis of step or ramp response.

• Sorry, still not clear. Jun 13, 2016 at 5:01

If you test the system with a complicated input, then the output will be the response of the system, convolved with something complicated. It will be difficult to figure out what is due to your system, and what is due to the complicated input.

On the other hand, if you put in a simple input, it will be easier to understand your system. There are several simple inputs to choose from.

One simple input would be a sine-wave source. However, this only excites a single frequency, so you would need many different test frequencies to explore the full range of behaviour of your system.

Another simple input is the impulse. This has a wide frequency range, so explores the behaviour of your system at all frequencies. Any input signal can be constructed from delayed and scaled impulses, so it is easy to work out what your system response will be to more complicated input signals.

The frequency response is simply the impulse response transformed into the frequency domain.