# Why impulse response

Why do people calculate impulse response? Generally systems are designed after modelling the possible input signals and calculating the appropriate filters for the desired outputs. Where does impulse response calculation become important?

• And how are you going to apply these "appropriate filters" if not by convolving their impulse response with input signal in general? Hence your system is described by... impulse response, isn't it? – jojek Jul 2 '14 at 6:38
• probably the simplest answer to your question is that you bang your system with an impulse and let it ring. you record the ringing response to the impulse and that tells you how the system will respond to any other input. this can be true of any input that is sufficiently broadbanded, but the dirac impulse is simple, completely flat in its spectrum, and has concentrated all of its energy at $t=0$. and it lends itself to the convolution integral better than, say, a unit step. – robert bristow-johnson Jul 2 '14 at 14:20

Having the impulse response allows one to simplify the input of a sampled linear time-invariant system down to one single unit-magnitude input sample.

It is usually fairly simple to build any time domain input from a single unit input (with a linear combination thereof), and thus, given the impulse response for that single input, the output for any time-domain-described input for that system. Other basis vectors are possible, but may require more or less work (computational effort) for a given type of description of the input.

The impulse response of an LTI system fully characterizes the system. It is unique to the LTI system.

Just to give you some examples:

• The impulse response lets you make statements about the stability of the system
• The impulse response convoluted with an input will give you the output of a system. This makes it especially easy in cases of DT signals.
• When cascading systems (i.e. constructing a series of systems) the complete impulse response is simple the convolution of all of the system impulse responses if the systems are LTI.