# Behaviour of System by Impulse Response

In a book by Alan Oppenheim, it is given that for a LTI system, its characteristics are completely determined by its impulse response. But if impulse response of a system is basically an input output relation and impulse response can be different for the same system depending upon our choice of output, so how does it completely characterize the system? Isn't it contradictory? And is complete characterization of system also include stability of system?

• Hi: The output of an LTI system is the convolution of the impulse response and the input so, if you have the impulse response and the input, you have the output. So, the book should have said that its characteristics are completely determined by its impulse response and its input. – mark leeds Jun 18 at 6:13
• Isn't Complete Characteristics of system also include stability? and evaluating output from input and impulse response doesn't tell about stability of system (let's say if system is non controllable or non observable), so how we can say impulse response completely characterise the system? – user215805 Jun 18 at 6:25
• Hi: Sounds like, based on matt's answer, you are correct. you need the equivalent state space formulation in order to understand stability. – mark leeds Jun 18 at 15:49

The impulse response characterizes an LTI system completely, in the sense that the response to any input can be computed from the input and from the impulse response by convolution. That's a consequence of the system being linear and time-invariant.

If we define stability as bounded-input bounded-output (BIBO) stability then stability can be determined from the impulse response. For the system to be stable, the impulse response must be absolutely integrable.

Note that when describing systems by impulse responses, or, equivalently, frequency responses or transfer functions, we focus on the input-output relation. We do not consider the internal dynamical behavior of the system. A state-space model is needed if we not only want to describe the system's input-output behavior but also its internal behavior.

But if impulse response of a system is basically an input output relation and impulse response can be different for the same system depending upon our choice of output, so how does it completely characterize the system? Isn't it contradictory?

Because any input signal can be form using impulses. If you know how the system may response to one impulse then you know how the system will response to many impulses, only true for LTI systems of course.

And is complete characterization of system also include stability of system?