# Basic question: Why is the output of a system the convolution between the impulse response and the input?

I forgot a very simple fact and I am now struggling to find reference that proves this basic property?

How would you prove that for a single in single out system, the system output is the impulse response convoluted with the input?

• Note that this is only true for linear time invariant systems. – MBaz Feb 11 '15 at 1:46
• This has been answered quite a few times on this site. Have a look at e.g. this and this answer. – Matt L. Feb 11 '15 at 9:21

Because it is the response of the system when an unit impulse (delta function) is applied to the input of the system. So basically, If you multiply this output impulse response of the system to each of the input samples and then add them all, you get the overall output of the system. And this is only true, If the system agrees with the linearity properties such as homogeneity, additivity and shift invariance. For more details and to understand the mathematics behind this, you can go through this link: http://www.dspguide.com/ch6.htm

I tried to put this as comment instead of answer, since its not detailed enough, however, due to my lack of reputation, I could not comment. Anyways, the link I have provided is very useful. Hope it helps!

• To understand the beauty and importance of convolution, just imagine - what if a systems impulse response is different for each incoming individual samples, then how predictable would the system be?. – PsychedGuy Feb 11 '15 at 9:35
• This is exactly the case for a linear but time-varying system, where the impulse response is a two-dimensional function. Note that this 2-D impulse response completely describes the system, i.e. such a system is actually completely predictable (as long as you have complete knowledge of the impulse response). – Matt L. Feb 11 '15 at 9:56
• Yeah,I agree that time varying systems are predictable, but still, at the end of the day the system has to be linear in nature right. – PsychedGuy Feb 11 '15 at 10:20