# historical context of impulse response-convolution

Hi All: I have zero background on the history of LTI systems so I was wondering when the concept of LTI and the fact that an LTI signal can be be completely described by its impulse response came about. I ask because, in econometrics, there are models called auto-regressive distributed lags (ARDLs). A specific ARDL called the koyck distributed lag came out in the early 1950's and was a pretty big deal in the econometrics community.

https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf ( koyck is on page 12 ).

But, now that I understand LTI, impulse response and convolution, my guess is that the explanation and theory behind the Koyck ADl could have been much more easily explained and understood using the LTI framework ( by using the impulse response and convolution). Yet, as far as I am aware, an explanation of the Koyck distributed lag using this approach does not exist. That's very odd to me since the LTI framework makes the understanding of the Koyck ADL so much clearer and easier to understand. Thanks for any historical context of the LTI concept. If it's after the 1950's, then it's understandable that Koyck didn't know about it yet. If it was well known before the 1950's, then my guess is that he was unaware of it. Otherwise, he would have explained his Koyck model using that approach.

• Heaviside. He was the first to use a step function to predict current in the context of passive circuit theory, – user28715 Jan 26 '18 at 20:44

Stanley: Thanks for your input. Your answer led me to the "brilliant" idea of googling myself. Based on the link below, I think one can define the birth of convolution as sometime between 1760-1900 depending on how one defines birth. https://pulse.embs.org/january-2015/history-convolution-operation/.

Either way, it seems that Koyck was unaware of it since he came up with his model in the 1950's. He would have had a much easier time using the DSP framework in his derivation but he came up with the model without impulse response or convolution which is quite impressive in itself. I'm gonna check this answer but your mention of Mr. Heavside is appreciated.