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# c25 - 18.03 Class 25 Convolution[1 We learn about a system...

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Convolution [1] We learn about a system by studying it responses to various input signals. I claim that the weight function w(t) --- the solution to p(D)x = delta(t) with rest initial conditions --- contains complete data about the LTI operator p(D) (and so about the system it represents). In fact there is a formula which gives the system response (with rest initial conditions) to any input signal q(t) as a kind of "product" of w(t) with q(t) . Suppose phosphates from a farm run off fields into a lake, at a rate q(t) which varies with the seasons. For definiteness let's say q(t) = 1 + cos(bt) Once in the lake, the phosphate decays: it's carried out of the stream at a rate proportional to the amount in the lake: x' + ax = q(t) , x(0) = 0 The weight function for this system is w(t) = e^{-at} for t > 0 = 0 for t < 0 . This tells you how much of each pound is left after t units of time have elapsed. If c pounds go in at time tau, then w(t-tau) c is the amount left at time t > tau. Fix a time t . We'll think about how x(t) gets built up from the contributions made during the time between 0 and t . We'll need another letter to denote that changing time; tau. We'll replace the continuous input represented by q(t) by a discrete input. Divide time into very small intervals, Delta tau (1 second maybe) . During the Delta tau time interval around time tau , the quantity of phosphate entering the lake is q(tau) Delta tau How much of that drop remains at time t? Well, the weight function tells you!

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c25 - 18.03 Class 25 Convolution[1 We learn about a system...

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