Divide input x(τ) into pulses. The system response at t is then determined by x(τ) weighted by h(t- τ) (i. x(τ) h(t- τ)) for the shaded pulse, PLUS the contribution from all the previous pulses of x(τ). The summation of all these weighted inputs is the convolution integral.

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function ( ). The term convolution refers to both the resulting function and to the process of computing it.

Table 1: Convolution Table No f1(t) f2(t) f1(t) f2(t) = f2(t) f1(t) 1 f(t) (t T) f(t T) 2 e tu(t) u(t) 1 e t u(t) 3 u(t) u(t) tu(t) 4 e 1tu(t) e 2tu(t) e 1t e 2t 1 2 u(t); 1 ̸= 2 5 e tu(t) e tu(t) te tu(t) 6 te tu(t) e tu(t) 1 2 t2e tu(t) 7 tnu(t) e tu(t) n!e t n+1u(t) ∑n j=0 n!tn j j+1(n j)! u(t) 8 tmu(t) tnu(t) m!n! (m+n+1)! tm+n+1u(t) 9 ...

Table 1: Convolution Table (1 (t) is a unit-step function)No f1(t) f2(t) f1(t)∗f2(t) = f2(t)∗f1(t) 1 f(t) δ(t−T) f(t−T) 2 eλt1 (t) 1 (t) 1−eλt −λ 1 (t)3 1 (t) 1 (t) t1 (t)4 eλ1t1 (t) eλ2t1 (t) eλ1t −eλ2t λ1 −λ2 1 (t), λ1 ̸= λ25 eλt1 (t) eλt1 (t) teλt1 (t) 6 teλt1 (t) eλt1 (t) 1 2 t2eλt1 (t) 7 tn1 (t) eλt1 (t) n!eλt λn+1 1 (t)−∑n j=0 n!tn−j λj+1(n−j)!

Q: How do I tell MATLAB where to plot the convolution? A: If the time of the first element of is 0 and the time of the first element of h is h0 then the time of the first element of is 0 + h0.

CONVOLUTION As Applied to Linear Time-Invariant Systems The convolution integral occurs frequently in the physical sciences. The convolution integral of two functions f1(t) and f2(t) is denoted symbolically by f1(t) * f2(t). ∫∞ −∞ f1 (t)* f2 (t) ≡ f1 (τ) f2 (t− τ)dτ

For n < 0 h[n-i]x(i) = 0 ∀i ⇒ =[ ] 0 y n for n <0 − h i x i [ ] [ ] i For n = 0 ⇒y[0] = 1-3 -2 -1 1 2 3 4 5 6 7 8 9 x i [ ] i Notice that for n = 0, n = 1 ...

The last expression is a convolution. In fact, since w(t) = e kt, we have that the last integral is just wf(t). That is, we have found the formula x(t) = wf(t). This is the simplest case of Green’s formula. It gives the response as the convolution of the input and weight functions. We will see that this holds for all LTI systems.

Determine graphically y(t) = x(t)*h(t) for x(t) = e-tu(t) and h(t) = e-2tu(t). linearly from t=0 to zero at t=1. Divide input x(τ) into pulses. previous pulses of x(τ). inputs is the convolution integral. Let us put everything together, using our RLC circuit as an example. x(t) = 10e−3t u(t), y(0) = …

Properties of Convolution Transference: between Input & Output Suppose x[n] * h[n] = y[n] If L is a linear system, x1[n] = L{x[n]}, y1[n] = L{y[n]} Then x1[n] ∗ h[n]= y1[n]