Convolution

Compound influence of the input and the system's response to each of the input at the observation moment.

Model

$$ x = f \to \boxed{\text{System} = g} \to y $$

View I: System Response

$$ \begin{aligned} (f * g)(t) &= \int_{-\infty}^{t} f(\tau) g(t - \tau) d\tau \end{aligned} $$

• $ t $ : Observation Time
• $ \tau $ : Happening Time
• $ f $ : Impulse Function / Input Signal
• $ g $ : Response Function / System's Standard Response

The system's response to the input happening at $ \tau $ is $ g(t - \tau) $, since $ t - \tau $ time has elapsed at the observation moment. (Consider $ g(t) = e^{-t} $ as a valid diminishing system response example.)

View II: Joint Distribution

$X, Y$ are independent random variables. $$ \begin{aligned} Z &= X + Y \\ f_Z &= f_X * f_Y \\ f_Z(z) &= \int f_X(x) f_Y(z - x) dx \\ &= \int f_X(z-y) f_Y(y) dy \end{aligned} $$ The joint pdf if the joint effort of the individual pdfs.

by Jon