Week 9

LTI Circuits

Because [L]inear, [T]ime-[I]nvariant circuits possess these properties, then if

\[x(t) = a_1x_1(t-t_1)+a_2x_2(t-t_2)+\cdots\]

then it follows from its linearity that a system’s response to an input $x(t)$ is

\[y(t) = a_1y_1(t-t_1)+a_2y_2(t-t_2)+\cdots\]

Convolution

Convolution is a mathematical operation on two functions (e.g. $x(t)$ and $h(t)$) that produces a third function $y(t) = x(t) \mathord{*}h(t)$. Intuitively, convolution represents the amount of overlap between two functions.

We define convolution to be equal to the integral

\[y(t) = x(t) \mathord{*}h(t) = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\]

and, vice versa,

\[y(t) = h(t) \mathord{*}x(t) = \int_{-\infty}^{\infty}h(\tau)x(t-\tau) d\tau\]

Properties of Convolution

• Commutative: $ f\mathord{}g = g\mathord{}f $
• Associative
• Distributive