A function $\begin{array}{l}f * g\end{array}$ of real-number argument $\begin{array}{l}t\end{array}$ , compiled from the two functions $\begin{array}{l}f(t)\end{array}$ and $\begin{array}{l}g(t)\end{array}$ by specific integration:

(1)	$\begin{array}{l}\displaystyle (f * g) (t) = \int_{-\infty}^{\infty} f(t-\tau) \, g(\tau) \, d \tau\end{array}$

For functions $\begin{array}{l}f, \, g\end{array}$ supported on only $\begin{array}{l}[0, \infty)\end{array}$ (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:

(2)	$\begin{array}{l}\displaystyle (f * g) (t) = \int_0^t f(t-\tau) \, g(\tau) \, d \tau\end{array}$

Properties

(3)	$\begin{array}{l}\displaystyle \int_0^t f(t-\tau) \, g(\tau) \, d \tau = \int_0^t f(\tau) \, g(t-\tau) \, d \tau\end{array}$

(4)	$\begin{array}{l}\displaystyle \int_0^t f(t-\tau) \, \dot h(\tau) d\tau = f(0) h(t) - f(t) h(0) - \int_0^t \dot f(t-\tau) \, h(\tau) \, d\tau\end{array}$

where $\begin{array}{l}\dot g()\end{array}$ means derivative by the whole argument.

Discrete form

For the functions $\begin{array}{l}\{ f_n = f(t_n)\}, \ \{ g_n = g(t_n)\}, \ n=1..N\end{array}$ defined over the discrete time grid $\begin{array}{l}\{ t_n \}, n = 1..N\end{array}$ the convolution equation is taking the discrete form:

(5)	$\begin{array}{l}\displaystyle (f * g) (t_n) = \sum_{m=0}^n f(t_n-\tau_m) \, g(\tau_m) \, (t_n - \tau_m)\end{array}$

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Properties

Discrete form

See also

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Convolution @math

Properties

Discrete form

See also