Editing Causal system (section)

==Examples==
The following examples are for systems with an input <math>x</math> and output <math>y</math>.

=== Examples of causal systems ===
* Memoryless system
::<math>y \left( t \right) = 1 - x \left( t \right) \cos \left( \omega t \right)</math>

* Memory-enabled system
::<math>y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)</math>

* Autoregressive filter
::<math>y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau</math>

=== Examples of non-causal (acausal) systems ===
*
::<math>y(t)=\int_{-\infty}^\infty \sin (t+\tau) x(\tau)\,d\tau</math>

* Central moving average
::<math>y_n=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}</math>

=== Examples of anti-causal systems ===
*
::<math>y(t) =\int _0^\infty x (t+\tau)\,d\tau</math>

*Look-ahead
::<math>y_n=x_{n+1}</math>

=== Additional examples of causal systems ===
* Linear Time-Invariant (LTI) System
::<math>y(t) = \int_{-\infty}^{t} x(\tau) h(t - \tau) \, d\tau</math>

* Moving average filter
::<math>y[n] = \frac{1}{N} \sum_{k=0}^{N-1} x[n-k]</math>

=== Additional examples of non-causal (acausal) systems ===
* Smoothing Filter
::<math>y(t) = \frac{1}{T} \int_{t - T/2}^{t + T/2} x(\tau) \, d\tau</math>

* Ideal low-pass filter
::<math>y(t) = \int_{-\infty}^{\infty} x(\tau) \mathrm{sinc}(t - \tau) \, d\tau</math>

=== Additional examples of anti-causal systems ===
* Future Input Dependence
::<math>y(t) = \int_{t}^{\infty} x(\tau) \, d\tau</math>