Causal Filter - Example

Example

The following definition is a moving (or "sliding") average of input data . A constant factor of 1/2 is omitted for simplicity:

where x could represent a spatial coordinate, as in image processing. But if represents time, then a moving average defined that way is non-causal (also called non-realizable), because depends on future inputs, such as . A realizable output is

which is a delayed version of the non-realizable output.

Any linear filter (such as a moving average) can be characterized by a function h(t) called its impulse response. Its output is the convolution

$f(t) = (h*s)(t) = \int_{-\infty}^{\infty} h(\tau) s(t - \tau)\, d\tau. \,$

In those terms, causality requires

$f(t) = \int_{0}^{\infty} h(\tau) s(t - \tau)\, d\tau$

and general equality of these two expressions requires h(t) = 0 for all t < 0.

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