Causal filter

In signal processing, a causal filter is a linear and time-invariant causal system. The word "causal" indicates that the filter output depends only on past and present inputs. A filter whose output also depends on future inputs is acausal. A filter whose output depends "only" on future inputs is anti-causal. Systems (including filters) that are "realizable" (i.e. that operate in real time) must be causal because such systems cannot act on a future input. In effect that means the output sample that best represents the input at time $t,$ comes out slightly later. A common design practice is to create a realizable filter by shortening and/or time-shifting a non-causal impulse response. If shortening is necessary, it is often accomplished as the product of the impulse-response with a window function.

Example

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

:$f(x)\; =\; int\_\{x-1\}^\{x+1\}\; s(\; au),\; d\; au\; =\; int\_\{-1\}^\{+1\}\; s(x\; +\; au)\; ,d\; au,$

where "x" could represent a spatial coordinate, as in image processing. But if $x,$ represents time $(t),$, then a moving average defined that way is non-causal (also called "non-realizable"), because $f(t),$ depends on future inputs, such as $s(t+1),$. A realizable output is

:$f(t-1)\; =\; int\_\{-2\}^\{0\}\; s(t\; +\; au),\; d\; au\; =\; int\_\{0\}^\{+2\}\; s(t\; -\; au)\; ,\; d\; au,$

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

:$f(t)\; =\; (h*s)(t)\; =\; int\_\{-infty\}^\{infty\}\; h(\; au)\; s(t\; -\; au),\; d\; au.\; ,$

In those terms, causality requires

:$f(t)\; =\; int\_\{0\}^\{infty\}\; h(\; au)\; s(t\; -\; au),\; d\; au$

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

Characterization of causal filters in the frequency domain

Let "h"("t") be a causal filter with corresponding Fourier transform "H"(&omega;). Define the function

:$g(t)\; =\; \{h(t)\; +\; h^\{*\}(-t)\; over\; 2\}$

which is non-causal. On the other hand, "g"("t") is Hermitian and, consequently, its Fourier transform "G"(&omega;) is real-valued. We now have the following relation

:$h(t)\; =\; 2,\; operatorname\{step\}(t)\; cdot\; g(t),$

where step("t") is the unit step function.

This means that the Fourier transforms of "h"("t") and "g"("t") are related as follows

:$H(omega)\; =\; left(delta(omega)\; -\; \{i\; over\; pi\; omega\}\; ight)\; *\; G(omega)\; =G(omega)\; -\; icdot\; widehat\; G(omega)\; ,$

where $widehat\; G(omega),$ is a Hilbert transform done in the frequency domain (rather than the time domain).