Insignal processing, acausal filter is alinear and time-invariantcausal system. The wordcausal indicates that the filter output depends only on past and present inputs. Afilter whose output also depends on future inputs isnon-causal, whereas a filter whose output dependsonly on future inputs isanti-causal. Systems (including filters) that arerealizable (i.e. that operate inreal 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${\displaystyle t,}$ comes out slightly later. A common design practice fordigital filters 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 awindow function.

An example of an anti-causal filter is amaximum phase filter, which can be defined as astable, anti-causal filter whose inverse is also stable and anti-causal.

The following definition is asliding ormoving average of input data${\displaystyle s(x)}$. A constant factor of1⁄2 is omitted for simplicity:

${\displaystyle f(x)={\int }_{x-1}^{x+1}s(\tau )d\tau ={\int }_{-1}^{+1}s(x+\tau )d\tau }$

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

${\displaystyle f(t-1)={\int }_{-2}^{0}s(t+\tau )d\tau ={\int }_0^{+2}s(t-\tau )d\tau }$

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

Any linear filter (such as a moving average) can be characterized by a functionh(t) called itsimpulse response. Its output is theconvolution

${\displaystyle f(t)=(h\ast s)(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(\tau )s(t-\tau )d\tau .}$

In those terms, causality requires

${\displaystyle f(t)={\int }_0^{\mathrm{\infty }}h(\tau )s(t-\tau )d\tau }$

and general equality of these two expressions requiresh(t) = 0 for allt < 0.

Leth(t) be a causal filter with corresponding Fourier transformH(ω). Define the function

${\displaystyle g(t)=\frac{h(t)+h^{\ast }(-t)}{2}}$

which is non-causal. On the other hand,g(t) isHermitian and, consequently, its Fourier transformG(ω) is real-valued. We now have the following relation

${\displaystyle h(t)=2\mathrm{\Theta }(t)\cdot g(t)}$

where Θ(t) is theHeaviside unit step function.

This means that the Fourier transforms ofh(t) andg(t) are related as follows

${\displaystyle H(\omega )=\left(\delta (\omega )-\frac{i}{\pi \omega }\right)\ast G(\omega )=G(\omega )-i\cdot \hat{G}(\omega )}$

where${\displaystyle \hat{G}(\omega )}$ is aHilbert transform done in the frequency domain (rather than the time domain). The sign of${\displaystyle \hat{G}(\omega )}$ may depend on the definition of the Fourier Transform.

Taking the Hilbert transform of the above equation yields this relation between "H" and its Hilbert transform:

${\displaystyle \hat{H}(\omega )=iH(\omega )}$

• Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (September 2007),Numerical Recipes (3rd ed.), Cambridge University Press, p. 767,ISBN 9780521880688
• Rowell (January 2009),Determining a System's Causality from its Frequency Response(PDF), MIT OpenCourseWare

• Signal processing
• Filter theory