In signal processing, a causal filter is a linear and time-invariantcausal 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 non-causal, whereas 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 for digital 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 a window function.

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

Example

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

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

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(ω). 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(ω) is real-valued. We now have the following relation

where ${\widehat {G))(\omega )\,$ is a Hilbert transform done in the frequency domain (rather than the time domain). The sign of ${\widehat {G))(\omega )\,$ 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:

${\widehat {H))(\omega )=iH(\omega )$

References

Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (September 2007), Numerical Recipes (3rd ed.), Cambridge University Press, p. 767, ISBN9780521880688