In signal processing, a causal filter is a linear time-invariant (LTI) causal system. The word causal indicates that the filter output depends only on past and present inputs. A filter whose output depends partly or fully on future inputs is non-causal (also, acausal), 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 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 by defining the shortened impulse response as the product of the non-causal 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. A constant factor of is omitted for simplicity:

, then a moving average defined that way is non-causal (also called non-realizable. 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

In those terms, causality requires

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

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 Θ(t) is the Heaviside unit step function.

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

is a Hilbert transform 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:

References