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 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 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. The following definition is a sliding or moving average of input data . A constant factor of is omitted for simplicity: where 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 In those terms, causality requires and general equality of these two expressions requires h(t) = 0 for all t < 0. 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 where is a Hilbert transform done in the frequency domain (rather than the time domain).