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 $t,$ 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 is omitted for simplicity:

:$f(x) = \int_^ s(\tau)\, d\tau\ = \int_^ s(x + \tau) \,d\tau\,$

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

:$f(t-1) = \int_^ s(t + \tau)\, d\tau = \int_^ s(t - \tau) \, d\tau\,$

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

Any linear filter (such as a moving average) can be characterized by a function ''h''(''t'') called its impulse response. Its output is the convolution

:$f(t) = (h*s)(t) = \int_^ h(\tau) s(t - \tau)\, d\tau. \,$

In those terms, causality requires

:$f(t) = \int_^ h(\tau) s(t - \tau)\, d\tau$

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

:$g(t) =$

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

:$h(t) = 2\, \Theta(t) \cdot g(t)\,$

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

:$H(\omega) = \left(\delta(\omega) - \right) * G(\omega) = G(\omega) - i\cdot \widehat G(\omega) \,$

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:

:$\widehat H(\omega) = i H(\omega)$

References

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{{DEFAULTSORT:Causal Filter
Signal processing
Filter theory