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.

File:Illustration of causal and non-causal filters.svg

Each component of the causal filter output begins when its stimulus begins. The outputs of the non-causal filter begin before the stimulus begins.

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:

$$\begin{gathered} f(x)=\underset{x-1}{\overset{x+1}{\int }}s(\tau )d\tau \\ =\underset{-1}{\overset{+1}{\int }}s(x+\tau )d\tau \end{gathered}$$

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

$f(t-1)=\underset{-2}{\overset{0}{\int }}s(t+\tau )d\tau =\underset{0}{\overset{+2}{\int }}s(t-\tau )d\tau$

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

$f(t)=(h*s)(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(\tau )s(t-\tau )d\tau .$

In those terms, causality requires

$f(t)=\underset{0}{\overset{\mathrm{\infty }}{\int }}h(\tau )s(t-\tau )d\tau$

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

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

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

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

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

$H(\omega )=(\delta (\omega )-\frac{i}{\pi \omega })*G(\omega )=G(\omega )-i\cdot \hat{G}(\omega )$

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

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

References

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