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 ${\displaystyle t,}$ 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 ${\displaystyle s(x)\,}$. A constant factor of 1⁄2 is omitted for simplicity:

${\displaystyle f(x)=\int _{x-1}^{x+1}s(\tau )\,d\tau \ =\int _{-1}^{+1}s(x+\tau )\,d\tau \,}$

where ${\displaystyle x}$ could represent a spatial coordinate, as in image processing. But if ${\displaystyle x}$ represents time ${\displaystyle (t)\,}$, then a moving average defined that way is non-causal (also called non-realizable), because ${\displaystyle f(t)\,}$ depends on future inputs, such as ${\displaystyle s(t+1)\,}$. A realizable output is

${\displaystyle f(t-1)=\int _{-2}^{0}s(t+\tau )\,d\tau =\int _{0}^{+2}s(t-\tau )\,d\tau \,}$

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

${\displaystyle f(t)=(h*s)(t)=\int _{-\infty }^{\infty }h(\tau )s(t-\tau )\,d\tau .\,}$

In those terms, causality requires

${\displaystyle f(t)=\int _{0}^{\infty }h(\tau )s(t-\tau )\,d\tau }$

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

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

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

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

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

${\displaystyle H(\omega )=\left(\delta (\omega )-{i \over \pi \omega }\right)*G(\omega )=G(\omega )-i\cdot {\widehat {G}}(\omega )\,}$

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

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

References

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