Causal filter

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In signal processing, a causal filter is one whose output depends only on past and present inputs. A filter whose output also depends on future inputs is non-causal. Filters that operate in real time are causal. In effect that means the output sample that best represents the input at time t\, comes out slightly later. A common design practice 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.

[edit] Example

The following definition is a moving (or "sliding") average of input data s(x)\,. A constant factor of 1/2 is omitted for simplicity:

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

f(t) = (h*s)(t) = \int_{-\infty}^{\infty} h(\tau) s(t - \tau)\, d\tau. \,

In those terms, causality requires

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.

[edit] 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) = {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

h(t) = 2\, \operatorname{step}(t) \cdot g(t)\,

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

H(\omega) = \left(\delta(\omega) - {i \over \pi \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).