In signal processing, a causal filter is a linear and timeinvariant 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 noncausal, whereas a filter whose output depends only on future inputs is anticausal. 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 timeshifting a noncausal impulse response. If shortening is necessary, it is often accomplished as the product of the impulseresponse with a window function.
An example of an anticausal filter is a maximum phase filter, which can be defined as a stable, anticausal filter whose inverse is also stable and anticausal.
Each component of the causal filter output begins when its stimulus begins. The outputs of the noncausal filter begin before the stimulus begins.
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_{x1}^{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 noncausal (also called nonrealizable), because f(t)\, depends on future inputs, such as s(t+1)\,. A realizable output is

f(t1) = \int_{2}^{0} s(t + \tau)\, d\tau = \int_{0}^{+2} s(t  \tau) \, d\tau\,
which is a delayed version of the nonrealizable 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.
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 noncausal. On the other hand, g(t) is Hermitian and, consequently, its Fourier transform G(ω) is realvalued. 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)  {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). 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

Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (September 2007),
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