Causal filter

 ( Signal Processing ) 

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 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 BIBO stability, anti-causal filter whose inverse is also stable and anti-causal.

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Example The following definition is a sliding or moving average of input data $s(x)\backslash ,$. A constant factor of is omitted for simplicity:

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

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

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

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.

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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\, \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 $\backslash widehat\; G(\backslash omega)\backslash ,$ is a Hilbert transform done in the frequency domain (rather than the time domain). The sign of $\backslash widehat\; G(\backslash omega)\backslash ,$ 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)

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