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Causal filter

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Title: Causal filter  
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Subject: Filter design, Filter theory, Causal system, Pulse shaping, Digital filter
Collection: Filter Theory, Signal Processing
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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.

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 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.

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 \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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