Causal filter - Alchetron, The Free Social Encyclopedia

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.

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 ) = ∫ x − 1 x + 1 s ( τ ) d τ = ∫ − 1 + 1 s ( x + τ ) d τ

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 ) = ∫ − 2 0 s ( t + τ ) d τ = ∫ 0 + 2 s ( t − τ ) d τ

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 ) = ∫ − ∞ ∞ h ( τ ) s ( t − τ ) d τ .

In those terms, causality requires

f ( t ) = ∫ 0 ∞ h ( τ ) s ( t − τ ) d τ

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 ) 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 Θ ( t ) ⋅ 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 ( ω ) = ( δ ( ω ) − i π ω ) ∗ G ( ω ) = G ( ω ) − i ⋅ G ^ ( ω )

where G ^ ( ω ) is a Hilbert transform done in the frequency domain (rather than the time domain). The sign of G ^ ( ω ) 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:

H ^ ( ω ) = i H ( ω )

References

Causal filter Wikipedia

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Contents

Example

Characterization of causal filters in the frequency domain

References