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Causal filter

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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 , {\displaystyle 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 sliding or moving average of input data s ( x ) {\displaystyle 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 {\displaystyle f(x)=\int _{x-1}^{x+1}s(\tau )\,d\tau \ =\int _{-1}^{+1}s(x+\tau )\,d\tau \,}

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

f ( t − 1 ) = ∫ − 2 0 s ( t + τ ) d τ = ∫ 0 + 2 s ( t − τ ) d τ {\displaystyle 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 ) = ∫ − ∞ ∞ h ( τ ) s ( t − τ ) d τ . {\displaystyle f(t)=(h*s)(t)=\int _{-\infty }^{\infty }h(\tau )s(t-\tau )\,d\tau .\,}

In those terms, causality requires

f ( t ) = ∫ 0 ∞ h ( τ ) s ( t − τ ) d τ {\displaystyle 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 ) 2 {\displaystyle 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 Θ ( t ) ⋅ g ( t ) {\displaystyle 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 ( ω ) = ( δ ( ω ) − i π ω ) ∗ G ( ω ) = G ( ω ) − i ⋅ G ^ ( ω ) {\displaystyle H(\omega )=\left(\delta (\omega )-{i \over \pi \omega }\right)*G(\omega )=G(\omega )-i\cdot {\widehat {G}}(\omega )\,}

where G ^ ( ω ) {\displaystyle {\widehat {G}}(\omega )\,} is a Hilbert transform done in the frequency domain (rather than the time domain). The sign of G ^ ( ω ) {\displaystyle {\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:

H ^ ( ω ) = i H ( ω ) {\displaystyle {\widehat {H}}(\omega )=iH(\omega )}

References

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