|(One intermediate revision by the same user not shown)|
| || |
except possibly at a finite collection of values of <math>
/omega .</math> |+|
except possibly at a finite collection of values of <math> omega </math>
<math> F(\omega) </math> analytic.
|−|<!--T:44--> | |
|−|Where <math> F(\omega) </math> is analytic. | |
| || |
| || |
The property of a sequence such that there is zero energy before some finite starting time. Minimum-phase wavelets are causal but zero-phase wavelets are not.
Causality and the Fourier Transform
The issue of causality affects data both in the time domain and the frequency domain. This fact is apparent when considering the Fourier transform
of causal functions.
A common Fourier Transform convention for forward and inverse temporal transforms is, respectively,
Here the limits of imply that is causal and, therefore for .
Thus the inverse Fourier transform must yield a result that agrees with causality.
The inverse Fourier transform as a contour integral
If we interpret the inverse Fourier transform as a contour integration
and consider only problems where has poles or branch points that
exist on or near the real axis, then there is only one particular choice of integration contour that will
yield a causal result for
This is the case of the portion of parallel to the real axis passing over the poles.
To perform a contour integration requires that the integration contour be closed, and that Cauchy's Theorem and, for Fourier integrals, a result
known as Jordan's lemma be applied to show that contribution from the contour at infinity vanishes.
Closure in the upper half-plane of will yield the result of (because no pole is included inside
the contour) and corresponds to the case. Closure of the cuntour in the lower half plane of (where one or more poles
are included inside the contour) will yield a nonzero value of and will correspond to the values of for
To see why this is so, consider as a complex variable and rewrite the exponential in the inverse Fourier
transform definition as
The integral can only converge when the integrand is decaying, so (corresponding to closure in the upper half-plane of ) implies that and (corresponding to closure in the lower half-plane of ) implies that
"Figure 1: For the Fourier transform exponent sign conventions here, this is the integration contour that yields a causal result."
For this exponent sign convention, a contour that passes over the singularities of the integrand. Because there are no singularities
in the upper half plane of causality is identified exactly with analyticity in some half-plane of
which, in the case of these Fourier transform exponent sign conventions is the upper half-plane.
Causality and the Hilbert transform
We formally consider a causal function to be where is the Heaviside step function
We want to find the Fourier transform of which may be shown to be a frequency domain convolution
The Heaviside step is neither even nor odd but may be written in terms of the constant function and the
We need to compute the Fourier transform of
The first two integrals may be performed by elementary means (with the appropriate sign chosen for the imaginary part of to insure exponential decay of the integrand),
and may be combined based on the property of analytic continuation,
and the third follows from the definition of the Dictionary:Dirac function
Forming the frequency domain convolution
The first integral makes sense only as a Cauchy principal value integral, and the second may be performed by applying the sifting property of the delta function
Because is a complex-valued function we may write this function in terms of its real and imaginary parts
However, because is analytic in the upper half-plane of the Hilbert transform relations exist between
the real and imaginary parts of
except possibly at a finite collection of values of
where fails to be analytic.
The Kramers-Kronig Relations
Thus, the Fourier transform of a causal function may be written in terms of Hilbert transform pairs
This result is often called the Kramers-Kroenig relation.