Editing Hilbert transform (section)

===Causality===
The function <math>h(t) = 1/(\pi t)</math> presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0):
* Its duration is infinite (technically ''infinite [[support (mathematics)|support]]'').  Finite-length ''[[Window function|windowing]]'' reduces the effective frequency range of the transform; shorter windows result in greater losses at low and high frequencies. See also [[quadrature filter]].
* It is a [[causal filter|non-causal filter]].  So a delayed version, <math>h(t-\tau),</math> is required.  The corresponding output is subsequently delayed by <math>\tau.</math>  When creating the imaginary part of an [[analytic signal]], the source (real part) must also be delayed by <math>\tau</math>.