文档介绍：W236: THE DISCRETE HILBERT TRANSFORM: A BRIEF TUTORIAL
Introduction
OK. You've read about the discrete Hilbert transform in the DSP literature. You've
plodded through the mathematical descriptions of analytic functions, with the constraints on
their z-transforms in their regions of convergence. It's likely that you've also encountered
the Cauchy integral theorem used in the definition of the Hilbert transform. Unfortunately,
the author(s) did not supply a "Hilbert transform psychic hotline" phone number you can call
to help make sense of it all. That's where this DSP Workshop es in.
Here we'll gently introduce the Hilbert transform (HT) from a practical standpoint, and
explain some of the mathematics behind its description. In addition to providing some of
the algebraic steps that are "missing" from many textbooks, we'll illustrate the time and
frequency domain characteristics of the transform with an emphasis on the physical
meaning of the quadrature (complex) signals associated with HT applications. With that
said, let's get started.
Hilbert Transform Definition
The HT is mathematical process performed on a real signal xr(t) yielding a new real
signal xht(t), as shown in Figure 1.
x (t)
x r (t) Hilbert transform, ht
h(t), H(w)
X r ( w) X ht (w)
Figure 1. The notation used to define the Hilbert transform.
o
Our goal here is to ensure that xht(t) is a 90 phase-shifted version of xr(t). So, before
we carry on, let's make sure we understand the notation used in Figure 1. The above
variables are defined as:
xr(t) - a real continuous time-domain input signal,
h(t) - the time impulse response of a Hilbert transformer,
xht(t) - the HT of xr(t), (xht(t)is also a real time-domain signal),
w
Xr( ) - the Fourier transform of input xr(t),
H(w) - the frequency response (complex) of a Hilbert transformer,
w
Xht( ) - the Fourier transform of output xht(t),
w - continuous frequency measured in radians/second, and
t - continuous