03-09-2012, 11:18 AM
The Hilbert transform
1The Hilbert.pdf (Size: 506.53 KB / Downloads: 638)
Abstract
The information about the Hilbert transform is often scattered in
books about signal processing. Their authors frequently use mathematical
formulas without explaining them thoroughly to the reader.
The purpose of this report is to make a more stringent presentation of
the Hilbert transform but still with the signal processing application
in mind.
Introduction
Signal processing is a fast growing area today and a desired e®ectiveness in utilization
of bandwidth and energy makes the progress even faster. Special signal
processors have been developed to make it possible to implement the theoretical
knowledge in an e±cient way. Signal processors are nowadays frequently used in
equipment for radio, transportation, medicine and production etc.
Mathematical motivations for the Hilbert transform
In this chapter we motivate the Hilbert transform in three di®erent ways. First
we use the Cauchy integral in the complex plane and second we use the Fourier
transform in the frequency domain and third we look at the §¼=2 phase-shift
which is a basic property of the Hilbert transform.
Numerical calculations of the Hilbert transform
The purpose of this chapter is to study di®erent types of numerical calculation
methods for the Hilbert transform.
Continuous time
Numerical integration.
Numerical integration works ¯ne on smooth functions that decrease rapidly at
in¯nity. When we want to calculate the Hilbert transform by De¯nition 1.1 we
are facing some problems. In numerical integration we use ¯nite intervals and it
is therefore important to consider the integration region to control the calculation
error. This is the reason why a rapid decrease at in¯nity is an advantage. Another
problem is that the integrand in De¯nition 1.1 is in¯nite when the nominator vanishes.
However, by using more integration grid points in the numerical integration
close to this value we get a better approximation.
Hermite polynomials
The numerical integration is ine±cient when a function decreases in a slow rate
at in¯nity. It is sometimes better to use a series of orthogonal polynomials where
the function does not have to decrease rapid at in¯nity. In this section we use the
Hermite polynomials to calculate the Hilbert transform. First we need to take a
look at the de¯nition of the Hermite polynomials.
An application
In signal systems we often make use of modulated signals. One big problem when
we modulate a signal is that its spectrum mirrors itself on both sides of the carrier
frequency. Only one of these spectra, or sidebands, is needed to demodulate the
signal and therefore we have redundant information. There are di®erent techniques
to remove one of the sidebands and thereby create a single sideband signal
(SSB). The most simple one is to use a low-pass ¯lter with the middle of the
transitionband on the carrier frequency. This technique demands that we have a
narrow transitionband and a good suppression which is hard to obtain. Another
technique is to use the Hilbert transform where we remove one of the sidebands
by adding the signal to itself. In theory we will get a transitionband that is zero
and a suppression that is in¯nite.