📄 an introduction to wavelets fourier analysis.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!-- saved from url=(0049)http://www.amara.com/IEEEwave/IW_fourier_ana.html -->
<!-- Amara Graps' IEEE Paper: An Intro to Wavelets --><HTML><HEAD><TITLE>An Introduction to Wavelets: Fourier Analysis</TITLE>
<META http-equiv=Content-Type content="text/html; charset=gb2312"><LINK rev=made
href="mailto:amara@amara.com"><!-- Make Background color GhostWhite and all links color DarkBlue-->
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY link=#00008b bgColor=#f8f8ff>
<BLOCKQUOTE>
<HR align=center noShade SIZE=2>
<P>
<H2><FONT size=8>F</FONT>ourier <FONT size=6>A</FONT>nalysis</H2>
<P>
<HR align=center noShade SIZE=2>
<P>Fourier's representation of functions as a superposition of sines and
cosines has become ubiquitous for both the analytic and numerical solution of
differential equations and for the analysis and treatment of communication
signals. Fourier and wavelet analysis have some very strong links.
<H2>Fourier Transforms</H2>The Fourier transform's utility lies in its ability
to analyze a signal in the time domain for its frequency content. The
transform works by first translating a function in the time domain into a
function in the frequency domain. The signal can then be analyzed for its
frequency content because the Fourier coefficients of the transformed function
represent the contribution of each sine and cosine function at each frequency.
An inverse Fourier transform does just what you'd expect, transform data from
the frequency domain into the time domain.
<H3>Discrete Fourier Transforms</H3>The discrete Fourier transform (DFT)
estimates the Fourier transform of a function from a finite number of its
sampled points. The sampled points are supposed to be typical of what the
signal looks like at all other times.
<P>The DFT has symmetry properties almost exactly the same as the continuous
Fourier transform. In addition, the formula for the inverse discrete Fourier
transform is easily calculated using the one for the discrete Fourier
transform because the two formulas are almost identical.
<H3>Windowed Fourier Transforms</H3>If <EM>f(t)</EM> is a nonperiodic signal,
the summation of the periodic functions, sine and cosine, does not accurately
represent the signal. You could artificially extend the signal to make it
periodic but it would require additional continuity at the endpoints. The
windowed Fourier transform (WFT) is one solution to the problem of better
representing the nonperiodic signal. The WFT can be used to give information
about signals simultaneously in the time domain and in the frequency domain.
<P>With the WFT, the input signal <EM>f(t)</EM> is chopped up into sections,
and each section is analyzed for its frequency content separately. If the
signal has sharp transitions, we window the input data so that the sections
converge to zero at the endpoints <A
href="http://www.amara.com/IEEEwave/IW_ref.html#three">(3)</A>. This windowing
is accomplished via a weight function that places less emphasis near the
interval's endpoints than in the middle. The effect of the window is to
localize the signal in time.
<H3>Fast Fourier Transforms</H3>To approximate a function by samples, and to
approximate the Fourier integral by the discrete Fourier transform, requires
applying a matrix whose order is the number sample points <EM>n.</EM> Since
multiplying an <IMG alt="n x n"
src="An Introduction to Wavelets Fourier Analysis.files/IW_eqnxn.gif"
align=top> matrix by a vector costs on the order of <IMG alt=n^2
src="An Introduction to Wavelets Fourier Analysis.files/IW_eqn2.gif"
align=top> arithmetic operations, the problem gets quickly worse as the number
of sample points increases. However, if the samples are uniformly spaced, then
the Fourier matrix can be factored into a product of just a few sparse
matrices, and the resulting factors can be applied to a vector in a total of
order <IMG alt="n log n"
src="An Introduction to Wavelets Fourier Analysis.files/IW_eqnlogn.gif"
align=top> arithmetic operations. This is the so-called <EM>fast Fourier
transform</EM> or FFT <A
href="http://www.amara.com/IEEEwave/IW_ref.html#four">(4)</A>.
<P>
<HR align=center noShade SIZE=2>
<P><B><A href="http://www.amara.com/agraps.html">[Home]</A> <A
href="http://www.amara.com/current/wavelet.html">[Wavelet Page]</A> <A
href="http://www.amara.com/IEEEwave/IEEEwavelet.html#contents">[Contents]</A>
<A href="http://www.amara.com/IEEEwave/IW_basis.html">[Previous]</A> <A
href="http://www.amara.com/IEEEwave/IW_wave_vs_four.html">[Next]</A> </B>
<P>
<HR align=center noShade SIZE=2>
<P>
<H5>You may <A
href="http://www.amara.com/ftpstuff/IEEEwavelet.ps.gz">download</A> this
paper: "Introduction to Wavelets" </H5>
<P>
<HR align=center noShade SIZE=2>
<CENTER>
<TABLE>
<TBODY>
<TR><!-- Miscellaneous Contact Information -->
<TD><BASEFONT size=2>
<ADDRESS>Last Modified by <A href="mailto:amara@amara.com">Amara
Graps</A> on 8 October 1997.<BR>© Copyright Amara Graps, 1995-1997.
</ADDRESS></BASEFONT></TD></TR></TBODY></TABLE></CENTER></BLOCKQUOTE></BODY></HTML>
⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -