📄 an introduction to wavelets wavelet versus fourier transforms.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!-- saved from url=(0050)http://www.amara.com/IEEEwave/IW_wave_vs_four.html -->
<!-- Amara Graps' IEEE Paper: An Intro to Wavelets --><HTML><HEAD><TITLE>An Introduction to Wavelets: Wavelet versus Fourier Transforms</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>W</FONT>avelet vs <FONT size=6>F</FONT>ourier <FONT
size=6>T</FONT>ransforms</H2>
<P>
<HR align=center noShade SIZE=2>
<P>
<H3>Similarities between Fourier and Wavelet Transforms</H3>The fast Fourier
transform (FFT) and the discrete wavelet transform (DWT) are both linear
operations that generate a data structure that contains <IMG alt="log_2 n"
src="An Introduction to Wavelets Wavelet versus Fourier Transforms.files/IW_eqlog2n.gif"
align=top> segments of various lengths, usually filling and transforming it
into a different data vector of length <IMG alt=2^n
src="An Introduction to Wavelets Wavelet versus Fourier Transforms.files/IW_eq2n.gif"
align=top>.
<P>The mathematical properties of the matrices involved in the transforms are
similar as well. The inverse transform matrix for both the FFT and the DWT is
the transpose of the original. As a result, both transforms can be viewed as a
rotation in function space to a different domain. For the FFT, this new domain
contains basis functions that are sines and cosines. For the wavelet
transform, this new domain contains more complicated basis functions called
wavelets, mother wavelets, or analyzing wavelets.
<P>Both transforms have another similarity. The basis functions are localized
in frequency, making mathematical tools such as power spectra (how much power
is contained in a frequency interval) and scalegrams (to be defined later)
useful at picking out frequencies and calculating power distributions.
<H3>Dissimilarities between Fourier and Wavelet Transforms</H3>The most
interesting dissimilarity between these two kinds of transforms is that
individual wavelet functions are <EM>localized in space.</EM> Fourier sine and
cosine functions are not. This localization feature, along with wavelets'
localization of frequency, makes many functions and operators using wavelets
"sparse" when transformed into the wavelet domain. This sparseness, in turn,
results in a number of useful applications such as data compression, detecting
features in images, and removing noise from time series.
<P>One way to see the time-frequency resolution differences between the
Fourier transform and the wavelet transform is to look at the basis function
coverage of the time-frequency plane <A
href="http://www.amara.com/IEEEwave/IW_ref.html#five">(5)</A>. Figure 1 shows
a windowed Fourier transform, where the window is simply a square wave. The
square wave window truncates the sine or cosine function to fit a window of a
particular width. Because a single window is used for all frequencies in the
WFT, the resolution of the analysis is the same at all locations in the
time-frequency plane.
<P>
<UL><IMG alt=Fig1
src="An Introduction to Wavelets Wavelet versus Fourier Transforms.files/IW_fig1.gif"
align=top></UL>
<P><B>Fig. 1. Fourier basis functions, time-frequency tiles, and coverage of
the time-frequency plane.</B>
<P>An advantage of wavelet transforms is that the windows <EM>vary.</EM> In
order to isolate signal discontinuities, one would like to have some very
short basis functions. At the same time, in order to obtain detailed frequency
analysis, one would like to have some very long basis functions. A way to
achieve this is to have short high-frequency basis functions and long
low-frequency ones. This happy medium is exactly what you get with wavelet
transforms. Figure 2 shows the coverage in the time-frequency plane with one
wavelet function, the Daubechies wavelet.
<P>
<UL><IMG alt=Fig2
src="An Introduction to Wavelets Wavelet versus Fourier Transforms.files/IW_fig2.gif"
align=middle></UL>
<P><B>Fig. 2. Daubechies wavelet basis functions, time-frequency tiles, and
coverage of the time-frequency plane.</B>
<P>One thing to remember is that wavelet transforms do not have a single set
of basis functions like the Fourier transform, which utilizes just the sine
and cosine functions. Instead, wavelet transforms have an infinite set of
possible basis functions. Thus wavelet analysis provides immediate access to
information that can be obscured by other time-frequency methods such as
Fourier analysis.
<P>
<P>
<HR align=center noShade SIZE=2>
<P><B><A href="http://www.amara.com/index.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_fourier_ana.html">[Previous]</A> <A
href="http://www.amara.com/IEEEwave/IW_see_wave.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 + -