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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">WAVELETS EXTENSION PACK</p>
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        <p style="Section Title" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Fractional Brownian Motion</p>
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        <phrase>Brownian motion, fractional</phrase>
        <phrase>fractional Brownian motion</phrase>
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          <c val="#000">In this worksheet, the orthonormal wavelet transform is used to synthesize fractional Brownian motion (fBm). The theoretical underpinnings are developed in a paper of P. Flandrin (1992) (see also </c>
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                <link href="./Bibliog.xmcd" title="Links to section" popup="false">G. W. Wornell</link>
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          <c val="#000">).</c>
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        <phrase>Flandrin, P.</phrase>
        <phrase>orthonormal wavelet transform</phrase>
        <phrase>wavelet transform, orthonormal</phrase>
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          <c val="#000">H is the fBm scaling parameter and is called the </c>
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        <phrase>fBm scaling parameter</phrase>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">sd is the standard deviation.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">len is the length of the fBm time series to be generated.</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This length must be a power of two.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The wavelet algorithm for generating fBm time series requires a power of</p>
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          <c val="#000">The following function creates the approximate fBm from white noise in the wavelet transform domain by scaling the noise deviates at the level i <tab /><tab />by</c>
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          <c val="#000">times the standard deviation</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now you have an approximate fractional Brownian motion in the wavelet transform domain. To get the representation in the time domain, simply perform the inverse wavelet transform:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Due to the periodic nature of the wavelet transform, the above result is not a very good approximation to fBm. For example, the end points of the series are quite close to each other, which is not usually the case for correctly simulated fBm series. To obtain a better approximation, you can generate a series of length <region region-id="127" left="35.25" top="1125.75" width="23.25" height="15.75" align-x="42.75" align-y="1137" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="default" /><ml:real>4</ml:real><ml:id xml:space="preserve">len</ml:id></ml:apply></math><rendering item-idref="19" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sp /></c></inlineAttr></f>and then subsample any len consecutive values.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The first difference of an fBm series is known as fractional Gaussian noise.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Note that, when H=1/2, a fractional Brownian motion becomes ordinary Brownian motion. You can obtain a simulation of samples of a Brownian motion by constructing a random walk process.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">For comparison, here is what you get if you use the wavelet based approximate method with H=1/2</p>
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        <ml:define>
          <ml:id xml:space="preserve">more_noise</ml:id>
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              <ml:apply>
                <ml:mult style="default" />
                <ml:real>16</ml:real>
                <ml:id xml:space="preserve">len</ml:id>
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              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
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          </ml:apply>
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          <ml:id xml:space="preserve">mx</ml:id>
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              <ml:id xml:space="preserve">v</ml:id>
              <ml:real>12</ml:real>
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          </ml:apply>
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      <rendering item-idref="56" />
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      <pageBreak />
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      <plot disable-calc="false" item-idref="57" />
      <rendering item-idref="58" />
    </region>
    <region region-id="671" left="36" top="2538.75" width="52.5" height="15.75" align-x="51" align-y="2550" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="4424" left="162" top="2586.75" width="154.5" height="42.75" align-x="162" align-y="2598" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The power spectrum should have a slope close to -2 when plotted on a log-log scale.</p>
      </text>
      <indexes>
        <phrase>power spectrum</phrase>
      </indexes>
    </region>
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        </ml:define>
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    <region region-id="458" left="1062" top="2623.5" width="92.25" height="21" align-x="1082.25" align-y="2640" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">len</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
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            <ml:real>1000000</ml:real>
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    <region region-id="4472" left="144" top="3066.75" width="180.75" height="42.75" align-x="144" align-y="3078" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The power spectrum of the first differences should have a slope close to 0 when plotted on a log-log scale.</p>
      </text>
      <indexes>
        <phrase>power spectrum</phrase>
      </indexes>
    </region>
    <region region-id="4473" left="24" top="3072.75" width="42" height="17.25" align-x="49.5" align-y="3084" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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          <ml:real>3</ml:real>
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    <region region-id="4474" left="6" top="3132" width="342" height="235.5" align-x="6" align-y="3132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now extend the synthesis to create a 2-dimensional (2D) fractional Brownian Surface. </p>
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        <phrase>2D fractional Brownian surface</phrase>
        <phrase>Brownian surface, 2D fractional</phrase>
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