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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">Local Cosine Transforms</p>
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        <phrase>cosine transforms, local</phrase>
        <phrase>local cosine transforms</phrase>
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                <c val="#000">lct(a, n, f)</c>
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        <phrase>lct function</phrase>
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                <c val="#000">ilct(w, n, b, f)</c>
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        <phrase>ilct function</phrase>
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                <c val="#000">cpt(a, n)</c>
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        <phrase>cpt function</phrase>
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        <phrase>icpt function</phrase>
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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">These functions perform a level <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> forward and inverse local cosine transform (LCT) on one-dimensional (1D) or two-dimensional (2D) data. The arguments to these functions are:</p>
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        <phrase>cosine transforms, forward and inverse local</phrase>
        <phrase>local cosine transforms, forward and inverse</phrase>
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          <b>,</b> the array of data to transform.</p>
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          <b>,</b> a cosine packet basis constructed from a cosine packet table.</p>
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            <b>
              <c val="#008080">• </c>
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          <b>n</b>
          <b>,</b> the number of levels in the transform.</p>
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          <b>b</b>, a 2- or 3-column matrix representing the cosine packet basis on which to compute the inverse transform.</p>
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          <b>f,</b> the data taper to use in the transform. </p>
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        <phrase>taper, data, to use in transform</phrase>
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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">All tapers are defined in the file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b>. As several functions used in this worksheet are defined in <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">wavedef.xmcd</inlineAttr></b></f>, a reference to this file is included. </p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Computing the Local Cosine Transform </p>
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        <phrase>cosine transform, computing local</phrase>
        <phrase>local cosine transform, computing</phrase>
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        <phrase>lct function</phrase>
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        <phrase>cpt function</phrase>
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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 function <b>lct </b>computes a level <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> local cosine transform of the data in <b>a</b> using data taper <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">f</inlineAttr></b></f>. The local cosine transform is essentially a fancy version of a blocked discrete cosine transform. In a blocked transform, the data array <b>a</b> is broken into consecutive evenly sized blocks of data which are then tapered and individually transformed with a Type IV DCT. The function <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">cpt</f></inlineAttr></b> is a shortcut version of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">lct</inlineAttr></b></f> with a <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">trig</inlineAttr></f> taper (defined below).</p>
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        <phrase>blocked discrete cosine transform</phrase>
        <phrase>cosine transform, blocked discrete</phrase>
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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">Block at level 0:</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">Blocks at level 2:</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">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">What distinguishes the </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">lct</inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false"> from other blocked DCTs is the fact that the decomposition of the data is not done by simply chopping off a chunk of data. Instead, very special smooth windows are used to delineate the chosen blocks.</inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12"> Several tapers are defined in the</f> file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b><f family="Times New Roman" charset="0" size="12">. Valid tapers are: </f><inlineAttr line-through="false"><f family="Times New Roman" charset="0" size="12"><b>box</b></f></inlineAttr><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">poly1</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">poly2</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">poly3</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">poly4</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, and </inlineAttr></f><b><inlineAttr line-through="false"><f family="Times New Roman" charset="0" size="12">trig</f></inlineAttr></b><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">.</inlineAttr></f><f family="Courier New" charset="0" size="12"><inlineAttr line-through="false"><sp /></inlineAttr></f></p>
      </text>
      <indexes>
        <phrase>trig taper</phrase>
        <phrase>box taper</phrase>
        <phrase>poly1 taper</phrase>
        <phrase>poly2 taper</phrase>
        <phrase>poly3 taper</phrase>
        <phrase>poly4 taper</phrase>
        <phrase>valid tapers</phrase>
        <phrase>tapers, valid</phrase>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">The following example uses </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">lct</inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false"> to define a test signal and compute its transform:</inlineAttr>
          </f>
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    <region region-id="4397" left="6" top="1038" width="338.25" height="181.5" align-x="6" align-y="1038" 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">This signal is a concatenation of four sinusoids having different frequencies. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Compute and plot the local cosine transform of the signal. First, define a taper:</p>
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        <phrase>plot local cosine transform of signal</phrase>
        <phrase>signal, plot local cosine transform of</phrase>
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          <ml:function>
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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">Define a tapering function.</p>
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          <ml:id xml:space="preserve">T</ml:id>
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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">Compute the local cosine transform.</p>
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              <ml:id xml:space="preserve">y</ml:id>
              <ml:real>2</ml:real>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">trig</inlineAttr></b></f> taper is used with <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">lct</inlineAttr></b></f> the</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">transform can equivalently be computed </p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">using the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">cpt</inlineAttr></b></f> function.</p>
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        <phrase>cpt function</phrase>
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            <ml:real>5.4949604629671311E-15</ml:real>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The LCT of this function is very enlightening. Note that like the <b><u><link href="./Wpt.xmcd" title="Links to section" popup="false">wavelet packet transform</link></u></b>, <b>lct</b> returns an array containing multiple levels of the transform. Also like the wavelet packet transforms, to extract a particular level, you simply take the appropriate row from the table.</p>
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      <pageBreak />
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    <region region-id="4677" left="6" top="1530.75" width="323.25" height="28.5" align-x="6" align-y="1542" 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 zero level of the LCT<b><sp /></b>contains the DCT of the single tapered block of data at level 0.</p>
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    <region region-id="4420" left="6" top="1728.75" width="335.25" height="28.5" align-x="6" align-y="1740" 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 first half of the first level of the LCT is the DCT of the first block on level 1, the second half is the DCT of the second block on level 1.</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 difference between these two levels in the LCT is that the 0th level is the DCT of the whole signal. Thus, you see four peaks which correspond to the four distinct frequencies in the signal. The 1st level is two DCT's, one for the first half of the data and one for the second half of the data. You see that the first DCT on the first level has only two peaks corresponding to the fact that there are only two frequencies present in the first half of the signal. A similar statement holds for the second DCT.</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">You can also compute a best basis from a cosine packet table. For example, compute the best basis for a ramp signal:</p>
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      <indexes>
        <phrase>ramp signal</phrase>
        <phrase>signal, ramp</phrase>
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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">Define the ramp test signal.</p>
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              <ml:real>1</ml:real>
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    <region region-id="4432" left="30" top="2250.75" width="127.5" height="15.75" align-x="42.75" align-y="2262" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">J</ml:id>
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            <ml:minus />
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              <ml:id xml:space="preserve">MaxDWTLevel</ml:id>
              <ml:id xml:space="preserve">r</ml:id>
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    <region region-id="4783" left="246" top="2250.75" width="153.75" height="42.75" align-x="246" align-y="2262" 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">Select a decomposition level using <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">MaxDWTLevel</c></inlineAttr></b></f>, a function defined in the file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b>.</p>
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      <indexes>
        <phrase>MaxDWTLevel function</phrase>
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    <region region-id="4787" left="246" top="2304.75" width="160.5" height="28.5" align-x="246" align-y="2316" 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">Create a cosine packet table using the local cosine transform function.</p>
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    <region region-id="4789" left="246" top="2388.75" width="146.25" height="14.25" align-x="246" align-y="2400" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Define the entropy cost function.</p>
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          <ml:id xml:space="preserve">b</ml:id>
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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">Find the minimum entropy basis using <b><u><link href="./best.xmcd" title="Links to section" popup="false">best basis</link></u></b>, which should give the optimal compression.</p>
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    <region region-id="4680" left="6" top="2640.75" width="333" height="42.75" align-x="6" align-y="2652" 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 look at the degree of concentration achieved by the best basis. As with wavelet packets, use the <b><u><link href="./Iwpt.xmcd" title="Links to section" popup="false">extract_basis</link></u></b> function to select a basis from the cosine packet table.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Computing the Inverse Local Cosine Transform </p>
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      <indexes>
        <phrase>cosine transform, computing inverse local</phrase>
        <phrase>inverse local cosine transform, computing</phrase>
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    <region region-id="4691" left="36" top="3060.75" width="63.75" height="14.25" align-x="36" align-y="3072" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
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          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">ilct(w, n, b, f)</inlineAttr>
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      <indexes>
        <phrase>ilct function</phrase>
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          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">icpt(w, n, b)</c>
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          </f>
        </p>
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      <indexes>
        <phrase>icpt function</phrase>
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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 <b>ilct</b> function takes the inverse cosine packet transform of a basis of local cosine functions contained in a cosine packet table<b> w</b> and described by the basis matrix <b>b</b>. Recall that the forward local cosine transform <b>lct</b> gives a cosine packet table as its return value. This table is a redundant description of the signal, so you often want to find a non-redundant subset of coefficients from the table, i.e. a basis of cosine packet coefficients. This is an identical situation to that with wavelet packets; therefore the same functions can be used to select the basis.</p>
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      <indexes>
        <phrase>basis of local cosine functions</phrase>
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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 function <b>ilct</b> requires that you pass an argument <b>n</b> which is the number of levels used in the original <b>lct</b> function. This may be larger than the number of levels present in the basis <b>b</b>. <b>ilct</b> also requires that you specify the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">taper </inlineAttr></f><b>f</b> which was used in the original <f family="Times New Roman" charset="0"><inlineAttr line-through="false"><b>lct</b></inlineAttr></f> function. The function <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">icpt</f></inlineAttr></b> is the shortcut version of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">ilct</inlineAttr></b></f> with a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">trig</inlineAttr></b></f> taper.</p>
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        <phrase>trig taper</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">As an example, compute the inverse local cosine transform of the best basis for the ramp signal computed above:</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">Again with a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">trig</inlineAttr></b></f> taper the function</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">You can test the validity of the basis by measuring the reconstruction error between the original and reconstructed signals:</p>
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