Beispiel: Lokale Kosinus-Transformation

Funktionen > Transformationen und Filter > Beispiel: Lokale Kosinus-Transformation

1. Definieren Sie ein Testsignal durch Verketten von vier Sinuskurven, die vier unterschiedliche Frequenzen aufweisen.

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2. Plotten Sie das Testsignal.

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          <axisLine position="origin" positionticmark="5" legendWidth="61.0033333333333" />
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            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
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            <tickMarks display="true" />
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          <axisLabel />
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          <numberFormat>
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                <minus />
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                <real>1</real>
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        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-1" end="1">
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3. Verwenden Sie die Funktion cpt, um die lokale Vorwärts-Kosinustransformation der Ebene 2 zu berechnen.

Die zurückgegebene Matrix enthält die drei Ebenen der lokalen Kosinustransformation.

Block auf Ebene 0:	<region id="ID0E4BYDB" top="921.60000000000014" left="307.20000000000005" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="183"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>0</real> <real>0</real> </sequence> </apply> </math> </region>
Block auf Ebene 1:	<region id="ID0ECDYDB" top="960.00000000000011" left="230.40000000000003" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="185"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>1</real> <real>0</real> </sequence> </apply> </math> </region>		<region id="ID0EYDYDB" top="960.00000000000011" left="384.00000000000006" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="187"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>1</real> <real>1</real> </sequence> </apply> </math> </region>
Block auf Ebene 2:	<region id="ID0E2EYDB" top="998.40000000000009" left="192.00000000000003" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="189"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>2</real> <real>0</real> </sequence> </apply> </math> </region>	<region id="ID0EPFYDB" top="998.40000000000009" left="268.80000000000007" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="191"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>2</real> <real>1</real> </sequence> </apply> </math> </region>	<region id="ID0EDGYDB" top="998.40000000000009" left="345.6" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="193"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>2</real> <real>2</real> </sequence> </apply> </math> </region>	<region id="ID0EXGYDB" top="998.40000000000009" left="422.40000000000009" height="36.800000000000004" width="40.436666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="195"> <apply xmlns="http://schemas.mathsoft.com/math50"> <indexer /> <id xml:space="preserve">W</id> <sequence> <real>2</real> <real>3</real> </sequence> </apply> </math> </region>

4. Plotten Sie Ebene 0 der Transformation, die die diskrete Kosinustransformation (DCT) des einzelnen verjüngten Datenblocks enthält.

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        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="550">
          <axisLine position="origin" positionticmark="4" legendWidth="61.0033333333333" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
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            <tickMarks display="true" />
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          <axisLabel />
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Die Transformation enthält vier Spitzenwerte, die den Frequenzen der vier Sinuskurven entsprechen.

5. Plotten Sie Ebene 1 der Transformation, die zwei diskrete Kosinustransformationen enthält, eine für die erste Hälfte der Daten und eine für die zweite Hälfte.

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                    <real>10</real>
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                    </id>
                  </apply>
                  <real>2</real>
                </apply>
              </apply>
            </marker>
          </markers>
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="242">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">p</id>
              </math>
              <math resultRef="243">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue resultRef="241">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <minus />
                <id xml:space="preserve">n</id>
                <real>1</real>
              </apply>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-5" end="6">
          <axisLine position="origin" positionticmark="0" legendWidth="85.76" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="244">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <parens>
                    <apply>
                      <indexer />
                      <id xml:space="preserve">T</id>
                      <real>1</real>
                    </apply>
                  </parens>
                  <id xml:space="preserve">p</id>
                </apply>
              </math>
              <math resultRef="245">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

Die Transformation enthält zwei Spitzenwerte in jeder der beiden diskreten Kosinustransformationen mit n/2 Datenpunkten. Wenn Sie auf die vertikalen Markierungen klicken, wird angezeigt, dass die ersten beiden auf bei der halben Frequenz der ersten beiden Sinuskurven und die beiden zweiten bei (n/2) + der Hälfte der Frequenz der beiden zweiten Sinuskurven liegen. Die blaue Markierung zeigt die Grenze zwischen den beiden diskreten Kosinustransformationen an.

6. Plotten Sie Ebene 2 der Transformation, die die vier diskreten Kosinustransformationen enthält, jeweils eine für jedes der vier Datenviertel.

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    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="257">
          <traceStyle color="#FFFF0000" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="457.200012946129" height="182.400005435944" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="550">
          <axisLine position="origin" positionticmark="5" legendWidth="61.0033333333333" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="258" display="true" color="#FF068149" line-style="Dot" value="8">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <div />
                <id xml:space="preserve">
                  <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">f<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                </id>
                <real>4</real>
              </apply>
            </marker>
            <marker resultRef="260" display="true" color="#FF068149" line-style="Dot" value="168">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <plus />
                <apply>
                  <div />
                  <id xml:space="preserve">n</id>
                  <real>4</real>
                </apply>
                <apply>
                  <mult />
                  <real>5</real>
                  <apply>
                    <div />
                    <id xml:space="preserve">
                      <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">f<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                    </id>
                    <real>4</real>
                  </apply>
                </apply>
              </apply>
            </marker>
            <marker resultRef="262" display="true" color="#FF068149" line-style="Dot" value="336">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <plus />
                <apply>
                  <div />
                  <id xml:space="preserve">n</id>
                  <real>2</real>
                </apply>
                <apply>
                  <mult />
                  <real>10</real>
                  <apply>
                    <div />
                    <id xml:space="preserve">
                      <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">f<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                    </id>
                    <real>4</real>
                  </apply>
                </apply>
              </apply>
            </marker>
            <marker resultRef="264" display="true" color="#FF068149" line-style="Dot" value="504">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <plus />
                <apply>
                  <div />
                  <apply>
                    <scale />
                    <real>3</real>
                    <id xml:space="preserve">n</id>
                  </apply>
                  <real>4</real>
                </apply>
                <apply>
                  <mult />
                  <real>15</real>
                  <apply>
                    <div />
                    <id xml:space="preserve">
                      <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">f<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                    </id>
                    <real>4</real>
                  </apply>
                </apply>
              </apply>
            </marker>
            <marker resultRef="266" display="true" color="#FF2E3192" line-style="Dash" value="128">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <div />
                <id xml:space="preserve">n</id>
                <real>4</real>
              </apply>
            </marker>
            <marker resultRef="268" display="true" color="#FF2E3192" line-style="Dash" value="256">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <div />
                <id xml:space="preserve">n</id>
                <real>2</real>
              </apply>
            </marker>
            <marker resultRef="270" display="true" color="#FF2E3192" line-style="Dash" value="384">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <div />
                <apply>
                  <scale />
                  <real>3</real>
                  <id xml:space="preserve">n</id>
                </apply>
                <real>4</real>
              </apply>
            </marker>
          </markers>
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="274">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">p</id>
              </math>
              <math resultRef="275">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue resultRef="273">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <minus />
                <id xml:space="preserve">n</id>
                <real>1</real>
              </apply>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-5" end="7">
          <axisLine position="origin" positionticmark="0" legendWidth="85.76" />
          <axisGrid>
            <gridFrequency>13</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="276">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <parens>
                    <apply>
                      <indexer />
                      <id xml:space="preserve">T</id>
                      <real>2</real>
                    </apply>
                  </parens>
                  <id xml:space="preserve">p</id>
                </apply>
              </math>
              <math resultRef="277">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

Die Transformation enthält einen Spitzenwert in jeder der vier diskreten Kosinustransformationen mit n/4 Datenpunkten. Wenn Sie auf die vertikalen Markierungen klicken, wird angezeigt, dass der erste Spitzenwert beim Viertel der Frequenz der ersten Sinuskurve liegt, der zweiten Spitzenwert bei n/4 + ein Viertel der Frequenz der zweiten Sinuskurve, der dritten Spitzenwert bei 2n/4 + ein Viertel der Frequenz der dritten Sinuskurve und der vierten Spitzenwert bei 3n/4 + ein Viertel der Frequenz der vierten Sinuskurve. Die blauen Markierungen zeigen die Grenzen zwischen den vier diskreten Kosinustransformationen.

Zugehörige Themen

Lokale Kosinustransformationen

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