範例︰複數函數的 DFT

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範例︰複數函數的 DFT

使用 dft 與 idft 函數計算複數或實數的傅立葉轉換或傅立葉逆轉換。

1. 建立若干具有 N 個數據點的模仿複數數據。

2. 定義取樣間距。

3. 使用 exp 函數定義以下信號。

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  <math resultRef="64">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <indexer />
        <id labels="VARIABLE" xml:space="preserve">s</id>
        <id labels="*" xml:space="preserve">n</id>
      </apply>
      <apply>
        <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">exp</id>
        <apply>
          <mult />
          <apply>
            <mult />
            <imag symbol="i">1</imag>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
              <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">ω<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
            </id>
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            <indexer />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">t</id>
            <id xml:space="preserve">n</id>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
    </resultFormat>
  </math>
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4. 定義取樣頻率及對應至已轉換向量內第 n 項的頻率。

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    <define xmlns="http://schemas.mathsoft.com/math50">
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        <indexer />
        <id labels="*" xml:space="preserve">freq</id>
        <id labels="*" xml:space="preserve">n</id>
      </apply>
      <apply>
        <mult />
        <apply>
          <div />
          <id xml:space="preserve">n</id>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id>
        </apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
          <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">f<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">s</Subscript></Span>
        </id>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
    </resultFormat>
  </math>
</region>

5. 套用 dft 函數，將數據轉換成頻域。

6. 將陣列 D 的元素量值儲存到新的陣列。

7. 使用 match 與 max 函數計算尖峰振幅和頻率。

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  <math resultRef="76">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve">p</id>
      <eval>
        <apply>
          <indexer />
          <parens>
            <apply>
              <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">match</id>
              <sequence>
                <apply>
                  <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">max</id>
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">S</id>
                </apply>
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">S</id>
              </sequence>
            </apply>
          </parens>
          <real>0</real>
        </apply>
        <unitOverride>
          <placeholder />
        </unitOverride>
      </eval>
    </define>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
    </resultFormat>
  </math>
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8. 將 DFT 的量值以頻率函數圖表表示。

按一下以複製此運算式

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        <trace resultRef="80">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">線條</traceStyle>
        </trace>
      </traces>
      <graph-size width="480.4" height="230.4" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="1">
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            <gridLabels display="true" />
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          <axisLabel />
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              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">m</id>
            </marker>
          </markers>
          <numberFormat>
            <general precision="2" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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            </secondTickValue>
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              <real xmlns="http://schemas.mathsoft.com/math50">100</real>
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          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

9. 顯示對應至 ω0/2π 之尖峰值的頻率。

10. 使用 idft 函數顯示信號的 DFT IDFT 傳回信號本身。

相關主題

數據的離散傅立葉轉換

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