範例︰向量的 dft 和 idft

函數 > 轉換與濾波器 > 範例︰向量的 dft 和 idft

範例︰向量的 dft 和 idft

使用 dft 與 idft 函數來尋找向量的離散傅立葉正轉換或離散傅立葉逆轉換。

1. 定義數據向量的長度。

2. 使用 exp 函數建立長度 N 的實數數據向量。

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  <math resultRef="66">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <indexer />
        <id labels="VARIABLE" xml:space="preserve">V</id>
        <id labels="*" xml:space="preserve">n</id>
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      <apply>
        <id labels="FUNCTION" xml:space="preserve">exp</id>
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              <real>2</real>
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          <real>8</real>
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    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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3. 繪製數據。

按一下以複製此運算式

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            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="75">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
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4. 使用 dft 函數，以計算 V 的離散傅立葉轉換。

5. 繪製 Z 的絕對值。

按一下以複製此運算式

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                <apply xmlns="http://schemas.mathsoft.com/math50">
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                    <indexer />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Z</id>
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
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6. 顯示 dft 函數的定義。

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        <summation />
        <lambda>
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              <mult />
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                <indexer />
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">V</id>
                <id xml:space="preserve">k</id>
              </apply>
              <apply>
                <pow />
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">ω</id>
                <parens>
                  <apply>
                    <mult />
                    <apply>
                      <neg />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id>
                    </apply>
                    <id xml:space="preserve">k</id>
                  </apply>
                </parens>
              </apply>
            </apply>
          </parens>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <minus />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
    </resultFormat>
  </math>
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7. 使用上述定義來尋找特定頻率元素，並將其與 dft 函數輸出中的對應元素進行比較。

高斯 idft

idft 函數是 dft 的逆轉換。它接受實數或複數向量作為其引數，並傳回相同長度的向量。

1. 顯示 dft 函數的逆轉換是函數本身。

2. 使用先前定義的向量 V，然後顯示 V 的 dft 的逆轉是 V 本身。

按一下以複製此運算式

按一下以複製此運算式

3. 顯示 idft 函數的定義。

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          <real>1</real>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id>
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        <apply>
          <summation />
          <lambda>
            <boundVars>
              <id xml:space="preserve">k</id>
            </boundVars>
            <parens>
              <apply>
                <mult />
                <apply>
                  <indexer />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Z</id>
                  <id xml:space="preserve">k</id>
                </apply>
                <apply>
                  <pow />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">ω</id>
                  <parens>
                    <apply>
                      <mult />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id>
                      <id xml:space="preserve">k</id>
                    </apply>
                  </parens>
                </apply>
              </apply>
            </parens>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <apply>
              <minus />
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id>
              <real>1</real>
            </apply>
          </upperBound>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
    </resultFormat>
  </math>
</region>

4. 使用上述定義來尋找特定頻率元素，並將其與 idft 函數和向量 V 輸出中的對應元素進行比較。

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