示例：使用 nfact 和 ofact 修饰符

符号 > 关键字和修饰符 > 示例：使用 nfact 和 ofact 修饰符

归一化乘数 (nfact) 和振荡因子 (ofact) 的表达式如下：

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            <id xml:space="preserve">fourier</id>
            <apply>
              <equal />
              <id xml:space="preserve">nfact</id>
              <id xml:space="preserve">a</id>
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          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <id xml:space="preserve">b</id>
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        </sequence>
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                  <id xml:space="preserve" labels="VARIABLE">t</id>
                </apply>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="CONSTANT">e</id>
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">ω</id>
                        <id xml:space="preserve" labels="VARIABLE">b</id>
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                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                    <imag symbol="i">1</imag>
                  </apply>
                </apply>
              </apply>
            </lambda>
            <lowerBound>
              <apply>
                <neg />
                <id xml:space="preserve" labels="CONSTANT">∞</id>
              </apply>
            </lowerBound>
            <upperBound>
              <id xml:space="preserve" labels="CONSTANT">∞</id>
            </upperBound>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
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      <apply>
        <id xml:space="preserve">f</id>
        <id xml:space="preserve">ω</id>
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      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">invfourier</id>
            <apply>
              <equal />
              <id xml:space="preserve">nfact</id>
              <id xml:space="preserve">a</id>
            </apply>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <id xml:space="preserve">b</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <mult />
          <id xml:space="preserve" labels="VARIABLE">a</id>
          <apply>
            <integral />
            <lambda>
              <boundVars>
                <id xml:space="preserve" labels="VARIABLE">ω</id>
              </boundVars>
              <apply>
                <mult />
                <apply>
                  <id xml:space="preserve" labels="VARIABLE">f</id>
                  <id xml:space="preserve" labels="VARIABLE">ω</id>
                </apply>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="CONSTANT">e</id>
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">ω</id>
                        <id xml:space="preserve" labels="VARIABLE">b</id>
                      </apply>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                    <imag symbol="i">1</imag>
                  </apply>
                </apply>
              </apply>
            </lambda>
            <lowerBound>
              <apply>
                <neg />
                <id xml:space="preserve" labels="CONSTANT">∞</id>
              </apply>
            </lowerBound>
            <upperBound>
              <id xml:space="preserve" labels="CONSTANT">∞</id>
            </upperBound>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

傅立叶变换和逆傅立叶变换仅归一化和振荡因子不同。这两种因子的选择通常是为了后续的直接变换和逆变换能够得出原函数。例如：

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        <command>
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        <symResult>
          <apply>
            <plus />
            <apply>
              <mult />
              <id xml:space="preserve" labels="CONSTANT">π</id>
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                <id xml:space="preserve" labels="VARIABLE">ω</id>
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            <apply>
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              <imag symbol="i">1</imag>
              <id xml:space="preserve" labels="VARIABLE">ω</id>
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          </apply>
        </symResult>
      </symEval>
      <command>
        <sequence>
          <id xml:space="preserve">invfourier</id>
          <id xml:space="preserve">simplify</id>
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      <symResult>
        <apply>
          <id xml:space="preserve" labels="FUNCTION">Φ</id>
          <id xml:space="preserve" labels="VARIABLE">t</id>
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      </symResult>
    </symEval>
  </math>
</region>

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        <command>
          <sequence>
            <sequence>
              <id xml:space="preserve">fourier</id>
              <apply>
                <equal />
                <id xml:space="preserve">nfact</id>
                <real>1</real>
              </apply>
            </sequence>
            <apply>
              <equal />
              <id xml:space="preserve">ofact</id>
              <real>1</real>
            </apply>
          </sequence>
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <mult />
              <id xml:space="preserve" labels="CONSTANT">π</id>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">Δ</id>
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            <apply>
              <div />
              <imag symbol="i">1</imag>
              <id xml:space="preserve" labels="VARIABLE">ω</id>
            </apply>
          </apply>
        </symResult>
      </symEval>
      <command>
        <sequence>
          <sequence>
            <sequence>
              <id xml:space="preserve">invfourier</id>
              <apply>
                <equal />
                <id xml:space="preserve">nfact</id>
                <apply>
                  <div />
                  <real>1</real>
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
                  </apply>
                </apply>
              </apply>
            </sequence>
            <apply>
              <equal />
              <id xml:space="preserve">ofact</id>
              <real>-1</real>
            </apply>
          </sequence>
          <id xml:space="preserve">simplify</id>
        </sequence>
      </command>
      <symResult>
        <apply>
          <id xml:space="preserve" labels="FUNCTION">Φ</id>
          <id xml:space="preserve" labels="VARIABLE">t</id>
        </apply>
      </symResult>
    </symEval>
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在不同的数学和工程字段中选择不同的归一化和振荡因子对以将不同类型的信号进行归一化。

1. 概率论 (无单位模型)

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          <apply>
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                  <imag symbol="i">1</imag>
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          <lowerBound>
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              <id xml:space="preserve" labels="CONSTANT">∞</id>
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                    <id xml:space="preserve" labels="VARIABLE">t</id>
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                  <imag symbol="i">1</imag>
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          <lowerBound>
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              <id xml:space="preserve" labels="CONSTANT">∞</id>
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          <upperBound>
            <id xml:space="preserve" labels="CONSTANT">∞</id>
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                <id xml:space="preserve" labels="CONSTANT">∞</id>
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            <upperBound>
              <id xml:space="preserve" labels="CONSTANT">∞</id>
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            <mult />
            <real>2</real>
            <id xml:space="preserve" labels="CONSTANT">π</id>
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2. 电气工程 (频率单位为赫兹，Wolfram 集之一)

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          </apply>
        </sequence>
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      <symResult>
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          <lambda>
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              <apply>
                <id xml:space="preserve" labels="VARIABLE">f</id>
                <id xml:space="preserve" labels="VARIABLE">t</id>
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              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <apply>
                  <neg />
                  <parens>
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <apply>
                          <mult />
                          <apply>
                            <mult />
                            <real>2</real>
                            <id xml:space="preserve" labels="VARIABLE">ω</id>
                          </apply>
                          <id xml:space="preserve" labels="VARIABLE">t</id>
                        </apply>
                        <id xml:space="preserve" labels="CONSTANT">π</id>
                      </apply>
                      <imag symbol="i">1</imag>
                    </apply>
                  </parens>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <apply>
              <neg />
              <id xml:space="preserve" labels="CONSTANT">∞</id>
            </apply>
          </lowerBound>
          <upperBound>
            <id xml:space="preserve" labels="CONSTANT">∞</id>
          </upperBound>
        </apply>
      </symResult>
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            <id xml:space="preserve">invfourier</id>
            <apply>
              <equal />
              <id xml:space="preserve">nfact</id>
              <real>1</real>
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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <apply>
              <mult />
              <real>2</real>
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
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      <symResult>
        <apply>
          <integral />
          <lambda>
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              <id xml:space="preserve" labels="VARIABLE">ω</id>
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              <apply>
                <id xml:space="preserve" labels="VARIABLE">f</id>
                <id xml:space="preserve" labels="VARIABLE">ω</id>
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              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <apply>
                  <mult />
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <real>2</real>
                        <id xml:space="preserve" labels="VARIABLE">ω</id>
                      </apply>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <id xml:space="preserve" labels="CONSTANT">π</id>
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                  <imag symbol="i">1</imag>
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              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <apply>
              <neg />
              <id xml:space="preserve" labels="CONSTANT">∞</id>
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          </lowerBound>
          <upperBound>
            <id xml:space="preserve" labels="CONSTANT">∞</id>
          </upperBound>
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3. 电气工程 (采用角频率)

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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <real>-1</real>
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          <apply>
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              <apply>
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                <id xml:space="preserve" labels="CONSTANT">e</id>
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                  <neg />
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                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">ω</id>
                        <id xml:space="preserve" labels="VARIABLE">t</id>
                      </apply>
                      <imag symbol="i">1</imag>
                    </apply>
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                </apply>
              </apply>
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          </lambda>
          <lowerBound>
            <apply>
              <neg />
              <id xml:space="preserve" labels="CONSTANT">∞</id>
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          <upperBound>
            <id xml:space="preserve" labels="CONSTANT">∞</id>
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              <equal />
              <id xml:space="preserve">nfact</id>
              <apply>
                <div />
                <real>1</real>
                <apply>
                  <mult />
                  <real>2</real>
                  <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
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              </apply>
            </apply>
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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <real>1</real>
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        <apply>
          <div />
          <apply>
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            <lambda>
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                <id xml:space="preserve" labels="VARIABLE">ω</id>
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              <apply>
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                  <id xml:space="preserve" labels="VARIABLE">f</id>
                  <id xml:space="preserve" labels="VARIABLE">ω</id>
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                  <id xml:space="preserve" labels="CONSTANT">e</id>
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                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">ω</id>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <imag symbol="i">1</imag>
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                <neg />
                <id xml:space="preserve" labels="CONSTANT">∞</id>
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            <upperBound>
              <id xml:space="preserve" labels="CONSTANT">∞</id>
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          <apply>
            <mult />
            <real>2</real>
            <id xml:space="preserve" labels="CONSTANT">π</id>
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4. 变换在直接对称和逆对称中至关重要。相同的输入 (具有不同的变换变量) 可通过使用适当的 nfact 和 ofact 值集返回相同的输出结果。

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                <real>1</real>
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                    <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <real>-1</real>
          </apply>
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      <symResult>
        <apply>
          <div />
          <apply>
            <nthRoot />
            <placeholder />
            <real>2</real>
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          <apply>
            <mult />
            <real>2</real>
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              <id xml:space="preserve" labels="CONSTANT">π</id>
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                <real>1</real>
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          <apply>
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            <id xml:space="preserve">ofact</id>
            <real>1</real>
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        <apply>
          <div />
          <apply>
            <nthRoot />
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            <real>2</real>
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          <apply>
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            <real>2</real>
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve" labels="CONSTANT">π</id>
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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <real>-1</real>
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            <mult />
            <apply>
              <nthRoot />
              <placeholder />
              <real>2</real>
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              <lambda>
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                  <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <id xml:space="preserve" labels="VARIABLE">f</id>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <id xml:space="preserve" labels="CONSTANT">e</id>
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                            <id xml:space="preserve" labels="VARIABLE">t</id>
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                          <imag symbol="i">1</imag>
                        </apply>
                      </parens>
                    </apply>
                  </apply>
                </apply>
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              <lowerBound>
                <apply>
                  <neg />
                  <id xml:space="preserve" labels="CONSTANT">∞</id>
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              <upperBound>
                <id xml:space="preserve" labels="CONSTANT">∞</id>
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          <apply>
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            <real>2</real>
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              <id xml:space="preserve" labels="CONSTANT">π</id>
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                    <id xml:space="preserve" labels="VARIABLE">ω</id>
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                        <id xml:space="preserve" labels="VARIABLE">ω</id>
                        <id xml:space="preserve" labels="VARIABLE">t</id>
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                      <imag symbol="i">1</imag>
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              <lowerBound>
                <apply>
                  <neg />
                  <id xml:space="preserve" labels="CONSTANT">∞</id>
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              <upperBound>
                <id xml:space="preserve" labels="CONSTANT">∞</id>
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            </apply>
          </apply>
          <apply>
            <mult />
            <real>2</real>
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve" labels="CONSTANT">π</id>
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傅立叶和逆傅立叶之间的可逆性

若对输入表达式应用关键字 fourier，则对结果应用关键字 invfourier 必将返回初始表达式。当关键字 fourier 或 invfourier 与默认修饰符 nfact 和 ofact 搭配使用时具有可逆性。

在使用修饰符 nfact 和 ofact 时，以下条件适用：

• 关键字名称后紧跟修饰符 nfact 和 ofact。可分别使用这两个修饰符或将二者纳入同一表达式。

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                  <real>1</real>
                  <id xml:space="preserve" labels="VARIABLE">ω</id>
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          </parens>
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        <apply>
          <mult />
          <apply>
            <mult />
            <apply>
              <mult />
              <real>2</real>
              <id xml:space="preserve" labels="CONSTANT">π</id>
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            <imag symbol="i">1</imag>
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          <apply>
            <id xml:space="preserve" labels="FUNCTION">Δ</id>
            <sequence>
              <real>1</real>
              <id xml:space="preserve" labels="VARIABLE">ω</id>
            </sequence>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 不得将矩阵用作因子。

<region id="ID0EFLRLB" actualWidth="224.26666666666668" actualHeight="60.800000000000004" top="2457.6000000000004" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="86">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve">t</id>
      <command>
        <sequence>
          <sequence>
            <sequence>
              <id xml:space="preserve">fourier</id>
              <apply>
                <equal />
                <id xml:space="preserve">nfact</id>
                <matrix rows="1" cols="1">
                  <apply>
                    <div />
                    <real>1</real>
                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve">π</id>
                    </apply>
                  </apply>
                </matrix>
              </apply>
            </sequence>
            <id xml:space="preserve">t</id>
          </sequence>
          <id xml:space="preserve">s</id>
        </sequence>
      </command>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0ETLRLB" actualWidth="246.79666666666668" actualHeight="60.800000000000004" top="2534.4000000000005" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="87">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve">t</id>
      <command>
        <sequence>
          <sequence>
            <sequence>
              <id xml:space="preserve">invfourier</id>
              <apply>
                <equal />
                <id xml:space="preserve">nfact</id>
                <matrix rows="1" cols="1">
                  <apply>
                    <div />
                    <real>1</real>
                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve">π</id>
                    </apply>
                  </apply>
                </matrix>
              </apply>
            </sequence>
            <id xml:space="preserve">t</id>
          </sequence>
          <id xml:space="preserve">s</id>
        </sequence>
      </command>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 这两个修饰符支持逆傅立叶变换。

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  <math resultRef="88">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve">t</id>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">fourier</id>
            <apply>
              <equal />
              <id xml:space="preserve">ofact</id>
              <apply>
                <div />
                <real>1</real>
                <apply>
                  <mult />
                  <real>2</real>
                  <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
                </apply>
              </apply>
            </apply>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">nfact</id>
            <real>1</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <neg />
          <parens>
            <apply>
              <mult />
              <apply>
                <mult />
                <apply>
                  <mult />
                  <real>8</real>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="CONSTANT">π</id>
                    <real>3</real>
                  </apply>
                </apply>
                <imag symbol="i">1</imag>
              </apply>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">Δ</id>
                <sequence>
                  <real>1</real>
                  <id xml:space="preserve" labels="VARIABLE">ω</id>
                </sequence>
              </apply>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0ERNRLB" actualWidth="405.39333333333337" actualHeight="60.800000000000004" top="2726.4000000000005" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="89">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve">ω</id>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">invfourier</id>
            <apply>
              <equal />
              <id xml:space="preserve">ofact</id>
              <apply>
                <div />
                <real>1</real>
                <apply>
                  <mult />
                  <real>2</real>
                  <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">π</id>
                </apply>
              </apply>
            </apply>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">nfact</id>
            <real>1</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <neg />
          <parens>
            <apply>
              <mult />
              <apply>
                <mult />
                <apply>
                  <mult />
                  <real>8</real>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="CONSTANT">π</id>
                    <real>3</real>
                  </apply>
                </apply>
                <imag symbol="i">1</imag>
              </apply>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">Δ</id>
                <sequence>
                  <real>1</real>
                  <id xml:space="preserve" labels="VARIABLE">t</id>
                </sequence>
              </apply>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

这对您有帮助吗?