例: nfact および ofact 変更因子の使用

シンボリック > キーワードと変更因子 > 例: nfact および ofact 変更因子の使用

規格化乗数 (nfact) と振動因子 (ofact) は以下のように表されます。

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            <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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            <lambda>
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                <id xml:space="preserve" labels="VARIABLE">t</id>
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              <apply>
                <mult />
                <apply>
                  <id xml:space="preserve" labels="VARIABLE">f</id>
                  <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>
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      <apply>
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        <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>
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フーリエ変換と逆フーリエ変換の違いは、規格化因子と振動因子によってのみ決定されます。通常、これらの因子は、直接および逆変換の後続のパフォーマンスが元の関数を提供するように選択されます。例:

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        <symResult>
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              <id xml:space="preserve" labels="VARIABLE">ω</id>
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          <id xml:space="preserve">simplify</id>
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          <id xml:space="preserve" labels="FUNCTION">Φ</id>
          <id xml:space="preserve" labels="VARIABLE">t</id>
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            <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 />
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              <mult />
              <id xml:space="preserve" labels="CONSTANT">π</id>
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            </apply>
            <apply>
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              <imag symbol="i">1</imag>
              <id xml:space="preserve" labels="VARIABLE">ω</id>
            </apply>
          </apply>
        </symResult>
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      <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>
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数学とエンジニアリングのさまざまな分野において、規格化因子と振動因子のペアは、さまざまなタイプの信号を正規化するために異なる方法で選択されます。

1. 確率論 (単位のないモデル)

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                <real>1</real>
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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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              <neg />
              <id xml:space="preserve" labels="CONSTANT">∞</id>
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                        <imag symbol="i">1</imag>
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2. 電気工学 (ヘルツ単位の周波数、ウルフラムセットの 1 つ)

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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>
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              <id xml:space="preserve" labels="CONSTANT">∞</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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              <id xml:space="preserve" labels="CONSTANT">∞</id>
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3. 電気工学 (角周波数を使用)

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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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              <id xml:space="preserve" labels="CONSTANT">∞</id>
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                    <imag symbol="i">1</imag>
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4. 直接および逆の対称では、変換は重要です。同じ入力 (変換の変数は異なる) が適切な nfact 値と ofact 値のセットを使用して同じ出力結果を返します。

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                          <imag symbol="i">1</imag>
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                      <imag symbol="i">1</imag>
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                <id xml:space="preserve" labels="CONSTANT">∞</id>
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フーリエと逆フーリエの間の可逆性

入力数式でキーワード fourier を適用するとき、結果でキーワード invfourier を適用することにより最初の式が返されなければなりません。キーワード fourier または invfourier をデフォルトの nfact および ofact 変更因子とともに使用するとき、可逆性がサポートされます。

nfact および ofact 変更因子を使用するとき、以下の条件が適用されます。

• nfact および ofact 変更因子は、キーワード名の直後に含められます。2 つの変更因子を別々に使用することも、両方を定義式に含めることもできます。

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

• 行列を因子として使用することはできません。

<region id="ID0EPPPLB" 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="ID0E4PPLB" 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>

• 変更因子の順序を逆にすることは可能です。

<region id="ID0ENRPLB" actualWidth="382.86333333333334" actualHeight="60.800000000000004" top="2649.6000000000004" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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="ID0E2RPLB" 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>

これは役に立ちましたか?