Beispiel: Verwenden von nfact-und ofact-Modifizierern

Symbolische Mathematik > Schlüsselwörter und Modifizierer > Beispiel: Verwenden von nfact-und ofact-Modifizierern

Die Normalisierungsmultiplikatoren (nfact) und der Oszillationsfaktor (ofact) werden wie folgt ausgedrückt:

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              <id xml:space="preserve">a</id>
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          <apply>
            <equal />
            <id xml:space="preserve">ofact</id>
            <id xml:space="preserve">b</id>
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              <apply>
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                  <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>
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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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        </apply>
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          <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>
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              <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>
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                    <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>
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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>
              </apply>
            </lowerBound>
            <upperBound>
              <id xml:space="preserve" labels="CONSTANT">∞</id>
            </upperBound>
          </apply>
        </apply>
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    </symEval>
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Fourier- und inverse Fourier-Transformationen unterscheiden sich nur bei Normalisierungs- und Oszillationsfaktoren. Die Faktoren werden in der Regel so gewählt, dass die nachfolgende Leistung der direkten und inversen Transformation die ursprüngliche Funktion angibt. Beispiel:

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              <imag symbol="i">1</imag>
              <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">nfact</id>
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            </sequence>
            <apply>
              <equal />
              <id xml:space="preserve">ofact</id>
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        </command>
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            <plus />
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              <imag symbol="i">1</imag>
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              <id xml:space="preserve">invfourier</id>
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                <id xml:space="preserve">nfact</id>
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                  <real>1</real>
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                    <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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In verschiedenen Bereichen der Mathematik und der technischen Entwicklung werden die Paare von Normalisierungs- und Oszillationsfaktoren unterschiedlich gewählt, um verschiedene Signaltypen zu normalisieren.

1. Wahrscheinlichkeitstheorie (Modelle ohne Einheiten)

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2. Elektrotechnik (Frequenz in Hertz, einer der Wolfram-Sätze)

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3. Elektrotechnik (Winkelfrequenz wird verwendet)

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                      <imag symbol="i">1</imag>
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4. Bei Symmetrie der direkten und inversen ist die Transformation wichtig. Die gleiche Eingabe (mit unterschiedlichen Variablen der Transformation) gibt das gleiche Ausgabeergebnis zurück, wenn man den entsprechenden Satz von nfact- und ofact-Werten verwendet.

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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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Invertierbarkeit zwischen Fourier und inversem Fourier

Wenn Sie das Schlüsselwort fourier auf einen Eingabeausdruck anwenden, muss die Anwendung des Schlüsselworts invfourier auf das Ergebnis zum ursprünglichen Ausdruck zurückführen. Die Invertierbarkeit wird bei Verwendung der Schlüsselwörter fourier oder invfourier mit standardmäßigen nfact- und ofact-Modifizierern unterstützt.

Bei Verwendung der Modifizierer nfact und ofact gelten die folgenden Bedingungen:

• Schließen Sie die Modifizierer nfact und ofact unmittelbar nach dem Schlüsselwortnamen ein. Sie können die beiden Modifizierer einzeln verwenden oder beide in einen Ausdruck einschließen.

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          </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>

• Matrizen sind nicht als Faktoren zulässig.

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    <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>

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  <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>

• Die inverse Reihenfolge von Modifizierern ist akzeptabel.

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve">t</id>
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            <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>

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  <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>

War dies hilfreich?