Esempio: trasformata di Hartley

Funzioni > Elaborazione dei segnali > Ulteriori trasformate > Esempio: trasformata di Hartley

Utilizzare la funzione dht per trovare la trasformata di Hartley dei segnali.

La somma caratterizzante della trasformata di Hartley è analoga alla somma della trasformata di Hartley discreta.

dove N è il numero di elementi nell'array di dati reali x.

Somma di onde sinusoidali

Trovare la trasformata di Hartley di una somma di onde sinusoidali e confrontarla con la grandezza della trasformata di Fourier complessa.

1. Definire il numero di elementi.

2. Utilizzare la funzione sin per definire il segnale di input.

3. Tracciare il grafico del segnale.

Fare clic per copiare questa espressione

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4. Utilizzare la funzione dht per mostrare le frequenze discrete rappresentate nelle due onde sinusoidali, quindi utilizzare la funzione center per spostare il componente DC al centro.

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5. Utilizzare le funzioni match e max per trovare le frequenze discrete a cui si verificano i picchi e contrassegnare uno dei punti con un marcatore orizzontale e verticale.

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6. Utilizzare le funzioni dft e center per ottenere e centrare la trasformata discreta di Fourier.

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7. Tracciare il grafico dei valori assoluti della DFT.

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8. Utilizzare le funzioni match e max per trovare le frequenze discrete a cui si verificano i picchi.

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La stretta relazione tra la trasformata di Hartley e la trasformata di Fourier può essere dimostrata mostrando come calcolare fase e grandezza utilizzando la trasformata di Hartley. Questa operazione può essere effettuata utilizzando un piccolo insieme di dati.

Segnale rumoroso e breve

1. Definire un segnale di input rumoroso di 7 punti di campionamento e tracciarne il grafico.

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            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="340">
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">n</id>
              </math>
              <math resultRef="341">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0.21" end="1.09">
          <axisLine position="origin" positionticmark="0" legendWidth="66.5833333333333" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="342">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">v</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n</id>
                </apply>
              </math>
              <math resultRef="343">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

2. Calcolare e centrare la trasformata di Hartley.

3. Calcolare le parti pari e dispari della trasformata di Hartley discreta (queste formule presuppongono un valore di N dispari).

4. Utilizzare la funzione angle per calcolare il vettore di fase.

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  <math resultRef="354">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <indexer />
        <id xml:space="preserve" labels="*">ϕ</id>
        <id xml:space="preserve" labels="*">n</id>
      </apply>
      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">angle</id>
        <sequence>
          <apply>
            <indexer />
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">E</id>
            <id xml:space="preserve">n</id>
          </apply>
          <apply>
            <neg />
            <apply>
              <indexer />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">O</id>
              <id xml:space="preserve">n</id>
            </apply>
          </apply>
        </sequence>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

5. Definire il vettore di grandezza.

6. Utilizzare le funzioni phase, phasecor e center per creare il vettore di fase.

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  <math resultRef="358">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve" labels="*">phs2</id>
      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">phasecor</id>
        <apply>
          <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">phase</id>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">center</id>
            <apply>
              <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">dft</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">v</id>
            </apply>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

7. Utilizzare la funzione augment per confrontare i risultati di fase con quelli ottenuti tramite la trasformata di Fourier.

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  <math resultRef="360">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">augment</id>
        <sequence>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">phasecor</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">ϕ</id>
          </apply>
          <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">phs2</id>
        </sequence>
      </apply>
      <unitOverride>
        <id xml:space="preserve" labels="*">π</id>
      </unitOverride>
    </eval>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

I risultati relativi alle fasi corrispondono.

8. Utilizzare le funzioni dft e center per creare il vettore di grandezza.

9. Utilizzare la funzione augment per confrontare i risultati di grandezza con quelli ottenuti tramite la trasformata di Fourier.

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  <math resultRef="364">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">augment</id>
        <sequence>
          <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
          <apply>
            <vectorize />
            <apply>
              <absval />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">R</id>
            </apply>
          </apply>
        </sequence>
      </apply>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

I risultati relativi alla grandezza corrispondono.

Per informazioni complete sulla trasformata di Hartley e le relative applicazioni, vedere il libro di Ronald Bracewell, The Hartley Transform (Oxford University Press).

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