Fractals > Fractals and Signal Analysis

Fractal signal analysis and generation represent a multifaceted field with diverse applications and implications; it present a rich area of exploration, blending mathematical theory, computational techniques, and practical applications. Of course, it's not all peaches and cream, there are numerous factors and problems that circle fractal models that warrant careful examination.

Techniques like fractional Brownian motion (fBm), fractional Gaussian noise (fGn), or multifractal processes offer ways to generate signals with self-similar or scale-invariant properties. However, the choice of method and the parameters involved, such as the Hurst exponent or the multifractal spectrum, can significantly impact the resulting signal's characteristics. Therefore, critical assessment of these techniques involves understanding their limitations, assumptions, and applicability to real-world phenomena.

Fractal signal analysis encompasses various methods for characterizing and quantifying fractal properties. Techniques like fractal dimension estimation, detrended fluctuation analysis (DFA), or wavelet analysis offer insights into different aspects of fractal behavior. Then again, the choice of analysis can influence the conclusions drawn about a signal's fractal nature. Fractal analysis has applications in data preprocessing, parameter selection, and statistical significance, highlighting their importance but also the challenges and limitations when drawing meaningful information.

Brownian Motion Power Spectrum

Fractal signal analysis example using JavaScript that generates a fractal signal and then presents a visualization.

Generating a fractal signal using fractional Brownian motion (fBm) and analyzing its power spectrum:

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Fractal Signal Analysis</title>
    <script src="https://cdnjs.cloudflare.com/ajax/libs/plotly.js/2.5.1/plotly.min.js"></script>
</head>
<body>
    <div id="signalPlot"></div>
    <div id="spectrumPlot"></div>
    <script>
        // Function to generate fractional Brownian motion (fBm) signal
        function generateFractalSignal(n, H) {
            const signal = [0];
            for (let i = 1; i < n; i++) {
                const delta = Math.random() * 2 - 1;
                signal[i] = signal[i - 1] + delta * Math.pow(i, -H);
            }
            return signal;
        }

// Function to calculate power spectrum of a signal
        function calculatePowerSpectrum(signal) {
            const N = signal.length;
            const spectrum = new Array(N / 2);
            for (let k = 0; k < N / 2; k++) {
                let sum = 0;
                for (let n = 0; n < N; n++) {
                    sum += signal[n] * Math.cos(2 * Math.PI * k * n / N);
                }
                spectrum[k] = Math.pow(sum, 2);
            }
            return spectrum;
        }

// Generate fBm signal
        const signalLength = 1024;
        const hurstExponent = 0.8;
        const fractalSignal = generateFractalSignal(signalLength, hurstExponent);

// Calculate power spectrum
        const powerSpectrum = calculatePowerSpectrum(fractalSignal);

// Plot signal
        const signalData = [{
            x: Array.from({ length: signalLength }, (_, i) => i),
            y: fractalSignal,
            mode: 'lines',
            type: 'scatter',
            name: 'Fractal Signal'
        }];
        Plotly.newPlot('signalPlot', signalData, { title: 'Fractal Signal' });

// Plot power spectrum
        const spectrumData = [{
            x: Array.from({ length: powerSpectrum.length }, (_, i) => i),
            y: powerSpectrum,
            mode: 'lines',
            type: 'scatter',
            name: 'Power Spectrum'
        }];
        Plotly.newPlot('spectrumPlot', spectrumData, { title: 'Power Spectrum' });
    </script>
</body>
</html>

$fractal brownian noise power spectrum$

Fractal Signal (Midpoint Displacement Algorithm)

This algorithm iteratively subdivides a line segment and displaces the midpoint by a random amount.

<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Fractal Signal Visualization</title>
  
  <script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
</head>
<body>
  
  <div id="plot"></div>

// Midpoint displacement
      function displace(start, end, amplitude) {
        if (end - start < 2) return;
        let mid = Math.floor((start + end) / 2);
        let displacement = (Math.random() - 0.5) * amplitude;
        signal[mid] = (signal[start] + signal[end]) / 2 + displacement;
        amplitude *= roughnessFactor;
        displace(start, mid, amplitude);
        displace(mid, end, amplitude);
      }

displace(0, n - 1, amplitude);
      return signal;
    }

// Generate fractal signal
    const nPoints = 1000; // Number of points
    const roughness = 0.7; // Roughness parameter (controls the fractal structure)
    const fractalSignal = generateFractalSignal(nPoints, roughness);

// Create trace for the fractal signal
    const trace = {
      x: Array.from({ length: nPoints }, (_, i) => i),
      y: fractalSignal,
      mode: 'lines',
      type: 'scatter'
    };

// Layout for the plot
    const layout = {
      title: 'Fractal Signal Visualization',
      xaxis: {
        title: 'Time'
      },
      yaxis: {
        title: 'Value'
      }
    };

// Plot the fractal signal
    Plotly.newPlot('plot', [trace], layout);
  </script>
</body>
</html>

$fractal signal generated using the midpoint algorithm$

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