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Fourier Analysis

Fourier Analysis. D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa. Why use Fourier Analysis?. In theory: Every periodic signal can be represented by a series (sometimes an infinite series) of sine waves of appropriate amplitude and frequency.

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Fourier Analysis

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  1. Fourier Analysis D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa

  2. Why use Fourier Analysis? • In theory: Every periodic signal can be represented by a series (sometimes an infinite series) of sine waves of appropriate amplitude and frequency. • In practice: Any signal can be represented by a series of sine waves. • The series is called a Fourier series. • The process of converting a signal to its Fourier series is called a Fourier Transformation. Biomechanics Laborartory, University of Ottawa

  3. Generalized Equation of aSinusoidal Waveform • w(t) = a0 + a1 sin (2pf t + q) • w(t) is the value of the waveform at time t Biomechanics Laborartory, University of Ottawa

  4. Generalized Equation of aSinusoidal Waveform • w(t) = a0 + a1 sin (2pf t + q) • a0 is an offset in units of the signal • Offset (also called DC level or DC bias): • mean value of the signal • AC signals, such as the line voltage of an electrical outlet, have means of zero Biomechanics Laborartory, University of Ottawa

  5. Offset Changes Biomechanics Laborartory, University of Ottawa

  6. Generalized Equation of aSinusoidal Waveform • w(t) = a0 + a1 sin (2pf t + q) • a1is an amplitude in units of the signal • Amplitude: • difference between mean value and peak value • sometimes reported as a peak-to-peak value (i.e., ap-p = 2 a) Biomechanics Laborartory, University of Ottawa

  7. Amplitude Changes Biomechanics Laborartory, University of Ottawa

  8. Generalized Equation of aSinusoidal Waveform • w(t) = a0 + a1 sin (2pf t + q) • f is the frequency in cycles per second or hertz (Hz) • Frequency: • number of cycles (n) per second • sometimes reported in radians per second • (i.e., w = 2pf) • can be computed from duration of the cycle or period (T): (f = n/T) Biomechanics Laborartory, University of Ottawa

  9. Frequency Changes Biomechanics Laborartory, University of Ottawa

  10. Generalized Equation of aSinusoidal Waveform • w(t) = a0 + a1 sin (2pf t + q) • q is phase angle in radians • Phase angle: • delay or phase shift of the signal • can also be reported as a time delay in seconds • e.g., if q = p/2, sine wave becomes a cosine Biomechanics Laborartory, University of Ottawa

  11. Phase Changes Biomechanics Laborartory, University of Ottawa

  12. Generalized Equation of a Fourier Series • w(t) = a0 + S ai sin (2pfit + qi) • since frequencies are measured in cycles per second and a cycle is equal to 2p radians, the frequency in radians per second, called the angular frequency, is: w = 2pf • therefore: w(t) = a0 + S ai sin (wit + qi) Biomechanics Laborartory, University of Ottawa

  13. Alternate Form of Fourier Transform • an alternate representation of a Fourier series uses sine and cosine functions and harmonics (multiples) of the fundamental frequency • the fundamental frequency is equal to the inverse of the period (T, duration of the signal): f1 = 1/period = 1/T • phase angle is replaced by a cosine function • maximum number in series is half the number of data points (number samples/2) Biomechanics Laborartory, University of Ottawa

  14. Fourier Coefficients • w(t) = a0 + S [ bi sin (wit) + ci cos (wit) ] • bi and ci, called the Fourier coefficients, are the amplitudes of the paired series of sine and cosine waves (i=1 to n/2); a0 is the DC offset • various processes compute these coefficients, such as the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) • FFTs compute faster but require that the number of samples in a signal be a power of 2 (e.g., 512, 1024, 2048 samples, etc.) Biomechanics Laborartory, University of Ottawa

  15. Fourier Transforms of Known Waveforms • Sine wave: w(t)=a sin(wt) • Square wave: w(t)=a [sin(wt) + 1/3 sin(3wt) + 1/5 sin(5wt) + ... ] • Triangle wave: w(t)=8a/p2 [cos(wt) + 1/9 cos(3wt) + 1/25 cos(5wt) + ...] • Sawtooth wave: w(t)=2a/p [sin(wt) – 1/2 sin(2wt) + 1/3 sin(3wt) – 1/4 sin(4wt) + 1/5 sin(5wt) + ...] Biomechanics Laborartory, University of Ottawa

  16. Pezzack’s Angular Displacement Data Biomechanics Laborartory, University of Ottawa

  17. Bias = a0 = 1.0055 Harmonic Freq. ci bi Normalized number (hertz) cos(q) sin(q) power 1 0.353 -0.5098 0.3975 100.0000 2 0.706 -0.5274 -0.3321 92.9441 3 1.059 0.0961 0.2401 16.0055 4 1.411 0.1607 -0.0460 6.6874 5 1.764 -0.0485 -0.1124 3.5849 6 2.117 -0.0598 0.0352 1.1522 7 2.470 0.0344 0.0229 0.4080 8 2.823 0.0052 -0.0222 0.1242 9 3.176 -0.0138 0.0031 0.0481 10 3.528 0.0051 0.0090 0.0258 11 3.881 -0.0009 -0.0043 0.0045 Fourier Analysis of Pezzack’s Angular Displacement Data Biomechanics Laborartory, University of Ottawa

  18. Reconstruction of Pezzack’s Angular Displacement Data 8 harmonics gave a reasonable approximation raw signal (green) 8 harmonics (cyan) 4 harmonics (red) 2 harmonics (magenta) Biomechanics Laborartory, University of Ottawa

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