View
224
Download
0
Category
Tags:
Preview:
Citation preview
Fourier AnalysisFourier Analysis
D. Gordon E. Robertson, PhD, FCSBD. Gordon E. Robertson, PhD, FCSB
School of Human Kinetics
University of Ottawa
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 22
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.
Why use Fourier Analysis?Why use Fourier Analysis?
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 33
Generalized Equation of aGeneralized Equation of aSinusoidal WaveformSinusoidal Waveform
w(t) = a0 + a1 sin (2 f t + )
w(t) is the value of the waveform at time t
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 44
Generalized Equation of aGeneralized Equation of aSinusoidal WaveformSinusoidal Waveform
w(t) = a0 + a1 sin (2 f t + )
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
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 55
Offset ChangesOffset Changes
offset > 0
offset < 0
zero
zero
zero
offset = 0
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 66
Generalized Equation of aGeneralized Equation of aSinusoidal WaveformSinusoidal Waveform
w(t) = a0 + a1 sin (2 f t + )
a1 is 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)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 77
Amplitude ChangesAmplitude Changes
smaller (a < 1)
larger (a > 1)
original (a = 1)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 88
Generalized Equation of aGeneralized Equation of aSinusoidal WaveformSinusoidal Waveform
w(t) = a0 + a1 sin (2 f t + ) 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., = 2 f ) can be computed from duration of the cycle or
period (T): (f = n/T)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 99
Frequency ChangesFrequency Changes
original (f = 1)
faster (f > 1)
slower (f < 1)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1010
Generalized Equation of aGeneralized Equation of aSinusoidal WaveformSinusoidal Waveform
w(t) = a0 + a1 sin (2 f t + )
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 , sine wave becomes a cosine
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1111
Phase ChangesPhase Changes
delayed (lag, > 0)
early (lead, < 0)
zero time
original ( = 0)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1212
Generalized Equation of aGeneralized Equation of a Fourier Series Fourier Series
w(t) = a0 + ai sin (2 fi t + i)
since frequencies are measured in cycles per second and a cycle is equal to 2 radians, the frequency in radians per second, called the angular frequency, is:
= 2 f therefore:
w(t) = a0 + ai sin (i t + i)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1313
Alternate Form of Fourier Alternate Form of Fourier TransformTransform
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)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1414
Fourier CoefficientsFourier Coefficients
w(t) = a0 + [ bi sin (i t) + ci cos (i t) ] 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.)
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1515
Fourier Transforms of Known Fourier Transforms of Known WaveformsWaveforms
Sine wave:w(t)=a sin(wt)
Square wave:w(t)=a [sin(t) + 1/3 sin(3t) + 1/5 sin(5t) + ... ]
Triangle wave:w(t)=8a/2 [cos(t) + 1/9 cos(3t) + 1/25 cos(5t) + ...]
Sawtooth wave:w(t)=2a/ [sin(t) – 1/2 sin(2t) + 1/3 sin(3t)
– 1/4 sin(4t) + 1/5 sin(5t) + ...]
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1616
Pezzack’s Angular Displacement Pezzack’s Angular Displacement DataData
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1717
Fourier Analysis of Pezzack’s Fourier Analysis of Pezzack’s Angular Displacement DataAngular Displacement Data
Bias = a0 = 1.0055 Harmonic Freq. ci bi Normalizednumber (hertz) cos() sin() power
1 0.353 -0.5098 0.3975 100.00002 0.706 -0.5274 -0.3321 92.94413 1.059 0.0961 0.2401 16.00554 1.411 0.1607 -0.0460 6.68745 1.764 -0.0485 -0.1124 3.58496 2.117 -0.0598 0.0352 1.15227 2.470 0.0344 0.0229 0.40808 2.823 0.0052 -0.0222 0.12429 3.176 -0.0138 0.0031 0.048110 3.528 0.0051 0.0090 0.025811 3.881 -0.0009 -0.0043 0.0045
04/19/2304/19/23Biomechanics Laborartory, Biomechanics Laborartory,
University of OttawaUniversity of Ottawa 1818
Reconstruction of Pezzack’s Reconstruction of Pezzack’s Angular Displacement DataAngular Displacement Data
raw signal (green)8 harmonics (cyan)4 harmonics (red)2 harmonics (magenta)
8 harmonics gave a reasonable approximation
Recommended