Transcript Slide 1
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.
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.
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Generalized Equation of a
Sinusoidal Waveform
= a0 + a1 sin (2p f t + q)
w(t) is the value of the waveform at time t
w(t)
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Generalized Equation of a
Sinusoidal Waveform
= a0 + a1 sin (2p f t + q)
a0 is an offset in units of the signal
Offset (also called DC level or DC bias):
w(t)
mean value of the signal
AC signals, such as the line voltage of an
electrical outlet, have means of zero
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Offset Changes
zero
zero
zero
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offset = 0
offset > 0
offset < 0
5
Generalized Equation of a
Sinusoidal Waveform
= a0 + a1 sin (2p f t + q)
a1 is an amplitude in units of the signal
Amplitude:
w(t)
difference between mean value and peak
value
sometimes reported as a peak-to-peak value
(i.e., ap-p = 2 a)
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Amplitude Changes
original
(a = 1)
smaller
(a < 1)
larger
(a > 1)
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Generalized Equation of a
Sinusoidal Waveform
= a0 + a1 sin (2p f t + q)
f is the frequency in cycles per second or
hertz (Hz)
Frequency:
w(t)
number of cycles (n) per second
sometimes reported in radians per second
(i.e., w = 2p f )
can be computed from duration of the cycle or
period (T): (f = n/T)
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Frequency Changes
original
(f = 1)
slower
(f < 1)
faster
(f > 1)
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Generalized Equation of a
Sinusoidal Waveform
= a0 + a1 sin (2p f t + q)
q is phase angle in radians
Phase angle:
w(t)
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
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Phase Changes
original
(q = 0)
delayed
(lag, q > 0)
early
(lead, q < 0)
zero time
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Generalized Equation of a
Fourier Series
= a0 + S ai sin (2p fi t + 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 = 2p f
therefore:
w(t) = a0 + S ai sin (wi t + qi)
w(t)
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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)
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Fourier Coefficients
w(t) = a0 + S [ bi sin (wi t) + ci cos (wi 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.)
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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) + ...]
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Pezzack’s Angular Displacement
Data
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Fourier Analysis of Pezzack’s
Angular Displacement Data
Bias = a0 = 1.0055
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Harmonic Freq.
number (hertz)
ci
cos(q)
bi
sin(q)
1
2
3
4
5
6
7
8
9
10
11
-0.5098
-0.5274
0.0961
0.1607
-0.0485
-0.0598
0.0344
0.0052
-0.0138
0.0051
-0.0009
0.3975 100.0000
-0.3321 92.9441
0.2401 16.0055
-0.0460
6.6874
-0.1124
3.5849
0.0352
1.1522
0.0229
0.4080
-0.0222
0.1242
0.0031
0.0481
0.0090
0.0258
-0.0043
0.0045
0.353
0.706
1.059
1.411
1.764
2.117
2.470
2.823
3.176
3.528
3.881
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Normalized
power
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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)
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