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
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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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