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Characteristics of time-dependent
measurand
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Classification of time dependent
measurands
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Examples of time-dependent functions
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Simple harmonic relation• Any two variables can have a simple harmonic relation if the
second derivative of one w.r.t. the other is related to the
function itself.
• e.g. if the distance of a circular motion is described by
2
sin
first derivative (velocity) is
cos
and the second derivative (acceleration) is
sin
where is called the circular frequency ( rad/s)
while, the cyclic frequency (cycle p
x A t
the
xv A t
t
va A t t
er second, Hertz, f)is given by
=2 f
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The scotch-Yoke mechanism
It illustrates the definition of both cyclic and
circular frequency
When the mass block rotates a complete
cycle in a time t
The number of cycles per time (cyclic
frequency) is calculated by
f=1/t
while the circular frequency is
=2f
Example:
given that the time of one cycle is 0.2 sec
what is the cyclic and circular frequency
F=1/t=5 cycle/sec (Hz)
= 2*3.14*5=31.4 rad/s=31.4*57.3 degree/sec
The mass is oscillating between two
peaks to form a harmonic motion.
One cycle t=1/f
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Phase difference
In this case both of the mass block and the piston
have the same frequency, but they may have a
phase shift, i. e. there is a difference between the
start and end points for their cycles.
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Different units for cyclic motion
Angular frequency ω (Ordinary) frequency
2π radians per second exactly 1 hertz (Hz)
1 radian per second approximately 0.159155 Hz
1 radian per secondapproximately 57.29578 degrees
per second
1 radian per second
approximately 9.5493 revolutions
per minute (rpm)
0.1047 radian per second approximately 1 rpm
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Fourier Analysis
• Putting any function in the form of the summation of its
trigonometric components (Sinusoidal functions).
• The special case is when the function is itself a periodic function.
Then the Fourier series is used to transform it to a summation of
infinity number of sines and cosines.
• The more general case is when the function is not periodic, then the
Fourier transform is the way to analyze it to its sinusoidal
constituents
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The analogy of Fourier transform in
nature
A complex function can be represented as a summation of several more simple
sinusoidal functions. This is analogous to the analysis of the white light to many
simpler sinusoidal frequency-dependent components.
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Mathematical representation of periodic
measurand
• Any periodic function can be represented in terms of
its sinusoidal components. This is the main purpose
of Fourier series analysis
01
0
cos sin
is the circular frequency
t is time
n is the number of the harmonic
is the constant component
is the variating component
,
n nn
n
f t A A n t B n t
where
A
A B
are the coefficients for each frequency component of the signal
n
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Trigonometric identities
Using the relations between the
trigonometric functions, therepresentation of the periodic
functions can be simplified to:
0 1
0
1
*
0
1
2 2
1 * 1
cos sin
can be written as
cos( )
sin( )
tan , tan
n nn
n
n
n
n
n n n
n n
n n
f t A A n t B n t
f t A C n t
or
f t A C n t
where
C A B
B A
A B
2
0
2
2
2
2
2
1
2 cos( )
2 sin( )
1 21,2,3,...... and T is the periof of i.e. T=
T
T
T
n
T
T
n
T
A f t dt T
A f t n t dt
T
B f t n t dt T
where
n f t f
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Even and odd functions
• The definition of even and odd functions:
• If f(-t) = f(t) …… it is an even function (e.g cos
• If f (-t)= - f(t) …… it is an odd function (e.g. sin
• It is helpful when you are going to calculate the coefficients of
Fourier series.• If the function is even then all sin’s part will be zero. On the other
hand, if the function is odd then all cosin’s part is zero.
1
1
So for even functions
( ) cos
for odd functions
( ) sin
n
n
nn
f t A n t
and
f t B n t
E l t fi d F i i ffi i t
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Example to find Fourier series coefficients
for a periodic function
From the first look This is an odd function, why?
Then the function can be represented by
while the frequency and the period can be concluded from the
drawing as
The coefficients (1, , 3, ……. ) are calculated as follows:
1
( ) sinn
n
f t B n t
1 10.1 Hertz
10
and=2 2 *3.14 *0.1 0.628 rad/sec
f T
f
2
2
2 sin( )
T
n
T
B f t n t dt T
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0 5
5 0
0 5
5 0
2 2 21 sin 1 sin
10 10 10
2 10 2 10 2 = cos cos
10 2 10 2 10
2 10 4 = 1 cos cos 110 2
the function can be
n
nt nt B dt dt
nt nt
n n
n nn n
then
written as
4 2 4 6 4 10sin sin sin .........
10 3 10 5 10 f t t t t
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Periodic Function interpretation
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Heterodyne and beat frequencies
This is a useful phenomenon that resulted when two signals that have the same
amplitude and close frequencies are mixed. Then two new frequencies are generated.
One important application based on this phenomenon is mixing the high frequency
from broadcast stations with a local oscillator signal that have a close frequency signal.
The output will contain a high and low frequency components. By filtering the high
frequency component, the lower frequency signal that is easier in dealing with can be
used to extract data.
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Frequency spectrum
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Frequency spectrum
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Steps for Fourier transform
the sampling frequency, Nyquist frequency and
resolution
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Effect of sampling frequency and number of samples on
frequency analysis
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ℎ > 2 ×
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FT of common function
• Sine wave is represented by a impulse in the frequency domain at its own frequency
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FT of common function• An impulse is represented by a constant value over all the frequency range (real part is a cosine, imaginary one
a sine)
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FT of common function• Rectangular windows are represented by a sinc wave in the frequency domain (this
explains a lot in frequency analysis…)
Leakage and windowing
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Leakage and windowing• To limit this effect a different set a windows can
be chosen, depending on what is required
• One commonly used is the “hanning” one