virtual instrumentation applications
TRANSCRIPT
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Fourier transforms:
Definition : refer to notes
Property and its proof refer to this linkhttp://fourier.eng.hmc.edu/e101/lectures/hand
out3/node1.html
orProperties of Fourier Transform
Ruye Wang 2009-07-05
http://fourier.eng.hmc.edu/e101/lectures/handout3/node1.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node1.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node2.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node2.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node1.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node1.htmlhttp://fourier.eng.hmc.edu/e101/lectures/handout3/node1.html -
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SAMPLING RATE
The sampling rate, sample rate, or sampling frequency(fs) defines the number of samplesper
unit of time(usually seconds) taken from a continuous signalto make a discrete signal.
For time-domainsignals, the unit for sampling rate is hertz(inverse seconds, 1/s, s1),
sometimes noted as Sa/s or S/s (samples per second). The reciprocal of the sampling frequencyis the sampling periodorsampling interval, which is the time between samples
Sampling theorem
The NyquistShannon sampling theoremstates that perfect reconstruction of a signal is
possible when the sampling frequency is greater than twice the maximum frequency of the
signal being sampled, or equivalently, when the Nyquist frequency(half the sample rate)exceeds the highest frequency of the signal being sampled. If lower sampling rates are used,
the original signal's information may not be completely recoverable from the sampled signal.
For example, if a signal has an upper band limitof 100 Hz, a sampling frequency greater than
200 Hz will avoid aliasingand would theoretically allow perfect reconstruction.
Dirichlet conditions FT:
the Dirichlet conditionsare sufficient conditionsfor a real-valued, periodic functionf(x) to be
equal to the sum of its Fourier seriesat each point wherefiscontinuous. Moreover, the
behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint
of the values of the discontinuity). These conditions are named after Peter Gustav Lejeune
Dirichlet.
http://en.wikipedia.org/wiki/Continuous_signalhttp://en.wikipedia.org/wiki/Sample_(signal)http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Continuous_signalhttp://en.wikipedia.org/wiki/Discrete_signalhttp://en.wikipedia.org/wiki/Time-domainhttp://en.wikipedia.org/wiki/Hertzhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Bandwidth_(signal_processing)http://en.wikipedia.org/wiki/Aliasinghttp://en.wikipedia.org/wiki/Sufficient_conditionhttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlethttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Sufficient_conditionhttp://en.wikipedia.org/wiki/Aliasinghttp://en.wikipedia.org/wiki/Bandwidth_(signal_processing)http://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Hertzhttp://en.wikipedia.org/wiki/Time-domainhttp://en.wikipedia.org/wiki/Time-domainhttp://en.wikipedia.org/wiki/Time-domainhttp://en.wikipedia.org/wiki/Discrete_signalhttp://en.wikipedia.org/wiki/Continuous_signalhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Sample_(signal) -
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The conditions are:
f(x) must be absolutely integrableover a period.
f(x) must have a finite number of extremain any given interval, i.e. there must be a finite
number of maxima and minima in the interval.
f(x) must have a finite number of discontinuitiesin any given interval, however the discontinuity
cannot be infinite.f(x) must be bounded
The last three conditions are satisfied iffis a function of bounded variationover a period.
http://en.wikipedia.org/wiki/Absolutely_integrablehttp://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Classification_of_discontinuitieshttp://en.wikipedia.org/wiki/Bounded_functionhttp://en.wikipedia.org/wiki/Bounded_variationhttp://en.wikipedia.org/wiki/Bounded_variationhttp://en.wikipedia.org/wiki/Bounded_functionhttp://en.wikipedia.org/wiki/Classification_of_discontinuitieshttp://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Absolutely_integrablehttp://en.wikipedia.org/wiki/Absolutely_integrable -
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The power spectrum is symmetric about the Nyquist frequency as the following illustration
shows
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CORRELATION in VI
1. 1D Cross Correlation (DBL)
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1D Cross Correlation (CDB)
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2D cross correlation (CDB) same as DBL but here input is complex value
l l
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Cross Correlation Details
1D Cross Correlation
The cross correlation Rxy(t) of the sequencesx(t) and y(t) is defined by the following equation:
where the symbol denotes correlation.
The discrete implementation of the Cross Correlation VI is as follows.
Let hrepresent a sequence whose indexing can be negative, let Nbe the number of elements in
the input sequence X, let Mbe the number of elements in the sequence Y, and assume that the
indexed elements of Xand Ythat lie outside their range are equal to zero, as shown by thefollowing equations:
xj= 0,j< 0 orj N
and
yj= 0,j< 0 orj M.
Then the CrossCorrelation VI obtains the elements of husing the following equation:
forj=(N1),(N2), , 1, 0, 1, , (M2), (M1)
h l f h l d h l i h h b
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The elements of the output sequence Rxyare related to the elements in the sequence hby
Rxyi= hi(N1)
for i= 0, 1, 2, ,N+M2.
In order to make the cross correlation calculation more accurate, normalization is required insome situations.
VI provides biased and unbiased normalization.
1. Biased normalization
If the normalizationis biased, LabVIEW applies biased normalization as follows:
Rxy(biased)j=
forj= 0, 1, 2, ,M+N2
where Rxyis the cross correlation betweenxand ywith no normalization.
2. Unbiased normalization
If the normalizationis unbiased, LabVIEW applies unbiased normalization as follows:
Rxy(unbiased)j=
forj= 0, 1, 2, ,M+N2
where Rxyis the cross correlation betweenxand ywith no normalization.
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AUTO CORRELATION VI
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AUTO CORRELATION VI
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WINDOWING AND FILTERING
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WINDOWING AND FILTERING
1. FIR Windowed Filter
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In filtered window type,the values for low cutofffreq: fland highcutofffreq: fhmust observe the following relationship:0 < f1< f2< 0.5fswhere f1is low cutofffreq: fl, f2is high cutofffreq: fh, and fsis sampling
freq: fs.
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0 Rectangle (default)
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g ( )
1 Hanning
2 Hamming
3 Blackman-Harris
4 Exact Blackman
5 Blackman6 Flat Top
7 4 Term B-Harris
8 7 Term B-Harris
9 Low Sidelobe
11 Blackman Nuttall
30 Triangle
31 Bartlett-Hanning
32 Bohman
33 Parzen
34 Welch
60 Kaiser61 Dolph-Chebyshev
62 Gaussian
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