virtual instrumentation applications

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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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