the frequency domain & fourier analysis chapter 3 me 392 february 6, 2012 week 5
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The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6, 2012 Week 5. Joseph Vignola. Assignment 3. Assignment 3 was good Please have just your name in the filename for your submission The zip files should contain everything related to your assignment or lab - PowerPoint PPT PresentationTRANSCRIPT
The Frequency Domain & Fourier AnalysisChapter 3
ME 392February 6, 2012
Week 5Joseph Vignola
Assignment 3Assignment 3 was good
Please have just your name in the filename for your submission
The zip files should contain everything related to your assignment or lab
Switch lab partners each week, you should not work with the same person twice
Things to RememberSignal that come in and out of a computer don’t have enough power to do much more that drive earphones
The 2120 box can put out 12 volts
That doesn’t mean that you can hook up jumper cables to it and start your car with it.
So SO if the input impedance of what you are trying to power is low, the current demand will be high
Things to RememberLabVIEW for controlling experiments and acquiring data
Things like:Sampling frequencyMic sensitivity
Should be stored in a data file along with the data
Always store unprocessed data(at least as unprocessed as possible)
Things to RememberLabVIEW for controlling experiments and acquiring data
Matlab for processing
You will use a Matlab Script load the data file and experimental parameters into Matlab for processing
This one script will do everything including making and saving final plots
Things to RememberLabVIEW for controlling experiments and acquiring data
Matlab for processing
Don’t ever make a plot without axis labels and units
Sampled Time HistoriesLast week we talked about samples time histories In this case it is recorded audio but it could be the temperature or pressure inside an engine or any other type of data
Sampled Time HistoriesLast week we talked about samples time histories In this case it is recorded audio but it could be the temperature or pressure inside an engine or any other type of data
The next thing we want to think about is the periodic or repetitive part of a signal
Sampled Time HistoriesLast week we talked about samples time histories In this case it is recorded audio but it could be the temperature or pressure inside an engine or any other type of data
The next thing we want to think about is the periodic or repetitive part of a signal
Fourier’s Theorem tells us that any sequences can be represented as a sum of sinusoids
Sampled Time HistoriesLast week we talked about samples time histories This is an important idea because we can then represent the same sequence in terms it’s periodic content
The next thing we want to think about is the periodic or repetitive part of a signal
Fourier’s Theorem tells us that any sequences can be represented as a sum of sinusoids
Sampled Time HistoriesLast week we talked about samples time histories This is an important idea because we can then represent the same sequence in terms it’s periodic content
Sampled Time HistoriesLast week we talked about samples time histories
The frequency spectrum is a plot that tells you just how much of each frequency is in the original time history
Frequency SpectrumSome signal are dominated by a single tone (a single frequency)
Frequency SpectrumSome signal are dominated by a single tone (a single frequency)
Some have a discrete set of frequencies…
Frequency SpectrumSome signal are dominated by a single tone (a single frequency)
Some have a discrete set of frequencies…
and others have a little bit of a lot of frequencies
Fourier Transform The Fourier transform is the tool we use to generate a frequency spectrum from a time history
Fourier Transform The Fourier transform is the tool we use to generate a frequency spectrum from a time history
The Fourier transform is a lot like the Laplace transform
Fourier Transform The Fourier transform is the tool we use to generate a frequency spectrum from a time history
The Fourier transform is a lot like the Laplace transform
Fourier transform Laplace transform
Fourier Transform The Fourier transform is the tool we use to generate a frequency spectrum from a time history
The Fourier transform is a lot like the Laplace transform
Fourier transform Laplace transform
Except the transform variable is restricted to
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
Integrals become summations when working with discrete data
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
The most common DFT is the fast Fourier transform (FFT)
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
The most common DFT is the fast Fourier transform (FFT)This is what we will use on LabView and Matlab
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
Before we go any further we should note that measured sequence of values, xn are real numbers but the frequency amplitudes Xk are not.
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
Before we go any further we should note that measured sequence of values, xn are real numbers but the frequency amplitudes Xk are not.
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
It means that each of the sinusoidal components of the time history has both a magnitude and phase.
Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)
In other words both how big it is and how much it might be shifted right of left.
AliasingIf the phenomena you are measuring is changing faster that you are sampling the measured data won’t adequately represent what is happening
AliasingIf the phenomena you are measuring is changing faster that you are sampling the measured data won’t adequately represent what is happening
In fact it’s worse that not capturing the changes in the phenomena at all because will appear to have frequency content that is not real.
AliasingIf the phenomena you are measuring is changing faster that you are sampling the measured data won’t adequately represent what is happening
In fact it’s worse that not capturing the changes in the phenomena at all because will appear to have frequency content that is not real. This is called aliasing
Frequency ResolutionImagine a measurement window where you sample at a rate of 1000 samples per second for 100 ms(The window is the part of the response you can see)
Frequency ResolutionImagine a measurement window where you sample at a rate of 1000 samples per second for 100 ms
The lowest frequency oscillation you resolve in one full cycle in the measurement window
Frequency ResolutionImagine a measurement window where you sample at a rate of 1000 samples per second for 100 ms
The lowest frequency oscillation you resolve in one full cycle in the measurement window
This corresponds to the first non zero entry in the FFT
Frequency ResolutionImagine a measurement window where you sample at a rate of 1000 samples per second for 100 ms
The second non-zero entry in the FFT corresponds to two oscillations in the window
Frequency ResolutionImagine a measurement window where you sample at a rate of 1000 samples per second for 100 ms
The third non-zero entry in the FFT corresponds to three oscillations in the window
Frequency ResolutionQuestion: what do you think the first one corresponds to?
The first entry corresponds to an offset or the average value of the collection of values in the time history.
Frequency ResolutionQuestion: what do you think the first one corresponds to?
The first entry corresponds to an offset or the average value of the collection of values in the time history.
Frequency ResolutionQuestion: what do you think the first one corresponds to?
The first entry corresponds to an offset or the average value of the collection of values in the time history.
Twice the mean value
Frequency ResolutionQuestion: what if you don’t an integer number of oscillation in the window?
Three oscillation in the window
Frequency ResolutionQuestion: what if you don’t an integer number of oscillation in the window?
Then you can’t make the time history with just one sinusoid
Frequency ResolutionYou can have more than one frequency in the disturbance at a time
Frequency ResolutionYou can have more than one frequency in the disturbance at a time
They both appear in the frequency spectrum
Frequency Resolution4 oscillations in the measurement window
Frequency Resolution6 oscillations in the measurement window
Frequency Resolution8 oscillations in the measurement window
Frequency Resolution10 oscillations in the measurement window
Frequency Resolution20 oscillations in the measurement window
Frequency Resolution30 oscillations in the measurement window
Frequency Resolution40 oscillations in the measurement window
Frequency Resolution45 oscillations in the measurement window
Frequency Resolution49 oscillations in the measurement window
Frequency Resolution50 oscillations in the measurement window
That didn’t work at all
The sampled date captured nothing
Frequency Resolution50 oscillations in the measurement window
That didn’t work at all
The sampled date captured nothing
Sampling frequency = 1000samples/sec
Frequency Resolution50 oscillations in the measurement window
That didn’t work at all
The sampled date captured nothing
Sampling frequency = 1000samples/sec
We got into trouble when the frequency of the data reached half the sampling frequency
Frequency Resolution50 oscillations in the measurement window
That didn’t work at all
The sampled date captured nothing
Sampling frequency = 1000samples/sec
We got into trouble when the frequency of the data reached half the sampling frequency
This frequency is called the Nyquest frequency
Frequency ResolutionThe frequency resolution is the distance between any two frequencies in the spectrum
Frequency ResolutionThe frequency resolution is the distance between any two frequencies in the spectrum
Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?
It looks like its at 400Hz rather than 600Hz
Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?
It looks like its at 250Hz rather than 750Hz
Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?
It looks like its at 50Hz rather than 950Hz
Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?
It looks like its at 50Hz rather than 950Hz
As a mater of fact it looks just like
Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?
It looks like its at 50Hz rather than 950Hz
As a mater of fact it looks just like
This is aliasingWhen frequency content of one signal appears to be something it isn’t
Aliasing The problem with aliasing is that if a signal has noise, harmonics or any other variations that change faster then half the sampling frequency they will alias into the band of the frequency spectrum.
Aliasing The problem with aliasing is that if a signal has noise, harmonics or any other variations that change faster then half the sampling frequency they will alias into the band of the frequency spectrum.
The solution is to filter the signal before it is digitize to eliminate any oscillation at frequencies greater than half the sampling frequency.
Anti-Aliasing Filter The problem with aliasing is that if a signal has noise, harmonics or any other variations that change faster then half the sampling frequency they will alias into the band of the frequency spectrum.
The solution is to filter the signal before it is digitize to eliminate any oscillation at frequencies greater than half the sampling frequency.
For the first lab you will build an anti-aliasing filter
Anti-Aliasing Filter The problem with aliasing is that if a signal has noise, harmonics or any other variations that change faster then half the sampling frequency they will alias into the band of the frequency spectrum.
The solution is to filter the signal before it is digitize to eliminate any oscillation at frequencies greater than half the sampling frequency.
For the first lab you will build an anti-aliasing filter
This is a low pass filter that suppress frequencies above a cutoff frequency and passes frequencies below
Transfer Function Imagine that a signal, function or sequence of numbers has a spectrum.
Then we pass that signal, function or sequence of numbers through a circuit, some mathematical process of a computer algorithm that turns it into something else
The ratio of the spectrums after and before process is called the transfer function
Transfer Function Ok so maybe that is a little abstract
Physical devices like speaker or microphones, electric circuits and mathematical function we can create respond differentially to different frequencies
The ratio of the spectrums after and before process is called the transfer function
Transfer Function