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

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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 Presentation

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Page 1: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

The Frequency Domain & Fourier AnalysisChapter 3

ME 392February 6, 2012

Week 5Joseph Vignola

Page 2: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 3: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 4: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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)

Page 5: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 6: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Things to RememberLabVIEW for controlling experiments and acquiring data

Matlab for processing

Don’t ever make a plot without axis labels and units

Page 7: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 8: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 9: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 10: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 11: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 12: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 13: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency SpectrumSome signal are dominated by a single tone (a single frequency)

Page 14: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency SpectrumSome signal are dominated by a single tone (a single frequency)

Some have a discrete set of frequencies…

Page 15: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 16: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Fourier Transform The Fourier transform is the tool we use to generate a frequency spectrum from a time history

Page 17: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 18: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 19: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 20: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Fast Fourier TransformWhen we have sampled data rather than a continuous signal we use a “Discrete Fourier Transform” (DFT)

Page 21: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 22: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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)

Joseph Vignola
Page 23: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Joseph Vignola
Page 24: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Joseph Vignola
Page 25: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Joseph Vignola
Page 26: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Joseph Vignola
Page 27: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Joseph Vignola
Page 28: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

AliasingIf the phenomena you are measuring is changing faster that you are sampling the measured data won’t adequately represent what is happening

Joseph Vignola
Page 29: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Joseph Vignola
Page 30: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Joseph Vignola
Page 31: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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)

Page 32: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 33: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 34: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 35: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 36: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Page 37: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Page 38: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 39: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionQuestion: what if you don’t an integer number of oscillation in the window?

Three oscillation in the window

Page 40: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 41: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionYou can have more than one frequency in the disturbance at a time

Page 42: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionYou can have more than one frequency in the disturbance at a time

They both appear in the frequency spectrum

Page 43: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution4 oscillations in the measurement window

Page 44: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution6 oscillations in the measurement window

Page 45: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution8 oscillations in the measurement window

Page 46: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution10 oscillations in the measurement window

Page 47: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution20 oscillations in the measurement window

Page 48: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution30 oscillations in the measurement window

Page 49: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution40 oscillations in the measurement window

Page 50: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution45 oscillations in the measurement window

Page 51: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution49 oscillations in the measurement window

Page 52: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution50 oscillations in the measurement window

That didn’t work at all

The sampled date captured nothing

Page 53: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency Resolution50 oscillations in the measurement window

That didn’t work at all

The sampled date captured nothing

Sampling frequency = 1000samples/sec

Page 54: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 55: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 56: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionThe frequency resolution is the distance between any two frequencies in the spectrum

Page 57: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionThe frequency resolution is the distance between any two frequencies in the spectrum

Page 58: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?

It looks like its at 400Hz rather than 600Hz

Page 59: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?

It looks like its at 250Hz rather than 750Hz

Page 60: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Frequency ResolutionWhat happens if the frequency of the signal exceeds half the Nyquest frequency?

It looks like its at 50Hz rather than 950Hz

Page 61: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 62: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 63: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Page 64: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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.

Page 65: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 66: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 67: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 68: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

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

Page 69: The Frequency Domain & Fourier Analysis Chapter 3 ME 392 February 6,  2012 Week 5

Transfer Function