Advanced Filtering

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Advanced Filtering

• Bala Muralikrishnan
• Dept. of MEES
• UNCC

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(No Transcript)
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Filtering output -not simply a smoothing process
but view as a way to suppress high frequency
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• In order to filter a profile, we assume that a
  profile is the superposition of a number of sine
  waves
• Filtering suppresses some of these surface
  wavelengths, while retaining others
• We are taking a more frequency domain approach
  to filtering here

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• In order to do this, we need to know first what
  wavelengths are present in a surface
• How do we commonly resolve a signal into its
  different sinusoidal components Fourier
  Transform
• In this presentation we will
• Review fourier transform
• Resolve a signal into its constituent sinusoids
• Perform filtering in frequency domain
• Compare this with last presentation where we
  did filtering in time or spatial domain

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Fourier Transform
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Basic Theory

• Read any Undergraduate Math book!
• Idea is that any signal can be represented as a
  sum of different sinusoids of varying amplitude
  and wavelength
• Sinwave is defined by Asin(lambaXphi)
• Where A is the amplitude and lamda is the phase.
  Phi is an additional phase term

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Fourier Series Illustration
(1/1)sin(2pi1x)
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(1/1)sin(2pi1x) (1/3)sin(2pi3x)
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(1/1)sin(2pi1x) (1/3)sin(2pi3x) (1/5)sin(2p
i5x)
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(1/1)sin(2pi1x) (1/3)sin(2pi3x) (1/5)sin(2p
i5x) (1/7)sin(2pi7x)
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(1/1)sin(2pi1x) (1/3)sin(2pi3x) (1/5)sin(2p
i5x) (1/7)sin(2pi7x) (1/9)sin(2pi9x)
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Fourier Series
F(x) (1/1)sin(2pi1x) (1/3)sin(2pi3x)
(1/5)sin(2pi5x) (1/7)sin(2pi7x)
(1/9)sin(2pi9x)
We have generated a square wave by adding
different sin waves. These sine waves have
different amplitudes and wavelengths. The
reverse is the Fourier series - Any signal can be
decomposed into many sine waves Each sine wave
is characterized by its amplitude and
wavelength(or frequency) The input signal is
time-dependant or is said to be in time domain
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Fourier Transform

• Fourier Transform is the process of transforming
  a signal from time-domain to frequency domain.
• In this example,

Amplitude and Wavelength of Different sine waves
That make up the signal
Fourier Transform
F(x) Square wave
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Fourier Transform
Fourier Transform
F(x)
Time domain
Frequency domain
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FFT

• Fast-Fourier-Transform (FFT) is an algorithm to
  perform Fourier transform in a fast and efficient
  manner
• Check Matlabs fft function
• Download myfft function

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So, going back to surface metrology, given any
profile, we can take its fourier transform and
see what frequencies are present in a surface.
Once we know this info, then we can talk
about filtering
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Do it yourself

• gtgt spacing 1/1000     in mm
• gtgt length 8                in mm
• gtgt X (0spacinglength-spacing)
• gtgt Y1 sin(2pi.X/2)
• gtgt plot(X,Y1)
• gtgt title('A sine wave data')
• gtgt xlabel('mm')
• gtgt ylabel('um')
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• gtgt Y2 sin(2pi.X)
• gtgt  myFFT(X,Y2)
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• gtgt Y3 Y1 Y2
• gtgt myFFT(X,Y3)
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• gtgt Y4 Y1 0.5Y2
• gtgt myFFT(X,Y4)

• Using this function myFFT (download it from the
  website), generate different signals and look at
  the FFT plots.
• Are the FFT plots giving you the same info as
  that is contained in the original signal?
• In other words, if the input is a sine wave of
  amplitude 1 and wavelength 1, when u do the FFT,
  do u see a peak at wavelength 1, whose amplitude
  is 1?

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Filtering
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A simple Filter

• A very simple filter is like a sieve it
  prevents some particles from falling through
  while letting other slip through.
• The size of mesh in the sieve decides what
  particles can go through
• Each particle is characterized by its size
• The sieve is a filter it is characterized by
  the size of the mesh cutoff

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So, how does the filter work?

• If the size of the data(sand particle) is
  smaller than the cutoff of the filter(sieve)
• the filter transmits the data(particle), in other
  words, the particle can fall through the sieve
• Else, the filters blocks the data

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3mm dia
20mm dia
sieve
10mm

• Diameter of the mesh hole 10mm. Thus the cutoff
  is 10mm
• This filter can transmit particles whose dia is
  less than 10mm

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Filters in Metrology

• Filters we need in Metrology follow the same
  basic idea.
• What are going to filter?
• Surface profiles
• Roundness data
• Cylindricity data
• Why do we need to filter data
• To suppress noise
• Extract the trends in data

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What exactly are we going to filter

• Any signal can be represented as sum of different
  sine waves fourier theorem
• Square wave F(x) (1/1)sin(2pi1x)
  (1/3)sin(2pi3x) (1/5)sin(2pi5x)
  (1/7)sin(2pi7x) (1/9)sin(2pi9x)
• Just as the sand particle is characterized by it
  size, a continuous time signal is characterized
  by the wavelengths of the different sinusoids
  that make up this signal
• So, the goal our filter is to allow some of the
  sinusoids to pass through, while blocking the
  others

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Output signal has only some of those
wavelengths Ouput(x) (A1,lamda1)
Input containing sinusoids of many Wavelengths F
(x) (A1,lamda1) (A2,lamda2)
Filter
The filter has suppressed the other sinusoids
(A2,lamda2)
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How do you filter a signal?

• There are two general approach to filters
• The intuitive way to this would be in the
  frequency domain
• Time domain convolution is another way to filter
  signals

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Time domain filtering

• Smoothing process using convolution
• Take the profile and the filter, convolve the
  two, extract the waviness profile

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Filtering in Frequency domain
Input signal In time domain
Look at the FFT and suppress all wavelengths
you do not want
FFT
Take the inverse FFT
Filtered output
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An example generating a data set that has known
wavelengths

• Generate a signal that has two sine waves
  superimposed on them

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y 1sin(2pix/2) 0.5sin(2pix/0.5)
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Suppress the FFT
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Suppress the FFT
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Inverse FFT
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To summarize

• We started with
• this signal
• Took the FFT of
• the signal
• Suppress fft
• did inverse fft

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Filter window

• When we suppressed the FFT, we multiplied the FFT
  array by a bunch of 0s and 1s. This array of
  0s and 1s is the filter window, commonly called
  as the filter.
• The width of the filter (the number of 1s) is
  the cutoff of the filter

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1mm
1
0
4mm
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Transmission characteristics of a filter
1mm
1
0.5
0
Wavelength of sine wave (mm)
1mm
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A real filter
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Input Y1 1sin(2pix/1) Y2
1sin(2pix/2) Y3 1sin(2pix/3)

Ouput Z1 0.8sin(2pix/1) Z2
0.2sin(2pix/2) Z3 0
Filter transmits 80 at the cutoff
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• Our input is any continuous time signal that we
  sample to produce a discrete time signal
• This signal is made up of many sinusoids (Fourier
  theorem)
• Each sinusoid is characterized by its amplitude
  and wavelength (this is obtained from the FFT)
• The filter decides which of these wavelengths it
  will allow and which it will reject (based on its
  cutoff/transmission characteristics)
• The filter may also modify the amplitudes of
  those sine wave it allows through

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How do we get the window function?

• Look at the signal in the frequency domain
• Figure out what frequencies you need decide the
  cutoff
• Come up the bunch of 1s and 0s to get the filter
  array
• Remember we are in the frequency domain so
  dont compare this with the Gaussian function
  equation which is in the time or spatial domain

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Summary

• We have looked at filtering as not simply a
  smoothing process, but from wavelength content
  perspective.
• This is important because filtering is a
  wavelength/frequency domain operation
• In order to simplify our implementation, we use
  convolution in time domain
• Underlying theory of filters is in frequency
  domain

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Next Task

• We will review all of this information again
  briefly
• Then compare Time/spatial domain filtering
  using convolution against frequency domain
  filtering using FFT
• Then thats about all for filtering we will
  move on to Parameters the last piece in the
  puzzle