Signal Processing Using MATLAB

1
Signal Processing Using MATLAB
2
Introduction

• Any signal can be represented as a sum of
  different sinusoids.
• The signals we are interested in Metrology, like
  surface finish, are no different from other
  signals found in nature, like voltages, currents
  etc.
• We use Fourier transform to convert the time
  domain signal (like surface finish) into the
  frequency domain where we can relate specific
  frequency bands to its sources. Our signals are
  spatial, i.e,  a function of distance in space.

3
Sampled data
y4
y3
Y(um) 12.2 13.4 15.5 14.6 12.8
y2
yn
y1
spacing
If the data is surface finish data, then we know
that X(0,1,2,3,4)spacing (in um or mm)
4
Data representation

• Sampled data is a sequence of numbers
• We have to deal with arrays/matrices of numbers
• Surface finish array of numbers represent the
  relative heights of points on a surface
• CMM matrix represents (x,y,z) coordinates of
  different points

5
Square Wave
(1/1)sin(2pi1x) (1/3)sin(2pi3x) (1/5)sin(2p
i5x) (1/7)sin(2pi7x) (1/9)sin(2pi9x)
6
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 sine 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
7
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
8
(No Transcript)
9
lamda1
A1,lamda1

A2,lamda2

A3,lamda3
180 deg

90
270
90
180
270
360
Sampling interval
0 deg
10
Sampling in terms of wavelength

• The larger the sampling interval, the more
  information we lose in other words, we cannot
  capture sinusoids whose wavelengths are smaller
  than the sampling interval
• If you know the smallest wavelength you are
  interested in capturing, then your sampling
  interval should be half that wavelength shannon
  sampling theorem
• All measuring instruments have a certain default
  sampling interval, below which we cannot go.

11
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

12
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(2pix/1)
  (1/3)sin(2pix/3) (1/5)sin(2pix/5)
  (1/7)sin(2pix/7) (1/9)sin(2pix/9)
• 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

13
Output signal has only some of those
wavelengths Output(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)
14
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

15
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
16
An example generating a data set that has known
wavelengths

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

17
y 1sin(2pix/2) 0.5sin(2pix/0.5)
18
Suppress the FFT
19
Suppress the FFT
20
Inverse FFT
21
To summarize

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

22
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

23
FFT Using MATLAB

• Type help fft for Fourier transform
• Type help ifft for inverse Fourier transform
• 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')
•  gtgt myFFT(X,Y1)

24
(No Transcript)
25

• gtgt Y2 sin(2pi.X)
• gtgt  myFFT(X,Y2)

26
(No Transcript)
27

• gtgt Y3 Y1 Y2
• gtgt myFFT(X,Y3)

28
(No Transcript)
29

• gtgt Y4 Y1 0.5Y2
• gtgt myFFT(X,Y4)

30
(No Transcript)
31
Example

• Consider a surface profile obtained from a sine
  wave artifact. Let us assume that data points
  were collected from a surface finish instrument
  over a trace of 8mm with a spacing of 1um. Let
  the points be collected so that the spacing
  between adjacent points be 1um.
• gtgt  spacing 1/1000                             
  in mm
• gtgt length 8                                    
        in mm
• gtgt X (0spacinglength-spacing)          
• gtgt Y sin(2pi.X)                              
  
• Let us now do the FFT
• gtgt F fft(Y)
• gtgt size(F)                        You will
  notice that F is of the same length as Y, 8000
  points in this case

32
What do you observe?

• 1. The first term is always the DC term
• 2. The wavelengths decrease from the second term
  to the mid and then increases again. Since there
  is a repetition, we can discard the second half
  of the result.
• 3. The largest wavelength we can measure is the
  length of measurement itself (in this case, 8mm)
• 4. The smallest wavelength we can capture is
  twice the sampling interval (21um).

33
HW

• Implement Ra, Rq, Rz