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Fourier Transform Help Needed

1

$begingroup$

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one.

The only way I could think of to solve this was by considering it as a six-part piecewise function, but this seemed very long and complicated. Another user on this site asked me to write my function as sum of a triangle and two rectangles.

I tried adding the following waveforms
Triangular - -3 to +3 with height 3
Rectangular - -2 to 2 with height 1
Rectangular - -1 to 1 with height 1

I plotted them in MATLAB and ended up with this:

The vertical lines that connect the segments are a problem. Is there any way to just translate the required segments upwards without the lines? Also am I doing this wrong? Any help with solving this one would be really appreciated :)

functions fourier-analysis integral-transforms

share|cite|improve this question

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The lines are completely irrelevant. It's the very same function. MATLAB draws the lines because it generates the plot by simply evaluating the function on some number of points, and then "connects the dots". If, OTOH, you draw the plot yourself, you know that it has infinite slopy at $-2,-1,1,2$ and omit the line there...
  $endgroup$
  – fgp
  Mar 2 '14 at 13:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh, and the hand-drawn plot isn't strictly speaking well-defined either. It doesn't include the vertical lines, but it also fails to tell you what the function's value at the discontinuous points $-2,-1,1,2$ actually is. For $-2$ it could be $1$ or $2$. Which one it is is often indicated by placing a filled dot at the point corresponding to the value the function takes, and just a circle at the other end of the discontinuity.
  $endgroup$
  – fgp
  Mar 2 '14 at 14:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, so far that is right, $$f(x)=max(0,3-|x|)+rect(x/2)+rect(x).$$ Now the Fourier transforms are all linear, so that you can compute it for each term separately and then sum it up.
  $endgroup$
  – LutzL
  Mar 2 '14 at 14:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks so much :). So when finding the Transform can I use the lines y= 1 between -1 and +1 and y =1 between -2 and +2 for the rectangles? Something like this s18.postimg.org/iyoxji3c9/working.jpg
  $endgroup$
  – user2802349
  Mar 2 '14 at 15:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could someone tell me if that ^ method is right? :(
  $endgroup$
  – user2802349
  Mar 3 '14 at 16:03
  

  
  
  
  

add a comment | 

1

$begingroup$

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one.

The only way I could think of to solve this was by considering it as a six-part piecewise function, but this seemed very long and complicated. Another user on this site asked me to write my function as sum of a triangle and two rectangles.

I tried adding the following waveforms
Triangular - -3 to +3 with height 3
Rectangular - -2 to 2 with height 1
Rectangular - -1 to 1 with height 1

I plotted them in MATLAB and ended up with this:

The vertical lines that connect the segments are a problem. Is there any way to just translate the required segments upwards without the lines? Also am I doing this wrong? Any help with solving this one would be really appreciated :)

functions fourier-analysis integral-transforms

share|cite|improve this question

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The lines are completely irrelevant. It's the very same function. MATLAB draws the lines because it generates the plot by simply evaluating the function on some number of points, and then "connects the dots". If, OTOH, you draw the plot yourself, you know that it has infinite slopy at $-2,-1,1,2$ and omit the line there...
  $endgroup$
  – fgp
  Mar 2 '14 at 13:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh, and the hand-drawn plot isn't strictly speaking well-defined either. It doesn't include the vertical lines, but it also fails to tell you what the function's value at the discontinuous points $-2,-1,1,2$ actually is. For $-2$ it could be $1$ or $2$. Which one it is is often indicated by placing a filled dot at the point corresponding to the value the function takes, and just a circle at the other end of the discontinuity.
  $endgroup$
  – fgp
  Mar 2 '14 at 14:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, so far that is right, $$f(x)=max(0,3-|x|)+rect(x/2)+rect(x).$$ Now the Fourier transforms are all linear, so that you can compute it for each term separately and then sum it up.
  $endgroup$
  – LutzL
  Mar 2 '14 at 14:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks so much :). So when finding the Transform can I use the lines y= 1 between -1 and +1 and y =1 between -2 and +2 for the rectangles? Something like this s18.postimg.org/iyoxji3c9/working.jpg
  $endgroup$
  – user2802349
  Mar 2 '14 at 15:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could someone tell me if that ^ method is right? :(
  $endgroup$
  – user2802349
  Mar 3 '14 at 16:03
  

  
  
  
  

add a comment | 

$begingroup$

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one.

The only way I could think of to solve this was by considering it as a six-part piecewise function, but this seemed very long and complicated. Another user on this site asked me to write my function as sum of a triangle and two rectangles.

I tried adding the following waveforms
Triangular - -3 to +3 with height 3
Rectangular - -2 to 2 with height 1
Rectangular - -1 to 1 with height 1

I plotted them in MATLAB and ended up with this:

The vertical lines that connect the segments are a problem. Is there any way to just translate the required segments upwards without the lines? Also am I doing this wrong? Any help with solving this one would be really appreciated :)

functions fourier-analysis integral-transforms

share|cite|improve this question

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

$endgroup$

share|cite|improve this question

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

share|cite|improve this question

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

edited Mar 26 at 8:32

Glorfindel

3,41381930

edited Mar 26 at 8:32

Glorfindel

3,41381930

asked Mar 2 '14 at 13:48

user2802349user2802349

2416

asked Mar 2 '14 at 13:48

user2802349user2802349

2416