Differentiating a triangular wave
$begingroup$
I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.
But I couldn't work it out as I found those sins and cos dont match up
Square wave has this form
triangular wave has the following form
Can anyone show me how you differentiate triangular wave to get square? they are both sines.... I would imagine after differentiation you get a cos series for triangular wave.
Thanks every one for helping!
fourier-series
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
$endgroup$
add a comment |
$begingroup$
I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.
But I couldn't work it out as I found those sins and cos dont match up
Square wave has this form
triangular wave has the following form
Can anyone show me how you differentiate triangular wave to get square? they are both sines.... I would imagine after differentiation you get a cos series for triangular wave.
Thanks every one for helping!
fourier-series
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
$endgroup$
add a comment |
$begingroup$
I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.
But I couldn't work it out as I found those sins and cos dont match up
Square wave has this form
triangular wave has the following form
Can anyone show me how you differentiate triangular wave to get square? they are both sines.... I would imagine after differentiation you get a cos series for triangular wave.
Thanks every one for helping!
fourier-series
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
$endgroup$
I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.
But I couldn't work it out as I found those sins and cos dont match up
Square wave has this form
triangular wave has the following form
Can anyone show me how you differentiate triangular wave to get square? they are both sines.... I would imagine after differentiation you get a cos series for triangular wave.
Thanks every one for helping!
fourier-series
fourier-series
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
edited Dec 23 '18 at 21:14
Glorfindel
3,41981830
3,41981830
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
asked Sep 7 '13 at 3:42
el psy Congrooel psy Congroo
168111
168111
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why they are both $sin$ series, and the scaling explains the change from $8$ to $4$.
Alternatively, just compute the derivative of the triangular wave series and show that it is a transformed square wave.
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
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add a comment |
$begingroup$
The key observation is that a sine wave is the same as a cosine wave, but shifted by $frac pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why they are both $sin$ series, and the scaling explains the change from $8$ to $4$.
Alternatively, just compute the derivative of the triangular wave series and show that it is a transformed square wave.
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
$endgroup$
add a comment |
$begingroup$
Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why they are both $sin$ series, and the scaling explains the change from $8$ to $4$.
Alternatively, just compute the derivative of the triangular wave series and show that it is a transformed square wave.
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
$endgroup$
add a comment |
$begingroup$
Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why they are both $sin$ series, and the scaling explains the change from $8$ to $4$.
Alternatively, just compute the derivative of the triangular wave series and show that it is a transformed square wave.
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
$endgroup$
Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why they are both $sin$ series, and the scaling explains the change from $8$ to $4$.
Alternatively, just compute the derivative of the triangular wave series and show that it is a transformed square wave.
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
answered Sep 7 '13 at 3:52
Anthony CarapetisAnthony Carapetis
27.4k32865
27.4k32865
add a comment |
add a comment |
$begingroup$
The key observation is that a sine wave is the same as a cosine wave, but shifted by $frac pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
$endgroup$
add a comment |
$begingroup$
The key observation is that a sine wave is the same as a cosine wave, but shifted by $frac pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
$endgroup$
add a comment |
$begingroup$
The key observation is that a sine wave is the same as a cosine wave, but shifted by $frac pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
$endgroup$
The key observation is that a sine wave is the same as a cosine wave, but shifted by $frac pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
answered Sep 7 '13 at 3:53
Ross MillikanRoss Millikan
297k23198371
297k23198371
add a comment |
add a comment |
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