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tsunamiBTP

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Gibbs Phenomenon Peak Value at Discontin...

tsunamiBTP 292 Maple
series fourier discontinuity
December 06 2017
2 13

The Fourier series of waveforms with discontinuties experiences an overshoot near the discontinuity known as the "Gibbs phenomenon".  There is quite a bit of literature showing that the overshoot for a rectangle function is ~ 1.089.  What about other functions such as (1-x) or a decaying exponential for x positive?  Is there any reason to expect the overshoot ratio to be identical to the rectangle function?  I do know for a fact that the behavior of the overshoot is different for the triangle function (1-x) than for the rectangle function.  For low harmonics there is an undershoot for the triangle function case, but this is not the case for the rectangle function.  The overshoot occurs for the triangle function after a sufficient number of terms are included in the Fourier series.  The same is true for the decaying exponential.  This is illustrated in my worksheet linked below.

GIBBS_effect.mw

Does anyone know of MAPLE code that computes the theoretical overshoot if there is an infinite number of terms in the series for different waveforms or functions?

Fourier series representation of discont...

tsunamiBTP 292 Maple
fitting convergence fourier
December 01 2017
2 3

I have experimental data that in some respect can be modelled by an asymmetric triangle function.  In the link below is my worsheet examining the effect of making the asymmetric triangle symmetric.  The frequency content, 2*pi*k/T, for both triangles is still the same.  Of course the Fourier coefficients will be different, but it appears the convergence is superior for the symmetric triangle.

I could do this with my experimental data by mirroring it about t=0.  Is this a mistake?  If I can do the symmetric case I think I can fit the data with much fewer terms.  If anyone has experience on these matters I would appreciated your comments.

discontinuty_Fourier_representation.mw

Trouble with integration limits...

tsunamiBTP 292 Maple
integration
December 01 2017
1 2

Below is my attempt to evaluate an integral using the substition x=t/T.  MAPLE seems to be upset with my integration limits.  I do not know what I am doing wrong.

(int((1+t/T)*exp(-I*(2*Pi*n*t/T)), t = -T .. 0)+int((1-t/T)*exp(-I*(2*Pi*n*t/T)), t = 0 .. T))/T"(=)"-(1/2)*(cos(2*Pi*n)-1)/(n^2*Pi^2)

x := t/Tau;

1/Tau

 (1)

1/2*(int((1+x)*exp(-(I*2)*Pi*n*x), x = -1 .. 0)+int((1-x)*exp(-(I*2)*Pi*n*x), x = 0 .. 1))

Error, (in int) integration range or variable must be specified in the second argument, got t/Tau = -1 .. 0

  

``


 

Download integration_error.mw

Triangle function Fourier transform...

tsunamiBTP 292 Maple
fourier triangle inttrans
November 30 2017
2 3

I know that the Fourier transform of the triangle function is the sinc function squared, but I cannot seem to reproduce this in MAPLE.  How do I get from R2 to the square of the sinc function or do I have a screw up in defining the integral?  I need another set of eyes.  I have been at this for awhile & cannot seem to breakthrough.


 

plot(piecewise(-1 < x and x < 0, 1+x, 0 < x and x < 1, 1-x), x = -2 .. 2, linestyle = solid, thickness = 5, scaling = constrained, title = "Symmetric Triangle Wave")

  

(int((1+t)*exp(-I*(2*Pi*n*t/T)), t = -(1/2)*T .. 0)+int((1-t)*exp(-I*(2*Pi*n*t/T)), t = 0 .. (1/2)*T))/T = ((-(2*I)*n*Pi-T+I*T*n*Pi)*exp(I*n*Pi)+(2*I)*n*Pi+T-(2*I)*n*Pi+T+((2*I)*n*Pi-T-I*T*n*Pi)*exp(-I*n*Pi))/(2^2*n^2*Pi^2)
"(->)"true"(->)"true

(int((1+t)*exp(-I*(2*Pi*n*t/T)), t = -(1/2)*T .. 0)+int((1-t)*exp(-I*(2*Pi*n*t/T)), t = 0 .. (1/2)*T))/T = ((-T+I*T*n*Pi)*exp(I*n*Pi)+2*T-(T+I*T*n*Pi)*exp(-I*n*Pi))/(2^2*n^2*Pi^2)
"(->)"true"(->)"true

R1 := ((-T+I*T*n*Pi)*exp(I*n*Pi)+2*T-(T+I*T*n*Pi)*exp(-I*n*Pi))/(2^2*n^2*Pi^2)

(1/4)*((-T+I*T*n*Pi)*exp(I*n*Pi)+2*T-(T+I*T*n*Pi)*exp(-I*n*Pi))/(n^2*Pi^2)

 (1)

R1 = (1/4)*(2*T-T*(exp(-I*n*Pi)+exp(I*n*Pi))-I*(exp(-I*n*Pi)-exp(I*n*Pi))*Pi*T*n)/(n^2*Pi^2)"(->)"true

NULL

exp(-I*n*Pi)+exp(I*n*Pi)"(=)"2*cos(Pi*n)

exp(-I*n*Pi)+exp(I*n*Pi) = 2*cos(Pi*n)"(->)"false

exp(-I*n*Pi)-exp(I*n*Pi)"(=)"-(2*I)*sin(Pi*n)

exp(-I*n*Pi)-exp(I*n*Pi) = -(2*I)*sin(Pi*n)"(->)"false

R2 := (1/4)*(-2*cos(Pi*n)*T-2*sin(Pi*n)*T*n*Pi+2*T)/(n^2*Pi^2)

(1/4)*(-2*cos(Pi*n)*T-2*sin(Pi*n)*T*n*Pi+2*T)/(n^2*Pi^2)

 (2)

R2 = 2*T*(1-cos(Pi*n)-Pi*n*sin(Pi*n))/(2^2*n^2*Pi^2)"(->)"true"(->)"true"(->)"true"(->)"true"(->)"true

-2*cos(Pi*n)*T-2*sin(Pi*n)*T*n*Pi+2*T"(=)"-2*T*(cos(Pi*n)+Pi*n*sin(Pi*n)-1)``

1-cos(a) = 2*sin((1/2)*a)^2"(->)"true

(1/2)*(-exp(I*n*Pi)+1+I*n*Pi)/(Pi^2*n^2)-(1/2)*(I*exp(I*n*Pi)*Pi*n-exp(I*n*Pi)+1)*exp(-I*n*Pi)/(Pi^2*n^2) = (1/2)*(2-exp(I*n*Pi)-exp(-I*n*Pi))/(Pi^2*n^2)"(->)"true

2-exp(I*n*Pi)-exp(-I*n*Pi) = -exp(-I*n*Pi)*(exp(I*n*Pi)-1)^2"(->)"true

2-exp(I*n*Pi)-exp(-I*n*Pi) = -exp(-I*n*Pi)*(exp(I*n*Pi)-exp(I*n*Pi)*exp(-I*n*Pi))^2"(->)"true

2-exp(I*n*Pi)-exp(-I*n*Pi) = -exp(-I*n*Pi)*(exp(I*n*Pi)*(exp(I*n*Pi)-exp(-I*n*Pi)))^2"(->)"true"(->)"true

2-exp(I*Pi*n)-exp(-I*Pi*n) = -exp(-I*Pi*n)*exp((2*I)*Pi*n)*(exp(I*Pi*n)-exp(-I*Pi*n))^2"(->)"true"(->)"true

exp(I*Pi*n)-exp(-I*Pi*n)"(=)"(2*I)*sin(Pi*n)

``


 

Download symmetric_triangle_fourier_coeffs.mw

Euler's formula QUIRK...

tsunamiBTP 292
context-menu
November 30 2017
1 2

Why does MAPLE do this?  Check the link
 

exp(I*n*Pi)+exp(-I*n*Pi)"(=)"2*cos(Pi*n)

exp(-I*n*Pi)+exp(I*n*Pi) = 2*cos(n*Pi)"(->)"false

exp(-I*n*Pi)-exp(I*n*Pi)"(=)"-(2*I)*sin(Pi*n)

exp(-I*n*Pi)-exp(I*n*Pi) = -(2*I)*sin(n*Pi)"(->)"false``

``


 

Download Eulers_formula.mw

 

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