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Question: Fourier Transform result not understood

Question:Fourier Transform result not understood

Jan GvK 20 Maple 17
fourier-transform
January 27 2014
0

Hello,

I understand that the question is not really Maple related, but I still hope for some help.


See the worksheet below. I defined a pure sine wave and determined the complex Fourier coefficients for it which I used to plot the amplitude and power spectra. It is easy to see the relations in terms of amplitude and power between the time and frequency signal.

The Fourier Transform of the sine wave logically shows the Dirac distribution, but I can't see the relation in terms of amplitude and power to the original time signal. Taking the integral of the transformed signal (A) wil result in a step of Pi at w=-1 and again at w=1. What am I missing here?

Thanks

> 

restart; with(inttrans); with(plots); with(DynamicSystems)

 

Define a signal:

 

> 

T := 2*Pi;

2*Pi

  

sin(t)

 (1)

 

Determine the waveform power:

 

> 

F := (int(f^2, t))/T+C:
> 

C := simplify(solve(subs(t = 0, F) = 0, C)):
> 

eval((int(f^2, t = -(1/2)*T .. (1/2)*T))/T)

1/2

 (2)
> 

plot([f, f^2, F], t = -Pi .. Pi, gridlines = true)

 

 

Determine the complex Fourier series coefficients and plot the spectra:

 

> 

q := proc (n) options operator, arrow; (int(f*exp(-(2*I)*n*Pi*t/T), t = -(1/2)*T .. (1/2)*T))/T end proc:
> 

simplify(q(n))

I*sin(Pi*n)/(Pi*(n^2-1))

 (3)
> 

ComplexCoefficients := evalf(`<,>`(seq(q(n), n = -1 .. 1)))

ComplexCoefficients := Vector(3, {(1) = .5000000000*I, (2) = 0., (3) = -.5000000000*I})

 (4)
> 

B := evalf(`<,>`(seq(sqrt(Re(q(n))^2+Im(q(n))^2), n = -3 .. 3))):

 
> 

C := evalf(`<,>`(seq(Re(q(n))^2+Im(q(n))^2, n = -3 .. 3))); -1; DiscretePlot(C, -3, 1, titlefont = ["ARIAL", "bold", 14], title = "Power Spectrum", color = "Red", gridlines = true, style = stem)

 

So, the signal power for f of 1/2 can be found directly within the power spectrum plot "(2*1/(4))."

 

I would expect to be able to directly see the amplitude and power relation to the time signal from the Fourier Transform of f but i can't.

 

> 

A := fourier(f, t, w);

I*Pi*(Dirac(w+1)-Dirac(w-1))

 (5)


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