# Fourier series spectrum argument and absolute value [closed]

Started playing with Mathematica a week ago. Now I'm on some more "advanced" stuff and really can't figure out what is happening.

I'm trying to get the spectrum of this signal, $x(t) = 110 \sin{(120\pi t)} + 50 \cos{(360\pi t + \pi/3)}$. I have solved this by hand and got the coefficients, but I can't get them at all using Mathematica.

My code:

y[t_] = 110 Sin[120 Pi t] + 50 Cos[360 Pi t + Pi/3];
X[k_] = Integrate[y[t]*Exp[-2 Pi I t k/T], {t, -T/2, T/2}]


Integration is rather slow but that is expected of symbolic integration. What I'm wondering is the amplitude of the spectrum. It just disappears. Abs[X[2]] should be 55 but it's not. Same for every other defined value of the spectrum.

I used

DiscretePlot[Abs[Limit[X[k], k -> i]], {i, -10, 10}]


to get the plot but the amplitudes aren't correct. I used Table on with the limit and complex numbers were defined for correct $k$ but their values were wrong (missing amplitude).

Tried the same with MATLAB and I could see the exact values of amplitude appearing in the "prettified" version of the equation. In Mathematica, the spectrum function seems to be lacking these constants (I did substitute T for a value at one point).

Could this be a precision problem? How can I do it right?

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## closed as too localized by Sjoerd C. de Vries, m_goldberg, Yves Klett, whuber, Oleksandr R.Apr 17 '13 at 18:09

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maybe FourierTransform will be useful. Be sure to look at the "Details and Options" bit in the docs to see how to fix the constants you want. –  acl Apr 13 '13 at 22:15
Nope, fourier transform would be used mostly for the non-periodic functions, this one is periodic, series is what I want. But!, I'm not really looking for the series, I'm just integrating by the definition but the spectrum values just aren't right. I am reading the documentation and have checked it for the functionality I want and couldn't find something similiar. –  Looft Apr 13 '13 at 22:21
Do you know the difference between = and :=? See this question for things you ought to know. –  m_goldberg Apr 13 '13 at 22:45
Well it seems to me that you can get a Fourier series using FourierSeries[110 Sin[120 Pi t] + 50 Cos[360 Pi t + Pi/3], t, 400, FourierParameters -> {1, -Pi}] (you can change the parameters as desired). –  acl Apr 13 '13 at 23:21
Your signal has a period of 1/60, meaning a frequency of 60. X[2] would correspond to the component with frequency 120 in the spectrum. However your signal has no component of that frequency. So X[2]=0. If that's what you mean by "it just disappears" then that's why. (acl's answer shows the correct spectrum upto a normalizing constant.) –  Aky Apr 14 '13 at 9:43

If for pedagogical purposes you do not wish to use FourierSeries, here's one approach:

ClearAll[x, y];
y[t_] = 110 Sin[120 Pi t] + 50 Cos[360 Pi t + Pi/3];
x[k_] := With[{T = 1/60},
Integrate[y[t]*Exp[-2 Pi I t k/T], {t, 0, T}]
]

DiscretePlot[Abs[x[i]], {i, -10, 10}]


You're not specifying T anywhere in your example. I don't actually know what you are trying to do, so modify this as appropriate.

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This is entirely my mistake, I apologize for the inconvenience. Your answer is correct and it definitely helped me discover my own mistake :D For some reason I have forgotten the 1/T in the integral. –  Looft Apr 14 '13 at 9:28
@Leolinus no problem –  acl Apr 14 '13 at 10:14