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The value of this is <math>\frac{2\pi}{2\pi}\!</math>, which coincidently, by no planning of mine, turns out to be <math>1\!</math>. | The value of this is <math>\frac{2\pi}{2\pi}\!</math>, which coincidently, by no planning of mine, turns out to be <math>1\!</math>. | ||
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| + | ==Solution== | ||
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| + | We know that the equation for signal coefficients is as follows: | ||
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| + | <math>a_k=\frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt</math>. | ||
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| + | And the equation for fourier series of a function is as follows: | ||
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| + | <math>x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t}</math> | ||
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| + | We first put our signal into the first equation, and we get this monster: | ||
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| + | <math>a_0=\frac{1}{2\pi}\int_0^{2\pi}[4cos(2t) + (3j)sin(3t)]e^{0}dt</math> | ||
Revision as of 17:39, 24 September 2008
The signal
The signal I chose to use is as follows:
$ x(t) = 4cos(2t) + (3j)sin(3t)\! $
The fundamental period, denoted as $ T\! $, of this signal is $ 2\pi\! $. The fundamental frequency, denoted $ \omega_0\! $, is defined as:
$ \omega_0 = \frac{T}{2\pi}\! $
The value of this is $ \frac{2\pi}{2\pi}\! $, which coincidently, by no planning of mine, turns out to be $ 1\! $.
Solution
We know that the equation for signal coefficients is as follows:
$ a_k=\frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt $.
And the equation for fourier series of a function is as follows:
$ x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t} $
We first put our signal into the first equation, and we get this monster:
$ a_0=\frac{1}{2\pi}\int_0^{2\pi}[4cos(2t) + (3j)sin(3t)]e^{0}dt $
