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Find the exponential Fourier series of x(t) = 2 + 0.5cos(t+45)+2cos(3t)-2sin(4t+30)

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Oct7-07, 04:47 PM
P: 492
1. The problem statement, all variables and given/known data

For the periodic signal

[tex]x(t)\,=\,2\,+\,\frac{1}{2}\,cos\left(t\,+\,45^{\circ}\right)\,+\,2\,cos \left(3\,t\right)\,-\,2\,sin\left(4\,t\,+\,30^{\circ}\right)[/tex]

Find the exponential Fourier series.

2. Relevant equations

Euler’s Formula
[tex]x(t)\,=\,A\,cos\left(\omega_0\,t\,+\,\phi\right)\,=\,A\,\left[e^{j\,\left(\omega_0\,t\,+\,\phi\right)}\,+\, e^{-j\,\left(\omega_0\,t\,+\,\phi\right)}\right][/tex]

3. The attempt at a solution

To get [itex]\omega_0[/itex], we need to find the least common denominator between the following periods…


Which is [itex]2\,\pi[/itex].

So, now I use the formula [itex]\omega_0\,=\,\frac{2\,\pi}{T}[/itex]…


Now, I use Euler’s formula to convert the cos and sin to exponentials…


I don’t know if the last term (sin) is supposed to be kept as [tex]4\,t\,+\,30^{\circ}[/tex]

OR changed to a cosine to fit Euler’s formula by subtracting ninety degrees: [tex]4\,t\,-\,60^{\circ} [/tex]

I assumed the latter, is that correct?
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