
Complex Fourier series
Hi there;
I have a function;
$\displaystyle x(t) = cos (\displaystyle\frac{2\pi}{5}t) + 3sen (\displaystyle\frac{2\pi}{7}t)$
Where T=35 (period)
First is asking me to develope de complex Fourier series. Because of the type of function, I know that I can directly substitude by the Euler formula,
$\displaystyle x(t)=\dfrac{1}{2}\left(e^{\frac{2\pi i}{5} t}+e^{\frac{2\pi i}{5} t}\right)+\dfrac{3}{2i}\left(e^{\frac{2\pi i}{5} t}e^{\frac{2\pi i}{5} t}\right)=$
$\displaystyle =\dfrac{1}{2}\left(e^{7\cdot \frac{2\pi i}{35} t}+e^{7\cdot \frac{2\pi i}{35} t}\right)+\dfrac{3}{2i}\left(e^{7\cdot \frac{2\pi i}{35} t}e^{7\cdot \frac{2\pi i}{5} t}\right)=$
$\displaystyle =\dfrac{1}{2}e^{7\cdot \frac{2 \pi i t}{35}}+\dfrac{1}{2}e^{7\cdot \frac{2 \pi i t}{35}}\dfrac{3i}{2}e^{5\cdot \frac{2 \pi i t}{35}}+\dfrac{3i}{2}e^{5\cdot \frac{2 \pi i t}{35}}$
Second part of the problem is asking me to find de transfer function H(s) which represents the relation between the input X(t) and the output Y(t) represented by the ODE:
$\displaystyle y''(t)+4y'(t)+3y(t)=x(t)$
knowing that I have that $\displaystyle H(s)=\displaystyle\frac{1}{s^2+4s+3}$
the last part of the problem is asking me to find the complex Fourier series of Y(t).
and here is where I'm a bit lost, I'm not too sure if I have to follow this approach:
$\displaystyle L^{1}(L(Y(t))=L^{1}H(s)*L^{1}(L(X(t))$
if I have to, I find my self having to operate complex exponentials and real ones, can I just operate them as normal?
Any suggestions?
Thank you