Example of Computation of Fourier series of a CT SIGNAL
A practice problem on "Signals and Systems"
Fourier sum definition
The function as defined by summing fourier coefficients $ \,a_k $ is defined as:
$ x(t)=\sum^{\infty}_{k=-\infty} a_k e^{jk\omega_0 t}\, $
Example of a periodic CT signal
The following is a periodic signal:
$ \,x(t)=(1+j)cos(\pi t)+sin(2\pi t) $
Using Eulers formula, we can interpret this function in terms of exponentials which can then be used to compute the $ \,a_k $ values for a Fourier series:
$ \,x(t)=(1+j)\frac {e^{j\pi t}+e^{-j \pi t}}{2} + \frac {e^{j2 \pi t}-e^{-j2 \pi t}}{2j} $
Now splitting up:
$ x(t)=\frac{1+j}{2}e^{j\pi t}+\frac{1+j}{2}e^{-j\pi t}+\frac{1+j}{2j}e^{j2\pi t}+\frac{-1-j}{2j}e^{-j2\pi t} $
choose $ \,\omega_0 $ as $ \,\pi $, the smallest period between the two parts.
so this function becomes:
$ x(t)=\sum^{\infty}_{k=-\infty} a_k e^{jk\pi t}\, $
Which very nearly matches our function, we only need solve or point out our $ \,a_k $ values.
$ a_1=\frac{1+j}{2} $
$ a_{-1}=\frac{1+j}{2} $
$ a_2=\frac{1+j}{2j} $
$ a_{-2}=\frac{-1-j}{2j} $
All other $ \,a_k $ values are zero.
