Fourier Series for DT signals
Let $ x[n]\, $ be a periodic DT signal with fundamental period N.
Then $ x[n]=\sum_{k=0}^{N-1} a_k e^{jk\frac{2\pi}{N} n} $
where $ a_k=\frac{1}{N}\sum_{n=0}^{N-1} x[n] e^{-jk\frac{2\pi}{N} n} $
note that $ \frac{2\pi}{N} =\omega_0 $
Now consider the signal $ x[n]=sin(3 \pi n)\, $
It's periodic because $ \frac{\omega_0}{2\pi} = \frac{3\pi}{2\pi} =1.5 $ is a rational number.
Notice that $ x[0]= 1, x[1]=-1, x[2]=1, x[3]=-1 \, $ etc
Thus the fundamental period is 2.
Then...
$ a_0=\frac{1+(-1)}{2}=0 $
$ a_1=\frac{1}{2}\sum_{n=0}^{1} x[n] e^{-j1\frac{2\pi}{2} n} $
$ a_1=\frac{1}{2}\sum_{n=0}^{1} x[n] e^{-j\pi n} $
$ a_1=\frac{1}{2}(x[0] e^{0} + x[1]e^{-j\pi})=1 $
Finally,
$ a_{k_{even}}=0\, $
$ a_{k_{odd}}=1\, $
