DT signal
$ X[n] = 3\cos(3 \pi n + \pi)\, $
Find a value for k that makes N an integer:
$ N = \frac{2\pi}{3\pi} K \, $
K equals 3 results in a integer.
$ N = \frac{2\pi}{3\pi} 3 \, $
$ N = 2 \, $
Prove that it is periodic:
x(0)= -3 ; x(1)=3 ; x(2)=-3 ; x(3) = 3 etc.
$ a_k = \frac{1}{N} \sum^{N-1}_{n = 0} X[n] e^{-jk\frac{2\pi}{N} n} $
$ a_k = \frac{1}{2} \sum^{1}_{n = 0} X[n] e^{-jk\pi n} $
$ a_0 = \frac{1}{2} \sum^{1}_{n = 0} X[n] e^{0} $
$ a_0 = \frac{1}{2} (-3 + 3) = 0 $
$ a_1 = \frac{1}{2} \sum^{1}_{n = 0} X[n] e^{-j\pi n} $
$ a_1 = \frac{1}{2} (-3 * e^0 + 3 * e^{-j\pi}) $
$ a_1 = \frac{1}{2} (-3 * 1 + 3 * -1) = -3 $
