DT Fourier Series
If input is $ x[n] = (2 + j)cos(\frac{\pi}{3}n) + sin(\pi n - \pi) $
$ = (2 + j)(\frac{e^{j \frac{\pi}{3}n} + e^{-j \frac{\pi}{3}n}}{2})(\frac{e^{j \pi n - j \pi} - e^{-j \pi n + j \pi}}{2j}) $
$ = (\frac{2 + j}{4j})(e^{j \frac{\pi}{3}n} + e^{-j \frac{\pi}{3}n})(e^{j \pi n}e^{ - j \pi} - e^{-j \pi n}e^{j \pi}) $
$ = (\frac{2 + j}{4j})(e^{j \frac{\pi}{3}n} + e^{-j \frac{\pi}{3}n})(e^{j \pi n}(-1) - e^{-j \pi n}(1)) $
$ = (\frac{2 + j}{4j})(-e^{j \frac{4 \pi}{3}n} - e^{- \frac{2 \pi}{3}n} - e^{j \frac{2 \pi}{3}n} - e^{-j \frac{4 \pi}{3}n}) $
$ = (\frac{2 + e^{\frac{\pi}{2}}}{4j})(-e^{j \frac{4 \pi}{3}n} - e^{- \frac{2 \pi}{3}n} - e^{j \frac{2 \pi}{3}n} - e^{-j \frac{4 \pi}{3}n}) $
$ = (\frac{1}{4j})(-2e^{j \frac{4 \pi}{3}n} - 2e^{- \frac{2 \pi}{3}n} - 2e^{j \frac{2 \pi}{3}n} - 2e^{-j \frac{4 \pi}{3}n} - e^{j\frac{11 \pi}{6}n} - e^{-j \frac{\pi}{6}n} - e^{j\frac{7 \pi}{6}n} - e^{-j \frac{5 \pi}{6}n}) $
The fundamental frequency is $ \frac{\pi}{6} $ and the fundamental period is 12
Therefore $ a_{8} = \frac{-2}{4j}, a_{-4} = \frac{-2}{4j}, a_{4} = \frac{-2}{4j}, a_{-8} = \frac{-2}{4j}, a_{11} = \frac{-1}{4j}, a_{-1} = \frac{-1}{4j}, a_{7} = \frac{-1}{4j}, a_{-5} = \frac{-1}{4j} $
The other values of $ \ a_{k} = 0 $ within the fundamental period.
