A Periodic DT Signal
$ x[n] = 1 + sin({2\pi \over N})n + 5cos({2\pi \over N})n + 7sin({4\pi \over N}n - {\pi \over 2}) $
The signal above x[n] is periodic with period N.
$ x[n] = 1 + {1 \over 2j}[e^{j({2\pi \over N})n}-e^{-j({2\pi \over N})n}] + {5 \over 2}[e^{j({2\pi \over N})n}+e^{-j({2\pi \over N})n}] + {7 \over 2j}[e^{j({4\pi \over N}n-{\pi \over 2})}-e^{-j({4\pi \over N}n - {\pi \over 2})}] $
wrong ---> The Fourth term on the right hand goes away since $ e^{j{\pi \over 2}n} \text{ and } e^{-j{\pi \over 2}n} $ are zero.
$ x[n] = 1 + {1 \over 2j}[e^{j({2\pi \over N})n}-e^{-j({2\pi \over N})n}] + {5 \over 2}[e^{j({2\pi \over N})n}-e^{-j({2\pi \over N})n}] $
$ x[n] = 1 + e^{j({2\pi \over N})n}[{1 \over 2j} + {5 \over 2}] - e^{-j({2\pi \over N})n}[{1 \over 2j} - {5 \over 2}] - ({7 \over 2j}e^{-j{\pi \over 2}})e^{-j({4\pi \over N}n)} + ({7 \over 2j}e^{j{\pi \over 2}})e^{j({4\pi \over N}n)} $
