Computing the Fourier series coefficients for a Discrete Time signal x[n]
$ x[n] = 7sin(7\pi n + \frac{\pi}{8})\, $
$ x[n] = \frac{7}{2j}(e^{j(7\pi n + \frac{\pi}{8})} - e^{-j(7\pi n + \frac{\pi}{8})})\, $
$ x[n] = \frac{7}{2j}e^{j3\pi n}e^{j\frac{\pi}{8}} - \frac{7}{2j}e^{-j3\pi n}e^{-j\frac{\pi}{8}}\, $
Using the fact that $ e^{j\frac{\pi}{8}} $ and $ e^{-j\frac{\pi}{8}}\, $ are equal to j...
$ x[n] = \frac{7}{2}e^{j3\pi n} + \frac{7}{2}e^{-j3\pi n}\, $
Also, $ e^{j3\pi n} = e^{j\pi n} \rightarrow e^{-j3\pi n} \times 1 = e^{-j3\pi n}e^{2j2\pi n} = e^{j\pi n}\, $
$ x[n] = \frac{7}{2}e^{j\pi n} + \frac{7}{2}e^{j\pi n} = 7e^{j\pi n}\, $
$ x[n] = a_0e^{0j\omega_0 n} + a_1e^{1j\omega_0 n}\, $
$ \rightarrow a_1 = 7, a_0 = 0\, $
$ x[n] = 0e^{0j\omega_0 n} + 7e^{j\omega_0 n}\, $
