I tried to solve the below problem without success:
Let $x[n]$ be a periodic signal with period $N=8$ and Fourier series coefficients $a_k = -a_{k-4}$. A signal: $$ y[n] = (\frac{1 + (-1)^n}{2})x[n-1] $$ with period $N=8$ is generated. Denoting the Fourier series coefficients of $y[n]$ by $b_k$, find a function $f[k]$ such that $$ b_k = f[k]a_k $$
The solution mentions that $x[0] = x[+-2] = x[+-4] ... = 0$, but I haven't understood the reason. Can anyone help me please?
When I tried to solve $a_k = -a_{k-4}$, by applying Fourier series on both sides ($a_k=(1/8) \sum_{n=0}^7 x[n]e^{(-jk(\pi/4)n)}$ and $-a_{k-4}=-(1/8) \sum_{n=0}^7 x[n]e^{(-jk(\pi/4)n + j\pi n)}$), I came up with $x[0] + x[2]e^{(-j(\pi/2)k)} + x[4]e^{(-j(\pi/2)k)} = 0$ and got stuck.