Question of the Day 20-11-2018

Consider a causal and stable LTI system with rational transfer function $ H(z) $, whose corresponding impulse response begins at $ n=0.$ Furthermore, $H\left(1\right)=\frac{5}{1}$. The poles of $ H(z) $ are $P_k=\frac{1}{\sqrt{2}}exp\left(j\frac{(2k-1)\mathrm{\pi }}{4}\right)$ for $ k=1,2,3,4. $ The zeros of $ H(z) $ are all at $ z=0. $ Let $ g(n)=j^n=h\lbrack n\rbrack. $ The value of $ g\lbrack8\rbrack $ equals ___________. (Give the answer up to three decimal places.)