$(30\%)$ To design an FIR filter by windowing, let the window function $w(n)$ with length $M+1$ and main lobe width, ${}_M,$ be one of the functions mentioned in class to approximate the desired filter $H_d(e^j).$ If the impulse response of $H_d(e^j)$ is $h_d[n]=h_d[M-n],$ then both $h_d[n]$ and $w[n]$ are causal GLP$.$ Let $h[n]=h_d[n]W[n],$ then $h[n]$ is also GLP$.$

$($a$)(5\%)$ Find the relationship of $W_e(e^j),H_e(e^j)$ and $A_e(e^j)$ if the DTFTs of $w[n],h_d[n]$ and $h[n]$ are written as the forms: $W_e(e^j)e^{-j\frac{M}{2}},H_e(e^j)e^{-j\frac{M}{2}},$ and $A_e(e^j)e^{-j\frac{M}{2}},$ respectively.

$($b$)(5\%)$ Show that $A_e(e^j)$ is even.

$($c$)(5\%)$ Find ${\displaystyle \underset{-}{\overset{}{}}}{W}_e(e^j)d.$

$($d$)(5\%)$ If $H_d(e^j)=\{\begin{array}{cc}{e}^{-j\frac{M}{2}},& ||{}_c\text{;}\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}\text{.}& \end{array}$ response is in the right. Determine the frequencies at the maximum overshoot and the minimum undershoot, respectively. No credit will be given without explanation.

$($e$)(10\%)$ Find the expression for $$ in terms of $W_e(e^j)$ and ${}_M.$