Design a maximally flat lowpass filter satisfying the specification below:

\[\begin{aligned}& \omega_{\mathrm{c}}=0.4 \pi \mathrm{rad} / \mathrm{sample} \\& T_{\mathrm{r}}=0.2 \pi \mathrm{rad} / \mathrm{sample}\end{aligned}\]

where \(T_{\mathrm{r}}\) is the transition band as defined in Section 5.5. Find also the direct-form coefficients \(h(n)\), for \(n=0,1, \ldots, M\), and the alternative coefficients \(\hat{d}(n)\), for \(n=\) \(0,1, \ldots,(K-1)\).