(Solved): The figure below shows a simple digitization process Figure 1: A simple digitizer. where fT(t)={ ...

The figure below shows a simple digitization process Figure 1: A simple digitizer. where $f_T(t)=\{\begin{array}{ll}f(kT)& \text{ if }t?[kT(k+1)T)\\ 0& \text{ otherwise }\end{array}$ is the sampled signal with period $T$, The interpolated output signal $f_d,d?\{0,1\}$ of the signal is given by $f_0(t)=f_T(t)=f(kT)$ for $t?[kT(k+1)T)$ for Zero-Order-Hold (ZOH) interpolation, and $f_1(t)=(1+T?t)f(kT)+(t?T)f((k+1)T)$ for $t?[kT(k+1)T)$ for First-Order-Hold (FOH) interpolat Let $e_d(t)=f(t)?f_d(t)$ be the interpolation error. Then, for ZOH, $\begin{array}{rl}{?}_0(t)& =f(t)?f_0(t)\\ & =f(t)?f(kT),\text{ for }t?[kT\\ & =f(kT)+(t?KT)f^{?}(kT)+\mathcal{O}\left(T^2\right)?f(kT),\text{ for }t?[kT(k+1)T)\text{ [ after Taylor series expansion of }\\ & =(t?KT)f^{?}(kT)+\mathcal{O}\left(T^2\right)\end{array}$ Define the reconstruction error as $E_0={\left({?}_{kT}^{(k+1)T}{e}_0^2(t)dt\right)}^{\frac{1}{2}}$ If the input signal $f(t)$ satisfies the smoothness requirement $?f^{?}(t)??M$ for all $t?{\mathbb{R}}_{+}$and some $M>0$, (a) show that $E_0?\frac{T^{1.5}}{\sqrt{3}}M.$ (b) Following similar procedure as above, derive an upper bound for $E_1$. (c) If a reconstruction error of $?$ is desired, give sampling periods for both $\mathrm{ZOH}$ and $\mathrm{FOH}$ needed to achieve this.