\begin{exercise}{Discrete-time systems and stability}
Consider the system in the picture below. Assume a causal input ($x[n] = 0 $ for $n < 0$) and zero initial conditions.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\columnwidth]{system}
\end{figure}
\begin{enumerate}
\item Find the constant-coefficients difference equations linking $y_0[n]$,$y_1[n]$
and $y[n]$ to the input $x[n]$.
\item Find $H_0(z)$, $H_1(z)$ and $H(z)$, the transfer functions relating the input
$x[n]$ to the signals $y_0[n]$, $y_1[n]$ and $y[n]$, respectively.
\item Consider the relationship between the input and the output; is the
system BIBO stable?
\item Is the system stable internally? (i.e. are the subsystems described by
$H_0(z)$ and $H_1(z)$ stable?)
\item Consider the input $x[n] = u[n]$, where, as usual, $u[n] = 1$ for $n \geq 0$ and $u[n]=0$ for $n <0$. How do $y_0[n]$,$y_1[n]$ and $y[n]$ evolve over time? Sketch their values.
\end{enumerate}
\end{exercise}
\begin{exercise}{Filter properties I}
Assume G is a stable, causal IIR filter with impulse response $g[n]$ and transfer function $G(z)$. Which of the following statements is/are true for any choice of $G(z)$?
\begin{enumerate}
\item The inverse filter, $1/G(z)$ ,is stable.
\item The inverse filter is FIR.
\item The DTFT of $g[n]$ exists.
\item The cascade $G(z)G(z)$ is stable.
\end{enumerate}
\end{exercise}
\begin{exercise}{Filter properties II}
Consider $G(z)$, the transfer function of a causal stable LTI system. Which of the following statements is/are true for any such $G(z)$?
\begin{enumerate}
\item The zeros of $G(z)$ are inside the unit circle.
\item The ROC of $G(z)$ includes the curve $|z|=0.5$.
\item The system $H(z)=(1-3z^{-1})G(z)$ is stable.
\item The system is an IIR filter.
\end{enumerate}
\end{exercise}
\begin{exercise}{FIR Filters}
Consider the following set of complex numbers
\[
z_k = e^{j\pi(1 - 2^{-k})} \qquad k = 1, 2, \ldots, M
\]
For $M = 4$,
\begin{enumerate}
\item Plot $z_k$, $k = 1, 2, 3, 4$, on the complex plane. \item
Consider an FIR whose transfer function $H(z)$ has the following
zeros:
\[
\{z_1, z_2, z^*_1, z^*_2, -1\}
\]
and write out explicitly the expression for $H(z)$. \item How many
nonzero taps will the impulse response $h[n]$ have at most? \item
Sketch the magnitude of $H(e^{j\omega})$. \item What can you say
about this filter:
\begin{enumerate}
\item What FIR type is it? (I, II, etc.) \item Is it lowpass,
bandpass, highpass? \item Is it equiripple?
\item Is this a
``good'' filter? (By ``good'' we mean a filter which is close to 1
in the passband, close to zero in the stopband and which has a
narrow transition band).
\end{enumerate}
\end{enumerate}
\end{exercise}
\begin{exercise}{Linear Phase FIR Filters}
The equation that describes an FIR system is:
\[ y[n]=\sum_{k=0}^{M-1}b_kx[n-k], \]
where the coefficients $b_k$ represent the impulse response of the
FIR system, namely,
\[
h[n]= \left\{ \begin{array}{ll} b_n, & 0\leq n \leq M-1 \\
0,& \mbox{otherwise.} \end{array} \right .
\]
Therefore, the previous equation can be written in the following
form:
\[y[n]=\sum_{k=0}^{M-1}h[k]x[n-k], \]
while in the frequency domain the frequency response of the FIR