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\section{Exercises}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Interleaving]
Consider two two-sided sequences $\mathbf{h}$ and $\mathbf{g}$ and consider a third sequence $\mathbf{x}$ which is built by interleaving the values of $\mathbf{h}$ and $\mathbf{g}$ as:
\[
\mathbf{x} = \bigl[ \ldots \,\,\, h[-2] \,\,\, g[-2] \,\,\, h[-1] \,\,\, g[-1] \,\,\, h[0] \,\,\, g[0] \,\,\, h[1] \,\,\, g[1] \,\,\, h[2] \,\,\, g[2] \,\,\, \ldots \bigr]
\]
with $x[0] = h[0]$.
\begin{enumerate}
\item Express the $z$-transform of $\mathbf{x}$ in terms of the $z$-transforms $H(z)$ and $G(z)$.
\item Assume that the ROC for $H(z)$ is $0.64 < |z| < 4$ and that the ROC for $G(z)$ is $0.25 < |z| < 9$. What is the ROC of $X(z)$?
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[ROC subtleties]
Remember from classic calculus that, while the harmonic series is divergent, the harmonic series with alternating sign is convergent:
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln\frac{1}{2}.
\]
\begin{enumerate}
\item Using the sequence $\mathbf{x}$ defined by $x[n] = (1/n) u[n-1]$, show that we need to consider the \textit{absolute} convergence in determining the ROC if we want it to have circular symmetry.
\item Compute the transfer function of a system whose impulse response is equal to $\mathbf{x}$.
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\begin{enumerate}
\item Consider the \ztrans\ of $\mathbf{x}$
\[
X(z) = \sum_{n=1}^{\infty} \frac{z^{-n}}{n};
\]
the transform converges in $z=-1$ but not absolutely, since $X(1) = \infty$; if we required simple convergence to establish if a point belongs to the ROC, in this case the ROC wouldn't be circularly symmetric.
\item We have
\[
X(z) = \sum_{n=1}^{\infty} \frac{z^{-n}}{n} = -\log z;
\]
this is a rare instance of an irrational transfer function (but which has no practical use).
\end{enumerate}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Properties of the z-transform]
Let $\mathbf{x}$ be a discrete-time sequence and $X(z)$ be its corresponding $z$-transform with appropriate ROC.
\begin{enumerate}
\item Prove that the following relation holds:
\[
\mathcal{Z}\{nx[n]\} = -z\frac{d}{d z}\, X(z)
\]
\item Using (a), show that
\[
\mathcal{Z}\{(n+1)\alpha^{n}u[n]\} = \frac{1}{(1-\alpha z^{-1})^2}\,,\quad \mbox{ROC: } |z|>|\alpha|
\]
\item Suppose that the above expression corresponds to the impulse response of an LTI system. What can you say about the causality of such a system? What about its stability?
\item Let $\alpha=0.8$: what is the spectral behavior of the corresponding filter? What if $\alpha=-0.8$?
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Filter Stability]
Consider a causal discrete system represented by the following difference equation:
\[
y[n]-3.25\,y[n-1]+0.75\,y[n-2]=x[n-1]+3x[n-2]
\]
\begin{enumerate}
\item Compute the transfer function and check the stability of this system both analytically and graphically.
\item If the input signal is $x[n] = \delta[n]-3\delta[n-1]$, compute the $z$-transform of the output signal and discuss the stability.
\item Take an arbitrary input signal that does not cancel the unstable pole of the transfer function and repeat (b).
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Filter Stability]
Consider a causal LTI system with the following transfer function:
\[
H(z) = \frac{3 + 4.5\,z^{-1}}{1+1.5\,z^{-1}} - \frac{2}{1-0.5\,z^{-1}}
\]
Sketch the pole-zero plot of the transfer function and specify its region of convergence. Is the system stable?
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Filter Stability]
Consider the transfer function of an anticausal LTI system
\[
H(z) = (1-z^{-1}) - \frac{1}{1-0.5\,z^{-1}}
\]
Sketch the pole-zero plot of the transfer function and specify its region of convergence. Is the system stable?
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Pole-zero plot and magnitude response]
In the following pole-zero plots, in which the circle shows the unit circle, multiple zeros at the same location are indicated by the multiplicity number shown to the upper right of the zero. Sketch the magnitude of each frequency response and determine the type of filter (highpass, lowpass, bandpass).
\begin{tabular}{cc}
\begin{dspPZPlot}[clabel=~,cunits=false]{1.4}
\dspPZ[type=zero,label=3]{-1,0}
\dspPZ[label=none]{0.73,0}
\dspPZ[label=none]{0.8,0.2}
\dspPZ[label=none]{0.8,-0.2}
\end{dspPZPlot}
&
\begin{dspPZPlot}[clabel=~,cunits=false]{1.4}
\dspPZ[type=zero,label=3]{1,0}
\dspPZ[label=none]{-0.3,0}
\dspPZ[label=none]{-0.36,0.55}
\dspPZ[label=none]{-0.36,-0.55}
\end{dspPZPlot}
\\
\begin{dspPZPlot}[clabel=~,cunits=false]{1.4}
\dspPZ[type=zero,label=3]{0.342,0.93969}
\dspPZ[type=zero,label=3]{0.342,-0.93969}
\dspPZ[label=none]{0.73,0}
\dspPZ[label=none]{0.8,0.2}
\dspPZ[label=none]{0.8,-0.2}
\dspPZ[label=none]{-0.4,0}
\dspPZ[label=none]{-0.66,0.4}
\dspPZ[label=none]{-0.66,-0.4}
\end{dspPZPlot}
&
\end{tabular}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Z-transform and magnitude response]
Consider a causal LTI system described by the following transfer function:
\[
H(z)=\frac{ \frac{1}{6}+\frac{1}{2}\,z^{-1}+\frac{1}{2}\,z^{-2} +\frac{1}{6}\,z^{-3}}{ 1+\frac{1}{3}\,z^{-2}}
\]
\begin{enumerate}
\item Sketch the magnitude response $H(e^{j\omega})$ from the \ztrans . You can use a numerical package to find the
poles and the zeros of the transfer function. What type of filter is $H(z)$?
\item Sketch the pole-zero plot. Is the system stable?
\end{enumerate}
Now consider a length-$128$ input signal $\mathbf{x}$ defined as:
\[
x[n]= \begin{cases}
0 & \mbox{for } 0 \leq n \leq 50 \\
1 & \mbox{for } 51 \leq n \leq 127
\end{cases}
\]
\begin{enumerate}
\setcounter{enumi}{2}
\item Plot the magnitude of the DTFT $X(e^{j\omega})$.
\item Call $\mathbf{y}$ the signal obtained by filtering $\mathbf{x}$ with the system described by $H(z)$; plot the signal $\mathbf{y}$ and the magnitude of its DTFT.
\item Explain qualitatively the shape of $y[n]$.
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[DFT and z-transform]
It is immediately obvious that the DTFT of a sequence is simply its $z$-transform evaluated on the unit circle, i.e.\
for $z = e^{j\omega}$. Equivalently, for a finite-length signal $\mathbf{x}$, the DFT $\mathbf{X}$ is simply the $z$-transform of
the finite support extension of the signal evaluated at $z = e^{j\frac{2\pi}{N}k}$ for $k = 0, \ldots, N-1$:
\[
X[k] = \left. \sum_{n=0}^{N-1}x[n]z^{-n} \right|_{z = e^{j\frac{2\pi}{N}k}} = \sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}nk}
\]
By taking advantage of this fact, derive a simple expression for the DFT of the time-reversed signal
\[
\mathbf{x}_r = \bigl[\begin{matrix}x[N-1] & x[N-2] & \ldots & x[1] & x[0] \end{matrix} ]^T
\]
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[A CCDE]
Consider an LTI system described by the following constant-coefficient difference equation:
\[
y[n-1] + 0.25\,y[n-2] = x[n]
\]
\begin{enumerate}
\item Write the transfer function of the system.
\item Plot its poles and zeros, and show the ROC.
\item Compute the impulse response of the system.
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Inverse transform]
Write out the discrete-time signal $\mathbf{x}$ whose $z$-transform is
\[
X(z) = (1 + 2z) (1+3z^{-1})
\]
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifexercises{%
\begin{exercise}[Signal transforms]
Consider a discrete-time signal $\mathbf{x}$ whose $z$-transform is
\[
X(z) = 1 + z^{-1} + z^{-3} + z^{-4}
\]
\begin{enumerate}
\item Compute $X(e^{j\omega})$. Your final result should be in the form of a real function of $\omega$ times a pure phase term.
\item Sketch the magnitude of $X(e^{j\omega})$ as accurately as you can.
\end{enumerate}
Consider now the length-$4$ signal $\mathbf{y}$ defined by:
\[
y[n] = x[n], \qquad \quad n = 0, 1, 2, 3
\]
\begin{enumerate}
\setcounter{enumi}{2}
\item Compute the four DFT coefficients $Y[k]$, $k = 0, 1, 2, 3$.
\end{enumerate}
\end{exercise}
}\fi
\ifanswers{%
\begin{solution}{}
\end{solution}
}\fi
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