\documentclass{tripos}     % Do not add any commands here
\begin{document}           % outside the "question" environment!
\begin{question}[CSTpartII,year=2011,paper=8,question=6,author=MGK]{Digital Signal Processing}

\newcommand{\E}{\mathrm{e}}     % base of natural logarithm
\newcommand{\micro}{\ensuremath{\upmu}}  % 10^-6
\newcommand{\I}{\hspace{1pt}\mathrm{j}}     % square-root of -1
\newcommand{\PI}{\uppi}         % half-circumference of unit circle
\newcommand{\rs}{\mathrm{s}}    % sample

\begin{enumerate}

\item \topic{Fourier transform properties}
What can you say about the Fourier transform $X(f)$ if

\begin{enumerate}

\item $x(t)$ is real;
\fullmarks{2}

\begin{answer}
$X(-f) = [X(f)]^*$  (where $^*$ denotes the complex conjugate)
\end{answer}

\item $x(t) = -x(-t)$.
\fullmarks{2}

\begin{answer}
$X(f) = -X(-f)$
\end{answer}

\end{enumerate}

\item \topic{Fourier transform} Give the result of the Fourier
  transform $X(f) = \int_{-\infty}^\infty x(t)\,{\mathrm
    e}^{-2\pi{\mathrm j}ft}\,{\mathrm d}t$, using Dirac's delta where
  appropriate, of

\begin{enumerate}

\item $x(t) = 1$;
\fullmarks{1}

\begin{answer}
$X(f) = \delta(f)$
\end{answer}

\item $x(t) = \cos(2\pi t)$;
\fullmarks{2}

\begin{answer}
$X(f) = \frac12[\delta(f-1)+\delta(f+1)]$
\end{answer}

\item $x(t) = \mathrm{rect}(t)$;
\fullmarks{2}

\begin{answer}
$X(f) = \mathrm{sinc}(f) = \frac{\sin\PI f}{\PI f}$
\end{answer}

\item \topic{convolution theorem}
  $x(t) = [\frac12 + \frac12 \cdot \cos(2\pi t)] \cdot \mathrm{rect}(t)$.
  \fullmarks{3}

\begin{answer}
$X(f) = \frac12 \mathrm{sinc}(f)  +
        \frac14 \mathrm{sinc}(f-1) +
        \frac14 \mathrm{sinc}(f+1)$
\end{answer}

\end{enumerate}

\item \topic{stochastic signals} When is a random sequence $\{x_n\}$
  called a ``white noise'' signal? \fullmarks{2}

\begin{answer}
A random sequence $\{x_n\}$ is called ``white noise'' if its
autocorrelation sequence $\phi_{xx}(k) = \mathcal E(x_{n+k}\cdot x_n^*)
= 0$ for all $k\neq 0$.
\end{answer}

\item \topic{Karhunen-Lo\`eve transform}
Consider an $n$-dimensional random vector variable $\textbf{X}$.

\begin{enumerate}

\item How is its covariance matrix defined? \fullmarks{2}

\begin{answer}
$(\mathrm{Cov}(\mathbf{X}))_{i,j} = \mathcal E((X_i - \mathcal E(X_i))\cdot(X_j
  - \mathcal E(X_j)))$
\end{answer}

\item How can you change its representation without loss of
information into a random vector of equal dimensionality in which all
elements are mutually uncorrelated? \fullmarks{4}

\begin{answer}
Since $\mathrm{Cov}(\mathbf{X})$ is symmetric, it can be diagonalized
into $\mathrm{Cov}(\mathbf{X}) = BDB^T$, where $B$ is an orthonormal
matrix containing the eigenvectors of $\mathrm{Cov}(\mathbf{X})$ and
$D$ is a diagonal matrix containing the corresponding eigenvalues.
Then $B^T\mathbf{X}$ is the decorrelated representation with
$\mathrm{Cov}(B^T\mathbf{X}) = D$.
\end{answer}

\end{enumerate}

\end{enumerate}

\end{question}
\end{document}