\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}