\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{EOD waveform}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
{email: jan.grewe@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
Weakly electric fish employ their self-generated electric field for
prey-capture, navigation and also communication. In many of these fish
the {\em electric organ discharge} (EOD) is well described by a
combination of a sine-wave and a few of its harmonics (integer
multiples of the fundamental frequency).

\begin{questions}
  \question In the data file {\tt EOD\_data.mat} you find two
  variables. The first contains the time at which the EOD was sampled
  and the second the acutal EOD recording of a weakly electric fisch
  of the species {\em Apteronotus leptorhynchus}.
  \begin{parts}
    \part Load the data and create a plot showing the data. Time is given in
    seconds and the voltage is given in mV/cm.
    \part Fit the following curve to the EOD (select a \textbf{small} time
    window, containing only two or three electric organ discharges, for
    fitting, not the entire trace) using least squares:
    $$f_{\omega_0,b_0,\varphi_1, ...,\varphi_n}(t) = b_0 +
    \sum_{j=1}^n \alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j
    ).$$ $\omega_0$ is called the {\em fundamental frequency}. The single
    terms $\alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j )$
    are called the {\em harmonic components}. The variables $\varphi_j$
    are called {\em phases}, the $\alpha_j$ are the amplitudes. For
    the beginning choose $n=3$.
    \part Try different choices of $n$ and see how the fit
    changes. Plot the fits and the section of the original curve that
    you used for fitting for different choices of $n$.
    \part \label{fiterror} Plot the fitting error as a function of $n$.
    What do you observe?
    \part Another way to quantify the quality of the fit is to compute
    the correlation coefficient between the fit and the
    data. Illustrate this correlation for a few values of $n$. Plot
    the correlation coefficient as a function of $n$.  What is the
    minimum $n$ needed for a good fit? How does this compare to the
    results from (\ref{fiterror})?
    \part Plot the amplitudes $\alpha_j$ and phases $\varphi_j$ as a
    function of the frequencies $\omega_j$ --- the amplitude and phase
    spectra, also called ``Bode plot''.
    \part Why does the fitting fail when you try to fit the entire recording?
    \part (optional) If you want you can also play the different fits
    and the original as sound (check the help).

  \end{parts}
\end{questions}

\end{document}