64 lines
2.7 KiB
TeX
64 lines
2.7 KiB
TeX
\documentclass[a4paper,12pt,pdftex]{exam}
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\newcommand{\ptitle}{EOD waveform}
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\input{../header.tex}
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\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
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{email: jan.grewe@uni-tuebingen.de}
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\begin{document}
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\input{../instructions.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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Weakly electric fish employ their self-generated electric field for
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prey-capture, navigation and also communication. In many of these fish
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the {\em electric organ discharge} (EOD) is well described by a
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combination of a sine-wave and a few of its harmonics (integer
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multiples of the fundamental frequency).
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\begin{questions}
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\question In the data file {\tt EOD\_data.mat} you find two
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variables. The first contains the time at which the EOD was sampled
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and the second the acutal EOD recording of a weakly electric fisch
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of the species {\em Apteronotus leptorhynchus}.
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\begin{parts}
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\part Load the data and create a plot showing the data. Time is given in
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seconds and the voltage is given in mV/cm.
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\part Fit the following curve to the EOD (select a \textbf{small} time
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window, containing only two or three electric organ discharges, for
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fitting, not the entire trace) using least squares:
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$$f_{\omega_0,b_0,\varphi_1, ...,\varphi_n}(t) = b_0 +
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\sum_{j=1}^n \alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j
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).$$ $\omega_0$ is called the {\em fundamental frequency}. The single
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terms $\alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j )$
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are called the {\em harmonic components}. The variables $\varphi_j$
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are called {\em phases}, the $\alpha_j$ are the amplitudes. For
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the beginning choose $n=3$.
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\part Try different choices of $n$ and see how the fit
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changes. Plot the fits and the section of the original curve that
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you used for fitting for different choices of $n$.
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\part \label{fiterror} Plot the fitting error as a function of $n$.
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What do you observe?
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\part Another way to quantify the quality of the fit is to compute
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the correlation coefficient between the fit and the
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data. Illustrate this correlation for a few values of $n$. Plot
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the correlation coefficient as a function of $n$. What is the
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minimum $n$ needed for a good fit? How does this compare to the
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results from (\ref{fiterror})?
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\part Plot the amplitudes $\alpha_j$ and phases $\varphi_j$ as a
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function of the frequencies $\omega_j$ --- the amplitude and phase
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spectra, also called ``Bode plot''.
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\part Why does the fitting fail when you try to fit the entire recording?
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\part (optional) If you want you can also play the different fits
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and the original as sound (check the help).
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\end{parts}
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\end{questions}
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\end{document}
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