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\title{SP1 Oral Questions}
\author{Simon Hellmayr}
\date{February 2016}

\begin{document}

\maketitle

\section{Introduction}
This is a list of questions adapted from FET's DSV-Ausarbeitung-muendliche-pruefung.pdf from German to English. If questions arise, you found something that doesn't make sense, or have additional questions to add, feel free to fork this at http://www.github.com/shellmayr or contact me at simon.hellmayr[at]gmail.com
\section{Questions}
\begin{enumerate}
    \item What is Bezout's theorem? Explain using an equalizing problem, i.e. under-/over-determined system of equations $\Rightarrow$ solve using least squares.
    \item Bezout's theorem with regard to equalizers, an example of an equalizer, how can it be designed?
    \item How to construct an allpass from a transfer function $H(jw)$?
    \item What does an allpass look like in the time-variant case, how to check whether a given system is an allpass?
    \item Prove the allpass-property in state-space.
    \item Split up a system into an allpass and a minimal phase system.
    \item Explain general properties of allpasses using an example system $$ y_k = x_k \cdot q^{-3} $$
    \item Explain the Circle of Apollonius
    \item State-space description: Explain equations, matrices and show why it is being used (advantages/disadvantages). What changes when the system is time-variant? Discuss stability for both time-variant and time-invariant cases. 
    \item Payley-Wiener Theorem incl. example problem
    \item Linear phase filters
    \item Sampling: derivation, relation to least squares approximation, is sampling a linear transformation?
    \item Sampling theorem with derivation, interpolation function (lowpass, highpass), equidistance, linear operation?, relation to least squares approximation
    \item What is robustness? How is it defined? Small gain theorem as opposed to LTI stability criteria
    \item What is a linear vector space?
    \item Define: subspace, orthogonal subspace, orthogonal complement, relation to least squares
    \item Define: dimension, basis. Example problem in $GF(2)^3$, apply definitions to see if it is a linear vector space. Explain difference between normed vector and normed vector space.
    \item System of equations with 2 equations and 3 unknowns (2x3 matrix A, 3x1 vector x, 2x1 vector b) and its solution $x=\icol{-1\\0\\1}+t\cdot \icol{1\\1\\1}$. What are the respective column-space, row-space, null-space? How are these spaces related?
    \item Galois field $GF(2)^3$ vectors: $\icol{0\\0\\0},\icol{1\\0\\1},\icol{0\\1\\1},\icol{1\\1\\0}$: is this a subspace? Find the basis. Determine if there is a complement space and if it is orthogonal.
    \item Gramian matrix, relation to least squares
    \item Definiteness of a matrix
    \item Metric, norm, induced norm: explain concepts and show an example. Pseudoinverse, alternative pseudo-inverse, relation to least squares
    \item Euclidian norm and matrix norm. Weighted matrix norm and iterative algorithm to calculate any p-norm from the p=2 matrix norm.
    \item Metric, norm: differences, similarities. Examples for metrics, norms and induced norms. Is a metric always a norm? 
    \item Properties of eigenvalues and eigenvectors of unitary and hermitian matrices. Show why eigenvalues of hermitian matrices and real-valued.
    \item Calculate/show column-space/row-space of a given matrix, as well as left and right null-space.
    \item Pseudoinverse: which pseudoinverses do you know? How are they used in least squares? When does a pseudo-inverse exist? (determinant, eigenvalues, singular values)
    \item Name and explain 3 least squares applications
    \item Projection theorem $P=P^2$
    \item Linear dependence and independence
    \item Cauchy-Schwarz inequality + 2 examples
    \item Rayleigh-Quotient-problems $min(\frac{x^HAA^Hx}{x^Hx})$ and $min(\frac{x^HAA^Hx}{x^HBB^Hx})$
    \item Bessel inequality. Parseval's theorem. Why is a complete basis needed? Show using counterexample.
    \item Eigenfilter
    \item Singular value decomposition: general properties, what are the matrices composed of, what are their dimensions? What are singular values? Which norms  use the singular values? 
    \item Subspace-techniques: PhD, MUSIC, ESPRIT. Show principles. What are you trying to achieve? What is given? Why is Pisarenko's equation so hard to solve?
    \item Explain subspace-techniques and guiding principles, how are they used in UMTS? Orthogonal basis for ${\rm I\!R}^4$ without using zeros. 
    \item Subspace, orthogonal subspace, orthogonal complement, relation to least squares approximation
    \item Explain Parseval and Bessel inequality. When is it complete/incomplete? Example: no way to display $cos(t)$ using $sin(t)$
    \item Fast Fourier Transform: What is it? How is it used?
    \item Blind equalizer
    \item Kronecker product: show properties, example problem
    \item Kronecker product: Given a separable 4x4 matrix, what are the eigenvalues and eigenvectors?
    
\end{enumerate}


\end{document}