Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.1

Author: Panagiotis

21 Sep

In exercise [1, p. 34 exercise 2.1] we are asked to prove that the inverse of a lower triangular matrix is also lower triangular by using relation [1, p. 23, (2.30)]. read the conclusion >

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Filed under: Kay: Modern Spectral Estimation, Theory and Application, Solved Problems

Hard- und Software Cosimulation mit Visual Micro Lab

Author: Panagiotis

6 Sep

Oft ist es geschickt Code für einen eingebetteten Controller direkt mit seiner Hardwareumgebung zu simulieren. Ein hierfür frei erhältliches Tool, VMLAB (Visual Micro Lab) für die AVR Familie von ATMEL wird in dem beiliegenden Tutorial beschrieben.

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Filed under: German, Tutorials, VMLAB

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.5

Author: Panagiotis

13 Jun

In exercise [1, p. 34 exercise 2.5] we are asked to show that the matrix $\mathbf{R}=\sigma^2 \mathbf{I}+P_1\mathbf{e_1}\mathbf{e_1}^H+P_2\mathbf{e_2}\mathbf{e_2}^H$ is circulant when $\mathbf{e_i}^H=\left[ {\begin{array}{*{20}c} 1 & {e^{-j2\pi f_i } } & {e^{-j2\pi (2f_i) } } & \cdots & {e^{-j2\pi (n - 1)f_i } } \end{array}} \right], i=1,2$ . read the conclusion >

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Filed under: Kay: Modern Spectral Estimation, Theory and Application, Solved Problems

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.4

Author: Panagiotis

1 Feb

The exercise [1, p. 34 exercise 2.4] asks to show that if $\mathbf{F}$ is a full rank $m \times n$ matrix with $m$ > $n$ , $\mathbf{x}$ is a $m \times 1$ vector, and $\mathbf{y}$ is an $m \times 1$ vector, that the effect of the linear transformation

$\mathbf{y}=\mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H\mathbf{x}$

(1)

is to project $\mathbf{x}$ onto the subspace spanned by the columns of $\mathbf{F}$ . Specifically, if $\{\mathbf{f_1},\mathbf{f_2},...,\mathbf{f_n}\}$ are the columns of $\mathbf{F}$ , the exercise [1, p. 34 exercise 2.4] asks to show that

$(\mathbf{x-y})^{H}\mathbf{f_i}=0, \textrm{ i=1,2,...,n.}$

(2)

Furthermore it is asked why the transform $\mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H$ must be idempotent. read the conclusion >

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Filed under: Kay: Modern Spectral Estimation, Theory and Application, Solved Problems

Liboff: “Introductory Quantum Mechanics”, 2nd edition p.86 exercise 3.16

Author: Panagiotis

2 Jan

The exercise [1, p. 86 ex. 3.16] asks to prove that if the eigenfunctions and eigenvalues of an operator $\widehat{A}$ are $\{\phi_{n}\}$ and $\{a_{n}\}$ , respectively ( $\ensuremath{\widehat{A}}\phi_{n}=a_{n}\phi_{n}$ ) then the eigenfunctions of a function $f(x)$ having an expansion of the form:

$f(x)=\sum\limits^{\infty}_{l=0}b_{l}x^{l}$