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diff --git a/tabor/delavnica/predst/predstavitev.lyx b/tabor/delavnica/predst/predstavitev.lyx new file mode 100644 index 0000000..4e49773 --- /dev/null +++ b/tabor/delavnica/predst/predstavitev.lyx @@ -0,0 +1,3275 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass beamer +\begin_preamble +\usepackage[ + type={CC}, + modifier={by-sa}, + version={3.0}, +]{doclicense} + +\usepackage{bera}% optional: just to have a nice mono-spaced font +\usepackage{listings} +\usepackage{xcolor} +\lstset{ + extendedchars=true, + literate={č}{{\v{c}}}1 {ž}{{\v{z}}}1 {š}{{\v{s}}}1, +} + +\colorlet{punct}{red!60!black} +\definecolor{background}{HTML}{EEEEEE} +\definecolor{delim}{RGB}{20,105,176} +\colorlet{numb}{magenta!60!black} + +\lstdefinelanguage{json}{ + basicstyle=\normalfont\ttfamily, + numbers=left, + numberstyle=\scriptsize, + stepnumber=1, + numbersep=8pt, + showstringspaces=false, + breaklines=true, + frame=lines, + backgroundcolor=\color{background}, + literate= + *{0}{{{\color{numb}0}}}{1} + {1}{{{\color{numb}1}}}{1} + {2}{{{\color{numb}2}}}{1} + {3}{{{\color{numb}3}}}{1} + {4}{{{\color{numb}4}}}{1} + {5}{{{\color{numb}5}}}{1} + {6}{{{\color{numb}6}}}{1} + {7}{{{\color{numb}7}}}{1} + {8}{{{\color{numb}8}}}{1} + {9}{{{\color{numb}9}}}{1} + {:}{{{\color{punct}{:}}}}{1} + {,}{{{\color{punct}{,}}}}{1} + {\{}{{{\color{delim}{\{}}}}{1} + {\}}{{{\color{delim}{\}}}}}{1} + {[}{{{\color{delim}{[}}}}{1} + {]}{{{\color{delim}{]}}}}{1}, +} + + + + + + + +\usetheme{Warsaw} +% or ... + +\setbeamercovered{transparent} +% or whatever (possibly just delete it) +\usepackage{multimedia} + +\usetheme{Warsaw} + +\setbeamercolor{normal text}{fg=white,bg=black!90} +\setbeamercolor{structure}{fg=white} + +\setbeamercolor{alerted text}{fg=red!85!black} + +\setbeamercolor{item projected}{use=item,fg=black,bg=item.fg!35} + +\setbeamercolor*{palette primary}{use=structure,fg=structure.fg} +\setbeamercolor*{palette secondary}{use=structure,fg=structure.fg!95!black} +\setbeamercolor*{palette tertiary}{use=structure,fg=structure.fg!90!black} +\setbeamercolor*{palette quaternary}{use=structure,fg=structure.fg!95!black,bg=black!80} + +\setbeamercolor*{framesubtitle}{fg=white} + +\setbeamercolor*{block title}{parent=structure,bg=black!60} +\setbeamercolor*{block body}{fg=black,bg=black!10} +\setbeamercolor*{block title alerted}{parent=alerted text,bg=black!15} +\setbeamercolor*{block title example}{parent=example text,bg=black!15} +\usepackage{multicol} +\usepackage{listings} +\usepackage{siunitx} +\end_preamble +\options aspectratio=169 +\use_default_options false +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding utf8-plain +\fontencoding global +\font_roman "lmodern" "default" +\font_sans "lmss" "default" +\font_typewriter "lmtt" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures false +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 2 +\use_package amssymb 2 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 0 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 2 +\tocdepth 2 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +This file is a solution template for: +\end_layout + +\begin_layout Itemize +Talk at a conference/colloquium. + +\end_layout + +\begin_layout Itemize +Talk length is about 20min. + +\end_layout + +\begin_layout Itemize +Style is ornate. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>. + +\end_layout + +\begin_layout Plain Layout +In principle, this file can be redistributed and/or modified under the terms + of the GNU Public License, version 2. + However, this file is supposed to be a template to be modified for your + own needs. + For this reason, if you use this file as a template and not specifically + distribute it as part of a another package/program, the author grants the + extra permission to freely copy and modify this file as you see fit and + even to delete this copyright notice. + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Title +Fourierova transformacija +\begin_inset Argument 1 +status open + +\begin_layout Plain Layout +Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subtitle +Predavanje na naravoslovnem taboru +\end_layout + +\begin_layout Author +Vasja Žorž +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Flex InstituteMark +status open + +\begin_layout Plain Layout +1 +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +and +\end_layout + +\end_inset + + Anton Luka Šijanec +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Flex InstituteMark +status collapsed + +\begin_layout Plain Layout +2 +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Itemize +Give the names in the same order as the appear in the paper. +\end_layout + +\begin_layout Itemize +Use the +\begin_inset Quotes eld +\end_inset + +Institute mark +\begin_inset Quotes erd +\end_inset + + inset ( +\family sans +Insert\SpecialChar menuseparator +Custom Insets\SpecialChar menuseparator +InstituteMark +\family default +) only if the authors have different affiliations. +\end_layout + +\end_inset + + +\begin_inset Argument 1 +status open + +\begin_layout Plain Layout +Vasja Žorž +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +and +\end_layout + +\end_inset + + Anton Luka Šijanec +\begin_inset Note Note +status open + +\begin_layout Plain Layout +- optional, use only with lots of authors +\end_layout + +\begin_layout Plain Layout +- if there are really lots of authors, use +\begin_inset Quotes eld +\end_inset + +Author et al. +\begin_inset Quotes erd +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Institute +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Flex InstituteMark +status collapsed + +\begin_layout Plain Layout +1 +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Mentor: xxx +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +and +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + Gimnazija Bežigrad +\begin_inset Newline newline +\end_inset + +Medvedje Brdo +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Flex InstituteMark +status collapsed + +\begin_layout Plain Layout +2 +\end_layout + +\end_inset + +Department of Theoretical Philosophy +\begin_inset Newline newline +\end_inset + +University of Elsewhere +\begin_inset Note Note +status collapsed + +\begin_layout Itemize +Use the +\begin_inset Quotes eld +\end_inset + +Institute mark +\begin_inset Quotes erd +\end_inset + + inset ( +\family sans +Insert\SpecialChar menuseparator +Custom Insets\SpecialChar menuseparator +InstituteMark +\family default +) only if there are several affiliations. +\end_layout + +\begin_layout Itemize +Keep it simple, no one is interested in your street address. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Argument 1 +status open + +\begin_layout Plain Layout +Gimnazija Bežigrad +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +optional, but mostly needed +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Date +V sredo, 28. + junija 2023 +\begin_inset Note Note +status collapsed + +\begin_layout Itemize +Either use conference name or its abbreviation. + +\end_layout + +\begin_layout Itemize +Not really informative to the audience, more for people (including yourself) + who are reading the slides online +\end_layout + +\end_inset + + +\begin_inset Argument 1 +status open + +\begin_layout Plain Layout +28. + maja 2023 +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +optional, should be abbreviation of conference name +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +If you have a file called "institution-logo-filename.xxx", where xxx is a + graphic format that can be processed by latex or pdflatex, resp., then you + can add a logo by uncommenting the following: +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% +\backslash +pgfdeclareimage[height=0.5cm]{institution-logo}{institution-logo-filename} +\end_layout + +\begin_layout Plain Layout + +% +\backslash +logo{ +\backslash +pgfuseimage{institution-logo}} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +The following causes the table of contents to be shown at the beginning + of every subsection. + Delete this, if you do not want it. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% +\backslash +AtBeginSubsection[]{% enejble to inflate +\end_layout + +\begin_layout Plain Layout + +% +\backslash +frame<beamer>{ +\end_layout + +\begin_layout Plain Layout + +% +\backslash +frametitle{Pregled vsebine} +\end_layout + +\begin_layout Plain Layout + +% +\backslash +tableofcontents[currentsection, currentsubsection] +\end_layout + +\begin_layout Plain Layout + +% } +\end_layout + +\begin_layout Plain Layout + +%} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +If you wish to uncover everything in a step-wise fashion, uncomment the + following command: +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% +\backslash +beamerdefaultoverlayspecification{<+->} +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Pregled vsebine +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\begin_inset CommandInset toc +LatexCommand tableofcontents + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Structuring a talk is a difficult task and the following structure may not + be suitable. + Here are some rules that apply for this solution: +\end_layout + +\begin_layout Itemize +Exactly two or three sections (other than the summary). +\end_layout + +\begin_layout Itemize +At *most* three subsections per section. +\end_layout + +\begin_layout Itemize +Talk about 30s to 2min per frame. + So there should be between about 15 and 30 frames, all told. +\end_layout + +\begin_layout Itemize +A conference audience is likely to know very little of what you are going + to talk about. + So *simplify*! +\end_layout + +\begin_layout Itemize +In a 20min talk, getting the main ideas across is hard enough. + Leave out details, even if it means being less precise than you think necessary. +\end_layout + +\begin_layout Itemize +If you omit details that are vital to the proof/implementation, just say + so once. + Everybody will be happy with that. +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Uvod +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Motivacija +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename waveform.png + width 100text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Zvočni signal posnetka glasbenega inštrumenta +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Uvod +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Motivacija +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename spekter.png + width 100text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Frekvenčni spekter periodičnega signala +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Uvod +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Motivacija: Plimovanje +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename tideforces.jpg + width 40text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Višina gladine morja kot posledica vplivov nebesnih teles +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Uvod +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Motivacija: Plimovanje +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename tidalsum.gif + width 40text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Valovne komponente, ki se sestavijo v plimovanje +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Osnovni matematični pojmi +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Valovanje +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Formula $\lambda$ +\end_inset + +, +\begin_inset Formula $t_{0},$ +\end_inset + + +\begin_inset Formula $\nu$ +\end_inset + +, +\begin_inset Formula $\omega$ +\end_inset + +, +\begin_inset Formula $k$ +\end_inset + +, +\begin_inset Formula $\phi$ +\end_inset + +, +\begin_inset Formula $A$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $f(x)=A\cdot\cos(\omega t-kx+\phi)$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\omega=2\pi\nu$ +\end_inset + +, +\begin_inset Formula $\nu=1/t_{0}$ +\end_inset + +, +\begin_inset Formula $k=2\pi/\lambda$ +\end_inset + + +\end_layout + +\begin_layout Itemize +primeri valovanja: zvok, svetloba, izmenični tok, potresni valovi, plimovanje +\begin_inset Note Note +status open + +\begin_layout Itemize +https://www.desmos.com/calculator/ptxc1dutf9 +\end_layout + +\begin_layout Itemize +vprašanja: itd +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Osnovni matematični pojmi +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Množica +\begin_inset Formula $\mathbb{C}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +imaginarna enota: +\begin_inset Formula $i^{2}+1=0$ +\end_inset + + +\end_layout + +\begin_layout Itemize +imaginarna enota je definirano kot takšno število, ki reši to enačbo za + i +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\mathbb{C}\coloneqq\{a+bi;\ a,b\in\mathbb{R}\}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +množico kompleksnih števil lahko ponazorimo s kompleksno ravnino +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nariševa jo na tablo, poveva operacije v C +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Osnovni matematični pojmi +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Eulerjeva formula +\end_layout + +\begin_layout Itemize +polarni zapis kompleksnega števila: +\begin_inset Formula $r\cdot e^{i\theta}=r(\cos\theta+i\sin\theta)$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nariši +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Fourierova vrsta +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +https://en.wikipedia.org/wiki/Square_wave?useskin=vector#/media/File:SquareWaveFou +rierArrows.gif +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Formula $f(t)=A_{1}\cos(\omega_{1}t)+A_{2}\cos(\omega_{2}t)+...$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $S(x)=\frac{4}{\pi}(\sin\omega x+\frac{1}{3}\sin3\omega x+$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\ \ \ \ \ \ \ \frac{1}{5}\sin5\omega x+...)$ +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +vfill +\backslash +null +\backslash +columnbreak +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename series.png + width 30text% + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Fourierova vrsta +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename wolfram.png + width 82text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Različna valovanja kot neskončna Fourierova vrsta +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename transformacija.png + width 78text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Fourierova transformacija je matematični postopek, ki nam omogoča razcep + signalov na sestavne frekvence. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Intuicija +\begin_inset Note Note +status open + +\begin_layout Plain Layout +https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +url{http://4a.si/fourier} +\end_layout + +\end_inset + + +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename trans.png + width 78text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Intuitivno si lahko FT predstavljamo kot navijanje amplitud po enotski krožnici. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Matematična oblika +\end_layout + +\begin_layout Standard +Kako to navijanje predstavimo matematično +\begin_inset Formula +\[ +\hat{f}(\nu)=\int_{-\infty}^{\infty}f(t)\cdot e^{-i2\pi\nu t}dt +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna funkcija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +funkcija +\begin_inset Formula $f(x)$ +\end_inset + + je zvezna, če +\begin_inset Formula +\[ +\forall x_{0}\in D_{f}\text{, velja }\lim_{x\to x_{0}}f(x)=f(x_{0})\text{.} +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +diskretna funkcija je funkcija, katere spremenljivka zavzema le nekatere + realne vrednosti. +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Vzorčeni signali +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Za praktično obdelavo zvezne signale iz narave periodično vzorčimo +\end_layout + +\begin_layout Itemize +Vzorčenje je shranjevanje amplitude signala na določen časovni interval +\end_layout + +\begin_deeper +\begin_layout Itemize +Dobimo časovno diskreten signal, ki ga predstavlja niz števil +\begin_inset Formula +\[ +x[n]\ \ \ n\in\mathbb{Z},\ x[n]\in I +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $n$ +\end_inset + + označuje zaporedno številko vzorca +\end_layout + +\end_deeper +\begin_layout Itemize +frekvenca vzorčenja +\begin_inset Formula $\nu_{vz}$ +\end_inset + +, interval vzorčanja +\begin_inset Formula $t_{vz}$ +\end_inset + +, kvantizacijski interval +\begin_inset Formula $I$ +\end_inset + + (določen s številom bitov za zapis številk: +\begin_inset Formula $[0,2^{b}-1]$ +\end_inset + +) +\end_layout + +\begin_layout Itemize +Primer: mikrofon, EKG, merilnik temperature, merilnik tlaka, radijski sprejemnik +, ... +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Vzorčeni signali +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename sampling.png + width 82text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Vzorčenje funkcije +\begin_inset Formula $S(t)$ +\end_inset + + v zaporedje +\begin_inset Formula $S_{i}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Vzorčeni signali +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Teorija Nyquist-Shannon o vzorčenju +\end_layout + +\begin_layout Itemize +Povezava med +\begin_inset Formula $\nu_{max}$ +\end_inset + + vzorčenega signala in +\begin_inset Formula $\nu_{vz}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Za reprezentacijo signala +\begin_inset Formula $v_{max}$ +\end_inset + + mora biti +\begin_inset Formula $\nu_{vz}\geq2\nu_{max}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename nyquistshannon.png + width 80text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset Formula $\lambda/2<t_{vz}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Ker ne gre za časovno zvezni signal, ne moremo uporabiti integracije, ampak + vsoto preko vseh vzorcev. +\end_layout + +\begin_layout Itemize +Namesto oblike za časovno zvezni signal +\begin_inset Formula +\[ +\hat{f}(\nu)=\int_{-\infty}^{\infty}f(t)\cdot e^{-i2\pi\nu t}\ dt +\] + +\end_inset + +za časovno diskretni signal z +\begin_inset Formula $N$ +\end_inset + + vzorci uporabimo +\begin_inset Formula +\[ +\hat{f}[\nu]=\sum_{n=0}^{N}f[n]\cdot e^{-i2\pi\nu nt_{vz}}\text{.} +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +DFT je frekvenčno diskretna, vrednosti amplitud izračunamo samo za diskretne + frekvenčne komponente. +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Značilnost +\end_layout + +\begin_layout Itemize +Ločljivost med frekvenčnimi komponentami je pogojena z +\begin_inset Formula $N$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout FragileFrame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Naivni algoritem v psevdokodi +\end_layout + +\begin_layout Standard +\begin_inset listings +lstparams "language=Python,basicstyle={\ttfamily},breaklines=true" +inline false +status open + +\begin_layout Plain Layout + +število vzorci[] = {...} +\end_layout + +\begin_layout Plain Layout + +število frekvence[] = {1000, 2000, 3000, 4000} +\end_layout + +\begin_layout Plain Layout + +število težišča[frekvence_length] +\end_layout + +\begin_layout Plain Layout + +for frekvenca in range(frekvence.length): +\end_layout + +\begin_layout Plain Layout + + for vzorec in range(vzorci.length): +\end_layout + +\begin_layout Plain Layout + + težišča[frekvenca] += vzorci[vzorec] * pow(e, -i*2*pi*frekvence[frekvenc +a]*vzorec/vzorčna_frekvenca) +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout FragileFrame + +\end_layout + +\begin_layout FragileFrame + +\end_layout + +\begin_layout FragileFrame + +\end_layout + +\begin_layout FragileFrame + +\end_layout + +\begin_layout FragileFrame + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Časovna zahtevnost naivnega algoritma +\end_layout + +\begin_layout Itemize +Ocena časovne zahtevnosti algoritma glede na vhodne podatke: +\begin_inset Formula $O(...)$ +\end_inset + + +\end_layout + +\begin_layout Itemize +Naivni DFT algoritem: +\begin_inset Formula $O(m\cdot n)$ +\end_inset + + oziroma +\begin_inset Formula $O(n^{2})$ +\end_inset + + — polinomski čas +\end_layout + +\begin_layout Itemize +Hitra diskretna Fourierova transformacija (FFT) pa postopek optimizira na + +\begin_inset Formula $O(n\log n)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Itemize +Obstaja več algoritmov hitre transformacije +\end_layout + +\begin_deeper +\begin_layout Itemize +Gauss je 1805 že optimiziral algoritem za izračun orbit asteroidov +\end_layout + +\begin_layout Itemize +Cooley and Tukey sta 1965 predstavila moderni DIT FFT +\end_layout + +\end_deeper +\end_deeper +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Diskretna Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Primerjava časa izvajanja +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Tabular +<lyxtabular version="3" rows="3" columns="3"> +<features tabularvalignment="middle"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +vzorcev +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10000$ +\end_inset + +; +\begin_inset Formula $\SI{1}{\second}$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10\cdot10^{4}$ +\end_inset + +; +\begin_inset Formula $\SI{10}{\second}$ +\end_inset + + +\end_layout + +\end_inset +</cell> +</row> +<row> +<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +naiven +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{8}$ +\end_inset + + (1 sekunda) +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{10}$ +\end_inset + +(1,6 minute) +\end_layout + +\end_inset +</cell> +</row> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +FFT +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{4}\cdot4$ +\end_inset + + (0,0004 sekunde) +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{5}\cdot5$ +\end_inset + + (0,0005 sekund) +\end_layout + +\end_inset +</cell> +</row> +</lyxtabular> + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\align center +\begin_inset Tabular +<lyxtabular version="3" rows="3" columns="3"> +<features tabularvalignment="middle"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +vzorcev +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $100\cdot10^{4}$ +\end_inset + +; +\begin_inset Formula $\SI{100}{\second}$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $1000\cdot10^{4}$ +\end_inset + +; +\begin_inset Formula $\SI{180}{\second}$ +\end_inset + + +\end_layout + +\end_inset +</cell> +</row> +<row> +<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +naiven +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{12}$ +\end_inset + + (2,7 ure) +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{14}$ +\end_inset + + (11 dni) +\end_layout + +\end_inset +</cell> +</row> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +FFT +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{6}\cdot6$ +\end_inset + + (0,06 sekunde) +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $10^{7}\cdot7$ +\end_inset + + (0,7 sekunde) +\end_layout + +\end_inset +</cell> +</row> +</lyxtabular> + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Primerjava časa izjavanja DFT in FDFT. + +\begin_inset Formula $\nu_{vz}=\SI{10000}{\per\second}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout FragileFrame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Hitra Fourierova transformacija +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Praktična uporaba v Pythonu +\end_layout + +\begin_layout Itemize +Knjižnica numpy ima vgrajen algoritem za FFT, uporaba: +\begin_inset listings +lstparams "language=Python,numbers=left,breaklines=true" +inline false +status open + +\begin_layout Plain Layout + +from numpy.fft import fft +\end_layout + +\begin_layout Plain Layout + +import numpy as np +\end_layout + +\begin_layout Plain Layout + +frekvencni_spekter = fft(amplituda_v_odvisnosti_od_casa) +\end_layout + +\begin_layout Plain Layout + +print( +\begin_inset Quotes gld +\end_inset + +dominantna frekvenca: +\begin_inset Quotes grd +\end_inset + +, np.max(frekvencni_spekter)) +\end_layout + +\begin_layout Plain Layout + +realne_vrednosti = abs(frekvencni_spekter) +\end_layout + +\end_inset + + +\begin_inset Separator plain +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Praktična uporaba DFT/FFT +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Obdelava zvoka — izenačevalnik/equaliser +\end_layout + +\begin_layout Itemize +Kompresija slik JPEG z izgubami z dvodimenzionalnim DFT +\end_layout + +\begin_layout Itemize +Radijska komunikacija — digitalni prenos informacij preko radijskih valov +\end_layout + +\begin_layout Itemize +DTMF +\end_layout + +\begin_layout Itemize +Krovni pojem: digitalna obdelava signalov +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +DTMF — Dual tone multi frequency +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Izbiranje telefonske številke na telefonski liniji poteka preko zvoka +\end_layout + +\begin_layout Itemize +Pritiski na gumbe telefonske številčnice izdelujejo piske +\end_layout + +\begin_layout Itemize +Telefonska centrala na drugi strani s Fourierovo transformacijo piskov določi + pritisnjeno tipko +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename dtmf.png + width 72text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Pripadajoče frekvence na telefonski DTMF številčnici +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +DTMF — Dual tone multi frequency +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename dtmffreq.png + width 80text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Frekvenčni spekter piska, ki ga proizvaja številčnica +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Splošno načelo nedoločenosti +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Signal, zelo koncentriran v času, mora imeti široko Fourierovo transformacijo. +\end_layout + +\end_deeper +\begin_layout Frame + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Splošno načelo nedoločenosti +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Dopplerjev radar +\end_layout + +\begin_layout Itemize +Dolg signal korelira z natančno hitrostjo in široko pozicijo +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename doppler.png + width 80text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Dopplerjev radar +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Heisenbergovo načelo nedoločenosti +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Valovna funkcija +\end_layout + +\begin_layout Itemize +V kvantni mehaniki lahko delec opišemo s kompleksno valovno funkcijo, ki + je funkcija prostora in časa +\end_layout + +\end_deeper +\begin_layout Frame +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename wavelet.png + width 62text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Valovna funkcija +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Heisenbergovo načelo nedoločenosti +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout FrameSubtitle +Gibalna količina prostega delca +\end_layout + +\begin_layout Itemize +\begin_inset Formula $p=\hbar k$ +\end_inset + + ( +\begin_inset Formula $k$ +\end_inset + + je prostorska frekvenca), +\begin_inset Formula $k=2\pi/\lambda$ +\end_inset + + +\end_layout + +\begin_layout Itemize +Če izvedemo Fourierovo transformacijo na valovni funkciji po prostoru, dobimo + razporeditev amplitud po +\begin_inset Formula $k$ +\end_inset + +-ju. +\end_layout + +\begin_layout Itemize +Zaradi tega sta si nesigurnost v prostorski frekvenci in nesigurnost v poziciji + obratno sorazmerni +\begin_inset Formula +\[ +\Delta x\cdot\Delta p=\text{konst.} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename uncertainty.png + width 80text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Nedoločenost pozicije in gibalne količine delca +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Konec predstavitve +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +Hvala za pozornost! +\end_layout + +\end_deeper +\begin_layout Section* +\start_of_appendix +\begin_inset Note Note +status open + +\begin_layout Plain Layout +All of the following is optional and typically not needed. +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Section* +\start_of_appendix +Priloge +\end_layout + +\begin_layout Subsection* +Zahvala +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Zahvala +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Itemize +idk ... + šoli ig +\end_layout + +\end_deeper +\end_inset + + +\end_layout + +\begin_layout Subsection* +\begin_inset Note Note +status open + +\begin_layout Subsection* +Literatura in dodatno branje +\end_layout + +\begin_layout Frame +\begin_inset Argument 3 +status collapsed + +\begin_layout Plain Layout +allowframebreaks +\end_layout + +\end_inset + + +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Literatura in dodatno branje +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +ne navajaj vseh virov, samo najzanimivejše +\end_layout + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +beamertemplatebookbibitems +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Start with overview books. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "Author1990" +literal "true" + +\end_inset + +A. + Author. + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +newblock +\end_layout + +\end_inset + + +\emph on +Handbook of Everything +\emph default +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +newblock +\end_layout + +\end_inset + + Some Press, 1990. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +beamertemplatearticlebibitems +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Followed by interesting articles. + Keep the list short. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "Someone2002" +literal "true" + +\end_inset + +S. + Someone. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +newblock +\end_layout + +\end_inset + + On this and that +\emph on +. + +\emph default + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +newblock +\end_layout + +\end_inset + + +\emph on +Journal on This and That +\emph default +. + 2(1):50–100, 2000. +\end_layout + +\end_deeper +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Subsection* +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Frame +\begin_inset Argument 4 +status open + +\begin_layout Plain Layout +Viri slik in dovoljenje za uporabo +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Frame + +\end_layout + +\begin_deeper +\begin_layout Itemize +Uporaba generiranega dokumenta je dovoljena v skladu s CC BY-SA ( +\begin_inset CommandInset href +LatexCommand href +target "http://4a.si./bysa" +literal "true" + +\end_inset + +) +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Dovoljenja za uporabo LyX kode pa nisem podal! +\end_layout + +\end_inset + + +\begin_inset Newline newline +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +doclicenseImage[imagewidth=2cm] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Itemize +Slika +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Shematski-prikaz-povezav" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +: Scott Martin: spremenjena, izvorna pod CC BY-SA +\end_layout + +\begin_layout Itemize +Slika +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Drevesno-iskanje-soležnikov" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +: Limaner: izvorna pod CC BY-SA +\end_layout + +\end_deeper +\end_inset + + +\end_layout + +\end_body +\end_document |