path: root/tabor/delavnica/predst/predstavitev.lyx

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This file is a solution template for:
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Talk at a conference/colloquium.
 
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Talk length is about 20min.
 
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Style is ornate.
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Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
 
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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.
 
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Fourierova transformacija
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Fourierova transformacija
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Predavanje na naravoslovnem taboru
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Vasja Žorž
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1
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and
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 Anton Luka Šijanec
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2
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Give the names in the same order as the appear in the paper.
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Vasja Žorž
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and
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 Anton Luka Šijanec
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Mentor: xxx 
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and
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 Gimnazija Bežigrad
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Medvedje Brdo
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2
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Department of Theoretical Philosophy
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Gimnazija Bežigrad
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V sredo, 28.
 junija 2023
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Either use conference name or its abbreviation.
 
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Not really informative to the audience, more for people (including yourself)
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28.
 maja 2023
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Exactly two or three sections (other than the summary).
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Uvod
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Motivacija
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Zvočni signal posnetka glasbenega inštrumenta
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Motivacija
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Frekvenčni spekter periodičnega signala
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Motivacija: Plimovanje
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Višina gladine morja kot posledica vplivov nebesnih teles
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Motivacija: Plimovanje
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Valovne komponente, ki se sestavijo v plimovanje
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Valovanje
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, 
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, 
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, 
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, 
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, 
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primeri valovanja: zvok, svetloba, izmenični tok, potresni valovi, plimovanje
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https://www.desmos.com/calculator/ptxc1dutf9
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vprašanja: itd
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Osnovni matematični pojmi
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Množica 
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imaginarna enota: 
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imaginarna enota je definirano kot takšno število, ki reši to enačbo za
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množico kompleksnih števil lahko ponazorimo s kompleksno ravnino
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nariševa jo na tablo, poveva operacije v C
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Eulerjeva formula
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nariši
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Fourierova vrsta
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https://en.wikipedia.org/wiki/Square_wave?useskin=vector#/media/File:SquareWaveFou
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Fourierova vrsta
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Različna valovanja kot neskončna Fourierova vrsta
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Fourierova transformacija
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Fourierova transformacija je matematični postopek, ki nam omogoča razcep
 signalov na sestavne frekvence.
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Fourierova transformacija
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Intuicija
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https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html
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Intuitivno si lahko FT predstavljamo kot navijanje amplitud po enotski krožnici.
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\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

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\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


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\begin_layout Frame
\begin_inset Argument 4
status open

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Viri slik in dovoljenje za uporabo
\end_layout

\end_inset


\end_layout

\begin_layout Frame

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\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