Shannon-Nyquistov izrek vzorčenja in zgoščeno zaznavanje

Ritovšek, Tanja (author), Saksida, Pavle (mentor)

V magistrskem delu se seznanimo s področjem teorije signalov -- osnovnimi pojmi in tehniko obdelave signalov. Osredotočimo se na popolno rekonstrukcijo signala iz serije meritev. Ob tem se srečamo s teorijo Fourierove transformacije, ki nam omogoča predstavitev (originalnega) signala v frekvenčni domeni. Največ pozornosti nato posvetimo izreku vzorčenja, imenovanemu tudi Shannon-Nyquistov izrek, ki velja za eno osnovnih načel teorije informacij in diskretne obdelave signalov. Določa spodnjo mejo za število meritev, ki jo je potrebno doseči, da lahko splošni signal popolnoma rekonstruiramo. Z različnimi matematičnimi sredstvi ga tudi dokažemo. Ob koncu naloge si ogledamo še tehniko zgoščenega zaznavanja, ki predstavlja nov pogled na klasično teorijo vzorčenja. Ta pravi, da lahko signal, ki je v določeni bazi redek, popolnoma rekonstruiramo tudi iz manjšega števila meritev, kot nam to zapoveduje Shannon-Nyquistov izrek.

• Language: Slovenian
• Keywords: Teorija signalov, Fourierova analiza, Poissonova sumacijska formula, teorija vzorčenja, Shannon-Nyquistov izrek, zgoščeno zaznavanje, linearni sistemi in programiranje
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-121650
• COBISS.SI-ID: 33238787
• Publication date in RUL: 21.10.2020

Secondary language

• Language: English
• Title: Shannon-Nyquist sampling theorem and compressed sensing

Abstract

In this work, we first introduce the field of signal theory -- basic concepts and technique of signal processing. We mostly focus on complete signal reconstruction from a series of measurements. In doing so, we come across Fourier transform theory, which allows us to represent the (original) signal in the frequency domain. We then give most attention to the sampling theorem, also called the Shannon-Nyquist theorem, which is considered one of the basic principles of information theory and discrete signal processing. It sets the lower limit for the number of measurements that must be achieved in order to fully reconstruct the general signal. From various mathematical points of view, we also prove it. At the end of this paper, we take a closer look at the technique of compressed sensing, which represents a new perspective on classical sampling theory. It says that a signal, that is sparse in a given base, can also be completely reconstructed from a smaller number of measurements than the Shannon-Nyquist theorem dictates.

• Keywords: Signal theory, Fourier analysis, Poisson summation equation, sampling theory, Shannon-Nyquist theorem, compressed sensing, linear systems and programming