Central limit theorems for weakly dependent stochastic processes
Author
Rubak, Ege
Term
4. term
Education
Publication year
2007
Abstract
Dette speciale undersøger, hvor godt en trådløs kommunikationsmodtager kan genskabe et sendt signal, når radiokanalen forvrænger det, og modtageren bruger en equalizer (udligningsfilter) til at modvirke forvrængningen. Fordi trådløse systemer ofte er mobile, ændrer kanalen sig tilfældigt over tid og frekvens. Derfor varierer systemets ydeevne fra situation til situation. Ydelsen måles med den mindste gennemsnitlige kvadratiske fejl (MMSE), som udtrykker den gennemsnitlige afvigelse mellem det oprindelige og det genskabte signal. MMSE afhænger af den konkrete kanal og er derfor en tilfældig størrelse. Hvis man kender fordelingen af MMSE for en given kanalmodel, kan man sætte konfidensbånd på ydelsen. MMSE påvirkes også af systemets båndbredde B. I ultrabredbånd (UWB) bruges meget store båndbredder, så det er vigtigt at forstå, hvordan MMSE opfører sig, når B bliver stor (asymptotisk adfærd). Specialet opstiller en matematisk model med to equalizere: en ideel, uendelig equalizer og en praktisk, endelig equalizer. De giver to mål for ydeevne, MMSE_inf og MMSE_N. Kanalen beskrives ved sin overførselsfunktion H (Fourier-transformen af kanalens impulsrespons). Da kanalen er tilfældig, er H en stokastisk proces. I modellen bestemmes MMSE_inf entydigt af en integralformel, der involverer H. En kort simuleringsstudie peger på, at en central grænseværdisætning (CGS) kan gælde for MMSE_inf. Resten af specialet udvikler den nødvendige teori til at undersøge dette. Der gives baggrund i stokastiske processer, herunder en detaljeret bevisførelse af en CGS for kontinuertidsprocesser, der viser, at en normaliseret integralstørrelse bliver tilnærmelsesvis gaussisk, når integrationsintervallet går mod uendeligt—forudsat at processen kun er svagt afhængig over tid (beskrevet via mixing-egenskaber). En udfordring her er, at H i problemstillingen afhænger af båndbredden B, mens klassisk teori antager uafhængighed af integrationsgrænsen. For at håndtere dette studeres H’s grænseopførsel, når B vokser, ved hjælp af svag konvergens i metriske rum. Til sidst analyseres egenskaberne af MMSE_inf, MMSE_N og H. Der bestemmes en svag grænse for H, og det vises, at en CGS gælder for MMSE_inf under passende betingelser. Endvidere bevises, at MMSE_N konvergerer mod MMSE_inf, når N går mod uendeligt, hvilket giver en betingelse, hvorefter CGS-opførselen for MMSE_inf overføres til MMSE_N. Samlet giver resultaterne et grundlag for at kvantificere variationen i systemets ydelse ved meget store båndbredder og dermed for at angive konfidensbånd.
This thesis asks how well a wireless receiver can recover a transmitted signal when the radio channel distorts it, and studies receivers that use an equalizer to counteract that distortion. Because wireless systems are often mobile, the channel changes randomly over time and frequency, so performance varies from one situation to another. Performance is measured by the minimum mean square error (MMSE), which captures the average mismatch between the original and the recovered signal. MMSE depends on the specific channel and is therefore a random quantity. Knowing its distribution for a given channel model would allow confidence bands on performance. MMSE also depends on the system bandwidth B. In ultra-wideband (UWB) systems, B can be very large, making it important to understand how MMSE behaves as B grows (asymptotic behavior). The thesis presents a mathematical model with two equalizers: an idealized infinite-length equalizer and a practical finite-length equalizer, leading to two performance measures, MMSE_inf and MMSE_N. The channel is described by its transfer function H (the Fourier transform of the impulse response). Because the channel is random, H is a stochastic process. In the model, MMSE_inf is uniquely determined by an integral expression involving H. A brief simulation study suggests that a central limit theorem (CLT) may hold for MMSE_inf. The remainder of the thesis develops the theory needed to study this. It reviews stochastic-process tools and provides a detailed proof of a CLT for continuous-time processes, showing that a normalized integral becomes approximately Gaussian as the integration range grows—provided the process is only weakly dependent over time (formalized via mixing properties). A key complication is that, in this problem, H depends on the bandwidth B, whereas classical CLTs assume the process does not depend on the integration limit. To address this, the thesis analyzes the limiting behavior of H as B increases using weak convergence in metric spaces. Finally, the properties of MMSE_inf, MMSE_N, and H are analyzed. A weak limit of H is identified, and under suitable conditions a CLT is proved for MMSE_inf. It is also shown that MMSE_N converges to MMSE_inf as N grows, which yields a condition under which MMSE_N inherits the same CLT behavior. Together, these results provide a basis for quantifying performance variability at very large bandwidths and for attaching confidence bands.
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