In this work, first the Cramer-Rao lower bound (CRLB) of the signal-to-noise ratio (SNR) estimate for binary phase shift keying (BPSK) modulated signals in additive white Gaussian noise (AWGN) channels is derived. All the steps and results of this CRLB derivation are shown in a detailed manner. Two major estimation scenarios are considered herein: the non-data-aided (NDA) and data-aided (DA) frameworks, respectively. The non-data-aided scenario does not assume the periodic transmission of known data symbols (pilots) to limit the system throughput, while the data-aided scenario assumes the transmission of known transmit data symbols or training sequences to estimate the channel parameters. The Cramer-Rao lower bounds for the non-data-aided and data-aided scenarios are derived. In addition, the modified Cramer-Rao lower bound (MCRLB) is also calculated and compared to the true CRLBs. It is shown that in the low SNR regime the true CRLB is tighter than the MCRLB in the non-data-aided estimation scenario. Second, the Bayesian Cramer-Rao lower bound (BCRLB) for SNR estimate is considered for BPSK modulated signals in the presence of time-selective fading channels. Only the data-aided scenario is considered, and the time-selective fading channel is modeled by means of a polynomial function. A BCRLB on the variance of the SNR estimate is found and the simulation results are presented.

This thesis is a part of a research project performed by two MS students, Zhouchi Li and the author. The overall objective of the project is the design, implementation and performance evaluation of algorithms for newborns localization and tracking in hospitals using Apple iBeacon technology. Although we were working on the project together, I lead performance evaluation of the in-room localization system using Cramer Rao Lower Bound (CRLB). My partner, Zhouchi Li, leads modeling the path-loss of iBeacons and presence detection algorithms. This thesis describes the project with a focus on my individual contributions in CRLB analysis under different iBeacon deployment patterns as well as performance evaluation using practical characteristics of shadow fading. Today, Wi-Fi localization is the most popular indoor localization technique, which provides an accuracy of a few meters to distinguish the presences in different rooms of a building. With the recent introduction of iBeacon by Apple, possibility of more accurate in-room localization has emerged for specific applications such as locating newborns inside a hospital. The iBeacon uses Bluetooth Low Energy (BLE) technology that broadcasts beacons with unique information to the nearby receivable devices such as iPhone and android smart phones. The RSS of these beacons can be used to estimate the location and to construct an in-room localization system. In this thesis, we investigate in-room localization system using iBeacon for the newborns in hospitals with an accuracy of about 1 meter. We firstly present an in-room localization system using RSS from iBeacon. Then, based on the traditional Cramer-Rao Lower Bound (CRLB) we analyze the optimal deployment strategy for different iBeacon deployment patterns in the nursery room. Finally, we introduce a novel approach for calculation of the CRLB which includes practical conditions to analyze the influence of variable variance of shadow fading and coverage probability.

This thesis is a part of a research project performed by two MS students Yang Yang and the author. The overall objective of the project is the design, implementation, and performance evaluation of algorithms for newborn localization and tracking in hospitals using Apple iBeacon technology. In the research project, I lead the path-loss modeling of iBeacon, design of algorithms for in-room presence detection system, and analysis of the accelerometer sensor. My partner, Yang Yang, leads the performance evaluation of the localization system using Cramer Rao Lower Bound (CRLB). This manuscript describes the project with a focus on my contributions in modeling the behavior of sensors and presence detection algorithms. Today, RFID detection is the most popular indoor detection technique. It provides high precision detection rate to distinguish the number of people in certain rooms of a building. However, special scanners and manual operations are required. This increases the cost and operation complexity. With the recent introduction of iBeacon by Apple, possibility of more efficient in-room presence detection has emerged for specific applications. An example of these applicatons is recording the number of visitors and newborns in a nursery room inside a hospital. The iBeacon uses Bluetooth Low Energy (BLE) technology for proximity broadcasting. Additionally, iBeacon carries a motion detection sensor, which can be utilized for counting the number of people and newborns entering and leaving a room. In this thesis we introduce a novel intelligent in-room presence detection system using iBeacon for the newborns in hospitals to determine the number of visitors and newborns' location in the nursery room. We first develop a software application on iPhone to receive and extract the necessary data from iBeacon for further analysis. We build the path-loss model for the iBeacon based on the received signal strength (RSS) of the iBeacon, which is used for performance evaluation using CRLB in Yang Yang's project. We also utilize the accelerometer in the smart phones to improve the performance of our detection system.

"Precise Tracking of Things (PToT) using RF signals has posed a serious challenge in an indoor environment. The precision localization information is an enabler for better coordinated-tasks and is essential for a successful launch of many emerging applications. PToT relies on two principal components, a novel navigation (tracking) algorithm, and a hybrid 3D fingerprint database. In this dissertation, we begin by using the two widely known ranging techniques, Time Of Arrival (TOA) associated with Ultra-wideband (UWB) and Received Signal Strength (RSS) with WiFi signals. First, we use the theoretical models derived from empirical measurement of TOA and RSS to analyze the performance of hybrid (WiFi & UWB) cooperative localization accuracy in a multi-robot operation in a typical office environment. To measure the performance of this hybrid localization, we derive a mathematical formulation for the Crame ́r-Rao-Lower- Bound (CRLB). The hybrid method shows more accuracy over WiFi-only approach. In achieving more precision, we extend our work. Second, we introduce a novel approach, a Kernel Method Particle Filter (KMPF) for tracking and predicting the position by accessing the information created by hybrid 3D fingerprint database. We derive the mathematical and statistical framework for the Particle Filter based on the statistical assumptions about the behavior of channel models. We also describe the formation of one of the necessary PToT component, a 3D fingerprint database. We compare the performance of the KMPF against the CRLB using WiFi signal channel models."

Diffusion weighted imaging (DWI) is a branch within the field of magnetic resonance imaging (MRI) that relies on the diffusion of water molecules for its contrast. Its clinical applications include the early diagnosis of ischemic stroke and mapping of the nerve tracts of the brain. The recent development of filter exchange imaging (FEXI) and the introduction of the apparent exchange rate (AXR) present a new DWI based technique that uses the exchange of water between compartments as contrast. FEXI could offer new clinical possibilities in diagnosis, differentiation and treatment follow-up of conditions involving edema or altered membrane permeability, such as tumors, cerebral edema, multiple sclerosis and stroke. Necessary steps in determining the potential of AXR as a new biomarker include running comparative studies between controls and different patient groups, looking for conditions showing large AXR-changes. However, before designing such studies, the experimental protocol of FEXI should be optimized to minimize the experimental variance. Such optimization would improve the data quality, shorten the scan time and keep the required study group sizes smaller.  Here, optimization was done using an active imaging approach and the Cramer-Rao lower bound (CRLB) of Fisher information theory. Three optimal protocols were obtained, each specialized at different tissue types, and the CRLB method was verified by bootstrapping. A test-retest study of 18 volunteers was conducted in order to investigate the reproducibility of the AXR as measured by one of the protocols, adapted for the scanner. Group sizes required were calculated based on both CRLB and the variability of the test-retest data, as well as choices in data analysis such as region of interest (ROI) size. The result of this study is new protocols offering a reduction in coefficient of variation (CV) of around 30%, as compared to previously presented protocols. Calculations of group sizes required showed that they can be used to decide whether any patient group, in a given brain region, has large alterations of AXR using as few as four individuals per group, on average, while still keeping the scan time below 15 minutes. The test-retest study showed a larger than expected variability however, and uncovered artifact like changes in AXR between measurements. Reproducibility of AXR values ranged from modest to acceptable, depending on the brain region. Group size estimations based on the collected data showed that it is still possible to detect AXR difference larger than 50% in most brain regions using fewer than ten individuals. Limitations of this study include an imprecise knowledge of model priors and a possibly suboptimal modeling of the bias caused by weak signals. Future studies on FEXI methodology could improve the method further by addressing these matters and possibly also the unknown source of variability. For minimal variability, comparative studies of AXR in patient groups could use a protocol among those presented here, while choosing large ROI sizes and calculating the AXR based on averaged signals.

Recently, indoor localization becomes a hot topic no matter in industry or academic field. Smart phones are good candidates for localization since they are carrying various sensors such as GPS, Wi-Fi, accelerometer, barometer and etc, which can be used to estimate the current location. But there are still many challenges for 3D indoor geolocation using smart phones, among which the map selection and 3D performance evaluation problems are the most common and crucial. In the indoor environment, the popular outdoor Google maps cannot be utilized since we need maps showing the layout of every individual floor. Also, layout of different floors differ from one another. Therefore, algorithms are required to detect whether we are inside or outside a building and determine on which floor we are located so that an appropriate map can be selected accordingly. For Wi-Fi based indoor localization, the performance of location estimation is closely related to the algorithms and deployment that we are using. It is difficult to find out a general approach that can be used to evaluate any localization system. On one hand, since the RF signal will suffer extra loss when traveling through the ceilings between floors, its propagation property will be different from the empirical ones and consequently we should design a new propagation model for 3D scenarios. On the other hand, properties of sensors are unique so that corresponding models are required before we analyze the localization scheme. In-depth investigation on the possible hybrid are also needed in case more than one sensor is operated in the localization system. In this thesis, we firstly designed two algorithms to use GPS signal for detecting whether the smart device is operating inside or outside a building, which is called outdoor-indoor transition detection. We also design another algorithm to use barometer data for determining on which floor are we located, which is considered as a multi-floor transition detection. With three scenarios designed inside the Akwater Kent Laboratory building (AK building) at Worcester Polytechnic Institute (WPI), we collected raw data from an Android phone with a version of 4.3 and conducted experimental analysis based on that. An efficient way to quantitatively evaluate the 3D localization systems is using Cramer-Rao Lower Bound (CRLB), which is considered as the lower bound of the estimated error for any localization system. The characteristics of Wi-Fi and barometer signals are explored and proper models are introduced as a foundation. Then we extended the 2D CRLB into a 3D format so that it can fit the our 3D scenarios. A barometer-assisted CRLB is introduced as an improvement for the existing Wi-Fi Receive Signal Strength (RSS)-only scheme and both of the two schemes are compared with the contours in every scenario and the statistical analysis.

<p>Many methods used for estimation and detection consider only the mean and variance of the involved noise instead of the full noise descriptions. One reason for this is that the mathematics is often considerably simplified this way. However, the implications of the simplifications are seldom studied, and this thesis shows that if no approximations are made performance is gained. Furthermore, the gain is quantified in terms of the useful information in the noise distributions involved. The useful information is given by the intrinsic accuracy, and a method to compute the intrinsic accuracy for a given distribution, using Monte Carlo methods, is outlined.</p><p>A lower bound for the covariance of the estimation error for any unbiased estimator s given by the Cramér-Rao lower bound (CRLB). At the same time, the Kalman filter is the best linear unbiased estimator (BLUE) for linear systems. It is in this thesis shown that the CRLB and the BLUE performance are given by the same expression, which is parameterized in the intrinsic accuracy of the noise. How the performance depends on the noise is then used to indicate when nonlinear filters, e.g., a particle filter, should be used instead of a Kalman filter. The CRLB results are shown, in simulations, to be a useful indication of when to use more powerful estimation methods. The simulations also show that other techniques should be used as a complement to the CRLB analysis to get conclusive performance results.</p><p>For fault detection, the statistics of the asymptotic generalized likelihood ratio (GLR) test provides an upper bound on the obtainable detection performance. The performance is in this thesis shown to depend on the intrinsic accuracy of the involved noise. The asymptotic GLR performance can then be calculated for a test using the actual noise and for a test using the approximative Gaussian noise. Based on the difference in performance, it is possible to draw conclusions about the quality of the Gaussian approximation. Simulations show that when the difference in performance is large, an exact noise representation improves the detection. Simulations also show that it is difficult to predict the exact influence on the detection performance caused by substituting the system noise with Gaussian noise approximations.</p> / <p>Många metoder som används i estimerings- och detekteringssammanhang tar endast hänsyn till medelvärde och varians hos ingående brus istället för att använda en fullständig brusbeskrivning. En av anledningarna till detta är att den förenklade brusmodellen underlättar många beräkningar. Dock studeras sällan de effekter förenklingarna leder till. Denna avhandling visar att om inga förenklingar görs kan prestandan förbättras och det visas också hur förbättringen kan relateras till den intressanta informationen i det involverade bruset. Den intressanta informationen är den inneboende noggrannheten (eng. intrinsic accuracy) och ett sätt för att bestämma den inneboende noggrannheten hos en given fördelning med hjälp av Monte-Carlo-metoder presenteras.</p><p>Ett mått på hur bra en estimator utan bias kan göras ges av Cramér-Raos undre gräns (CRLB). Samtidigt är det känt att kalmanfiltret är den bästa lineära biasfria estimatorn (BLUE) för lineära system. Det visas här att CRLB och BLUE-prestanda ges av samma matematiska uttryck där den inneboende noggrannheten ingår som en parameter. Kunskap om hur informationen påverkar prestandan kan sedan användas för att indikera när ett olineärt filter, t.ex. ett partikelfilter, bör användas istället för ett kalmanfilter. Med hjälp av simuleringar visas att CRLB är ett användbart mått för att indikera när mer avancerade metoder kan vara lönsamma. Simuleringarna visar dock också att CRLB-analysen bör kompletteras med andra tekniker för att det ska vara möjligt att dra definitiva slutsatser.</p><p>I fallet feldetektion ger de asymptotiska egenskaperna hos den generaliserade sannolikhetskvoten (eng. generalized likelihood ratio, GLR) en övre gräns för hur bra detektorer som kan konstrueras. Det visas här hur den övre gränsen beror på den inneboende noggrannheten hos det aktuella bruset. Genom att beräkna asymptotisk GLR-testprestanda för det sanna bruset och för en gaussisk brusapproximation går det att dra slutsatser om huruvida den gaussiska approximationen är tillräckligt bra för att kunna användas. I simuleringar visas att det är lönsamt att använda sig av en exakt brusbeskrivning om skillnaden i prestanda är stor mellan de båda fallen. Simuleringarna indikerar också att det kan vara svårt att förutsäga den exakta effekten av en gaussisk brusapproximation.</p> / Report code: LiU-Tek-Lic-2005:54

This thesis presents how to estimate the length of a power cable using the MaximumLikelihood Estimate (MLE) technique by using Matlab. The model of the power cableis evaluated in the time domain with additive white Gaussian noise. The statistics havebeen used to evaluate the performance of the estimator, by repeating the experiment fora large number of samples where the random additive noise is generated for each sample.The estimated sample variance is compared to the theoretical Cramer Raw lower Bound(CRLB) for unbiased estimators. At the end of thesis, numerical results are presentedthat show when the resulting sample variance is close to the CRLB, and hence that theperformance of the estimator will be more accurate.

Thanks to its improved capabilities, the Multiple Input Multiple Output (MIMO) radar is attracting the attention of researchers and practitioners alike. Because it transmits orthogonal or partially correlated waveforms, this emerging technology outperformed the phased array radar by providing better parametric identifiability, achieving higher spatial resolution, and designing complex beampatterns. To avoid jamming and enhance the signal to noise ratio, it is often interesting to maximize the transmitted power in a given region of interest and minimize it elsewhere. This problem is known as the transmit beampattern design and is usually tackled as a two-step process: a transmit covariance matrix is firstly designed by minimizing a convex optimization problem, which is then used to generate practical waveforms. In this work, we propose simple novel methods to generate correlated waveforms using finite alphabet constant and non-constant-envelope symbols. To generate finite alphabet waveforms, the proposed method maps easily generated Gaussian random variables onto the phase-shift-keying, pulse-amplitude, and quadrature-amplitude modulation schemes. For such mapping, the probability density function of Gaussian random variables is divided into M regions, where M is the number of alphabets in the corresponding modulation scheme. By exploiting the mapping function, the relationship between the cross-correlation of Gaussian and finite alphabet symbols is derived. The second part of this thesis covers the topic of target parameter estimation. To determine the reflection coefficient, spatial location, and Doppler shift of a target, maximum likelihood estimation yields the best performance. However, it requires a two dimensional search problem. Therefore, its computational complexity is prohibitively high. So, we proposed a reduced complexity and optimum performance algorithm which allows the two dimensional fast Fourier transform to jointly estimate the spatial location and Doppler shift. To assess the performance of the proposed estimators, the Cramér-Rao Lower Bound (CRLB) is derived. Simulation results show that the mean square estimation error of the proposed estimators achieve the CRLB. Keywords: Collocate antennas, multiple-input multiple-output (MIMO) radar, Finite alphabet waveforms, Hermite polynomials, Reflection coefficient, Doppler, Spatial location, Cramér-Rao Lower Bound.

Many methods used for estimation and detection consider only the mean and variance of the involved noise instead of the full noise descriptions. One reason for this is that the mathematics is often considerably simplified this way. However, the implications of the simplifications are seldom studied, and this thesis shows that if no approximations are made performance is gained. Furthermore, the gain is quantified in terms of the useful information in the noise distributions involved. The useful information is given by the intrinsic accuracy, and a method to compute the intrinsic accuracy for a given distribution, using Monte Carlo methods, is outlined. A lower bound for the covariance of the estimation error for any unbiased estimator s given by the Cramér-Rao lower bound (CRLB). At the same time, the Kalman filter is the best linear unbiased estimator (BLUE) for linear systems. It is in this thesis shown that the CRLB and the BLUE performance are given by the same expression, which is parameterized in the intrinsic accuracy of the noise. How the performance depends on the noise is then used to indicate when nonlinear filters, e.g., a particle filter, should be used instead of a Kalman filter. The CRLB results are shown, in simulations, to be a useful indication of when to use more powerful estimation methods. The simulations also show that other techniques should be used as a complement to the CRLB analysis to get conclusive performance results. For fault detection, the statistics of the asymptotic generalized likelihood ratio (GLR) test provides an upper bound on the obtainable detection performance. The performance is in this thesis shown to depend on the intrinsic accuracy of the involved noise. The asymptotic GLR performance can then be calculated for a test using the actual noise and for a test using the approximative Gaussian noise. Based on the difference in performance, it is possible to draw conclusions about the quality of the Gaussian approximation. Simulations show that when the difference in performance is large, an exact noise representation improves the detection. Simulations also show that it is difficult to predict the exact influence on the detection performance caused by substituting the system noise with Gaussian noise approximations. / Många metoder som används i estimerings- och detekteringssammanhang tar endast hänsyn till medelvärde och varians hos ingående brus istället för att använda en fullständig brusbeskrivning. En av anledningarna till detta är att den förenklade brusmodellen underlättar många beräkningar. Dock studeras sällan de effekter förenklingarna leder till. Denna avhandling visar att om inga förenklingar görs kan prestandan förbättras och det visas också hur förbättringen kan relateras till den intressanta informationen i det involverade bruset. Den intressanta informationen är den inneboende noggrannheten (eng. intrinsic accuracy) och ett sätt för att bestämma den inneboende noggrannheten hos en given fördelning med hjälp av Monte-Carlo-metoder presenteras. Ett mått på hur bra en estimator utan bias kan göras ges av Cramér-Raos undre gräns (CRLB). Samtidigt är det känt att kalmanfiltret är den bästa lineära biasfria estimatorn (BLUE) för lineära system. Det visas här att CRLB och BLUE-prestanda ges av samma matematiska uttryck där den inneboende noggrannheten ingår som en parameter. Kunskap om hur informationen påverkar prestandan kan sedan användas för att indikera när ett olineärt filter, t.ex. ett partikelfilter, bör användas istället för ett kalmanfilter. Med hjälp av simuleringar visas att CRLB är ett användbart mått för att indikera när mer avancerade metoder kan vara lönsamma. Simuleringarna visar dock också att CRLB-analysen bör kompletteras med andra tekniker för att det ska vara möjligt att dra definitiva slutsatser. I fallet feldetektion ger de asymptotiska egenskaperna hos den generaliserade sannolikhetskvoten (eng. generalized likelihood ratio, GLR) en övre gräns för hur bra detektorer som kan konstrueras. Det visas här hur den övre gränsen beror på den inneboende noggrannheten hos det aktuella bruset. Genom att beräkna asymptotisk GLR-testprestanda för det sanna bruset och för en gaussisk brusapproximation går det att dra slutsatser om huruvida den gaussiska approximationen är tillräckligt bra för att kunna användas. I simuleringar visas att det är lönsamt att använda sig av en exakt brusbeskrivning om skillnaden i prestanda är stor mellan de båda fallen. Simuleringarna indikerar också att det kan vara svårt att förutsäga den exakta effekten av en gaussisk brusapproximation. / <p>Report code: LiU-Tek-Lic-2005:54</p>