Conventional tools in array signal processing have traditionally relied on the availability of a large number of samples acquired at each sensor or array element (antenna, hydrophone, microphone, etc.). Large sample size assumptions typically guarantee the consistency of estimators, detectors, classifiers and multiple other widely used signal processing procedures. However, practical scenario and array mobility conditions, together with the need for low latency and reduced scanning times, impose strong limits on the total number of observations that can be effectively processed. When the number of collected samples per sensor is small, conventional large sample asymptotic approaches are not relevant anymore. Recently, large random matrix theory tools have been proposed in order to address the small sample support problem in array signal processing. In fact, it has been shown that the most important and longstanding problems in this field can be reformulated and studied according to this asymptotic paradigm. By exploiting the latest advances in large random matrix theory and high dimensional statistics, a novel and unconventional methodology can be established, which provides an unprecedented treatment of the finite sample-per-sensor regime. In this talk, we will see that random matrix theory establishes a unifying framework for the study of array signal processing techniques under the constraint of a small number of observations per sensor, which has radically changed the way in which array processing methodologies have been traditionally established. We will show how this unconventional way of revisiting classical array processing has lead to major advances in the design and analysis of signal processing techniques for multidimensional observations.
Rayleigh fading occurs without a line of sight path between transmitter and receiver, as in heavily built-up urban environments, and results in fast fading. Ricean fading includes a stronger line of sight or reflection signal with multipath interference, causing slower fading. The document discusses three propagation mechanisms - reflection, diffraction, and scattering - and explains how physical factors like multipath propagation, mobile speed, surrounding object speed, and signal bandwidth influence fading in mobile radio channels. Doppler shift from motion causes frequency modulation, while coherence bandwidth characterizes the frequency range where the channel response remains flat.
Concept of Diversity & Fading (wireless communication)Omkar Rane
This document discusses concepts related to fading and diversity in wireless communication systems. It introduces fading as signal variations caused by multipath interference from multiple signal propagation paths. It describes two types of fading: large-scale fading due to path loss and shadowing, and small-scale fading which includes fast fading due to mobility and slow fading due to shadowing. It also discusses different diversity techniques that can be used to combat fading, including space, polarization, frequency and time diversity.
Hello everyone. This is a short presentation on path loss and shadowing. I have not covered all the topics but a brief idea is given on path loss and wireless channel propagation models.
Hope you find it useful.
Thanks
This document discusses aperture antennas. It begins by defining an aperture antenna as an antenna that uses an opening or closed surface as the radiating element. It then lists the main types of aperture antennas like horn antennas, reflector antennas, slot antennas, and microstrip antennas. The document focuses on analyzing aperture antennas using techniques like the current distribution method, aperture analysis, and the Fourier transform method. It explains key principles used in aperture analysis like the field equivalence principle, Huygens' principle, and Babinet's principle. The document provides examples of analyzing specific aperture antenna types and their radiation patterns.
There are 3 main propagation mechanisms in mobile communication systems:
1. Reflection occurs when signals bounce off surfaces like buildings and earth.
2. Diffraction is when signals bend around obstacles like hills and buildings.
3. Scattering is when signals are deflected in many directions by small obstacles like trees and signs. These 3 mechanisms impact the received power and must be considered in propagation models.
Introduction to multiple signal classifier (music)Milkessa Negeri
This document provides an introduction to the MUSIC algorithm, which is used to estimate the frequency content of a signal or autocorrelation matrix using an eigenspace method. It assumes a signal consists of complex exponentials in noise. MUSIC is a high-resolution algorithm that uses the eigenvectors of the autocorrelation matrix to separate the signal and noise subspaces. The document also describes how MUSIC can be used for adaptive beamforming to enhance a desired signal while suppressing interference using an array of sensors. It compares MUSIC to the ESPRIT algorithm for direction of arrival estimation.
Wireless communication systems are impacted by fading effects that cause fluctuations in signal strength. Fading occurs due to multipath propagation which results in multiple versions of the transmitted signal reaching the receiver at different times. This can cause either flat or frequency selective fading depending on the delay spread. Modulation techniques like BPSK can be used to combat fading. Simulation of a Rayleigh fading channel, which occurs when there is no dominant signal path, showed that it significantly impacts the bit error rate of a BPSK modulated signal. Future work could explore additional modulation techniques and integrating the model into a network simulator.
Rayleigh fading occurs without a line of sight path between transmitter and receiver, as in heavily built-up urban environments, and results in fast fading. Ricean fading includes a stronger line of sight or reflection signal with multipath interference, causing slower fading. The document discusses three propagation mechanisms - reflection, diffraction, and scattering - and explains how physical factors like multipath propagation, mobile speed, surrounding object speed, and signal bandwidth influence fading in mobile radio channels. Doppler shift from motion causes frequency modulation, while coherence bandwidth characterizes the frequency range where the channel response remains flat.
Concept of Diversity & Fading (wireless communication)Omkar Rane
This document discusses concepts related to fading and diversity in wireless communication systems. It introduces fading as signal variations caused by multipath interference from multiple signal propagation paths. It describes two types of fading: large-scale fading due to path loss and shadowing, and small-scale fading which includes fast fading due to mobility and slow fading due to shadowing. It also discusses different diversity techniques that can be used to combat fading, including space, polarization, frequency and time diversity.
Hello everyone. This is a short presentation on path loss and shadowing. I have not covered all the topics but a brief idea is given on path loss and wireless channel propagation models.
Hope you find it useful.
Thanks
This document discusses aperture antennas. It begins by defining an aperture antenna as an antenna that uses an opening or closed surface as the radiating element. It then lists the main types of aperture antennas like horn antennas, reflector antennas, slot antennas, and microstrip antennas. The document focuses on analyzing aperture antennas using techniques like the current distribution method, aperture analysis, and the Fourier transform method. It explains key principles used in aperture analysis like the field equivalence principle, Huygens' principle, and Babinet's principle. The document provides examples of analyzing specific aperture antenna types and their radiation patterns.
There are 3 main propagation mechanisms in mobile communication systems:
1. Reflection occurs when signals bounce off surfaces like buildings and earth.
2. Diffraction is when signals bend around obstacles like hills and buildings.
3. Scattering is when signals are deflected in many directions by small obstacles like trees and signs. These 3 mechanisms impact the received power and must be considered in propagation models.
Introduction to multiple signal classifier (music)Milkessa Negeri
This document provides an introduction to the MUSIC algorithm, which is used to estimate the frequency content of a signal or autocorrelation matrix using an eigenspace method. It assumes a signal consists of complex exponentials in noise. MUSIC is a high-resolution algorithm that uses the eigenvectors of the autocorrelation matrix to separate the signal and noise subspaces. The document also describes how MUSIC can be used for adaptive beamforming to enhance a desired signal while suppressing interference using an array of sensors. It compares MUSIC to the ESPRIT algorithm for direction of arrival estimation.
Wireless communication systems are impacted by fading effects that cause fluctuations in signal strength. Fading occurs due to multipath propagation which results in multiple versions of the transmitted signal reaching the receiver at different times. This can cause either flat or frequency selective fading depending on the delay spread. Modulation techniques like BPSK can be used to combat fading. Simulation of a Rayleigh fading channel, which occurs when there is no dominant signal path, showed that it significantly impacts the bit error rate of a BPSK modulated signal. Future work could explore additional modulation techniques and integrating the model into a network simulator.
The document describes the design and simulation of a basic half-wave dipole antenna. Key points:
1) The aim is to design a dipole antenna for a given frequency of 3.3 GHz and study the effects of varying the dielectric constant and substrate thickness on the radiation properties and frequency response.
2) Important antenna characteristics to consider include radiation patterns, gain, and frequency response.
3) The half-wave dipole antenna is designed with each arm measuring 22.725mm to operate at the target frequency, and each arm width is 4.545mm.
4) Simulation shows the antenna operates at 2.8GHz with a return loss of -14.50dB and gain of
This document summarizes an online seminar about antenna basics and design concepts. It discusses the historical development of antennas from the 19th century works of scientists like Maxwell and Hertz to modern applications. Key antenna topics are defined, like radiation patterns, polarization, directivity and gain. Specific antenna types are described, such as dipoles, loops, Yagi-Uda arrays. The presentation outlines antenna parameters that influence performance, including materials, size, efficiency and impedance matching.
The document discusses various image transforms. It begins by explaining why transforms are used, such as for fast computation and obtaining conceptual insights. It then introduces image transforms as unitary matrices that represent images using a discrete set of basis images. It proceeds to describe one-dimensional orthogonal and unitary transforms using matrices. It also discusses separable two-dimensional transforms and provides properties of unitary transforms such as energy conservation. Specific transforms discussed in more detail include the discrete Fourier transform, discrete cosine transform, discrete sine transform, and Hadamard transform.
This document discusses the concept of diffraction as it relates to wireless communication. It explains that diffraction allows radio signals to propagate behind obstacles between a transmitter and receiver. It presents Huygen's principle, which states that each point on a wavefront can be considered a secondary source of wavelets. These wavelets combine to form a new wavefront. The document also covers knife-edge diffraction geometry and how to calculate the excess path length and phase difference between the diffracted and direct paths. It defines Fresnel zones and introduces the Fresnel zone diffraction parameter used to determine whether interference will be constructive or destructive. Additionally, it explains diffraction loss that occurs when secondary waves are blocked, resulting in only partial energy being diffract
The document discusses the Yagi-Uda antenna, which consists of multiple parallel dipole elements including a reflector, driven element, and multiple directors. It operates in the HF to UHF bands and provides a directional radiation pattern with moderate gain. Key advantages are its directionality and ability to operate at high frequencies. Common applications include television reception and radar systems where its directional properties and moderate gain are beneficial.
Fading and the Doppler effect can impact wireless communication. Fading occurs when multipath signals interfere, causing fluctuations in signal strength over time. The Doppler effect is the change in frequency of a wave due to relative motion between the source and observer. In wireless communication, the Doppler effect causes shifts in the received carrier frequency due to motion between the transmitter and receiver. This Doppler spread must be accounted for in system design through techniques like Doppler compensation.
MicroStrip Antenna
Introduction .
Micro-Strip Antennas Types .
Micro-Strip Antennas Shapes .
Types of Substrates (Dielectric Media) .
Comparison of various types of flat profile printed antennas .
Advantages & DisAdvantages of MSAs .
Applications of MSAs .
Radiation patterns of MSAs .
How to Optimizing the Substrate Properties for Increased Bandwidth ?
Comparing the different feed techniques .
5. convolution and correlation of discrete time signals MdFazleRabbi18
This document discusses convolution and correlation of discrete time signals. It defines convolution as a mathematical way of combining two signals to form a third signal, which is equivalent to finite impulse response filtering. Convolution relates the input, output, and impulse response of a linear time-invariant system. The document also provides examples of discrete linear convolution and periodic convolution. It then defines correlation as a measure of similarity between signals, discussing cross-correlation and auto-correlation, and providing examples of calculating each.
This document discusses Friis transmission formula for free space path loss. It defines key terms like power density, effective aperture, and antenna gain. The Friis formula calculates received power as a function of transmitted power, transmitter and receiver gains, wavelength, and distance. It states that path loss increases with distance and is inversely proportional to the square of the distance. The document also notes some drawbacks of the Friis model and conditions for applying it in the far field region.
This document discusses different types of antennas used for transmitting and receiving electromagnetic waves. It describes log-periodic antennas, which work over a wide frequency range using a logarithmic size progression of elements. Specific types are described, including bow-tie antennas and log-periodic dipole arrays. Wire antennas like dipoles, monopoles, and loops are also covered. Travelling wave antennas transmit signals along their length, represented by helical and Yagi-Uda antennas. Microwave antennas and reflector antennas are used at higher frequencies for applications like communication and radar. Key antenna properties and a variety of applications are also summarized.
- Daubechies wavelets are a family of orthogonal wavelets that provide the highest number of vanishing moments for a given width, defined through recursive equations.
- They are approximately localized in both time and frequency domains. The wavelets and scaling functions are not defined by closed-form equations, but are instead generated numerically through an iterative process.
- Properties include orthogonality, localization, and a maximal number of vanishing moments for a given support width, with more coefficients providing more moments. They are widely used for problems involving signal discontinuities or self-similarity.
This power point presentation discusses cell splitting and sectoring techniques used to increase channel capacity in cellular networks. It explains that a large cellular area is divided into smaller hexagonal cells, each with its own base station and frequency set. To further increase capacity, cells can be split into smaller cells served by additional base stations. Alternatively, directional antennas can be used to sector each cell into three segments to reduce interference and allow frequency reuse over smaller areas. Both techniques aim to add channels by subdividing congested cells.
1) The document discusses small-scale fading in mobile radio channels caused by multipath propagation. Multipath signals interfere constructively and destructively, causing rapid fluctuations in received signal strength over small distances.
2) Key parameters that characterize multipath channels are delay spread (στ), coherence bandwidth (Bc), Doppler spread (BD), and coherence time (Tc). Delay spread and coherence bandwidth describe time dispersion, while Doppler spread and coherence time describe frequency dispersion from mobility.
3) There are different types of fading depending on how a signal's bandwidth compares to these channel parameters. Flat fading occurs when the signal bandwidth is narrow compared to the channel bandwidth, preserving the signal's spectral properties.
This document discusses various diversity techniques used in wireless communications to combat fading. It describes types of diversity including time, frequency, multiuser, and space diversity. It also outlines combining techniques such as selection combining, maximal ratio combining and equal gain combining that are used to improve the signal by combining signals from multiple diversity branches. The document concludes by discussing multiple input multiple output (MIMO) systems and orthogonal frequency division multiple access (OFDMA) schemes that exploit diversity and multiuser diversity.
1. The document discusses radiation from a two-wire transmission line connected to an antenna. It explains how electric and magnetic fields are created between the conductors when a voltage is applied. Electromagnetic waves travel along the transmission line and enter the antenna.
2. When part of the antenna structure is removed, free space waves are formed by connecting the open ends of the electric field lines. The constant phase point of these waves moves outward at the speed of light.
3. Key terms related to antennas like radial power flow, radiation resistance, uniform current distribution, principle planes, beam width, polarization, effective aperture area, directive gain, power gain, and dual characteristics are defined in the document.
Cellular concepts and system design fundamentalsKamal Sharma
The document discusses the cellular concept which aims to increase capacity by replacing single high-power transmitters with multiple low-power transmitters, each covering a small cell. Key aspects covered include:
- Cells are allocated different channel groups to minimize interference between nearby base stations.
- A cellular system reuses the same set of channels in different cells through frequency planning and by assigning different channel groups to neighboring cells.
- Hexagonal cell shapes help maximize coverage while minimizing gaps and support efficient frequency reuse patterns.
- Techniques like cell splitting, sectoring, and microcells help increase capacity by reducing cell sizes and reusing frequencies.
The document discusses capacitance variation in MEMS capacitors based on plate area, distance between plates, and different dielectric materials. It outlines the research significance, methodology, experimental details, results and discussion, and conclusion. The results show that capacitance increases with larger plate area, smaller distance between plates, and dielectric materials with higher relative dielectric constants. The capacitance variation can be utilized in applications such as wireless communications, sensing, and electronics.
This document discusses smart antenna technology. It defines smart antennas as antenna systems that combine multiple antenna elements with signal processing to optimize radiation and reception patterns in response to the signal environment. The document describes two main types of smart antennas: switched beam antennas which form fixed beams and adaptively switch between them, and adaptive array antennas which can form an infinite number of patterns in real-time to maximize desired signals and minimize interference. It compares the advantages and drawbacks of each type and discusses applications of smart antenna technology in fields like wireless networks and satellite systems.
1. The document discusses various image transforms including discrete cosine transform (DCT), discrete wavelet transform (DWT), and contourlet transform.
2. DCT transforms an image into frequency domain and organizes values based on human visual system importance. DWT analyzes images using wavelets of different scales and positions.
3. Contourlet transform is derived directly from discrete domain to capture smooth contours and edges at any orientation, decoupling multiscale and directional decompositions. It provides better efficiency than DWT for representing images.
Concepts of & cell sectoring and micro cellKundan Kumar
The document discusses concepts related to cellular network sectoring and microcells. It explains that cells can have square or hexagonal shapes, with hexagons providing equidistant antennas. Frequency reuse allows the same frequencies to be used in different cells by controlling base station power to limit interference. Common frequency reuse patterns include reuse factors of 1, 3, 7, etc. Capacity can be increased through methods like frequency borrowing, cell splitting, cell sectoring, and microcells which use smaller cell sizes.
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
The document describes the design and simulation of a basic half-wave dipole antenna. Key points:
1) The aim is to design a dipole antenna for a given frequency of 3.3 GHz and study the effects of varying the dielectric constant and substrate thickness on the radiation properties and frequency response.
2) Important antenna characteristics to consider include radiation patterns, gain, and frequency response.
3) The half-wave dipole antenna is designed with each arm measuring 22.725mm to operate at the target frequency, and each arm width is 4.545mm.
4) Simulation shows the antenna operates at 2.8GHz with a return loss of -14.50dB and gain of
This document summarizes an online seminar about antenna basics and design concepts. It discusses the historical development of antennas from the 19th century works of scientists like Maxwell and Hertz to modern applications. Key antenna topics are defined, like radiation patterns, polarization, directivity and gain. Specific antenna types are described, such as dipoles, loops, Yagi-Uda arrays. The presentation outlines antenna parameters that influence performance, including materials, size, efficiency and impedance matching.
The document discusses various image transforms. It begins by explaining why transforms are used, such as for fast computation and obtaining conceptual insights. It then introduces image transforms as unitary matrices that represent images using a discrete set of basis images. It proceeds to describe one-dimensional orthogonal and unitary transforms using matrices. It also discusses separable two-dimensional transforms and provides properties of unitary transforms such as energy conservation. Specific transforms discussed in more detail include the discrete Fourier transform, discrete cosine transform, discrete sine transform, and Hadamard transform.
This document discusses the concept of diffraction as it relates to wireless communication. It explains that diffraction allows radio signals to propagate behind obstacles between a transmitter and receiver. It presents Huygen's principle, which states that each point on a wavefront can be considered a secondary source of wavelets. These wavelets combine to form a new wavefront. The document also covers knife-edge diffraction geometry and how to calculate the excess path length and phase difference between the diffracted and direct paths. It defines Fresnel zones and introduces the Fresnel zone diffraction parameter used to determine whether interference will be constructive or destructive. Additionally, it explains diffraction loss that occurs when secondary waves are blocked, resulting in only partial energy being diffract
The document discusses the Yagi-Uda antenna, which consists of multiple parallel dipole elements including a reflector, driven element, and multiple directors. It operates in the HF to UHF bands and provides a directional radiation pattern with moderate gain. Key advantages are its directionality and ability to operate at high frequencies. Common applications include television reception and radar systems where its directional properties and moderate gain are beneficial.
Fading and the Doppler effect can impact wireless communication. Fading occurs when multipath signals interfere, causing fluctuations in signal strength over time. The Doppler effect is the change in frequency of a wave due to relative motion between the source and observer. In wireless communication, the Doppler effect causes shifts in the received carrier frequency due to motion between the transmitter and receiver. This Doppler spread must be accounted for in system design through techniques like Doppler compensation.
MicroStrip Antenna
Introduction .
Micro-Strip Antennas Types .
Micro-Strip Antennas Shapes .
Types of Substrates (Dielectric Media) .
Comparison of various types of flat profile printed antennas .
Advantages & DisAdvantages of MSAs .
Applications of MSAs .
Radiation patterns of MSAs .
How to Optimizing the Substrate Properties for Increased Bandwidth ?
Comparing the different feed techniques .
5. convolution and correlation of discrete time signals MdFazleRabbi18
This document discusses convolution and correlation of discrete time signals. It defines convolution as a mathematical way of combining two signals to form a third signal, which is equivalent to finite impulse response filtering. Convolution relates the input, output, and impulse response of a linear time-invariant system. The document also provides examples of discrete linear convolution and periodic convolution. It then defines correlation as a measure of similarity between signals, discussing cross-correlation and auto-correlation, and providing examples of calculating each.
This document discusses Friis transmission formula for free space path loss. It defines key terms like power density, effective aperture, and antenna gain. The Friis formula calculates received power as a function of transmitted power, transmitter and receiver gains, wavelength, and distance. It states that path loss increases with distance and is inversely proportional to the square of the distance. The document also notes some drawbacks of the Friis model and conditions for applying it in the far field region.
This document discusses different types of antennas used for transmitting and receiving electromagnetic waves. It describes log-periodic antennas, which work over a wide frequency range using a logarithmic size progression of elements. Specific types are described, including bow-tie antennas and log-periodic dipole arrays. Wire antennas like dipoles, monopoles, and loops are also covered. Travelling wave antennas transmit signals along their length, represented by helical and Yagi-Uda antennas. Microwave antennas and reflector antennas are used at higher frequencies for applications like communication and radar. Key antenna properties and a variety of applications are also summarized.
- Daubechies wavelets are a family of orthogonal wavelets that provide the highest number of vanishing moments for a given width, defined through recursive equations.
- They are approximately localized in both time and frequency domains. The wavelets and scaling functions are not defined by closed-form equations, but are instead generated numerically through an iterative process.
- Properties include orthogonality, localization, and a maximal number of vanishing moments for a given support width, with more coefficients providing more moments. They are widely used for problems involving signal discontinuities or self-similarity.
This power point presentation discusses cell splitting and sectoring techniques used to increase channel capacity in cellular networks. It explains that a large cellular area is divided into smaller hexagonal cells, each with its own base station and frequency set. To further increase capacity, cells can be split into smaller cells served by additional base stations. Alternatively, directional antennas can be used to sector each cell into three segments to reduce interference and allow frequency reuse over smaller areas. Both techniques aim to add channels by subdividing congested cells.
1) The document discusses small-scale fading in mobile radio channels caused by multipath propagation. Multipath signals interfere constructively and destructively, causing rapid fluctuations in received signal strength over small distances.
2) Key parameters that characterize multipath channels are delay spread (στ), coherence bandwidth (Bc), Doppler spread (BD), and coherence time (Tc). Delay spread and coherence bandwidth describe time dispersion, while Doppler spread and coherence time describe frequency dispersion from mobility.
3) There are different types of fading depending on how a signal's bandwidth compares to these channel parameters. Flat fading occurs when the signal bandwidth is narrow compared to the channel bandwidth, preserving the signal's spectral properties.
This document discusses various diversity techniques used in wireless communications to combat fading. It describes types of diversity including time, frequency, multiuser, and space diversity. It also outlines combining techniques such as selection combining, maximal ratio combining and equal gain combining that are used to improve the signal by combining signals from multiple diversity branches. The document concludes by discussing multiple input multiple output (MIMO) systems and orthogonal frequency division multiple access (OFDMA) schemes that exploit diversity and multiuser diversity.
1. The document discusses radiation from a two-wire transmission line connected to an antenna. It explains how electric and magnetic fields are created between the conductors when a voltage is applied. Electromagnetic waves travel along the transmission line and enter the antenna.
2. When part of the antenna structure is removed, free space waves are formed by connecting the open ends of the electric field lines. The constant phase point of these waves moves outward at the speed of light.
3. Key terms related to antennas like radial power flow, radiation resistance, uniform current distribution, principle planes, beam width, polarization, effective aperture area, directive gain, power gain, and dual characteristics are defined in the document.
Cellular concepts and system design fundamentalsKamal Sharma
The document discusses the cellular concept which aims to increase capacity by replacing single high-power transmitters with multiple low-power transmitters, each covering a small cell. Key aspects covered include:
- Cells are allocated different channel groups to minimize interference between nearby base stations.
- A cellular system reuses the same set of channels in different cells through frequency planning and by assigning different channel groups to neighboring cells.
- Hexagonal cell shapes help maximize coverage while minimizing gaps and support efficient frequency reuse patterns.
- Techniques like cell splitting, sectoring, and microcells help increase capacity by reducing cell sizes and reusing frequencies.
The document discusses capacitance variation in MEMS capacitors based on plate area, distance between plates, and different dielectric materials. It outlines the research significance, methodology, experimental details, results and discussion, and conclusion. The results show that capacitance increases with larger plate area, smaller distance between plates, and dielectric materials with higher relative dielectric constants. The capacitance variation can be utilized in applications such as wireless communications, sensing, and electronics.
This document discusses smart antenna technology. It defines smart antennas as antenna systems that combine multiple antenna elements with signal processing to optimize radiation and reception patterns in response to the signal environment. The document describes two main types of smart antennas: switched beam antennas which form fixed beams and adaptively switch between them, and adaptive array antennas which can form an infinite number of patterns in real-time to maximize desired signals and minimize interference. It compares the advantages and drawbacks of each type and discusses applications of smart antenna technology in fields like wireless networks and satellite systems.
1. The document discusses various image transforms including discrete cosine transform (DCT), discrete wavelet transform (DWT), and contourlet transform.
2. DCT transforms an image into frequency domain and organizes values based on human visual system importance. DWT analyzes images using wavelets of different scales and positions.
3. Contourlet transform is derived directly from discrete domain to capture smooth contours and edges at any orientation, decoupling multiscale and directional decompositions. It provides better efficiency than DWT for representing images.
Concepts of & cell sectoring and micro cellKundan Kumar
The document discusses concepts related to cellular network sectoring and microcells. It explains that cells can have square or hexagonal shapes, with hexagons providing equidistant antennas. Frequency reuse allows the same frequencies to be used in different cells by controlling base station power to limit interference. Common frequency reuse patterns include reuse factors of 1, 3, 7, etc. Capacity can be increased through methods like frequency borrowing, cell splitting, cell sectoring, and microcells which use smaller cell sizes.
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
The MAIN CONTRIBUTION is an on-line heuristic law to set the training process and to modify the NN topology based on the Levenberg-Marquardt method.
An Area Predictor Filter using nonlinear autoregressive model based on neural networks for time series forecasting is introduced.
The core of the proposal is to analyze the roughness (long or short term stochastic dependence) of time series evaluated by the Hurst parameter (H).
The proposed law adapts in real time the topology of the filter at each stage of time series, changing the number of pattern, the number of iterations and the input vector length.
The main results show a good performance of the predictor, considering in particular to time series whose H parameter has a high roughness of signal, which is evaluated by HS and HA, respectively.
These results encouraged to continue working on new adjustment algorithms for time series modeling natural phenomena.
Adaptive blind multiuser detection under impulsive noise using principal comp...csandit
The document describes an adaptive blind multiuser detection method for asynchronous code division multiple access (CDMA) systems using principal component analysis (PCA) in impulsive noise environments. PCA is used to extract the principal components from the received signal without requiring training sequences or prior knowledge of channel characteristics. The PCA blind multiuser detector provides robust performance compared to knowledge-based detectors when signature waveforms and timing offsets of users are unknown. Simulation results show the proposed PCA method offers substantial gains over traditional subspace methods for multiuser detection.
ADAPTIVE BLIND MULTIUSER DETECTION UNDER IMPULSIVE NOISE USING PRINCIPAL COMP...csandit
In this paper we consider blind signal detection for an asynchronous code division multiple access (CDMA) system with Principal component analysis (PCA) in impulsive noise. The blind multiuser detector requires no training sequences compared with the conventional multiuser detection receiver. The proposed PCA blind multiuser detector is robust when compared with knowledge based signature waveforms and the timing of the user of interest. PCA is a statistical method for reducing the dimension of data set, spectral decomposition of the covariance matrix of the dataset i.e first and second order statistics are estimated.
Principal component analysis makes no assumption on the independence of the data vectors PCA searches for linear combinations with the largest variances and when several linear combinations are needed, it considers variances in decreasing order of importance. PCA
improves SNR of signals used for differential side channel analysis. In different to other approaches, the linear minimum mean-square-error (MMSE) detector is obtained blindly; the detector does not use any training sequence like in subspace methods to detect multi user
receiver. The algorithm need not estimate the subspace rank in order to reduce the computational complexity. Simulation results show that the new algorithm offers substantial performance gains over the traditional subspace methods.
ADAPTIVE BLIND MULTIUSER DETECTION UNDER IMPULSIVE NOISE USING PRINCIPAL COMP...cscpconf
The document describes an adaptive blind multiuser detection method for an asynchronous code division multiple access (CDMA) system using principal component analysis (PCA) in impulsive noise. PCA is used to reduce the dimensionality of the received signal data set without much loss of information. The key steps are to calculate the covariance matrix of the received signal, perform singular value decomposition to obtain the principal components which are ordered by decreasing variance, and select the top principal components which describe most of the variance in the original data to reduce dimensionality. Simulation results showed the proposed PCA blind multiuser detection method offers substantial performance gains over traditional subspace methods.
Diagnosis of Faulty Sensors in Antenna Array using Hybrid Differential Evolut...IJECEIAES
In this work, differential evolution based compressive sensing technique for detection of faulty sensors in linear arrays has been presented. This algorithm starts from taking the linear measurements of the power pattern generated by the array under test. The difference between the collected compressive measurements and measured healthy array field pattern is minimized using a hybrid differential evolution (DE). In the proposed method, the slow convergence of DE based compressed sensing technique is accelerated with the help of parallel coordinate decent algorithm (PCD). The combination of DE with PCD makes the minimization faster and precise. Simulation results validate the performance to detect faulty sensors from a small number of measurements.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...Alexander Litvinenko
Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications.
We develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of complex targets.
To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method.
This document contains lecture notes on signals and systems for a course at Chadalawada Ramanamma Engineering College. It includes:
1. An introduction to signals, systems, and some common elementary signals like the unit step, unit impulse, ramp, sinusoid, and exponential signals.
2. A classification of signals as continuous/discrete, deterministic/non-deterministic, even/odd, periodic/aperiodic, energy/power, and real/imaginary.
3. A discussion of basic operations on signals like amplitude scaling, addition, and subtraction.
Ill-posedness formulation of the emission source localization in the radio- d...Ahmed Ammar Rebai PhD
To contact the authors : tarek.salhi@gmail.com and ahmed.rebai2@gmail.com
In the field of radio detection in astroparticle physics, many studies have shown the strong dependence of the solution of the radio-transient sources localization problem (the radio-shower time of arrival on antennas) such solutions are purely numerical artifacts. Based on a detailed analysis of some already published results of radio-detection experiments like : CODALEMA 3 in France, AERA in Argentina and TREND in China, we demonstrate the ill-posed character of this problem in the sens of Hadamard. Two approaches have been used as the existence of solutions degeneration and the bad conditioning of the mathematical formulation problem. A comparison between experimental results and simulations have been made, to highlight the mathematical studies. Many properties of the non-linear least square function are discussed such as the configuration of the set of solutions and the bias.
Bit Error Rate Performance of MIMO Spatial Multiplexing with MPSK Modulation ...ijsrd.com
Wireless communication is one of the most effective areas of technology development of our time. Wireless communications today covers a very wide array of applications. In this, we study the performance of general MIMO system, the general V-BLAST architecture with MPSK Modulation in Rayleigh fading channels. Based on bit error rate, we show the performance of the 2x2 schemes with MPSK Modulation in noisy environment. We also show the bit error rate performance of 2x2, 3x3, 4x4 systems with BPSK modulation. We see that the bit error rate performance of 2x2 systems with QPSK modulation gives us the best performance among other schemes analysed here.
Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko
Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamics
The document discusses stochastic processes and random signals. Some key points:
- Stochastic processes describe random experiments that vary over time or space, such as noise in an audio signal.
- Random signals have uncertainty and cannot be precisely defined at a given time, but their average properties can be described.
- Random processes (also called stochastic processes) model time-varying waveforms with randomness, like data transmitted over a noisy channel.
- Random processes can be classified as continuous or discrete, stationary or non-stationary, predictable or unpredictable, and real-valued or complex-valued.
- Random processes are defined mathematically as measurable functions that map outcomes of a random experiment to real
My data are incomplete and noisy: Information-reduction statistical methods f...Umberto Picchini
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Accompanying code is available at https://github.com/umbertopicchini/pomp-ricker and https://github.com/umbertopicchini/abc_g-and-k
Readership lecture given at Lund University on 7 June 2016. The lecture is of popular science nature hence mathematical detail is kept to a minimum. However numerous links and references are offered for further reading.
This document provides an overview of digital modulators and line codes. It begins by introducing common digital modulation techniques like ASK, PSK, and FSK and modeling them using complex representations and signal constellations. It then discusses QPSK modulation in more detail, showing its block structure and constellation diagram. Finally, it briefly covers performance evaluation criteria and line codes, discussing how they can be represented using power spectral density.
This document discusses signals and their classification. It defines signals, analog and digital signals, periodic and aperiodic signals. It also discusses representing signals in Matlab and Simulink. Key signal types covered include exponential, sinusoidal, unit impulse and step functions. Matlab is presented as a tool for programming and analyzing discrete signals while Simulink can be used to model and simulate continuous systems.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
A Novel Algorithm to Estimate Closely Spaced Source DOA IJECEIAES
In order to improve resolution and direction of arrival (DOA) estimation of two closely spaced sources, in context of array processing, a new algorithm is presented. However, the proposed algorithm combines both spatial sampling technic to widen the resolution and a high resolution method which is the Multiple Signal Classification (MUSIC) to estimate the DOA of two closely spaced sources impinging on the far-field of Uniform Linear Array (ULA). Simulations examples are discussed to demonstrate the performance and the effectiveness of the proposed approach (referred as Spatial sampling MUSIC SS-MUSIC) compared to the classical MUSIC method when it’s used alone in this context.
In this paper, a new algorithm for a high resolution
Direction Of Arrival (DOA) estimation method for multiple
wideband signals is proposed. The proposed method proceeds
in two steps. In the first step, the received signals data is
decomposed in a Toeplitz form using the first-order statistics.
In the second step, The QR decomposition is applied on the
constructed Toeplitz matrix. Compared with existing schemes,
the proposed scheme provides several advantages. First, it
requires computing the triangular matrix R or the orthogonal
matrix Q to find the DOA; these matrices can be computed
with O(n2) operation. However, most of the existing schemes
required eignvalue decomposition (EVD) for the covariance
matrix or singular value decomposition (SVD) for the data
matrix; using EVD or SVD requires much more complex
computational O(n3) operation. Second, the proposed scheme
is more suitable for high-speed communication since it
requires first-order statistics and a single snapshot. Third,
the proposed scheme can estimate the correlated wideband
signals without using spatial smoothing techniques; whereas,
already-existing schemes do not. Accuracy of the proposed
wideband DOA estimation method is evaluated through
computer simulation in comparison with a conventional
method.
Similar to Random Matrix Theory in Array Signal Processing: Application Examples (20)
A recent direction in Business Process Management studied methodologies to control the execution of Business Processes under several sources of uncertainty in order to always get to the end by satisfying all constraints. Current approaches encode business processes into temporal constraint networks or timed game automata in order to exploit their related strategy synthesis algorithms. However, the proposed encodings can only synthesize single-strategies and fail to handle loops. To overcome these limits I will discuss a recent approach based on supervisory control. The approach considers structured business processes with resources, parallel and mutually exclusive branches, loops, and uncertainty. I will discuss an encoding into finite state automata and prove that their concurrent behavior models exactly all possible executions of the process. After that, I will introduce tentative commitment constraints as a new class of constraints restricting the executions of a process. Finally, I will discuss a tree decomposition of the process that plays a central role in modular supervisory control.
In his ignite talk „The Digital Transformation of Education: A Hyper-Disruptive Era through Blockchain and Generative AI,“ Dr. Alexander Pfeiffer delves into the intricate challenges and potential benefits associated with integrating blockchain technologies and generative AI into the educational landscape. He scrutinizes consensus algorithms and explores sustainable methods of operating blockchain systems, while also examining how smart contracts and transactions can be tailored to meet the specific needs of the educational sector. Alexander underscores the importance of establishing secure digital identities and ensuring robust data protection, while simultaneously casting a critical eye on potential risks and vulnerabilities. The topic of digital identities, facilitated through tokenization, forms a bridge between storing data using blockchain-based databases and the increasingly urgent need for content verification of AI-generated material.
Alexander explores the profound alterations occurring in teaching methodologies, assignment creation, and evaluation processes, shedding light on the hyper-disruptive impact these changes are having on both research and practical applications in education. The production of textual content by educators and students is analyzed with a focus on ensuring clear traceability of content sources and editors, and its proper citation, a critical aspect in the responsible use of AI. In addition to generative text and graphics, AI plays a crucial role in future learning and assignment practices, particularly through adaptive game-based learning and assessment. Alexander will provide a brief glimpse into his game „Gallery-Defender,“ a prototype demonstrating how AI and blockchain can be effectively implemented in serious gaming scenarios.
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The simulation of melee combat is central to many contemporary and traditional strategic games and simulations. In order to elevate this element of play from mere exercises of stats-comparison and dice rolling to a meaningful experience of play, strategy games rely on a rich plethora of cultural motives as deciding factors of their mechanic design. On the example of Samurai-themed skirmishing games, my talk elaborates on the impact that (popular) culture and other inspirations have on gaming experiences. It provides concrete examples from Japanese history, its traditional cinema, and postmodern Western reflections of Japanese cultural practices. Based on these insights, it compares four tabletop strategy games, muses on which phenomena they have adapted in their mechanics, and asks why or why not they may succeed in capturing a cultural essence via their rules.
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How does a development team expand on an already existing game?
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The document discusses Ben Calvert-Lee's work developing miniatures for tabletop games. It begins with an introduction to Ben's background and current role as a freelance lead sculptor. It then outlines the typical development pipeline for miniatures, from initial concepts and artwork to production. The document also discusses different miniature production methods. A case study details Ben's process for developing the Tengu faction for a game, including exploring species archetypes and incorporating unexpected developments into the designs.
In recent years, we have experienced an exponential growth in the amount of data generated by IoT devices. Data have to be processed strict low latency constraints, that cannot be addressed by conventional computing paradigm and architectures. On top of this, if we consider that we recently hit the limit codified by the Moore’s law, satisfying low-latency requirements of modern applications will become even more challenging in the future. In this talk, we discuss challenges and possibilities of heterogeneous distributed systems in the Post-Moore era.
In the modern world, we are permanently using, leveraging, interacting with, and relying upon systems of ever higher sophistication, ranging from our cars, recommender systems in eCommerce, and networks when we go online, to integrated circuits when using our PCs and smartphones, security-critical software when accessing our bank accounts, and spreadsheets for financial planning and decision making. The complexity of these systems coupled with our high dependency on them implies both a non-negligible likelihood of system failures, and a high potential that such failures have significant negative effects on our everyday life. For that reason, it is a vital requirement to keep the harm of emerging failures to a minimum, which means minimizing the system downtime as well as the cost of system repair. This is where model-based diagnosis comes into play.
Model-based diagnosis is a principled, domain-independent approach that can be generally applied to troubleshoot systems of a wide variety of types, including all the ones mentioned above. It exploits and orchestrates techniques for knowledge representation, automated reasoning, heuristic problem solving, intelligent search, learning, stochastics, statistics, decision making under uncertainty, as well as combinatorics and set theory to detect, localize, and fix faults in abnormally behaving systems.
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Function-as-a-Service (FaaS) is the latest paradigm of cloud computing in which developers deploy their codes as serverless functions, while the entire underlying platform and infrastructure is completely managed by cloud providers. Each cloud provider offers a huge set of cloud services and many libraries to simplify development and deployment, but only inside their clouds, often in a single cloud region. With such „help“ of cloud providers, users are locked to use resources and services of the selected cloud provider, which are often limited. Moreover, such heterogeneous and distributed environment of multiple cloud regions and providers challenge scientists to engineer cloud applications, often in a form of serverless workflows. In this talk, I will present our design principle „code once, run everywhere, with everything“. In particular, I will present challenges and our approaches and techniques how to program, model, orchestrate, and run distributed serverless workflow applications in federated FaaS.
This document summarizes a presentation on machine learning and fluid network planes. It begins with an agenda and introduction to fluid network planes and instances. It then discusses the role of machine learning in fluid network planes, including applications such as optimization, virtual network embedding problems, run-time operations, and intent-based closed-loop automation. Recent research is presented on machine learning-based YouTube QoE estimation using real 4G/5G network traces to predict video quality and inform control actions. Results are shown comparing 4G and 5G networks in terms of radio parameters, stalling events, handovers, and video resolutions under different mobility conditions.
The dynamics of networks enables the function of a variety of systems we rely on every day, from gene regulation and metabolism in the cell to the distribution of electric power and communication of information. Understanding, steering and predicting the function of interacting nonlinear dynamical systems, in particular if they are externally driven out of equilibrium, relies on obtaining and evaluating suitable models, posing at least two major challenges. First, how can we extract key structural system features of networks if only time series data provide information about the dynamics of (some) units? Second, how can we characterize nonlinear responses of nonlinear multi-dimensional systems externally driven by fluctuations, and consequently, predict tipping points at which normal operational states may be lost? Here we report recent progress on nonlinear response theory extended to predict tipping points and on model-free inference of network structural features from observed dynamics.
When it comes to integrating digital technologies into the classroom in higher education, many teachers face similar challenges. Nevertheless, it is difficult for teachers to share experiences because it is usually not possible to transfer successful teaching scenarios directly from one area to another, as subject-specific characteristics make it difficult to reuse them. To address this problem, instructional scenarios can be described as patterns that have been used previously in educational contexts. Patterns can capture proven teaching strategies and describe instructional scenarios in a consistent structure that can be reused. Because priorities for content, methods, and tools are different in each domain, a consensus-tested taxonomy was first developed with the goal of modeling a domain-independent database to collect digital instructional practices. In addition, this presentation will present preliminary insights into a data-driven approach to identifying effective instructional practices from interdisciplinary data as patterns. A web-based application will be developed for this that can both collect teaching/learning scenarios and individually extract scenarios from patterns for a learning platform.
The document discusses performance characterization across a computing continuum from the edge to the cloud. It evaluates the performance of video encoding and machine learning tasks on different devices. For video encoding, older single-board computers had significantly higher encoding times than other resources but provided lower data transfer times. For machine learning, training a convolutional neural network took much longer than a simpler model. Cloud and fog resources generally outperformed edge devices for more complex tasks. The document recommends offloading large or complex tasks to more powerful resources when possible.
East-west oriented photovoltaic power system is a new trend in orienting photovoltaic system. This lecture presents an evaluation of east–west oriented photovoltaic power system. A comparison between east–west oriented photovoltaic system and south oriented photovoltaic system in terms of cost of energy and technical requirement is conducted is presented in this lecture. In addition to that, the benefits of using east–west oriented photovoltaic system are discussed in this paper.
The document discusses using randomized recurrent neural networks and signature-based methods for machine learning in finance. It proposes splitting the input-output map of a dynamical system into a "reservoir" part and a linear "readout" part. The signature of the input signal provides a natural candidate for the reservoir, as it is point-separating and linear functions on the signature can approximate continuous functionals via the universal approximation theorem. The goal of the talk is to prove how dynamical systems can be approximated using randomized recurrent networks, with precise convergence rates, and to view randomized deep networks through this lens.
We live in a “digital” world, the separation between physical and virtual makes (almost) no sense anymore. Here, the Corona pandemic has also acted as an accelerator/magnifier demonstrating that the future of our digital society is here with all its possibilities, but also shortcomings.
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In the latest years, we have witnessed a growing number of media transmitted and stored on computers and mobile devices. For this reason, there is an actual need to employ smart compression algorithms to reduce the size of our media files. However, such techniques are often responsible for severe reduction of user perceived quality. In this talk we present several approaches we have developed to restore degraded images and videos to match their original quality, making use of Generative Adversarial Networks. The aim of the talk is to highlight the main features of our research work, including the advantages of our solution, the current challenges and the possible directions for future improvements.
Recommendation systems today are widely used across many applications such as in multimedia content platforms, social networks, and ecommerce, to provide suggestions to users that are most likely to fulfill their needs, thereby improving the user experience. Academic research, to date, largely focuses on the performance of recommendation models in terms of ranking quality or accuracy measures, which often don’t directly translate into improvements in the real-world. In this talk, we present some of the most interesting challenges that we face in the personalization efforts at Netflix. The goal of this talk is to sunshine challenging research problems in industrial recommendation systems and start a conversation about exciting areas of future research.
The document discusses the evolution to 5G networks and their benefits. It covers 5G principles like enhanced mobile broadband, massive machine-type communication, and ultra-reliable low-latency communications. Statistics are provided on 5G subscriptions, deployments, and expected growth in mobile data traffic. Use cases like smart cities, VR/AR, and autonomous vehicles are described. The presentation outlines Ericsson's 5G experience and global footprint.
Dr. Sean Tan, Head of Data Science, Changi Airport Group
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However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
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A tale of scale & speed: How the US Navy is enabling software delivery from l...
Random Matrix Theory in Array Signal Processing: Application Examples
1. Random Matrix Theory in Signal Processing
Xavier Mestre
xavier.mestre@cttc.cat
Centre Tecnològic de Telecomunicacions de Catalunya (CTTC)
Klagenfurt University (Austria)
February 25, 2019
2. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Outline
• Introduction to RMT: Convergence of spectral statistics of the sample covariance matrix.
• First Application: Subspace-based estimation of directions-of-arrival (DoA).
• Second Application: Detection tests of correlation and sphericity.
• Third Application: Large multivariate time series analysis
• Fourth Application: Outlier production characterization in Conditional/Unconditional Maximum
Likelihood parametric estimation.
Xavier Mestre: Random Matrix Theory in Signal Processing. 2/41
3. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
The sample covariance matrix
We assume that we collect independent samples (snapshots) from an array of antennas:
Consider the × observation matrix Y = [y(1) y()] and the sample covariance matrix
ˆR =
1
YY
=
1
X
=1
y()y
()
Xavier Mestre: Random Matrix Theory in Signal Processing. 3/41
4. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Problem statement and objective of the talk
Typically, one expects ˆR to be a close approximation of R. Under conventional statistical
assumptions, we have
ˆR → R
almost surely when → ∞ for a fixed . Furthermore, if (·) is a reasonable function, we also
have
³
ˆR
´
→ (R).
Unfortunately, when have the same order of magnitude, this does not hold anymore: the Finite
Sample Size effect appears.
In these situations, the regime where → ∞ but → , 0 ∞ becomes much more
relevant. RMT will help us in solving the following two problems in this regime:
• To what
³
ˆR
´
does converge to?
• How do we design (·) so that
³
ˆR
´
→ (R).
Xavier Mestre: Random Matrix Theory in Signal Processing. 4/41
5. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Conditional versus unconditional model
Typically, the observations are superposition of some signals plus noise. For example, in array
processing the observation consists of the contribution from signals:
Y = A (Θ) S + N
where:
• Matrix S ∈ C×
contains the contribution of the signals (at each of its rows).
• Matrix A (Θ) ∈ C×
contains, at each of its columns, the spatial signature of each source,
namely
A (Θ) =
£
a (1) a (2) · · · a ()
¤
• Matrix N ∈ C×
contains the background noise samples. It is typically modeled as a matrix
with i.i.d. entries following a zero mean Gaussian distribution
{N} ∼ CN
¡
0 2
¢
Xavier Mestre: Random Matrix Theory in Signal Processing. 5/41
6. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Conditional versus unconditional model (II)
Depending on how the signals are modeled, we differentiate between the conditional and the
unconditional models. Let us denote
S = [s(1) s()]
• Conditional Model: The entries of S are modelled as deterministic unknowns. In this case, the
observation can be described as
y() ∼ CN
¡
A (Θ) s() 2
I
¢
• Unconditional Model: The entries of S are modelled as random variables. Typically, we assume
that the column vectors s() are independent, circularly symmetric Gaussian Random variables, i.e.
s ∼ CN (0 P), P 0. In this case, we have
y() ∼ CN (0 R) R = A (Θ) PA
(Θ) + 2
I
Xavier Mestre: Random Matrix Theory in Signal Processing. 6/41
7. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Conditional versus unconditional model (III)
Depending on the signal model, the structure of the sample covariance matrix model will be inherently
different. Let X be an × matrix of i.i.d. Gaussian standardized entries {X} ∼ CN (0 1).
The two most important models can be described as:
• Conditional Model (also known as Information plus Noise model), the SCM can be expressed as
ˆR =
1
(V + X) (V + X)
where V some deterministic matrix that contains the signal (information) contribution.
• Unconditional Model (also known as Single Side Correlation model), the SCM can be expressed as
ˆR = R
12
µ
1
XX
¶
R
12
where R
12
is the positive Hermitian square root of R.
In many of the results obtained by RMT, the Gaussian assumption can be dropped.
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Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Uncorrelated signals: the Marchenko-Pastur Law
Consider the simplest case where ˆR = 1
XX
, where the entries of X are zero mean i.i.d. with
unit variance. Consider the eigenvalue distribution for different , but fixed ratio .
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
Eigenvalues
Numberofeigenvalues
Histogram of the eigenvalues of the sample covariance matrix, M=80, N=800
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
14
Eigenvalues
Histogram of the eigenvalues of the sample covariance matrix, M=800, N=8000
Numberofeigenvalues
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Uncorrelated signals: the Marchenko-Pastur Law (II)
It turns out, that when → ∞, → , 0 ∞, the empirical density of eigenvalues
converges to a deterministic measure.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
14
Eigenvalues
Histogram of the eigenvalues of the sample covariance matrix, M=800, N=8000
Numberofeigenvalues
For 1 () = 1
2
q¡
− −
¢ ¡
+
−
¢
I[−
+
]() −
= (1 −
√
)
2
+
= (1 +
√
)
2
.
Xavier Mestre: Random Matrix Theory in Signal Processing. 9/41
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The sample covariance matrix
We assume that we collect independent samples (snapshots) from an array of antennas:
ˆR = 1
P
=1 y()y
(). Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity.
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
Histogram of the eigenvalues of the sample covariance matrix, M=80, N=800
lambda
Numberofeigenvalues
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
30
lambda
Numberofeigenvalues
Histogram of the eigenvalues of the sample covariance matrix, M=400, N=4000
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The sample covariance matrix: asymptotic properties
When both → ∞, → , 0 ∞, the e.d.f. of the eigenvalues of ˆR tends to a
deterministic density function. Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Eigenvalues
Aysmptotic density of eigenvalues of the sample correlation matrix
c=0.01
c=0.1
c=1
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1st Application: Subspace-based estimation of directions of arrival (DoA)
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13. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Introduction and signal model
We consider DoA detection based on subspace approaches (MUSIC), that exploit the orthogonality
between signal and noise subspaces.
Consider a set of sources impinging on an array of sensors/antennas. We work with a fixed
number of snapshots ,
{y(1) y()}
assumed i.i.d., with zero mean and covariance R.
The true spatial covariance matrix can be described as
R = A (Θ) ΦA (Θ)
+ 2
I
where A (Θ) is an × matrix that contains the steering vectors corresponding to the different
sources,
A (Θ) =
£
a (1) a (2) · · · a ()
¤
and 2
is the background noise power.
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Introduction and signal model (II)
The eigendecomposition of R allows us to differentiate between signal and noise subspaces:
R =
£
E E
¤
∙
Λ 0
0 2
I−
¸
£
E E
¤
It turns out that E
a () = 0, = 1 .
Since R is unknown, one must work with the sample covariance matrix
ˆR =
1
X
=1
y()y
()
The MUSIC algorithm uses the sample noise eigenvectors, and searches for the deepest local minima
of the cost function
MUSIC () = a
() ˆE
ˆE
a ()
It is interesting to investigate the behavior of MUSIC () when have the same order of magnitude.
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Asymptotic behavior of MUSIC
The MUSIC algorithm suffers from the breakdown effect. The performance breaks down when
the number of samples or the SNR falls below a certain threshold. Cause: ˆE is not a very good
estimator of E when have the same order of magnitude.
The performance breakdown effect can be easily analyzed using random matrix theory, especially under
a noise eigenvalue separation assumption: |MUSIC () − ¯MUSIC ()| → 0
¯MUSIC () = s
()
à X
=1
()ee
!
s ()
() =
⎧
⎨
⎩
1 − 1
−
P
=−+1
³
2
−2 −
1
−1
´
≤ −
2
−2 −
1
−1
−
where { = 1 } are the solutions to 1
P
=1
− = 1
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Asymptotic behavior of MUSIC: an example
We consider a scenario with two sources impinging on a ULA ( = 05, = 20) from DoAs:
35◦
, 37◦
.
−100 −80 −60 −40 −20 0 20 40 60 80 100
−35
−30
−25
−20
−15
−10
−5
0
MUSIC asymptotic pseudospectrum, M=20, DoAs=[35,37]deg
Azimuth (deg)
32 34 36 38 40
−32
−30
−28
−26
−24
−22
−20
−18
N=25
N=15
SNR=12dB
SNR=17dB
2 4 6 8 10 12 14 16 18 20
25
30
35
40
45
50
SNR (dB)
Azimuth(deg)
Position of the two deepest local minima of the asymptotic MUSIC cost function
10 12 14
35
36
37
N=15
N=25
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-consistent subspace detection: G-MUSIC
We can derive an -consistent estimator of the cost function () = s
() EE
s ():
G-MUSIC () = s
()
à X
=1
()ˆeˆe
!
s ()
() =
⎧
⎨
⎩
1 +
P
=−+1
³
ˆ
ˆ−ˆ
− ˆ
ˆ−ˆ
´
≤ −
−
P−
=1
³
ˆ
ˆ−ˆ
− ˆ
ˆ−ˆ
´
−
where now ˆ1 ˆ are the solutions to the equation
1
X
=1
ˆ
ˆ − ˆ
=
1
Xavier Mestre: Random Matrix Theory in Signal Processing. 17/41
18. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Performance evaluation MUSIC vs. G-MUSIC
Comparative evaluation of MUSIC and G-MUSIC via simulations. Scenarios with four
(−20◦
−10◦
35◦
, 37◦
) and two (35◦
, 37◦
) sources respectively, ULA ( = 20, = 05).
−80 −60 −40 −20 0 20 40 60 80
10
−4
10
−3
10
−2
10
−1
10
0
Example of MUSIC and GMUSIC cost function, SNR=18dB, M=20, N=15, DoAs=35, 37, −10, −20 deg.
Angle of arrival (azimuth), degrees
MUSIC
GMUSIC
34 36 38
10
−4
10
−3
10
−2
5 10 15 20 25
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
SNR (dB)
MSE
Mean Squared Error
MUSIC
GMUSIC
CRB
M=20, N=15
M=20, N=75
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2nd Application: characterization of sphericity and correlation tests
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Problem formulation
We consider two very important tests in signal processing, which try to establish the structure of the
covariance matrix of the received signal:
• Sphericity test: seeks to establish whether the received signal is spatio-temporal white noise:
H0 : R = 2
I
H1 : R 6= 2
I
• Correlation test: seeks to establish whether the signals received from multiple sensors is corre-
lated:
H0 : R = R ¯ I
H1 : R 6= R ¯ I
In both cases, the true covariance matrix is unknown, so one must work on the sampled version ˆR.
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Generalized Maximum Likelihood Ratio Test (GLRT)
In order to address this binary hypothesis problem, one may resort to the Generalized Likelihood Ratio
Test (GLRT):
supR
Y
=1
Φ (y; R)
sup2
Y
=1
Φ (y; 2I)
H1
≷
H0
supR
Y
=1
Φ (y; R)
supD
Y
=1
Φ (y; D)
H1
≷
H0
where Φ (y; R) is the pdf of a complex Gaussian with zero mean and covariance R.
For ≥ , the GLRT for sphericity and correlation respectively reject H0 for large values of
ˆsphr
= log
∙
1
tr
³
ˆR
´¸
−
1
log det
³
ˆR
´
ˆcorr
= log
∙
1
tr
³
ˆC
´¸
−
1
log det
³
ˆC
´
where ˆC =
³
ˆR ¯ I
´−12
ˆR
³
ˆR ¯ I
´−12
is the sample correlation/coherence matrix.
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Frobenius norm test
Other more ad-hoc tests can be constructed using a more intuitive reasoning:
• Non-sphericity will manifest in ˆR being far from proportional to the identity.
• Correlation will lead to high absolute values of the off-diagonal elements of ˆC .
Therefore, it seems reasonable to design the test to reject H0 for large values of
ˆsphr
=
1
°
°
°
°
ˆR −
1
tr
h
ˆR
i
I
°
°
°
°
2
ˆsphr
=
1
°
°
°ˆC − I
°
°
°
2
In both cases, we have
ˆ
=
1
X
=1
ˆ
2
−
Ã
1
X
=1
ˆ
!2
which are LSS with () = 2
and () = .
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23. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
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General study of tests
It is generally difficult to derive the distribution of these tests, so in practice the literature has focused
on the case → ∞ for fixed
We would like to know the asymptotic behavior of these tests, for having the same order of
magnitude, allowing for the possibility of (undersampled regime).
Fortunately, there is a direct relationship between LSS and Stieltjes transform:
ˆ =
1
X
=1
³
ˆ
´
=
1
2 j
I
C−
() ˆ()
where
ˆ() =
1
X
=1
1
ˆ −
and where the contour C−
enclosed all the positive eigenvalues and not zero.
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First order convergence
By replacing ˆ() with the asymptotic equivalent, we obtain the almost sure asymptotic behavior
of the test ˆ, in the sense that |ˆ − ¯| → 0 where
¯ =
1
2 j
I
C−
() ¯()
Most of the times, we can carry out the integral and find a closed form for ¯.
For example, for the ¯
, we can establish
¯
=
(
+
+ −
log
¯
¯1 −
¯
¯
+
− −
log |∗| + 1
P
=1 log
¯
¯
¯
−∗
¯
¯
¯
where ∗ ≤ 0 is a solution to a certain equation and
is the large- value of the GLRT
= log
"
1
X
=1
#
−
1
X
=1
log
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Second order convergence
Using RMT tools that establish how ˆ() fluctuates around ¯(), we may establish a CLT on
these tests. Under certain statistical conditions, the LSS ˆ will asymptotically fluctuate as Gaussian
random variable, in the sense that
−1
( (ˆ − ¯) − )
L
−→ N (0 1)
where
=
1
2 j
I
C−
() ()
2
=
−1
42
I
C−
I
C−
(1)(2)2
(1 2) 12
where () is the original test function () after some change of variable, and where the mean
() and variance 2
(1 2) are different for the Sphericity and Correlation tests.
These integrals can be computed in closed form, and one can generally approximate ˆ ≈
N
¡
¯ + 2
2
¢
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Numerical Results: correlation test
Simulations for 105
independent simulation runs (GLRT). Under H0, D takes uniform values between
0 and 1. Under H1, R = D + Ψ where {Ψ} = 09|−|
.
250 300 350 400 450 500 550 600
0
0.005
0.01
0.015
Density of the statistic under H
0
300 350 400 450 500 550 600
0
0.005
0.01
0.015
Density of the statistic under H1
Simulated
Theory (large M,N)
Theory (large N)
M=20,N=25
M=20,N=25
= 20 = 25
250 300 350 400 450 500 550
0
0.005
0.01
0.015
Density of the statistic under H
0
, M=20, N=100
350 400 450 500 550 600 650 700
0
0.005
0.01
0.015
Density of the statistic under H1
, M=20, N=100
Simulated
Theory (large M,N)
Theory (large N)
= 20 = 100
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3rd Application: Large multi-variate time series
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Introduction: testing independence of multiple time series
We consider an -variate zero-mean Gaussian time series
y() = [1() ()]
where = 1 , and ask ourselves whether the different components of the series are independent.
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Motivation
Consider a certain window of samples and the extended random vector
y() =
⎡
⎣1() 1( + − 1)
| {z }
samples
() ( + − 1)
| {z }
samples
⎤
⎦
and consider the second order statistics of this vector, namely
E
£
y()y
()
¤
= R =
⎡
⎢
⎢
⎢
⎣
R
(11)
R
(12)
· · · R
(1)
R
(21)
R
(22)
· · · R
(2)
... ... ... ...
R
(1)
R
(2)
· · · R
()
⎤
⎥
⎥
⎥
⎦
where R
(0
)
has dimensions × . If the different time series are independent, R becomes block
diagonal and
=
1
Ã
log det R −
X
=1
log det R
()
!
= 0
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30. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
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Spatio-temporal sample covariance matrix
In practice, R has to be estimated from the observations y(). Using the above formulation, we
can estimate the time covariance between series and 0
as
ˆR
(0
)
=
1
YY
0
where
Y =
⎡
⎢
⎢
⎢
⎣
(1) (2) · · · () · · · ()
(2) ... · · · ... ... ( + 1)
... () ... ... · · · ...
() · · · () ( + 1) · · · ( + − 1)
⎤
⎥
⎥
⎥
⎦
has a Hankel structure. Under the null hypothesis (uncorrelation), and assuming stationarity
E
£
()∗
0(0
)
¤
= ( − 0
) =0 () =
Z 1
0
S () e2i
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Tests from the spatio-temporal sample covariance matrix
We can therefore build the test
ˆ =
1
Ã
log det ˆR −
X
=1
log det ˆR
()
!
and ask ourselves how to choose (time window parameter) to make ˆ close to zero under the
uncorrelation hypothesis. There is some trade-off between choosing small (so that ˆR is close to
R in spectral norm) and testing independence in large time lags ar large as possible.
For this all this, it appears reasonable to investigate the behavior of the eigenvalues of ˆR when
→ ∞ and → ∞ at the same rate, so that =
→ , 0 +∞. We will assume
that 4
→ 0, the spectral densities are uniformly bounded above and away from zero, and that
sup
X
∈Z
Ã
1
X
=1
| ()|2
!12
+∞.
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Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Tests from the spatio-temporal sample covariance matrix
Let Q(), ∈ C+
, denote the resolvent matrix of ˆR, that is Q() =
³
ˆR − I
´−1
. Under the
above assumptions, Q() ³ T() (deterministic asymptotic equivalent), where T() is the unique
solution to
T() =
−1
µ
I +
µ
−1
¡
I + Ψ (T())
¢
¶¶−1
in the class of matrix valued Stieltjes transforms, where Ψ : C×
→ C×
and Ψ : C×
→
C×
are the operators
Ψ(A) =
Z 1
0
d
() Ad ()
¡
S () ⊗ d () d
()
¢
Ψ(B) =
1
X
=1
Z 1
0
S ()
d
() B()
d ()
d () d
()
where d () =
£
1 ei(−1)
¤
and S () = diag (S1 () S ()).
Xavier Mestre: Random Matrix Theory in Signal Processing. 32/41
33. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
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4th Application: outlier characterization of Maximum Likelihood estimation
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34. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
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Considered scenario
• Consider an array of sensors receiving the signal transmitted by sources from parameters
¯ =
£
¯(1) ¯()
¤
, where we assume .
• Let y() denote an × 1 complex vector containing the received samples. We model this obser-
vation vector as
y() = A(¯)s() + n()
where s() contains the signal transmitted by the sources, n() contains the received
noise (assumed i.i.d. and CN(0 2
)) and
A(¯) =
£
a
¡
¯(1)
¢
a
¡
¯()
¢ ¤
• We assume that a total of snapshots are available, and that .
• Problem: estimate the parameters ¯ from the observations {y() = 1 }.
• We investigate the use of Maximum Likelihood approaches =⇒ Highest resolution at the cost of
increased computational complexity (multidimensional search)
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Maximum likelihood
• The estimated angles are determined as ˆ = arg min∈Θ ˆ () where ˆ () is the negative (concen-
trated) log-likelihood function.
• The “conditional” (or deterministic) model: assumes that the signals s() are deterministic un-
knowns. In this situation, one must minimize
ˆ () =
1
tr
h
P⊥
()ˆR
i
where ˆR = 1
P
=1 y()y
() is the sample covariance matrix, P⊥
() = I − P(), and
P() = A()
¡
A
()A()
¢−1
A
()
is the orthogonal projection on the column space of A().
• The “unconditional” (or stochastic) model: assumes that the source signals are random variables,
typically s() ∼ CN(0 P) and i.i.d. in the time domain. In this situation,
ˆ () =
1
log det
h
ˆ () P⊥
() + P()ˆRP()
i
Xavier Mestre: Random Matrix Theory in Signal Processing. 35/41
36. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Breakdown effect in maximum likelihood (I)
• General nonlinear parametric estimators exhibit a threshold effect.
• At low SNR, or low , the MSE suddenly departs from the Cramér Rao Bound. The presence of
outliers is the main cause for this behavior.
MSE
Threshold
effect
B
CRB
d
SNR
dB
Xavier Mestre: Random Matrix Theory in Signal Processing. 36/41
37. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Breakdown effect in maximum likelihood (II)
At low values of the SNR and , there exist realizations of the cost functions for which local minima
corresponding to outliers become deeper than the intended one.
UML, SNR=0dB, M=5, N=20, uncorrelated signals,DoA=[16,18]deg
θ1
(deg)θ2
(deg)
−80 −60 −40 −20 0 20 40 60 80
−80
−60
−40
−20
0
20
40
60
80
UML cost function
Local Minima
Intended Minimum
Selected Minimum
Xavier Mestre: Random Matrix Theory in Signal Processing. 37/41
39. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
First order behavior
When → ∞, under several technical conditions, the two ML cost functions become (pointwise)
asymptotically close to two deterministic counterparts, namely
|ˆ () − ¯ ()| → 0 |ˆ () − ¯ ()| → 0
a.s. pointwise in as → ∞, where
¯ () =
1
tr
£
P⊥
()R
¤
and
¯ () =
1
log det
£
2
() P⊥
() + P()RP()
¤
+
−
log
µ
−
¶
−
respectively, where R is the true covariance matrix of the observations and 2
() =
1
− tr
£
P⊥
()R
¤
.
Xavier Mestre: Random Matrix Theory in Signal Processing. 39/41
40. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Second order behavior
Let 1 be a set of multidimensional points (e.g. local minima of ¯ () or ¯ ()). Let
ˆη = [ˆ (1) ˆ ()]
and ¯η = [¯ (1) ¯ ()]
and take the equivalent definitions for the UML cost function. Assume that y() ∼ CN (0 R).
Under certain technical conditions, as → ∞ , → , 0 1, we have
Γ−1
(ˆη − ¯η) → N (0 I) and Γ−1
(ˆη − ¯η) → N (0 I)
for some covariance matrices Γ, Γ given by
{Γ} =
1
tr
£
P⊥
P⊥
¤
and
{Γ} =
1
2
2
1
tr
£
P⊥
P⊥
¤
+
1
2
1
tr
£
P⊥
Q
¤
+
1
2
1
tr
£
P⊥
Q
¤
− log
¯
¯
¯
¯1 −
1
tr [QQ]
¯
¯
¯
¯
where P = R
12
P
³
()
´
R
12
,P⊥
= R − P, and Q = R
12
A
£
A
RA
¤−1
A
R
12
.
Xavier Mestre: Random Matrix Theory in Signal Processing. 40/41
42. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Simulation results
ULA of = 5 elements, two sources coming from 16 and 18 degrees with respect to the broadside.
−15 −10 −5 0 5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Prob.ofRes.
Prob. of res., M=5, Theta=[16,18] deg, corr=0
UML (Predicted)
UML (Simulated)
CML (Predicted)
CML (Simulated)
N=100
N=10
Uncorrelated sources
−15 −10 −5 0 5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)Prob.ofRes.
Prob. of res., M=5, Theta=[16,18] deg, corr=0.95
UML (Predicted)
UML (Simulated)
CML (Predicted)
CML (Simulated)
N=100
N=10
Highly correlated sources
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43. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Concluding remarks
Xavier Mestre: Random Matrix Theory in Signal Processing. 43/41
44. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Conclusions
Random Matrix Theory offers the possibility of analyzing the behavior of different quantities depending
on ˆR when the sample size and the number of sensors/antennas have the same order of magnitude.
The objective is to describe the asymptotic behavior of a certain scalar function of ˆR, namely
³
ˆR
´
.
• Traditional Approach: Assuming that the number of samples is high, we might establish that
³
ˆR
´
→ (R) in some stochastic sense as → ∞ while remains fixed.
• New Approach: In order to characterize the situation where have the same order of
magnitude, one might consider the limit → ∞, → , 0 ∞.
Results obtained under this asymptotic limit turn out to be extremely accurate, even for reasonably
low .
Xavier Mestre: Random Matrix Theory in Signal Processing. 44/41
45. Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Thank you for your attention!!!
Xavier Mestre: Random Matrix Theory in Signal Processing. 45/41