Cooperative Spectrum Sensing Using Free Probability Theory

In this paper, using free probability theory, which is a valuable tool for describing the asymptotic behavior of multiple systems, especially for large matrices, a new cooperative MIMO scheme for spectrum sensing is proposed, which shows how asymptotic free property of random matrices and the property of Wishart distribution can be used to assist spectrum sensing in MIMO system for cognitive radios. Simulations over Rayleigh fading and AWGN channels demonstrate the proposed scheme has better detection performance compared with the energy detection techniques even for the case of a small sample of observations.

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper, we use another version of the techniques to extend the results of [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] to the case where the entries of the data matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, justifies the mean field approximation in the infinite width limit for the deep untrained neural networks and the property of the macroscopic universality of random matrix theory in this case.

Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the so-called matrix models or in free probability by the definition of a non-commutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non free-probabilists.

This paper presents bit error rate (BER) analysis of wireless sensor networks (WSNs) consisting of sensor nodes based on an IEEE 802.15.4 RF transceiver. Closed-form expressions for BER are obtained for WSNs operating over AWGN, Rayleigh and Nakagami-m fading channels. For the purpose of analysis, we consider an IEEE 802.15.4 RF transceiver using direct sequence spread spectrum-offset quadrature phase shift keying (DSSS-OQPSK) modulation under 2.4 GHz frequency band in a WSN. Analytical expressions for BER are derived for a wireless link between sensor nodes that act as a transmitter unit and a base station without considering the effect of interferers in the wireless environment. Numerical results for BER are obtained by varying the IEEE 802.15.4 standard specific physical layer parameters, such as number of bits used to represent a Zigbee symbol, number of modulation levels used in an OQPSK modulator, and various values of Rayleigh and Nakagami-m fading parameters, denoted as $$\alpha $$ ? and $$m$$ m , respectively. Moreover, optimum values of physical layer parameters are identified for improved system performance. It is found that error performance analysis of WSN shows improvement when lower number of bits is used to represent a Zigbee symbol. Specifically, under a Rayleigh fading channel which reflects a real-time WSN environment, the network exhibits better performance only when it is operated at high SNR values, i.e., BER of order $$10^{-2}$$ 10 - 2 is achieved when SNR lies in the range 5---15 dB. Also, the effect of fading parameters on network performance shows that better results are obtained for higher values of $$\alpha $$ ? and $$m$$ m for Rayleigh and Nakagami-m fading channels, respectively.

We study asymptotic distributions of large dimensional random matrices of the\nform $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using\nfree probability and combinatorics of colored labeled noncrossing partitions.\nThese matrices are taken from the set of off-diagonal blocks of the family\n$\\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically\nfree, asymptotically free against the family of deterministic diagonal\nmatrices, and whose norms are uniformly bounded almost surely. This class\nincludes unitarily invariant Hermitian random matrices with limit distributions\ngiven by compactly supported probability measures $\\nu$ on the real line. We\nexpress the limit moments in terms of colored labeled noncrossing pair\npartitions, to which we assign weights depending on even free cumulants of\n$\\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of\n$p$ independent blocks, we show that the limit moments are linear combinations\nof a new family of polynomials called generalized multivariate Fuss-Narayana\npolynomials. In turn, the product of two blocks of the same matrix leads to an\nexample with rescaled Raney numbers.\n

In this thesis, we cope with the fundamental mathematical and physical problems of the space-time-Doppler UWB system design for fading channels: multi-input multi-output (MIMO) wireless communication with adaptive filtering. The estimation algorithm from equalizer providing an accurate parameters estimate is investigated. UWB radio is a fast emerging technology with uniquely attractively features inviting major advances in wireless communications, networking, radar, imaging, and positioning systems. By its rule-making proposal in 2002, the Federal Communications Commission (FCC) in the United States essentially unleashed huge new bandwidth (3.6-10.6 GHz) at the noise floor, where UWB radios overlaying coexistence RF systems cerate using low-power ultra-short information bearing pulses. Right now, there are three basic types of UWB: multiband OFDM UWB (MB-OFDM UWB), impulse-UWB (I-UWB), and direct sequence UWB (DS-UWB). We focus on I-UWB and MB-OFDM-UWB, and compare their performance. A simple means for spreading the spectrum of low duty cycle pulse trains is time hopping (TH), with pulse position modulation (PPM) for data modulation at the rate of many pulses per data bit. The main contribution of this thesis for I-UWB is the design of leading edge detectors, coherent correlation detector (CCD), and pilot performance enhancement compared with different Rayleigh and Ricean fading conditions. For MB-OFDM UWB, we have considered the performance analysis and design criteria of MIMO OFDM UWB systems for high data-rate wireless transmission. The signal processing of frame detection, time synchronization, frequency synchronization, channel estimation, synchronization tracking using pilot subcarriers and 2-D MIMO detection algorithms is discussed. The main contributions of this thesis are 1) Compared with receiver employing the Maximum Likelihood (ML) detector, the receiver employing a low-complexity linear Minimum Mean Square Error (MMSE) demodulator can perform a limited performance loss in spatially uncorrelated channels. 2) Due to the operations of spreading and dispreading combined with MMSE-based frequency domain equalization, the adverse effects of the low-SNR subcarriers on the average BER performance are potentially improved. This is a direct consequence of spreading, because even if the signal corresponding to a specific chip is obliterated by a deep frequency domain channel fade, after dispreading these effects are spread over the Walsh-Hadamard Transform (WHT) length. Hence there is a high chance of still recovering all the partially affected subcarrier symbols without errors. 3) As a test case, the OFDM-based 200 Mb/s MB-UWB network (IEEE 802.15.3a) are considered, but the simulation results are shown above section. 4) The simulation channel can be easily applied to NOS and NLOS situation with WHT transformation from Simulink model. The fading channels of UWB with low/high signal to noise ratio (SNR), multipath effects, and multiuser interference may introduce large errors in location. The location engine performs the following tasks: █ Data fusion techniques: Different data fusion techniques are implemented using Time Of Arrival (TOA), Angle Of Arrival (AOA), or a combination of both. █ Channel modeling: A multipath, multiuser channel environment is created that models path loss, shadowing, Rayleigh fading, and Doppler frequency effects. █ Parameter estimation: TOA and AOA estimation algorithms are implemented as part of the location finding engine. Different variations of algorithms are implemented for performance and comparison purposes. █ Configuring the physical layer for different wireless networks. Among the programmable parameters are spreading factor, packet size, training length, constellation type, modulation technique, carrier frequency, level of transmitted signal power, and the number of antennas at the transmitter for both sides. █ Configuring the mobile user conditions. The wireless channel models depend on the Doppler frequencies present in the environment. The Doppler frequency depends on the mobile speed and carrier frequencies as input and generates a Rayleigh fading channel with the U-shape Doppler spectrum. █ Configuring the environmental parameters and network geographical structure. One of the factors affecting the performance of the wireless location system is the environment type (e.g., bad urban, urban, suburban, or rural area). For example, in a bad urban area with many blocking objects and buildings, non-line of sight (NLOS) effects plays an important role in the estimation accuracy.

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we show how the free deterministic equivalent of a random matrix model from [6] can be treated and we thus recover in a conceptual way the results from [6]. On a technical level, we generalize a result from scalar-valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In Appendix A, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.

We present three methods of weighting N-bit μ-law Pulse Code Modulation (PCM) with binary modulation, and evaluate their performances theoretically for transmission over Rayleigh fading and Gaussian channels. The digital modulation methods considered are noncoherent frequency shift keying and coherent phase shift keying. Ideal selection combining and ideal maximal ratio combining diversity techniques are employed when the transmission is over fading channels. Weighting System 1 is the conventional weighted pulse code modulated system, where the bits in every PCM word have the same weighting profile. Weighting System 2 has 2N unique weighting profiles, while System 3 has an addendum to System 2, where every bit in a particular word is further weighted by a unique multiplicative factor. For Rayleigh fading channels and the encoder operating at an input level that provided maximum signal-to-quantization noise-ratio, we obtained gains in overall signal-to-noise ratio (s/n) over unweighted μ-law PCM of 3, 4.5, and 6 dB for Systems 1, 2, and 3, respectively. When the systems were used in conjunction with Gaussian channels, the corresponding gains were 10, 12, and 17 dB, for a channel s/n of 10 dB. In addition to the theoretical results, we conducted computer simulations using four concatenated speech sentences transmitted via our weighted μ-law PCM systems over mobile radio channels. The simulation performances were in good agreement with our theoretical results, which were based on input signals having an exponential distribution.

The workshop brought together leading experts, as well as promising young researchers, in areas related to recent developments in free probability theory. Some particular emphasis was on the relation of free probability with random matrix theory.

The asymptotic behaviour of the mean spectral density for the product of N × N random unitary matrices distributed according to the Haar measure and a fixed diagonal matrix is investigated. The large N limit of the exact expression derived by Wei and Fyodorov (2008 J. Phys. A: Math. Theor. 41 502001) is calculated by a modification of the saddle point method. It is shown that in the bulk the result coincides with the one obtained within the free probability theory by Haagerup and Larsen (2000 J. Funct. Anal. 176 331). Close to the edge points of the bulk asymptotics, the mean density for large N is described by a universal function depending on a certain scaling variable. The large deviation formulae valid outside the bulk are also derived. Obtained formulae agree well with the results of direct numerical calculations.

In this paper, we consider secure downlink transmission in a multi-cell massive multiple-input multiple-output (MIMO) system where the numbers of base station (BS) antennas, mobile terminals, and eavesdropper antennas are asymptotically large. The channel state information of the eavesdropper is assumed to be unavailable at the BS and hence, linear precoding of data and artificial noise (AN) are employed for secrecy enhancement. Four different data precoders (i.e., selfish zero-forcing (ZF)/regularized channel inversion (RCI) and collaborative ZF/RCI precoders) and three different AN precoders (i.e., random, selfish/collaborative null-space based precoders) are investigated and the corresponding achievable ergodic secrecy rates are analyzed. Our analysis includes the effects of uplink channel estimation, pilot contamination, multi-cell interference, and path-loss. Furthermore, to strike a balance between complexity and performance, linear precoders that are based on matrix polynomials are proposed for both data and AN precoding. The polynomial coefficients of the data and AN precoders are optimized respectively for minimization of the sum mean squared error of and the AN leakage to the mobile terminals in the cell of interest using tools from free probability and random matrix theory. Our analytical and simulation results provide interesting insights for the design of secure multi-cell massive MIMO systems and reveal that the proposed polynomial data and AN precoders closely approach the performance of selfish RCI data and null-space based AN precoders, respectively.

Unitary designs have become a vital tool for investigating pseudorandomness, since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relations to quantum chaotic evolution and, in particular, to the eigenstate thermalization hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and k designs through free probability theory. First, by introducing the more general notion of k-freeness, we show that it can be used as an alternative probe to designs. In turn, free probability theory comes with several tools, useful, for instance, for the calculation of mixed moments or the so-called k-fold quantum channels. Our second result is the connection to quantum dynamics. Quantum ergodicity and, correspondingly, ETH apply to a restricted class of physical observables, as already discussed in the literature. In this spirit, we show that unitary evolution with generic Hamiltonians always leads to freeness at sufficiently long times but only when the operators considered are restricted within the ETH class. Our results provide a direct link between unitary designs, quantum chaos, and the eigenstate thermalization hypothesis and shed new light on the universality of late-time quantum dynamics.

In this paper, we analyze the Bit Error Rate (BER) performance of Golden coded Multiple-Input Multiple-Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) system over Rician multipath fading channel. We also compare the performance of the MIMO-OFDM system using Golden code in Rayleigh and Rician multipath fading channels. We discuss the effects of the presence of line-of-sight (LoS) component in the multipath fading environment which renders the improvement in the overall performance of the Golden coded MIMO-OFDM. This paper discusses the performance of Golden codes in a frequency selective Rician fading channel. To deal with the frequency selective fading channel, we use the OFDM (Orthogonal Frequency Division Multiplexing) modulation. The BER performance of the Golden coded MIMO-OFDM system under several propagation conditions has been evaluated. Keywords—MIMO, OFDM, Golden STBC, Rayleigh fading, Rician fading.

The BER of DS/CDMA systems that use symbol-aided demodulation (SAD) over Rayleigh and Rician fading channels is evaluated. Known symbols are transmitted with a certain frequency together with the data symbols and are used by the receiver to extract a phase reference for demodulation. Different schemes for filtering of the known symbols to obtain the phase reference needed for demodulation are considered and BER comparisons are made. The BER of these SAD DS/CDMA schemes is also compared with that of fully coherent and differentially coherent DS/CDMA systems operating over Rician and Rayleigh fading channels. The important role of filtering in SAD for DS/CDMA networks is studied in detail and it is shown that under some channel and system conditions (such as terrestrial mobile with low data rates) optimum (Wiener) filtering is needed while for other channel and system conditions (such as GEO satellite with higher data rates) simpler low pass filters are sufficient.

We report a very efficient semianalytical approach for the performance evaluation of differential detection schemes for GMSK signals of the DECT standard. Precisely, for a given channel, the performance is determined by means of an analytical procedure which includes the saddlepoint approximation. We consider both static channels (with impulse response generated by the simulation program SIRCIM) and two-ray Rayleigh and log-normal fading channels. As a departure from previous works, our receiver includes an all-digital part after the analog differential detection scheme. The digital part includes: (1) a block for the estimation of both the optimum sampling phase and the nonlinear channel coefficients (by making use of the DECT training sequence), (2) a one-tap decision feedback (DF) equalizer, and (3) a block for the evaluation of the approximate optimum bias level (/spl gamma//sub e/) in the threshold detector. Both the DF equalizer coefficient and /spl gamma//sub e/ are based on the nonlinear channel coefficients estimate. For channels with a normalized delay spread up to 0.2, the use of the optimum threshold together with the DF equalizer permits a gain of about 2 dB at BER=10/sup -6/ with respect to a receiver without equalization and a zero-level decision threshold. In addition, we discover that, in indoor environments, the 2-bit GMSK detector performs roughly the same as the 1-bit detector. The threshold optimization is also effective in the presence of channels affected by fading. To support this statement, we report the performance of the 1-bit differential detection scheme combined with antenna selection diversity in the presence of a two-ray log-normal and Rayleigh fading channel.