An iterative maximum-likelihood-based parameter estimation algorithm for Nakagami-m distribution

Publication in the conference proceedings of EUSIPCO, Tampere, Finland, 2000

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

We propose a novel optical wireless (OW) system based on a power adaptive multibeam spot-diffusing transmitter serving multiple seven-segment maximum ratio combining (MRC) angle diversity receivers. A feedback link is assumed between the transceivers so that each receiver conveys to the multibeam transmitter the new beams transmit power weights to be used to achieve the best signal quality at a given receiver location. Two cases involving three and five receivers are considered. Four different configurations for the placement of the three-receiver case in the room are also examined. The system's performance in each case is evaluated in terms of signal-to-noise ratio (SNR) and is compared with the single receiver scenario with and without power adaptation. In the presence of one receiver, the transmit spot powers can be adjusted for optimum performance at that receiver location. For multiple receivers, there is conflict, and we propose spot power adaptation based on the average requirements (power distribution in spots), i.e., transmit equal gain combining (EGC) of spot power or MRC of transmit spot powers. The results show that the three receivers benefit most from an adaptive transmitter when each is placed at a corner of the room. In this case, an SNR increase of as much as 2.6 dB is achieved for all three receivers at the corners by both MRC and EGC. Moreover, when all receivers are placed away from the line of diffusing spots, our proposed MRC collaborative approach is 1 dB better than the noncollaborative system. This gain reduces the difference from the upper bound set by the single receiver adaptation, which is 3 dB. For a mobile transmitter, MRC also significantly improved the SNR for the farthest receivers at the opposite end from the transmitter located near one room corner.

We show that an $L$ -branch maximal ratio combining (MRC) receiver with an average signal-to-noise ratio (SNR) $\bar{\gamma}$ in Nakagami- $m$ fading channels is equivalent to an $mL$ -branch MRC receiver with average SNR $\bar{\gamma}/m$ in Rayleigh fading channels in terms of symbol error probability (SEP) when $mL$ is an integer. Moreover, the orthogonal space–time block code (OSTBC) employing $M_\mathrm{t}$ transmit antennas and $M_\mathrm{r}$ receive antennas with the average SNR $\bar{\gamma}$ in Nakagami- $m$ fading channels is shown to be equivalent to an $mM_\mathrm{t}M_\mathrm{r}$ -branch MRC single-input–multiple-output (SIMO) receiver with the average SNR $\bar{\gamma}/mM_\mathrm{t}$ in Rayleigh fading channels.

This paper addresses the performance analysis of an adaptive wireless link with one antenna transmitter and a multiple antenna maximum-ratio combining (MRC) receiver. Two main assumptions are used in this paper: (1) Rayleigh fading correlated channels (i.e., MRC branch correlation) and (2) imperfect (outdated) channel state information at the transmitter (CSIT) side. The main contribution of this work lies in the derivation of analytic expressions (in terms of a series expansion) of the statistics of correct packet reception conditional on the decisions made by the transmitter based on outdated CSIT. The novelty of this derivation is the joint modelling of spatially correlated branches, imperfect CSIT, and adaptive modulation based on threshold-trigger decision. Contrary to common belief, the results presented here suggest that spatial correlation not always affects the performance of the MRC receiver: at low signal-to-noise ratio (SNR), correlation can improve performance rather than degrading it. In contrast, at high SNR, correlation is found to always degrade performance. At high SNR, correlation tends to worse the degrading effects of imperfect CSIT, particularly when the number of antennas increases. Imperfect CSIT causes errors in the assignment of MCSs, thus reducing throughput performance. These errors become more evident in the high SNR regime, particularly when the values of branch correlation and the number of antennas increase.

Introduction: Phosphorus magnetic resonance spectroscopy ( 31 P-MRS) allows insight into cardiac energetics in vivo , but is a technique with an intrinsically low signal to noise ratio (SNR). As ultra-high magnetic field MRI scanners become more widely available this limitation will lessen, allowing detection of smaller changes in metabolite concentrations or measurement of changes in smaller patient groups, which will further the understanding of cardiac energetics. Hypothesis: We hypothesised that the 2.4x increase in SNR, predicted by theory at 7T compared to 3T, will allow more precise metabolite concentration measurements in dilated cardiomyopathy (DCM) patients. Methods: 26 idiopathic DCM patients (69% male, mean age 55±2 yr, mean LVEF 35±2%) underwent 31 P-MRS at 3T and 7T field strengths. Patients were NYHA Class I (46%), II (50%) or III (4%). Results: The mean SNR increased from 6.5 ± 0.5 to 16.2 ± 1.5 for PCr (p <0.0001, paired t-test) and from 3.4 ± 0.2 to 11.4 ± 0.8 for γATP (p <0.0001) as the field strength increased from 3T to 7T. The PCr/ATP ratio was consistent between the 2 field strengths; at 3T PCr/ATP was 1.5 ± 0.1 and at 7T 1.5 ± 0.1; there was as expected no statistically significant difference (p=0.67) (fig a). The Cramér-Rao lower bounds (CRLB) quantify the minimum precision of fitted parameters. A desirable decrease in the PCr/ATP CRLB was seen from 3T to 7T (47 ± 3% vs 25 ± 2% p<0.0001)(fig b), i.e.-metabolites are indeed quantified more precisely at 7T compared to 3T. Conclusion: This study represents the first cardiac 31 P-MRS patient data acquired at a 7T field strength. Compared to a 2.8x SNR increase from 3T to 7T previously reported in normal volunteers, we observed a 2.5x increase in this patient population. There is no significant bias at 7T as the PCr/ATP ratios are similar. In conclusion 7T 31 P-MRS is feasible in DCM patients and more accurate than at 3T, which in future should allow more accurate quantification and understanding of myocardial energetics.

In this paper, we study the effect of imperfect channel estimation (ICE) on the performance of M-ary phase shift keying (M-PSK) with maximum ratio combining (MRC) in generalized Rician fading channels. Using decision variable (DV)-based and signal-to-noise ratio (SNR)-based moment generating function (MGF) approaches, error and outage probability formulas, and SNR statistics for the M-PSK MRC receiver are derived, taking into account the effects of ICE and all relevant system and channel parameters. We analytically quantify the average SNR (ASNR) loss of M-PSK caused by ICE, and provided a unified MGF expression for ICE and different M's. In addition, we point out a major approximation for a popular approach used in the literature to evaluate the adverse effect of Gaussian weighting errors.

Considering the influence of acceleration and the Gaussian envelope for a laser Doppler velocimeter (LDV), parameter estimation of a Doppler signal with a Gaussian envelope was investigated based on introducing acceleration. According to the theory of mathematics statistics, the Cramer-Rao lower bounds (CRLBs) of Doppler circular frequency and its first order rate were analyzed, formulas of CRLBs were given, and the power spectrum estimation with adjustment was discussed. The results of theory and the simulation show that the CRLBs are related to the data length, the signal-to-noise ratio (SNR), and the width of the Gaussian envelope, and they can be decreased by increasing the data length or improving the SNR; the larger the acceleration is and the narrower the Gaussian envelope is, the larger the CRLBs of Doppler circular frequency and its first order rate are; the gap between the variances of the measuring results and the CRLBs narrows when the SNR of the signal is improved, and is almost eliminated when the SNR is higher than 6 dB. It is concluded that the model presented is much more suitable for a LDV than that acquired by Rife and Boorstyn [IEEE Trans. Inform. Theory 20, 591 (1974)].

A method is provided for determining necessary conditions on sample size or signal to noise ratio (SNR) to obtain accurate parameter estimates from remote sensing measurements in fluctuating environments. These conditions are derived by expanding the bias and covariance of maximum likelihood estimates (MLEs) in inverse orders of sample size or SNR, where the first-order covariance term is the Cramer-Rao lower bound (CRLB). Necessary sample sizes or SNRs are determined by requiring that (i) the first-order bias and the second-order covariance are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. An analytical expression is provided for the second-order covariance of MLEs obtained from general complex Gaussian data vectors, which can be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) it allows for both the mean and variance of the measurement to be functions of the estimation parameters. Here, conditions are derived to obtain accurate source localization estimates in a fluctuating ocean waveguide containing random internal waves, and the consequences of the loss of coherence on their accuracy are quantified.

Based on selective transmission (ST) and maximal ratio combining (MRC) schemes, this paper focuses on the analysis of downlink system capacity in distributed antenna systems (DAS) over composite channels. Firstly, this paper establishes a composite channel model which takes path loss, lognormal shadowing and Rayleigh fading into account. Then, the probability density function (PDF) of the output signal to-noise ratio (SNR) is derived after MRC at the receiver. Moreover, the distribution of the output SNR is approximated by a lognormal distribution. After that, by making use of ST scheme, an approximate expression of the capacity for a mobile station (MS) over a given position is derived. Further, considering the distribution of MSs in the system, a closed-form expression of the system capacity is obtained. Numerical results show that the closed-form expression can provide sufficient precision for evaluating the system capacity performance of DAS.

In this paper, we analyze the error probability performance of maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC) diversity schemes with coherent BPSK signaling on Rayleigh fading channels with Gaussian channel estimation errors. We first show that, with weighting errors, the conditional probability of error is not an explicit function of the signal-to-noise ratio (SNR) at the output of the diversity combiner. We then show that averaging the conditional probability of error with the density function of the SNR at the output of the combiner yields a lower bound on the exact probability of error, independent of the underlying diversity combining scheme. Later, we derive the exact probability of error for MRC, EGC, and SC diversity schemes and show that the exact probability of error with weighting errors is very similar to the case of perfect channel estimation, but with the average SNR per diversity branch, /spl gamma/_, for the case of perfect channel estimation, replaced by the effective SNR, /spl gamma/_/sub /spl rho//, due to weighting errors, which is a function of both /spl gamma/_ and /spl rho/, the magnitude of the normalized cross correlation between the actual and the estimated channel gains. Finally, we show that, as /spl rho/ /spl rarr/ 0, the average probability of error approaches 0.5, irrespective of the order of diversity and the diversity combining rule.

The authors investigated the performance of the iterative Steiglitz-McBride (SM) algorithm on an autoregressive moving average (ARMA) model of signals from a fast, sparsely sampled, multiecho, chemical shift imaging (CSI) acquisition using simulation, phantom, ex vivo, and in vivo experiments with a focus on its potential usage in magnetic resonance (MR)-guided interventions. The ARMA signal model facilitated a rapid calculation of the chemical shift, apparent spin-spin relaxation time (T2*), and complex amplitudes of a multipeak system from a limited number of echoes (< or equal 16). Numerical simulations of one- and two-peak systems were used to assess the accuracy and uncertainty in the calculated spectral parameters as a function of acquisition and tissue parameters. The measured uncertainties from simulation were compared to the theoretical Cramer-Rao lower bound (CRLB) for the acquisition. Measurements made in phantoms were used to validate the T2* estimates and to validate uncertainty estimates made from the CRLB. We demonstrated application to real-time MR-guided interventions ex vivo by using the technique to monitor a percutaneous ethanol injection into a bovine liver and in vivo to monitor a laser-induced thermal therapy treatment in a canine brain. Simulation results showed that the chemical shift and amplitude uncertainties reached their respective CRLB at a signal-to-noise ratio (SNR) > or =5 for echo train lengths (ETLs) > or =4 using a fixed echo spacing of 3.3 ms. T2* estimates from the signal model possessed higher uncertainties but reached the CRLB at larger SNRs and/or ETLs. Highly accurate estimates for the chemical shift (<0.01 ppm) and amplitude (<1.0%) were obtained with > or =4 echoes and for T2*(<1.0%) with > or =7 echoes. We conclude that, over a reasonable range of SNR, the SM algorithm is a robust estimator of spectral parameters from fast CSI acquisitions that acquire < or =16 echoes for one- and two-peak systems. Preliminary ex vivo and in vivo experiments corroborated the results from simulation experiments and further indicate the potential of this technique for MR-guided interventional procedures with high spatiotemporal resolution approximately 1.6 x 1.6 x 4 mm3 in < or =5 s.

This paper presents a new Transmitted Reference (TR) Ultra-WideBand (UWB) receiver based on Spatial Diversity (SD), which employs Multi-Antenna Technology (MAT) to improve the performance of TR-UWB receiver. According to the amplitude of correlator output of every antenna branch, this paper analyzed the performances of the proposed TR-UWB receiver employing three different kinds of combination strategies, i.e., Maximum Ratio Combination (MRC), Equal Gain Combination (EGC), and Selective Combination (SC), which are different from conventional ones, and theoretically proved that the performance of EGC is better than MRC. Simulation results verify that when EGC is adopted and BER=10−3, increasing three antennas provides Signal to Noise Ratio (SNR) gain of about 3 dB in CM4 channel and SNR gain of about 2 dB in CM2 channel.

It has recently been found that by jointly processing multiple (sum, azimuth-, and elevation-difference) matched filter samples, it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cramer-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator's efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB's implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse's length, five targets can be isolated between two matched filter samples given that the target's signal-to-noise ratios (SNRs) are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of oversampling are given.

This paper examines the effect on Outage probability and Detection probability by using Maximal Ratio Combining (MRC), Square Law Combining (SLC) along with the addition of two relay transmission protocols Amplify and Forward (AF) and Decode and Forward (DF) over Rayleigh Fading Channel. In MRC the co-phased signals with highest Signal to Noise Ratio (SNR) are taken where SLC does not rely on phase estimation. The AF protocol works on the principle that the relay receives the signal and amplifies it along with the noise. In DF, the received signal at relay is decoded and again it is re-encoded before sending it to the destination. Single and multiple relays are taken in different scenarios and the diversity techniques (MRC and SLC) along with the relay transmission protocols (AF and DF) are analyzed. The performance of receiver's parameters, i.e. detection probability (P d ) and Outage probability (P out ) are examined. The simulation results clearly show the performance of the efficient combining scheme with the appropriate relay transmission protocol.