An efficient method for chaotic signal parameter estimation

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.

Wavelet transform-based methodologies for both Automatic Modulation Recognition (AMR) and Demodulation of digitally modulated communications signals can be utilized in an enabling platform for the implementation of a new class of communications systems. In particular, such techniques could enable the development of agile radio transceivers for use in both commercial and military applications. Such radio transceivers would have the ability to transmit and receive signals using many different modulation schemes while employing a common receiver architecture based on a single demodulator. In this dissertation, the development of AMR and Demodulation techniques are based on the relatively new mathematical theory of Wavelet Transforms (WTs). Information-bearing signals acquired by communications receivers are transformed into the wavelet-domain using the Continuous Wavelet Transform (CWT) and then applied to signal processing algorithms that also use the CWT in conjunction with pattern recognition techniques. In particular, the method of template-matching is used for both the AMR and Demodulation processes. Signal templates characterizing various modulated signals are used for both processes. The signal templates are determined based on the signal features present in the fractal patterns of their corresponding scalograms for specific modulation schemes as they appear in the wavelet-domain. The algorithms developed in this work are capable of both classifying the method of modulation used in the acquired signal, as well as subsequently automatically demodulating the signal to recover the message. The classes of digitally modulated signals considered in this work include variants of the Amplitude-, Frequency-, Phase-Shift Keying modulation families, i.e., ASK, FSK, and PSK, respectively, and multiple-level Quadrature Amplitude Modulation (M-ary QAM) families. The AMR and Demodulation performances are evaluated in the presence of Additive White Gaussian Noise (AWGN) over a wide range of Signal-to-Noise Ratio (SNR) values. Through extensive Monte Carlo computer simulations it is determined that the average correct classification rates using wavelet-based AMR for PSK, ASK, and QAM are over 98%, and over 90% for FSK signals, all at an SNR of 0 dB. The Bit Error Rate (BER) performance obtained using wavelet-based Demodulation is at least one order of magnitude better than the matched filter-based BER performance realized for the modulation schemes considered.

Estimation of channel fading parameters is an important task in the design of communication links such as in maximum ratio combining (MRC), where the SNR of the link has to be estimated. The maximum combining weights are directly related to the SNR or the fading channel coefficients. In this paper, we propose iterative techniques based on Maximum Likelihood parameter estimation to estimate the parameters of Nakagami-m distribution in the presence of additive white Gaussian noise. We show that the proposed iterative algorithms converge to a unique solution independent of the initial condition. However, for the purpose of fast convergence, a method is used to find an initial condition close to the true solution. This initial condition is obtained by solving for the unique positive root of a polynomial. Comparisons of our proposed approaches are made with respect to the noise and initial conditions. The performance of the algorithm with respect to the Cramer-Rao bound (CRB) is investigated. Computer simulation results for different signal to noise ratios (SNR) are presented.

Nonlinear time series denoising is the prerequisite for extracting effective information from observation sequence. An effective chaotic signal denoising method not only has a good signal-to-noise ratio (SNR) enhancement performance, but also can remain as a good unpredictable denoised signal. However, the inherent characteristics of chaos, such as extreme sensitivity to initial values and broadband spectrum, pose challenges for noise reduction of polluted chaotic signals. To address these issues, an adaptive smoothing multiscale morphological filtering (ASMMF) is proposed to reconstruct chaotic signals. In the process of noise reduction for contaminated chaotic signals, firstly, a multiscale morphological filter is constructed adaptively according to the multiscale permutation entropy (MPE) of morphological filter residuals, and the contaminated signals are filtered. Secondly, the weight coefficients of scale structural element are calculated by the residual sum of squares operation, and the chaotic signals are reconstructed. Finally, the resampled filter signals are smoothed by cubic B-spline interpolation operation. In the experiment, the Lorenz signal with white Gaussian noise, the measured sunspot, and the chaotic vibration signal are reconstructed by four comparison methods. The test results show that the proposed ASMMF method has obvious advantages in noise suppression and topological trajectory restoration.

This paper deals with the problem of estimating the average signal-to-noise ratio (SNR) for a communication system employing non-coherent binary frequency shift keying (NCBFSK) over fading channels with white Gaussian noise (AWGN). The maximum likelihood (ML) estimator and one using data statistics have been derived and simulated for various scenarios including data-aided (DA), non-data aided (NDA) and joint estimation using both the data and pilot sequences. We also derive the Cramer-Rao bound (CRB) for the estimators. The results show that for a particular region of interest (e.g. high SNR or low SNR) and depending upon the availability of pilot sequence, a particular SNR estimation scheme is suitable.

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)].

This paper presents a data-aided frequency offset estimator for a single complex sinusoidal signal corrupted by complex additive white Gaussian noise (AWGN). Kay [4] developed a maximum likelihood (ML) estimator based on differential phase measurements which achieves Cramer-Rao lower bound (CRLB) at high signal-to-noise ratios (SNRs); however, it exhibits large outliers at SNRs lower than 10-15 dB. In this paper, we propose an idea to divide the observation vector into adjacent segments. The samples within the same segment are then added together to produce a new sample. The closed-form of the proposed estimator is derived by applying the ML estimation over the new set of samples in the same way as of Kay's estimator. Simulation results show that with the proposed estimator, the threshold reduces 3-4 dB as the segment length is doubled. The estimation variance remains close to the CRLB even at SNR values as low as 0 dB. The proposed estimator is also compared with another family of frequency estimators that exploits the sample correlations.

The increasing demand for high-density and high-speed digital optical recording systems has made the development of advanced coding and signal processing techniques for fast and reliable data recovery increasingly important. In recent years, the parity-check (PC)-code-based reception technique has been widely studied for magnetic recording systems, and it is projected to be highly promising for high-density optical recording. The PC code is an inner error correction code (ECC), which can detect dominant short error events of the system, using only a few parity bits. This reduces the loss in error correction capability of the outer ECC due to random short errors and results in a simple and efficient solution to improve the overall performance. This thesis is dedicated to the design and analysis of PCcode- based recording systems to achieve higher recording capacity with low implementation complexity, for high-density blue laser disc systems. In particular, most of the key components of the PC-code-based optical recording system have been designed and optimized for different recording densities, and different proportions of white noise and media noise. During the development of advanced coding and detection techniques, it becomes necessary to investigate the system’s performance with different coding schemes and recording densities. In Chapter 2 of the thesis, we propose a generalized Braat-Hopkins model for optical recording channels, which provides a fair basis for the performance comparison of detection schemes over different coding schemes and recording densities, for channels with additive, white Gaussian noise (AWGN) and media noise. Various basic issues associated with the PC-code-based systems are investigated in Chapter 3. These include bounds for bit error rates and error event probabilities, the dominant error events at channel detector output, different PC codes, simple and efficient post-processors, the impact of error events that are split across data block boundaries and the corresponding remedy, as well as the effects of code rates and recording densities. Simulation results show that a 4-bit PC code achieves the best performance. The corresponding bit error rates (BERs) are very close to the performance bounds, at both nominal and high densities. Constrained codes, which serve as a key component in the read-write channel of data storage systems, are desired to have a high code rate with low-complexity encoder/decoder. In Chapter 4, we investigate the design of certain capacity-approaching constrained codes for optical recording systems. In particular, we derive analytically the relationship between the number of encoder states and the maximum size of these codes. We identify the minimum number of encoder states that maximizes the code rate, for any desired codeword length. The design of constrained PC codes is key to the development of PCcode- based systems, and the systematic design of efficient constrained PC codes remains a challenging problem. In Chapter 5, we propose a general and systematic code design technique for constructing capacity-approaching constrained PC codes, which can detect any type of error events of the system, with the minimum code rate loss. Compared to the rate 2/3 code without parity, the newly designed constrained 4-bit PC code achieves a performance gain of 2 dB at nominal density, and 1.5 dB at high density, at BER = 10-5. As the dominant noise for high-density optical recording systems, media noise severely degrades the performance of channel detectors and postprocessors that are designed for AWGN. In Chapter 6, we propose two novel modifications to the bit detector to combat media noise. We further develop a data-dependent post-processing scheme for error correction. Compared to the system designed without considering media noise and without PC codes, the overall performance gain of the developed scheme can be more than 11 dB at high media noise levels. In data storage systems, the ECC failure rate (EFR) serves as the ultimate measure of the data recovery performance. In Chapter 7, we develop semi-analytical approaches for estimating the EFR, and analyze EFRs for the developed PC-code-based systems, for the cases without and with interleaving. Our analysis shows that with the optimum interleaving degrees, compared to the rate 9/13 code without parity, the 2-bit PC code achieves a gain of around 0.5 dB, and the 4-bit PC code gains 0.6 dB, at high record ing density and EFR = 10-16. The thesis concludes in Chapter 8 with some suggestions for further work.

This thesis introduces some topics about initial downlink synchronization, algorithm derivations, and program simulations of IEEE 802.16m system. When a mobile station entering to the network, it needs to perform initial synchronization, including of symbol timing offset, carrier frequency offset and preamble index. We utilize the trait which the power of preamble is larger than it of the common data symbol to compute the moving power sum, and then estimate the left boundary of preamble by finding out the peak value of moving power sum. A symbol period from this estimated boundary is regarded as the estimated preamble, which has a phase noise with the exact preamble. We derive the quasi maximum likelihood estimation from the likelihood function of the estimated preamble to obtain fractional carrier frequency offset (FCFO) and the formula of channel estimation. After compensating the estimated fractional carrier frequency offset to the formula of channel estimation, we substitute several reasonable integral carrier frequency offsets (ICFOs) and primary advanced preambles (PA-Preambles) into this formula and obtain channel impulse responses (CIRs). After that, we compute different fine timing offset index 64-points power sum of these CIRs and find out the peak value whose ICFO, PA-Preamble index, and fine timing offset index are regarded as the result of the joint estimation. In terms of simulations, above all, we present a reasonable received signal model of IEEE 802.16m, and then simulate our algorithm in floating point under AWGN (additive white Gaussian noise) channel to verify its performance. Moreover, we test this algorithm under different multipath fading channels, Pedestrian B and SUI-5, at different speed which cause Doppler shifts, and signal to noise ratio (SNR) to observe the effect on this algorithm.

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.

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 &lt;0.0001, paired t-test) and from 3.4 ± 0.2 to 11.4 ± 0.8 for γATP (p &lt;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&lt;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.

Range, radial velocity, and acceleration MLE using frequency modulation coded LFM pulse train

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.

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.

&lt;p&gt;Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.&lt;/p&gt;