Compact CramÉr–Rao Bound Expression for Independent Component Analysis

Despite of the increased interest in independent component analysis (ICA) during the past two decades, a simple closed form expression of the Cramer-Rao bound (CRB) for the demixing matrix estimation has not been established in the open literature. In the present paper we fill this gap by deriving a simple closed-form expression for the CRB of the demixing matrix directly from its definition. A simulation study comparing ICA estimators with the CRB is given.     

CramÉr–Rao Bound Analysis of Positioning Approaches in GNSS Receivers

Recently, direct position estimation (DPE) has arisen as a potential approach to deal with the positioning problem in global navigation satellite system receivers. The conventional navigation solution is obtained in two steps: synchronization parameters are estimated and then a trilateration procedure is in charge of computing user's position, based on those parameters. In contrast, DPE estimates receiver's position directly from digitized signal. DPE was seen to provide GNSS receivers with appealing capabilities, such as multipath mitigation. However, a theoretical bound for those estimates is still missing and the answer to ldquohow better can DPE perform compared to the conventional approach?rdquo has not been addressed in the literature. Aiming at clarifying those issues, this paper presents the derivation of the CramEacuter-Rao bound (CRB) of position for both conventional and DPE approaches. We present the derivation for a multiantenna receiver as a general case. In addition, a number of realistic scenarios are tested in order to compare the theoretical performance bounds of both alternatives and the actual root mean squared error performance of the corresponding maximum likelihood estimator.     

Analytic and Asymptotic Analysis of Bayesian CramÉr–Rao Bound for Dynamical Phase Offset Estimation

In this paper, we present a closed-form expression of a Bayesian Cramer-Rao lower bound for the estimation of a dynamical phase offset in a non-data-aided BPSK transmitting context. This kind of bound is derived considering two different scenarios: a first expression is obtained in an offline context, and then a second expression in an online context logically follows. The SNR-asymptotic expressions of this bound drive us to introduce a new asymptotic bound, namely the asymptotic Bayesian Cramer-Rao Bound. This bound is close to the classical Bayesian bound but is easier to evaluate.     

Maximum-Likelihood Estimation, the CramÉr–Rao Bound, and the Method of Scoring With Parameter Constraints

Maximum-likelihood (ML) estimation is a popular approach to solving many signal processing problems. Many of these problems cannot be solved analytically and so numerical techniques such as the method of scoring are applied. However, in many scenarios, it is desirable to modify the ML problem with the inclusion of additional side information. Often this side information is in the form of parametric constraints, which the ML estimate (MLE) must now satisfy. We unify the asymptotic constrained ML (CML) theory with the constrained Cramer-Rao bound (CCRB) theory by showing the CML estimate (CMLE) is asymptotically efficient with respect to the CCRB. We also generalize the classical method of scoring using the CCRB to include the constraints, satisfying the constraints after each iterate. Convergence properties and examples verify the usefulness of the constrained scoring approach. As a particular example, an alternative and more general CMLE is developed for the complex parameter linear model with linear constraints. A novel proof of the efficiency of this estimator is provided using the CCRB.     

Range Compression and Waveform Optimization for MIMO Radar: A CramÉr–Rao Bound Based Study

A multi-input multi-output (MIMO) radar system, unlike standard phased-array radar, can transmit via its antennas multiple probing signals that may be correlated or uncorrelated with each other. This waveform diversity offered by MIMO radar enables superior capabilities compared with a standard phased-array radar. One of the common practices in radar has been range compression. We first address the question of ldquoto compress or not to compressrdquo by considering both the Cramer-Rao bound (CRB) and the sufficient statistic for parameter estimation. Next, we consider MIMO radar waveform optimization for parameter estimation for the general case of multiple targets in the presence of spatially colored interference and noise. We optimize the probing signal vector of a MIMO radar system by considering several design criteria, including minimizing the trace, determinant, and the largest eigenvalue of the CRB matrix. We also consider waveform optimization by minimizing the CRB of one of the target angles only or one of the target amplitudes only. Numerical examples are provided to demonstrate the effectiveness of the approaches we consider herein.     

Analytical Analysis of Bayesian CramÉr–Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System

In this paper, we consider the Bayesian Cramer-Rao bound (BCRB) for the dynamical estimation of multipath Rayleigh channel complex gains in data-aided (DA) and non-data-aided (NDA) OFDM systems. This bound is derived in an online and offline scenarios for time-invariant and time-varying complex gains within one OFDM symbol, assuming the availability of prior information. In NDA context, whereas this true BCRB is hard to evaluate, we present a closed-form expression of a BCRB, i.e., the asymptotic BCRB (ABCRB) or the modified BCRB (MBCRB). We discuss, based on the theoretical and simulation results, the interest of using some past and future observations in terms of Doppler spread for the complex gains estimation.     

A CramÉr-Rao Bound Characterization of the EM-Algorithm Mean Speed of Convergence

This paper deals with the mean speed of convergence of the expectation-maximization (EM) algorithm. We show that the asymptotic behavior (in terms of the number of observations) of the EM algorithm can be characterized as a function of the Cramer-Rao bounds (CRBs) associated to the so-called incomplete and complete data sets defined within the EM-algorithm framework. We particularize our result to the case of a complete data set defined as the concatenation of the observation vector and a vector of nuisance parameters, independent of the parameter of interest. In this particular case, we show that the CRB associated to the complete data set is nothing but the well-known modified CRB. Finally, we show by simulation that the proposed expression enables to properly characterize the EM-algorithm mean speed of convergence from the CRB behavior when the size of the observation set is large enough.     

Fiber-Optic Distributed Sensor Based on Hybrid Raman and Brillouin Scattering Employing Multiwavelength Fabry–PÉrot Lasers

We propose the use of standard multiwavelength Fabry–PÉrot (FP) lasers for distributed strain and temperature measurement in hybrid Raman- and Brillouin-based fiber-optic sensors. The multiple longitudinal modes of FP lasers allow for accurate spontaneous Raman scattering measurement and for simultaneous high-sensitivity coherent detection of the Brillouin frequency shift parameter for all longitudinal modes. Experimental results confirm great performance improvement in simultaneous strain and temperature measurement using a single optical fiber.      

Enhanced Modulation Bandwidth of a Fabry–PÉrot Semiconductor Laser Subject to Light Injection From Another Fabry–PÉrot Laser

Variations in optical spectrum and modulation bandwidth of a modulated Fabry-Perot (FP) semiconductor laser subject to the external light injection from another FP Laser is investigated in this paper. Optimal wavelength matching conditions for two FP lasers are discussed. A series of experiments show that two FP lasers should have a central wavelength overlapping and a mode spacing difference of several gigahertz. Under these conditions both the magnitude and phase frequency responses can be improved significantly.     

Cramer–Rao Bound for MIMO Radar Target Localization With Phase Errors

A new template matching method accelerated by an integral image is proposed. In contrast to the conventional winner-update template matching algorithm, the proposed scheme uses an integral image instead of a block sum pyramid to represent the search area. When an integral image is used, block sums on the lowest level are evaluated very fast. As a result, the speed with which nonbest candidates are rejected is nearly double that of the conventional scheme. Moreover, the proposed scheme needs less memory than the conventional approach to maintain block sums of candidates and can be easily extended to nonsquare (rectanglar) template matching.      

Tunable Dual-Wavelength Fiber Laser Using Optical-Injection Fabry–PÉrot Laser

In this investigation, we propose and experimentally demonstrate a dual-wavelength fiber laser based on a self-injected Fabry–PÉrot laser diode and an erbium-doped fiber. The dual-wavelength tuning range is 39.49 nm, from 1526.27 to 1565.76 nm, and the mode spacing of the dual-wavelength can be tuned from 1.32 to 39.49 nm. In addition, the output power difference of the dual-wavelength can be controlled and smaller than 1 dB.      

1.25-Gb/s Operation of a Spectrum-Sliced Fabry–PÉrot Laser Diode With Intensity Noise Suppression by a Second Fabry–PÉrot Laser Diode

We propose a low-cost solution for the intensity noise suppression in the spectrum-sliced Fabry-Perot laser diode (F-P LD), which is achieved by placing an F-P LD at the receiver region. The F-P LD at the receiver region provides the intensity noise suppression of about 10 dB as well as the increase of the received optical power for the spectrum-sliced optical signal. The Q-factor is improved about 5.9 at a data rate of 1.25 Gb/s. As a result, we successfully demonstrate 10-km error-free transmission at 1.25-Gb/s signal with a transmission penalty of less than 0.5 dB. It is also found that the low spectrum-sliced power of -22 dBm achieves the relative intensity noise level of -112.5 dB/Hz, which is almost independent of the operation current.     

A Note on LÉvÊque and Telatar's Upper Bound on the Capacity of Wireless Ad Hoc Networks

An alternative proof of the Leveque-Telatar (2005) upper bound on the maximum achievable information rate per communication pair in a wireless ad hoc network is presented. The argument exploits the spatial structure of the random point process representing the location of the nodes, and it is based on a simple geometric ldquostrippingrdquo construction that avoids complicated large deviation estimates.     

Effects of In-Band Crosstalk in Wavelength-Locked Fabry–PÉrot Laser-Diode-Based WDM PONs

We investigated the impact of in-band crosstalk on the system's performance in wavelength-locked Fabry-Perot laser-diode (FP-LD)-based wavelength-division-multiplexed passive optical networks (WDM PONs). As expected, wavelength-locked FP-LD, which was injected by using an incoherent broadband light source, was more tolerant to in-band crosstalk than distributed-feedback (DFB) laser diode. A 1-dB power penalty in the wavelength-locked FP-LD-based PON system was observed when the crosstalk-to-signal ratio was ~-9 dB. We also compared the measured power penalties with the calculated power penalties. It has been shown that the in-band crosstalk-induced power penalty in wavelength-locked FP-LD-based WDM PON could be estimated properly by taking into account both effects of power addition and signal-crosstalk beat noise.     

Fabry–PÉrot Cavity Based on a Suspended-Core Fiber for Strain and Temperature Measurement

A fiber-optic Fabry–PÉrot sensing structure based on the utilization of a suspended-core fiber is presented. The interferometric structure is formed when a small length of the suspended-core fiber is spliced to the end of a standard single-mode fiber. The interfering waves are generated by the refractive-index mismatches between the two fibers in the splice region and at the end of the suspended-core fiber. Thermal and strain responses of two different sensing heads associated with suspended-core fibers with three and four holes are characterized.      

Measurement of Facet Reflectivity Through Reflection Gain in Fabry–PÉrot Laser Diode

We propose a simple and useful method to measure the facet reflectivity in a Fabry-Perot laser diode (LD). The new measurement method does not require information on LD parameters such as threshold current or optical spectrum. The measurement is based on a simple formula that describes the relationship of facet reflectivity and reflection gain with launching the probe beam of a narrow spectral width.     

Fourier Transform White-Light Interferometry for the Measurement of Fiber-Optic Extrinsic Fabry–PÉrot Interferometric Sensors

A Fourier transform white-light interferometry for the absolute measurement of fiber-optic extrinsic Fabry-Perot interferometric sensors is presented. The continuous test shows the variation is plusmn0.3 mum when measuring a cavity length of 2300 mum. By combining with an average calculation, the variation of the measured results is only plusmn10 nm.     

Multiple Dual-Wavelengths Erbium-Doped Fiber Ring Laser Using a Polarization-Maintaining Fabry–PÉrot Filter

A multiple dual-wavelengths erbium-doped fiber laser is demonstrated. It is designed by using a narrowband polarization-maintaining Fabry–PÉrot filter in a ring cavity. More than 30 power equalized dual-wavelengths are achieved in a 10-nm range. The switchability from multiple dual-wavelengths to normal multiple wavelengths laser is also demonstrated experimentally.      

Temperature-Insensitive Micro Fabry–PÉrot Strain Sensor Fabricated by Chemically Etching Er-Doped Fiber

Micro extrinsic Fabry–PÉrot interferometric (MEFPI) sensors are fabricated by chemically etching Er-doped fibers with mixed hydrochloric (HCl) and hydrofluoric (HF) acid and fusion splicing. Compared with MEFPI sensors fabricated by etched single-mode fibers, the sensor performance is greatly improved by the chemical reaction between HCl acid and doped ${hbox {Er}}_{2}hbox{O}_{3}$ . A maximum visibility of $sim$ 24 dB is obtained, comparable to that of MEFPI sensors fabricated by excimer lasers. Our MEFPI sensor has high mechanical strength as the etching rate difference between fiber core and cladding is enlarged. Preliminary results indicate that this kind of sensor is insensitive to temperature while highly sensitive to strain, with sensitivities of $sim$0.65 $hbox{pm}/^{circ}hbox{C}$ and $sim$3.15 $hbox{pm}/muvarepsilon$ , respectively.      

Single and Multiwavelength All-Optical Clock Recovery Using Fabry–PÉrot Semiconductor Optical Amplifier

We experimentally demonstrate an all-optical clock recovery scheme for the return-to-zero data with single- and multiwavelength. This scheme is mainly based on clock-like pulse generation from an active filter with a multiquantum-well Fabry–PÉrot semiconductor optical amplifier and amplitude-equalization with self-nonlinear polarization switching. We obtain stable and low jitter optical clock signals in single and dual channels using this scheme, which has some distinct advantages including easy integration, free preamplification, convenient tuning, good tolerance to long “0s” data, and greater tolerance to the stability of the input wavelength.