Adaptive recovery of a chirped sinusoid in noise. I. Performance ofthe RLS algorithm

The authors study the ability of the exponentially weighted recursive least square (RLS) algorithm to track a complex chirped exponential signal buried in additive white Gaussian noise (power P _n). The signal is a sinusoid whose frequency is drifting at a constant rate Ψ. lt is recovered using an M-tap adaptive predictor. Five principal aspects of the study are presented: the methodology of the analysis; proof of the quasi-deterministic nature of the data-covariance estimate R(k); a new analysis of RLS for an inverse system modeling problem; a new analysis of RLS for a deterministic time-varying model for the optimum filter; and an evaluation of the residual output mean-square error (MSE) resulting from the nonoptimality of the adaptive predictor (the misadjustment) in terms of the forgetting rate (β) of the RLS algorithm. It is shown that the misadjustment is dominated by a lag term of order β^-2 and a noise term of order β. Thus, a value β_opt exists which yields a minimum misadjustment. It is proved that β_opt={(M+1)ρΨ²} ^1/3, and the minimum misadjustment is equal to (3/4)P_n(M+1)β_opt, where ρ is the input signal-to-noise ratio (SNR)     

Normalized data nonlinearities for LMS adaptation

Properly designed nonlinearly-modified LMS algorithms, in which various quantities in the stochastic gradient estimate are operated upon by memoryless nonlinearities, have been shown to perform better than the LMS algorithm in system identification-type problems. The authors investigate one such algorithm given by W_k+l=W_k+μ(d_k-W_k^t X_k)X_kf(X_k) in which the function f(X_k) is a scalar function of the sum of the squares of the N elements of the input data vector X_k. This form of algorithm generalizes the so-called normalized LMS (NLMS) algorithm. They evaluate the expected behavior of this nonlinear algorithm for both independent input vectors and correlated Gaussian input vectors assuming the system identification model. By comparing the nonlinear algorithm's behavior with that of the LMS algorithm, they then provide a method of optimizing the form of the nonlinearity for the given input statistics. In the independent input case, they show that the optimum nonlinearity is a single-parameter version of the NLMS algorithm with an additional constant in the denominator and show that this algorithm achieves a lower excess mean-square error (MSE) than the LMS algorithm with an equivalent convergence rate. Additionally, they examine the optimum step size sequence for the optimum nonlinear algorithm and show that the resulting algorithm performs better and is less complex to implement than the optimum step size algorithm derived for another form of the NLMS algorithm. Simulations verify the theory and the predicted performance improvements of the optimum normalized data nonlinearity algorithm     

Convergence models for Rosenblatt's perceptron learning algorithm

Presents a stochastic analysis of the steady-state and transient convergence properties of a single-layer perceptron for fast learning (large step-size, input-power product). The training data are modeled using a system identification formulation with zero-mean Gaussian inputs. The perceptron weights are adjusted by a learning algorithm equivalent to Rosenblatt's perceptron convergence procedure. It is shown that the convergence points of the algorithm depend on the step size μ and the input signal power (variance) σ_x² , and that the algorithm is stable essentially for μ>0. Two coupled nonlinear recursions are derived that accurately model the transient behavior of the algorithm. The authors also examine how these convergence results are affected by noisy perceptron input vectors. Computer simulations are presented to verify the analytical models     

Convergence analysis of the sign algorithm for adaptive filtering

We consider the convergence analysis of the sign algorithm for adaptive filtering when the input processes are uncorrelated and Gaussian and a fixed step size μ>0 is used. Exact recursive equations for the covariance matrix of the deviation error are established for any step size μ>0. Asymptotic time-averaged convergence for the mean-absolute deviation error, mean-square deviation error, and for the signal mean-square estimation error are established. These results are shown to hold for arbitrary step size μ>0     

Capacity of the Gaussian arbitrarily varying channel

The Gaussian arbitrarily varying channel with input constraint Γ and state constraint Λ admits input sequences x=(x₁,---,X_n) of real numbers with Σx_i²⩽nΓ and state sequences s=(S₁,---,s_n ) of real numbers with Σs_i²⩽nΛ; the output sequence x+s+V, where V=(V₁,---,V_n) is a sequence of independent and identically distributed Gaussian random variables with mean 0 and variance σ². It is proved that the capacity of this arbitrarily varying channel for deterministic codes and the average probability of error criterion equals 1/2 log (1+Γ/(Λ+σ²)) if Λ<Γ and is 0 otherwise     

Statistics of the binary quantizer error in single-loop sigma-deltamodulation with white Gaussian input

Representations and statistical properties of the process e¯ defined by e¯_n+1=λ(e¯_n+ξ_n ), are given. Here λ(u):=u-b·sign(u)+m and {ξ_n}_n=0^+∞ is Gaussian white noise. The process e¯ represents the binary quantizer error in a model for single-loop sigma-delta modulation. The innovations variables are found and the existence and uniqueness of an invariant probability measure, ergodicity properties, as well as the existence of the exponential moment with respect to the invariant probability are proved using Markov process theory. We consider also e¯ as a random perturbation, for small values of the variance of ξ_n, Of the orbits of s_n+1=λ(s_n). Here s_n has the uniform invariant distribution on the interval [m-h, m+b]. Analytical approximations to the structure of the power spectrum of e¯ are obtained using a linear prediction in terms of the innovations variables and the perturbation approach     

Kernel estimation of the instantaneous frequency

Considers kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise. The expected estimation error consists of two terms. The systematic bias error grows as the kernel halfwidth increases while the random error decreases. For a nonmodulated signal g(t), the kernel halfwidth that minimizes the expected error is proportional to h~[(σ²)/(N|∂_t² g|2)]^1/5 where σ² is the noise variance and N is the number of measurements per unit time. The author shows that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal, A(t)exp(iφ(t)). For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: h_1,3~[(σ²)/(A²N|∂_t ³(e^iφ¯(t/))|sup 2/)]¹/$ u7. Since the optimal halfwidths depend on derivatives of the unknown function, the authors initially estimate these derivatives prior to estimating the actual signal     

Measuring soil moisture with imaging radars

An empirical algorithm for the retrieval of soil moisture content and surface root mean square (RMS) height from remotely sensed radar data was developed using scatterometer data. The algorithm is optimized for bare surfaces and requires two copolarized channels at a frequency between 1.5 and 11 GHz. It gives best results for kh⩽2.5, μ_υ⩽35%, and &thetas;⩾30°. Omitting the usually weaker hv-polarized returns makes the algorithm less sensitive to system cross-talk and system noise, simplifies the calibration process and adds robustness to the algorithm in the presence of vegetation. However, inversion results indicate that significant amounts of vegetation (NDVI>0.4) cause the algorithm to underestimate soil moisture and overestimate RMS height. A simple criteria based on the σ_hv⁰/σ_vv⁰ ratio is developed to select the areas where the inversion is not impaired by the vegetation. The inversion accuracy is assessed on the original scatterometer data sets but also on several SAR data sets by comparing the derived soil moisture values with in-situ measurements collected over a variety of scenes between 1991 and 1994. Both spaceborne (SIR-C) and airborne (AIRSAR) data are used in the test. Over this large sample of conditions, the RMS error in the soil moisture estimate is found to be less than 4.2% soil moisture     

A method to dramatically improve subcarrier tracking

A method is presented for achieving a considerable improvement in phase tracking of square-wave subcarriers or other square waves. The amplitude of the phase quadrature reference signal is set to zero, except near the zero crossings of the input signal. Without changing the loop bandwidth, the variance of the phase error can be reduced by approximately Wσ₀² where σ ₀² is the phase error variance without windowing and W is the fraction of cycle in which the reference signal has a nonzero value. Simulation results confirm the results of the analysis and establish minimum W versus SNR (signal-to-noise ratio). Typically, the window can be made so narrow as to achieve a phase error variance of 1.5 σ₀⁴     

Excited-state absorption studies of Cr4+ ions in severalgarnet host crystals

An experimental study of saturable absorption and excited-state absorption (ESA) in several inorganic saturable absorbers, Cr⁴⁺ :YAG, Cr⁴⁺:GGG, and Cr⁴⁺:YSGG, is presented. We provide the theoretical background of absorption characteristics in saturable absorbers that exhibit ESA, with some new results: approximate analytical solutions are proposed for the optical transmission in the case of a slow absorber, and for various light intensity conditions of spatially or temporally Gaussian beams in fast and slow absorbers. Experimentally, partial bleaching of the first excited state itself could be observed in Cr⁴⁺:YAG at λ=1064 nm, yielding the higher excited-state lifetime as τ*=(0.55±0.1) ns. The regular transmission bleaching curve was measured in Cr⁴⁺:GGG, for the first time in this material, yielding σ_ga=(58±5)×10^-1 cm², and σe_s=(13±2)×10^-19 cm² at λ=1064 nm, ESA spectra were measured for the three materials between ~700 and 900 nm. All three exhibit crossing between saturable absorption at longer wavelengths and inverse saturable absorption at shorter wavelengths     

Leaky mode propagation in planar multilayer birefringentwaveguides: longitudinal dielectric tensor configuration

In this paper we examine leaky mode propagation in a general five-layered c-rotated optical structure with longitudinal dielectric tenser configuration that can be considered a useful pattern for many actual waveguides. The dependence of the leaky mode propagation on the longitudinal angle φ (between the optical c-axis and laboratory axis) is shown and the dispersion characteristics for different types and thicknesses of buffer and metal layer are reported. The guided mode losses at the wavelength λ=0.633 μm assume the lowest values (about 1 dB/cm) for an Ag layer and for φ=0°. Furthermore, we investigate the variation in the propagation characteristics of the leaky and guided modes with respect to the source wavelength. We obtain the transition wavelength from (G) guided modes to lowest order (L¹ ) leaky mode, having the ordinary component that leaks into the substrate; the transition wavelength to a higher order (L²) leaky mode, which has both ordinary and extraordinary leaky components and the leaky cutoff wavelength. As an example, for φ=10° and an Ag metal layer, the first-order G₁₁ mode transforms from guided to leaky L₁₁¹ at λ_gl≃0.9 μm. The losses exhibit a change of several dB near the wavelength transition from guided to leaky mode (e.g. The attenuation constant of the G₁₁ mode changes from 0.26×10² dB/cm at λ=0.633 μm to 0.18×10 ⁵ dB/cm at λ=0.95 μm where its ordinary component is a leaky one). A similar change is found near the transition wavelength from a lowest-order mode to the highest-order leaky mode     

Feedback capacity of the first-order moving average Gaussian channel

Despite numerous bounds and partial results, the feedback capacity of the stationary nonwhite Gaussian additive noise channel has been open, even for the simplest cases such as the first-order autoregressive Gaussian channel studied by Butman, Tiernan and Schalkwijk, Wolfowitz, Ozarow, and more recently, Yang, Kavccaronicacute, and Tatikonda. Here we consider another simple special case of the stationary first-order moving average additive Gaussian noise channel and find the feedback capacity in closed form. Specifically, the channel is given by Y_i=X_i+Z_i, i=1,2,..., where the input {X _i} satisfies a power constraint and the noise {Z_i} is a first-order moving average Gaussian process defined by Z_i=alphaU_i-1+U_i, |alpha|les 1, with white Gaussian innovations U_i, i=0,1,.... We show that the feedback capacity of this channel is C_FB=-log x₀ where x₀ is the unique positive root of the equation rhox²=(1-x²)(1-|alpha|x)² and rho is the ratio of the average input power per transmission to the variance of the noise innovation U_i. The optimal coding scheme parallels the simple linear signaling scheme by Schalkwijk and Kailath for the additive white Gaussian noise channel-the transmitter sends a real-valued information-bearing signal at the beginning of communication and subsequently refines the receiver's knowledge by processing the feedback noise signal through a linear stationary first-order autoregressive filter. The resulting error probability of the maximum likelihood decoding decays doubly exponentially in the duration of the communication. Refreshingly, this feedback capacity of the first-order moving average Gaussian channel is very similar in form to the best known achievable rate for the first-order autoregressive Gaussian noise channel given by Butman     

A minimum misadjustment adaptive FIR filter

The performance of an adaptive filter is limited by the misadjustment resulting from the variance of adapting parameters. This paper develops a method to reduce the misadjustment when the additive noise in the desired signal is correlated. Given a static linear model, the linear estimator that can achieve the minimum parameter variance estimate is known as the best linear unbiased estimator (BLUE). Starting from classical estimation theory and a Gaussian autoregressive (AR) noise model, a maximum likelihood (ML) estimator that jointly estimates the filter parameters and the noise statistics is established. The estimator is shown to approach the best linear unbiased estimator asymptotically. The proposed adaptive filtering method follows by modifying the commonly used mean-square error (MSE) criterion in accordance with the ML cost function. The new configuration consists of two adaptive components: a modeling filter and a noise whitening filter. Convergence study reveals that there is only one minimum in the error surface, and global convergence is guaranteed. Analysis of the adaptive system when optimized by LMS or RLS is made, together with the tracking capability investigation. The proposed adaptive method performs significantly better than a usual adaptive filter with correlated additive noise and tracks a time-varying system more effectively     

Single-mode fibres with extremely high-Δ and small-dimensionpure GeO2 core for efficient nonlinear optical applications

Single-mode fibres with 8.2% Δ and a 1.4 μm diameter-GeO ₂ core have been prepared for efficient stimulated Raman effect. When pumping a 2.7 m-long fibre by a Q-switched and mode-locked Nd:YAG laser (λ=1.064 μm), the first Stokes light (λ=1.114 μm) and second Stokes light (λ=1.169 μm) have been observed at the input peak power levels of 24 W and 40 W, respectively. These results indicate that the above critical power levels are approximately 10^-2~10^-3 smaller than those for the high-silica single-mode fibre     

Low frequency noise in 6H-SiC MOSFET's

The noise spectra for n-channel, depletion-mode MOSFETs fabricated in 6H-SiC material were measured from 1-10⁵ Hz at room temperature. Devices were biased in the linear regime, where the noise spectra was found to be dependent upon the drain-to-source bias current density. At a drain-to-source current of 50 μA for MOSFETs with a W/L of 400 μm/4 μm, the measured drain-to-source noise power spectral density was found to be A/(f^λ), with A being 2.6×10^-12 V², and λ being between 0.73 and 0.85, indicating a nonuniform spatial trap density skewed towards the oxide-semiconductor interface. The measured Hooge parameter (α_H) was 2×10^-5. This letter represents the first reported noise characterization of 6H-SiC MOSFET's     

High-performance long-wavelength (λ~1.3 μm) InGaAsPNquantum-well lasers

We demonstrate high-performance InGaAsPN quantum well based long-wavelength lasers grown on GaAs substrates, nitrogen containing lasers emitting in the λ=1.2- to 1.3-μm wavelength range were grown by gas source molecular beam epitaxy using a RF plasma nitrogen source. Under pulsed excitation, lasers emitting at λ=1.295 μm exhibited a record low threshold current density (J_TH) of 2. 5 kA/cm². Lasers grown with less nitrogen in the quantum well exhibited significantly lower threshold current densities of J_TH =1.9 kA/cm² at λ=1.27 μm and J_TH=1.27 kA/cm² at λ=1.2 μm. We also report a slope efficiency of 0.4 W/A and an output power of 450 mW under pulsed operation for nitrogen containing lasers emitting at 1.2 μm     

Low-frequency noise in TFSOI lateral n-p-n bipolar transistors

Low-frequency (1/f) noise is characterized as a function of base current density (J_B) on thin-film-silicon-on-insulator (TFSOI) lateral bipolar transistors. In the low injection region of operation, the noise power spectral density was proportional to J_B ^1.8 for J_B<0.4 μA/μm², which suggest that the noise in these devices is primarily dominated by a uniform distribution of noise sources across the emitter-base area. However in the high current region of operation (J_B>0.4 μm²), the noise bias dependence shifts to J_B ^1.2, indicating current crowding effects, alter the contribution of noise sources near the extrinsic base link region of the device. In addition to the expected 1/f noise and shot noise, we have observed a bias dependent generation-recombination (Gm) noise source in some of the devices. This G/R noise is correlated to random-telegraph-signal (RTS) noise resulting from single trapping centers, located at or near the spacer oxide and/or the Si to SIMOX interface, which modulate the emitter-base space charge region     

The effects of geometrical scaling on the frequency response andnoise performance of SiGe HBTs

We examine the geometrical scaling issues in SiGe HBT technology. Width Scaling, length scaling, and stripe-number scaling are quantified from a radio frequency (RF) design perspective at 2 GHz. We conclude that a SiGe HBT with emitter area A_E=0.5×20×6 μm² is optimum for low noise applications at J_c=0.1 mA/μm² and f=2 GHz using the design methodology, which guarantees optimal noise and input impedance matching with the simplest matching network. Finally, the optimal device sizes at f=4 and 6 GHz for low noise applications are also obtained using the same method     

Emission cross sections and spectroscopy of Ho3+ laserchannels in KGd(WO4)2 single crystal

The spectroscopic properties of Ho³⁺ laser channels in KGd(WO₄)₂ crystals have been investigated using optical absorption, photoluminescence, and lifetime measurements. The radiative lifetimes of Ho³⁺ have been calculated through a Judd-Ofelt (JO) formalism using 300-K optical absorption results. The JO parameters obtained were Ω₂=15.35×10^-20 cm², Ω ₄=3.79×10^-20 cm², Ω₆ =1.69×10^-20 cm². The 7-300-K lifetimes obtained in diluted (8·10¹⁸ cm^-3) KGW:0.1% Ho samples are: τ(⁵F₃)≈0.9 μs, τ( ⁵S₂)=19-3.6 μs, and τ(⁵F₅ )≈1.1 μs. For Ho concentrations below 1.5×10²⁰ cm^-3, multiphonon emission is the main source of non radiative losses, and the temperature independent multiphonon probability in KGW is found to follow the energy gap law τ_ph ^-1(0)=βexp(-αΔE), where β=1.4×10^-7 s^-1, and α=1.4×10³ cm. Above this holmium concentration, energy transfer between Ho impurities also contributes to the losses. The spectral distributions of the Ho³⁺ emission cross section σ_EM for several laser channels are calculated in σ- and π-polarized configurations. The peak a σ_EM values achieved for transitions to the ⁵I₈ level are ≈2×10^-20 cm² in the σ-polarized configuration, and three main lasing peaks at 2.02, 2.05, and 2.07 μm are envisaged inside the ⁵I₇→⁵I₈ channel     

Reconciliation of different gate-voltage dependencies of 1/f noisein n-MOS and p-MOS transistors

We have examined the 1/f noise of 3 μm×16 μm, n- and p-MOS transistors as a function of frequency (f), gate-voltage (V_g ) and temperature (T). Measurements were performed for 3 Hz⩽f⩽50 kHz, 100 mV⩽|V_g-V_th|⩽4 V, and 77 K⩽T⩽300 K, where V_th is the threshold voltage. Devices were operated in strong inversion in their linear regimes. At room temperature we find that, for n-MOS transistors, S(V_d)∝V_d²/(V_g-V_th )², and for p-MOS transistors, we generally find that S(V_d)∝V_d²/(V_g-V_th , consistent with trends reported by others. At lower temperatures, however, the results can be very different. In fact, we find that the temperature dependence of the noise and the gate-voltage dependence of the noise show similar features, consistent with the idea that the noise at a given T and V_g is determined by the trap density, D_t(E), at trap energies E=E(T,V_g). Both the T- and V_g-dependencies of the noise imply that D_t (E) tends to be constant near the silicon conduction band edge, but increases as E approaches the valence band edge. It is evidently these differences in D_t(E) that lead to differences in the gate-voltage dependence of the noise commonly observed at room temperature for n- and p-MOS transistors