Additive Wideband Noise Research Articles

Controlling of the bidirectional amplifier chain for optical frequency distribution based on a two-dimensional noise detector.

In the paper we present a solution for controlling the chain of bidirectional optical amplifiers, intended for long-haul fiber links used to distribute signals produced by optical atomic clocks. The solution is based on a dedicated two-channel noise detector, which allows independent measurement of the noise contributions related to interferometric signal fading and additive wideband noise. New signal quality metrics, based on two-dimensional noise detector, allows to distribute properly the needed gain among the chained amplifiers. Experimental results performed both in laboratory conditions and on a real 600 km-long link, confirming proper operation of proposed solutions, are presented.

Methods for Detection of Bioimpedance Variations in Resource Constrained Environments.

Changes in a certain parameter are often a few magnitudes smaller than the base value of the parameter, specifying significant requirements for the dynamic range and noise levels of the measurement system. In case of electrical bioimpedance acquisition, the variations can be 1000 times smaller than the entire measured value. Synchronous or lock-in measurement of these variations is discussed in the current paper, and novel measurement solutions are presented. Proposed methods are simple and robust when compared to other applicable solutions. A common feature shared by all members of the group of the proposed solutions is differentiation. It is achieved by calculating the differences between synchronously acquired consecutive samples, with lock-in integration and analog differentiation. All these methods enable inherent separation of variations from the static component of the signal. The variable component of the bioimpedance can, thus, be acquired using the full available dynamic range of the apparatus for its detection. Additive disturbing signals and omnipresent wideband noise are considered and the method for their reduction is proposed.

Effect of windowing and noise on the amplitude and phase estimators returned by the Taylor-based Weighted Least Squares

This paper analyzes the effect of windowing and additive wideband noise on the estimates returned by the real-valued Taylor-based Weighted Least Squares (TWLS) algorithm when applied to noisy sine-wave with constant amplitude and phase. The Taylor's series of the signal time-varying phasor are truncated to the second order and the two-term or three-term Minimum Sidelobe Level (MSL) and Maximum Sidelobe Decay (MSD) windows are considered. Expressions for the estimation variances are derived that hold even when the sine-wave amplitude and phase exhibit slow and limited, but significant, time variations. Then they are compared with the unbiased Cramér–Rao Lower Bounds (CRLBs) for constant parameter sine-waves in order to analyze the estimator statistical efficiency. The accuracies of the derived expressions are verified through computer simulations and validated by experimental results.

Accurate frequency estimation of a noisy sine-wave by means of an interpolated discrete-time Fourier transform algorithm

This paper investigates the accuracy of a recently proposed sine-wave frequency estimator based on the interpolation of Discrete-Time Fourier Transform (DTFT) spectrum samples located one frequency bin apart. Specifically, an analytical expression for the variance of the Interpolated DTFT (IpDTFT) frequency estimator due to additive wide-band noise is derived in the case when the acquired sine-wave samples are weighted by a Maximum Sidelobe Decay (MSD) window. The derived expression enables the identification of an iterative algorithm, called IpDTFT-NR procedure, that ensures the maximum Noise Rejection on the IpDTFT frequency estimator. Moreover, the accuracy of the IpDTFT-NR procedure is compared with that of another algorithm, called IpDTFT-IR procedure, which ensures the maximum Image component interference Rejection on the IpDTFT frequency estimator. To this aim, accurate expressions for the Mean Square Errors (MSEs) of the two frequency estimators are derived. Also, the value of the Signal-to-Noise Ratio (SNR) up to which the IpDTFT-NR procedure outperforms the IpDTFT-IR one is determined as a function of the number of analysed sine-wave cycles. Once the SNR and the number of observed cycles of the analysed signal are known or estimated, the knowledge of this threshold enables the definition of an algorithm, called IpDTFT-IR/NR procedure, that estimates the sine-wave frequency by using the most accurate of the two procedures above. The accuracies of the IpDTFT-IR/NR procedure and of other state-of-the-art interpolated Fourier algorithms are compared with each other using both simulation and experimental data.

Accuracy analysis of complex sinusoid amplitude and phase estimation by means of the interpolated discrete-time Fourier transform algorithm

In this paper the accuracy of the complex sinusoid amplitude and phase estimators provided by the Interpolated Weighted Discrete-Time Fourier Transform (IpWDTFT) algorithm is analyzed. It is shown that the use of the WDTFT spectral sample corresponding to the estimated sinusoid frequency allows to minimize the estimator asymptotic Mean Square Error (MSE) due to additive wideband noise. Also, the expression of the asymptotic expected sum-squared fitting error in the reconstruction of the complex sinusoid by means of the IpWDTFT algorithm is derived. All the derived expressions are verified through computer simulations.

Costas circuit under the action of additive harmonic interferences and wideband noise

The Markovian mathematical model of the Kostas phase-locked loop subjected to the simultaneous actions of additive harmonic interferences and a white Gaussian noise, which are described by stochastic differential equations involving the influence of the low-pass filters of the control loop on the useful pulsed signal shape, is discussed. The statistical characteristics of the error signal are analyzed in the linear approximation.

Frequency estimation by two- or three-point interpolated Fourier algorithms based on cosine windows

This paper investigates the frequency estimation of a complex sinusoid obtained by interpolating two or three samples of the Discrete-Time Fourier Transform (DTFT) of a signal weighted by a suitable cosine window. Versions of the algorithms based on both DTFT complex values and modules are considered and the expressions for the related frequency estimators are provided. Iterative procedures are used in order to minimize the estimator variance due to additive wideband noise. Furthermore, the accuracies of the proposed estimators are verified and compared each other and with state-of-the art algorithms by means of computer simulations when applied to noisy or noisy and harmonically distorted complex sinusoids, respectively.

Accuracy analysis of the normalized frequency estimation of a discrete-time sine-wave by the average-based interpolated DFT method

The normalized frequency of a sine-wave can be estimated by the average of the two estimates achieved using the Interpolated Discrete Fourier Transform (IpDFT) method. This approach is called average-based IpDFT method. The paper investigates the effect of the spectral interference from the image component and additive wideband noise on the accuracy of the normalized frequency estimator provided by the average-based IpDFT method with maximum sidelobe decay windows in the case of a discrete-time sine-wave. The expressions of the maximum of the normalized frequency estimator error due to the spectral interference and variance of the normalized frequency estimator due to the wideband noise are derived. Also, the expression of the combined standard uncertainty of the normalized frequency estimator is derived. The accuracies of all derived expressions are confirmed by means of computer simulations. Moreover, the performance of the average-based IpDFT method is compared with those of a state-of-the-art three-point IpDFT method and IpDFT method through both computer simulations and experimental results.

Accuracy of the Normalized Frequency Estimation of a Discrete-Time Sine-Wave by the Energy-Based Method

This paper investigates the effect of algorithm error, spectral interference, and additive wideband noise on the accuracy of the normalized frequency estimation provided by the Energy-Based Method (EBM) in the case of a discrete-time sine-wave. The statistical efficiency of the EBM estimator with respect to the unbiased Cramér-Rao Lower Bound (CRLB) is determined. The accuracy of the derived expressions is confirmed by means of computer simulations. Moreover, the expression of the rms error of the normalized frequency estimate is derived. Finally, the performance of the normalized frequency estimators provided by the EBM and other state-of-the-art methods are compared by means of both simulations and experimental results.

Accuracy of Sine Wave Frequency Estimation by Multipoint Interpolated DFT Approach

This paper focuses on the frequency-domain estimation of the normalized frequency of a sine wave corrupted by a stationary white noise. The weighted multipoint interpolated discrete Fourier transform method is considered, and its effect on both the spectral interference due to the image component and the additive wideband noise is taken into account. In particular, the expression of the combined standard uncertainty of the estimator is derived in the case when the H-term maximum sidelobe decay window (H ≥ 2) is used, and the number of interpolation points is 2J + 1 (J ≥ 1). Based on this expression, the number of interpolation points that minimize the estimator-combined uncertainty can be determined. The derived results are validated by means of computer simulations and applied to experimental data.

Analytic behavior of the LMS adaptive line enhancer for sinusoids corrupted by multiplicative and additive noise

The least mean squares adaptive line enhancer (LMS ALE) has been widely used for the enhancement of coherent sinusoids in additive wideband noise. This paper studies the behavior of the LMS ALE when applied to the enhancement of sinusoids that have been corrupted by both colored multiplicative and white additive noise. The multiplicative noise decorrelates the sinusoid, spreads its power spectrum, and acts as an additional corrupting noise. Closed-form expressions are derived for the optimum (Wiener filter) ALE output SNR as a function of the residual coherent sine wave power, the noncoherent sine wave power spectrum, and the background additive white noise. When the coherent to noncoherent sine wave power ratio is sufficiently small, it is shown that a nonlinear (e.g., square law) transformation of the ALE input results in a larger optimum ALE output SNR.

A parallel implementation of a smoothed adaptive deconvolution filter

This paper considers a multiple processor adaptive approach to deconvolution estimation obtained by applying the extended least squares (ELS) technique. The approach adopted within the system only requires one input (the corrupted signal). This contrasts with the more common use of a FIR filter coupled with the least mean square estimation algorithm. The adaptive deconvolution smoothing filter has been implemented on a three-transputer system. The network uses a transputer to act as a digital deconvolution smoothing filter while a second transputer handles the extended least squares (ELS) parameter estimation algorithm. A third transputer calculates the gain factor for the deconvolution smoother. The three transputers operate in parallel. The example considered in this paper is of a speech signal which has passed through a distorting communication channel resulting in a narrow band signal. The distorting channel is assumed to be represented as an all-pole model. This signal is then corrupted by additive wide band noise. The transputer network is used to estimate the original speech signal.

Recursive linear smoothed Newton predictors for polynomial extrapolation

Newton predictors have considerable gain at the higher frequencies, which reduces their applicability to practical signal processing where the narrowband primary signal is often corrupted by additive wideband noise. Two modifications that can be used to extrapolate low-order polynomials have been proposed. In both approaches, the highest order difference of successive input samples, approximating the constant nonzero derivative, is smoothed before it is added to the lower order differences, reducing the undesired noise gain. The linear smoothed Newton (LSN) predictor is extended in this work by including a recursive term in the basic transfer function and cascading the rest of the successive difference paths with appropriately delayed extrapolation filters of corresponding polynomial orders. This leads to computationally efficient IIR predictors with significantly lowered gain at the higher frequencies. The recursive predictor is analyzed in the time and frequency domains and compared to the other predictors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

ULF magnetic field measurements near the epicenter of the Ms 7.1 Loma Prieta earthquake

Anomalous measurements of the ultra-low frequency (ULF) magnetic field fluctuations prior to the magnitude 7.1 Loma Prieta earthquake of October 17, 1989, have been studied. For the past few years we have been monitoring fluctuations of the magnetic field of the Earth in the ULF range at Corralitos, California; our instruments were located 7 km from the epicenter of the earthquake. We have observed four anomalies in our data which may turn out to be precursors to the earthquake. First we observe narrowband noise fluctuations centered at 0.1 Hz and estimated to have a bandwidth of 0.00143–0.00167 Hz, i.e. a Q value of 60–70. The narrowband fluctuations appear to have a maximum equivalent amplitude of over 1400 pT/√Hz which is roughly 31 dB higher than the typical quiet average amplitude background levels. These fluctuations begin around September 12 and last until October 5. Next we observe the appearance of additive wideband noise fluctuations beginning around October 5 and continuing until the occurrence of the earthquake. These wideband fluctuations, which cover almost the entire 0.01–10 Hz frequency range of the system, have an average amplitude that is approximately 19 dB larger than typical levels in the 0.01–0.02 Hz band. Thirdly, we observe an atypical decrease in noise levels in the 0.2–5 Hz band throughout the day prior to the earthquake. The fourth anomaly is a jump in the power of magnetic field fluctuations, mostly in the 0.01–0.5 Hz band, in the three hours preceding the earthquake. This activity reached its highest level in the lowest 0.01–0.02 Hz band, and had a magnitude of roughly 60000 pT/√Hz, which is about 40 dB larger than typical background noise levels in the band. Our anomalous measurements do not appear to be the result of any magnetic field fluctuations generated in the upper atmosphere or to movement of our sensor caused by shocks preceding the quake. In describing these anomalous magnetic field fluctuations, both electrokinetic and piezoelectric theories of the generation of magnetic field fluctuations prior to earthquakes suggest wideband fluctuations. In our measurements, we have observed both wideband and narrowband fluctuations, so that our narrowband measurements do not seem to be easily explicable by the above theories. However, the wideband fluctuations could turn out to be attributable to the above mechanisms, since at the monitored frequencies, the skin depths of electromagnetic waves are comparable with the distances from our sensors to the earthquake focus, and therefore our measurements would not be unexpected.

A note on the differential equations of conditional probability density functions

In practice we quite often have to deal with a noise-corrupted signal of the type V(t) = X(t) + b(l) which re resents an inaccurate observation p (or measurement) of the parameter X(t). Quite often s(t) is an additive wide-band noise and is usually described as “white.” The statistical properties of both X(t) and B(t) are known a priori. On observing q(t) we obtain more accurate information about the values of the parameter to be estimated, X(t). It is now described by the “posterior” probability density function (p.d.f.) which is conditional upon observation. The equation governing the evolution of the conditional p.d.f. contains the function of observation as a driving term and describes in mathematical terms the “updating” effect while observing the signal V(t). By choosing a criterion of optimality (rms, maximum posterior probability etc.) one can obtain another dynamic equation for the estimate of the parameter x(t). The optimal filter may be then designed on the basis of that equation. The differential equation of conditional p.d.f.‘s was first obtained for the case of a Markovian parameter and an additive Gaussian wide-band noise by R. L. Stratonovich [l, 21. A somewhat different equation has been obtained by H. J. Kushner in [3]. The latter paper contains one or two misleading statements upon the nature of contradictions between the results of [l, 21, on the one hand, and those of [3], on the other. It is therefore thought desirable to look more closely into both “conflicting” equations (96) of [l] and (2.8) of [3] and investigate if any “errors due to the omission of certain significant terms” have been committed in [l], as Kushner alleges.