On the Selection of Optimum Savitzky-Golay Filters

Savitzky-Golay (S-G) filters are finite impulse response lowpass filters obtained while smoothing data using a local least-squares (LS) polynomial approximation. Savitzky and Golay proved in their hallmark paper that local LS fitting of polynomials and their evaluation at the mid-point of the approximation interval is equivalent to filtering with a fixed impulse response. The problem that we address here is, ``how to choose a pointwise minimum mean squared error (MMSE) S-G filter length or order for smoothing, while preserving the temporal structure of a time-varying signal.'' We solve the bias-variance tradeoff involved in the MMSE optimization using Stein's unbiased risk estimator (SURE). We observe that the 3-dB cutoff frequency of the SURE-optimal S-G filter is higher where the signal varies fast locally, and vice versa, essentially enabling us to suitably trade off the bias and variance, thereby resulting in near-MMSE performance. At low signal-to-noise ratios (SNRs), it is seen that the adaptive filter length algorithm performance improves by incorporating a regularization term in the SURE objective function. We consider the algorithm performance on real-world electrocardiogram (ECG) signals. The results exhibit considerable SNR improvement. Noise performance analysis shows that the proposed algorithms are comparable, and in some cases, better than some standard denoising techniques available in the literature.

A Savitzky-Golay Filtering Perspective of Dynamic Feature Computation

We address the classical problem of delta feature computation, and interpret the operation involved in terms of Savitzky- Golay (SG) filtering. Features such as themel-frequency cepstral coefficients (MFCCs), obtained based on short-time spectra of the speech signal, are commonly used in speech recognition tasks. In order to incorporate the dynamics of speech, auxiliary delta and delta-delta features, which are computed as temporal derivatives of the original features, are used. Typically, the delta features are computed in a smooth fashion using local least-squares (LS) polynomial fitting on each feature vector component trajectory. In the light of the original work of Savitzky and Golay, and a recent article by Schafer in IEEE Signal Processing Magazine, we interpret the dynamic feature vector computation for arbitrary derivative orders as SG filtering with a fixed impulse response. This filtering equivalence brings in significantly lower latency with no loss in accuracy, as validated by results on a TIMIT phoneme recognition task. The SG filters involved in dynamic parameter computation can be viewed as modulation filters, proposed by Hermansky.

Taylor series expansion based multidimensional image reconstruction for confocal and 4pi microscopy

We propose and experimentally demonstrate a three-dimensional (3D) image reconstruction methodology based on Taylor series approximation (TSA) in a Bayesian image reconstruction formulation. TSA incorporates the requirement of analyticity in the image domain, and acts as a finite impulse response filter. This technique is validated on images obtained from widefield, confocal laser scanning fluorescence microscopy and two-photon excited 4pi (2PE-4pi) fluorescence microscopy. Studies on simulated 3D objects, mitochondria-tagged yeast cells (labeled with Mitotracker Orange) and mitochondrial networks (tagged with Green fluorescent protein) show a signal-to-background improvement of 40% and resolution enhancement from 360 to 240 nm. This technique can easily be extended to other imaging modalities (single plane illumination microscopy (SPIM), individual molecule localization SPIM, stimulated emission depletion microscopy and its variants).

Maximum likelihood algorithms for joint estimation of synchronisation impairments and channel in multiple input multiple output-orthogonal frequency division multiplexing system

Maximum likelihood (ML) algorithms, for the joint estimation of synchronisation impairments and channel in multiple input multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) system, are investigated in this work. A system model that takes into account the effects of carrier frequency offset, sampling frequency offset, symbol timing error and channel impulse response is formulated. Cramer-Rao lower bounds for the estimation of continuous parameters are derived, which show the coupling effect among different impairments and the significance of the joint estimation. The authors propose an ML algorithm for the estimation of synchronisation impairments and channel together, using the grid search method. To reduce the complexity of the joint grid search in the ML algorithm, a modified ML (MML) algorithm with multiple one-dimensional searches is also proposed. Further, a stage-wise ML (SML) algorithm using existing algorithms, which estimate less number of parameters, is also proposed. Performance of the estimation algorithms is studied through numerical simulations and it is found that the proposed ML and MML algorithms exhibit better performance than SML algorithm.

Delay optimized zero-forcing channel shortening algorithm for wireless communication

In Orthogonal Frequency Division Multiplexing and Discrete Multitone transceivers, a guard interval called Cyclic Prefix (CP) is inserted to avoid inter-symbol interference. The length of the CP is usually greater than the impulse response of the channel resulting in a loss of useful data carriers. In order to avoid long CP, a time domain equalizer is used to shorten the channel. In this paper, we propose a method to include a delay in the zero-forcing equalizer and obtain an optimal value of the delay, based on the location of zeros of the channel. The performance of the algorithms is studied using numerical simulations.

Joint estimation of synchronization impairments in MIMO-OFDM system

A joint Maximum Likelihood (ML) estimation algorithm for the synchronization impairments such as Carrier Frequency Offset (CFO), Sampling Frequency Offset (SFO) and Symbol Timing Error (STE) in single user MIMO-OFDM system is investigated in this work. A received signal model that takes into account the nonlinear effects of CFO, SFO, STE and Channel Impulse Response (CIR) is formulated. Based on the signal model, a joint ML estimation algorithm is proposed. Cramer-Rao Lower Bound (CRLB) for the continuous parameters CFO and SFO is derived for the cases of with and without channel response effects and is used to compare the effect of coupling between different estimated parameters. The performance of the estimation method is studied through numerical simulations.

Auditory-motivated Gammatone wavelet transform

The ability of the continuous wavelet transform (CWT) to provide good time and frequency localization has made it a popular tool in time-frequency analysis of signals. Wavelets exhibit constant-Q property, which is also possessed by the basilar membrane filters in the peripheral auditory system. The basilar membrane filters or auditory filters are often modeled by a Gammatone function, which provides a good approximation to experimentally determined responses. The filterbank derived from these filters is referred to as a Gammatone filterbank. In general, wavelet analysis can be likened to a filterbank analysis and hence the interesting link between standard wavelet analysis and Gammatone filterbank. However, the Gammatone function does not exactly qualify as a wavelet because its time average is not zero. We show how bona fide wavelets can be constructed out of Gammatone functions. We analyze properties such as admissibility, time-bandwidth product, vanishing moments, which are particularly relevant in the context of wavelets. We also show how the proposed auditory wavelets are produced as the impulse response of a linear, shift-invariant system governed by a linear differential equation with constant coefficients. We propose analog circuit implementations of the proposed CWT. We also show how the Gammatone-derived wavelets can be used for singularity detection and time-frequency analysis of transient signals. (C) 2013 Elsevier B.V. All rights reserved.

Joint Approximately Sparse Channel Estimation and Data Detection in OFDM Systems Using Sparse Bayesian Learning

It is well known that the impulse response of a wide-band wireless channel is approximately sparse, in the sense that it has a small number of significant components relative to the channel delay spread. In this paper, we consider the estimation of the unknown channel coefficients and its support in OFDM systems using a sparse Bayesian learning (SBL) framework for exact inference. In a quasi-static, block-fading scenario, we employ the SBL algorithm for channel estimation and propose a joint SBL (J-SBL) and a low-complexity recursive J-SBL algorithm for joint channel estimation and data detection. In a time-varying scenario, we use a first-order autoregressive model for the wireless channel and propose a novel, recursive, low-complexity Kalman filtering-based SBL (KSBL) algorithm for channel estimation. We generalize the KSBL algorithm to obtain the recursive joint KSBL algorithm that performs joint channel estimation and data detection. Our algorithms can efficiently recover a group of approximately sparse vectors even when the measurement matrix is partially unknown due to the presence of unknown data symbols. Moreover, the algorithms can fully exploit the correlation structure in the multiple measurements. Monte Carlo simulations illustrate the efficacy of the proposed techniques in terms of the mean-square error and bit error rate performance.

Robust Savitzky-Golay Filters

Local polynomial approximation of data is an approach towards signal denoising. Savitzky-Golay (SG) filters are finite-impulse-response kernels, which convolve with the data to result in polynomial approximation for a chosen set of filter parameters. In the case of noise following Gaussian statistics, minimization of mean-squared error (MSE) between noisy signal and its polynomial approximation is optimum in the maximum-likelihood (ML) sense but the MSE criterion is not optimal for non-Gaussian noise conditions. In this paper, we robustify the SG filter for applications involving noise following a heavy-tailed distribution. The optimal filtering criterion is achieved by l(1) norm minimization of error through iteratively reweighted least-squares (IRLS) technique. It is interesting to note that at any stage of the iteration, we solve a weighted SG filter by minimizing l(2) norm but the process converges to l(1) minimized output. The results show consistent improvement over the standard SG filter performance.

Joint Channel Estimation and Data Detection in MIMO-OFDM Systems: A Sparse Bayesian Learning Approach

The impulse response of wireless channels between the N-t transmit and N-r receive antennas of a MIMO-OFDM system are group approximately sparse (ga-sparse), i.e., NtNt the channels have a small number of significant paths relative to the channel delay spread and the time-lags of the significant paths between transmit and receive antenna pairs coincide. Often, wireless channels are also group approximately cluster-sparse (gac-sparse), i.e., every ga-sparse channel consists of clusters, where a few clusters have all strong components while most clusters have all weak components. In this paper, we cast the problem of estimating the ga-sparse and gac-sparse block-fading and time-varying channels in the sparse Bayesian learning (SBL) framework and propose a bouquet of novel algorithms for pilot-based channel estimation, and joint channel estimation and data detection, in MIMO-OFDM systems. The proposed algorithms are capable of estimating the sparse wireless channels even when the measurement matrix is only partially known. Further, we employ a first-order autoregressive modeling of the temporal variation of the ga-sparse and gac-sparse channels and propose a recursive Kalman filtering and smoothing (KFS) technique for joint channel estimation, tracking, and data detection. We also propose novel, parallel-implementation based, low-complexity techniques for estimating gac-sparse channels. Monte Carlo simulations illustrate the benefit of exploiting the gac-sparse structure in the wireless channel in terms of the mean square error (MSE) and coded bit error rate (BER) performance.

Fixed-lag blind equalization and sequence estimation in digital communications systems using sequential importance sampling

We present methods for fixed-lag smoothing using Sequential Importance sampling (SIS) on a discrete non-linear, non-Gaussian state space system with unknown parameters. Our particular application is in the field of digital communication systems. Each input data point is taken from a finite set of symbols. We represent transmission media as a fixed filter with a finite impulse response (FIR), hence a discrete state-space system is formed. Conventional Markov chain Monte Carlo (MCMC) techniques such as the Gibbs sampler are unsuitable for this task because they can only perform processing on a batch of data. Data arrives sequentially, so it would seem sensible to process it in this way. In addition, many communication systems are interactive, so there is a maximum level of latency that can be tolerated before a symbol is decoded. We will demonstrate this method by simulation and compare its performance to existing techniques.

How Does the New Keynesian Monetary Model Fit in the U.S. and the Eurozone? an Indirect Inference Approach

This paper estimates a standard version of the New Keynesian monetary (NKM) model under alternative specifications of the monetary policy rule using U.S. and Eurozone data. The estimation procedure implemented is a classical method based on the indirect inference principle. An unrestricted VAR is considered as the auxiliary model. On the one hand, the estimation method proposed overcomes some of the shortcomings of using a structural VAR as the auxiliary model in order to identify the impulse response that defines the minimum distance estimator implemented in the literature. On the other hand, by following a classical approach we can further assess the estimation results found in recent papers that follow a maximum-likelihood Bayesian approach. The estimation results show that some structural parameter estimates are quite sensitive to the specification of monetary policy. Moreover, the estimation results in the U.S. show that the fit of the NKM under an optimal monetary plan is much worse than the fit of the NKM model assuming a forward-looking Taylor rule. In contrast to the U.S. case, in the Eurozone the best fit is obtained assuming a backward-looking Taylor rule, but the improvement is rather small with respect to assuming either a forward-looking Taylor rule or an optimal plan.

Constrained high-order statistical blind deconvolution of spectral data

A constrained high-order statistical algorithm is proposed to blindly deconvolute the measured spectral data and estimate the response function of the instruments simultaneously. In this algorithm, no prior-knowledge is necessary except a proper length of the unit-impulse response. This length can be easily set to be the width of the narrowest spectral line by observing the measured data. The feasibility of this method has been demonstrated experimentally by the measured Raman and absorption spectral data.

Damage detection in civil structures using high-frequency seismograms

The dynamic properties of a structure are a function of its physical properties, and changes in the physical properties of the structure, including the introduction of structural damage, can cause changes in its dynamic behavior. Structural health monitoring (SHM) and damage detection methods provide a means to assess the structural integrity and safety of a civil structure using measurements of its dynamic properties. In particular, these techniques enable a quick damage assessment following a seismic event. In this thesis, the application of high-frequency seismograms to damage detection in civil structures is investigated.

Two novel methods for SHM are developed and validated using small-scale experimental testing, existing structures in situ, and numerical testing. The first method is developed for pre-Northridge steel-moment-resisting frame buildings that are susceptible to weld fracture at beam-column connections. The method is based on using the response of a structure to a nondestructive force (i.e., a hammer blow) to approximate the response of the structure to a damage event (i.e., weld fracture). The method is applied to a small-scale experimental frame, where the impulse response functions of the frame are generated during an impact hammer test. The method is also applied to a numerical model of a steel frame, in which weld fracture is modeled as the tensile opening of a Mode I crack. Impulse response functions are experimentally obtained for a steel moment-resisting frame building in situ. Results indicate that while acceleration and velocity records generated by a damage event are best approximated by the acceleration and velocity records generated by a colocated hammer blow, the method may not be robust to noise. The method seems to be better suited for damage localization, where information such as arrival times and peak accelerations can also provide indication of the damage location. This is of significance for sparsely-instrumented civil structures.

The second SHM method is designed to extract features from high-frequency acceleration records that may indicate the presence of damage. As short-duration high-frequency signals (i.e., pulses) can be indicative of damage, this method relies on the identification and classification of pulses in the acceleration records. It is recommended that, in practice, the method be combined with a vibration-based method that can be used to estimate the loss of stiffness. Briefly, pulses observed in the acceleration time series when the structure is known to be in an undamaged state are compared with pulses observed when the structure is in a potentially damaged state. By comparing the pulse signatures from these two situations, changes in the high-frequency dynamic behavior of the structure can be identified, and damage signals can be extracted and subjected to further analysis. The method is successfully applied to a small-scale experimental shear beam that is dynamically excited at its base using a shake table and damaged by loosening a screw to create a moving part. Although the damage is aperiodic and nonlinear in nature, the damage signals are accurately identified, and the location of damage is determined using the amplitudes and arrival times of the damage signal. The method is also successfully applied to detect the occurrence of damage in a test bed data set provided by the Los Alamos National Laboratory, in which nonlinear damage is introduced into a small-scale steel frame by installing a bumper mechanism that inhibits the amount of motion between two floors. The method is successfully applied and is robust despite a low sampling rate, though false negatives (undetected damage signals) begin to occur at high levels of damage when the frequency of damage events increases. The method is also applied to acceleration data recorded on a damaged cable-stayed bridge in China, provided by the Center of Structural Monitoring and Control at the Harbin Institute of Technology. Acceleration records recorded after the date of damage show a clear increase in high-frequency short-duration pulses compared to those previously recorded. One undamage pulse and two damage pulses are identified from the data. The occurrence of the detected damage pulses is consistent with a progression of damage and matches the known chronology of damage.

Toward Understanding Astrophysical Phenomena

Fast radio bursts (FRBs), a novel type of radio pulse, whose physics is not yet understood at all. Only a handful of FRBs had been detected when we started this project. Taking account of the scant observations, we put physical constraints on FRBs. We excluded proposals of a galactic origin for their extraordinarily high dispersion measures (DM), in particular stellar coronas and HII regions. Therefore our work supports an extragalactic origin for FRBs. We show that the resolved scattering tail of FRB 110220 is unlikely to be due to propagation through the intergalactic plasma. Instead the scattering is probably caused by the interstellar medium in the FRB's host galaxy, and indicates that this burst sits in the central region of that galaxy. Pulse durations of order $\ms$ constrain source sizes of FRBs implying enormous brightness temperatures and thus coherent emission. Electric fields near FRBs at cosmological distances would be so strong that they could accelerate free electrons from rest to relativistic energies in a single wave period. When we worked on FRBs, it was unclear whether they were genuine astronomical signals as distinct from `perytons', clearly terrestrial radio bursts, sharing some common properties with FRBs. Recently, in April 2015, astronomers discovered that perytons were emitted by microwave ovens. Radio chirps similar to FRBs were emitted when their doors opened while they were still heating. Evidence for the astronomical nature of FRBs has strengthened since our paper was published. Some bursts have been found to show linear and circular polarizations and Faraday rotation of the linear polarization has also been detected. I hope to resume working on FRBs in the near future. But after we completed our FRB paper, I decided to pause this project because of the lack of observational constraints.

The pulsar triple system, J0733+1715, has its orbital parameters fitted to high accuracy owing to the precise timing of the central $\ms$ pulsar. The two orbits are highly hierarchical, namely $P_{\mathrm{orb,1}}\ll P_{\mathrm{orb,2}}$, where 1 and 2 label the inner and outer white dwarf (WD) companions respectively. Moreover, their orbital planes almost coincide, providing a unique opportunity to study secular interaction associated purely with eccentricity beyond the solar system. Secular interaction only involves effect averaged over many orbits. Thus each companion can be represented by an elliptical wire with its mass distributed inversely proportional to its local orbital speed. Generally there exists a mutual torque, which vanishes only when their apsidal lines are parallel or anti-parallel. To maintain either mode, the eccentricity ratio, $e_1/e_2$, must be of the proper value, so that both apsidal lines precess together. For J0733+1715, $e_1\ll e_2$ for the parallel mode, while $e_1\gg e_2$ for the anti-parallel one. We show that the former precesses $\sim 10$ times slower than the latter. Currently the system is dominated by the parallel mode. Although only a little anti-parallel mode survives, both eccentricities especially $e_1$ oscillate on $\sim 10^3\yr$ timescale. Detectable changes would occur within $\sim 1\yr$. We demonstrate that the anti-parallel mode gets damped $\sim 10^4$ times faster than its parallel brother by any dissipative process diminishing $e_1$. If it is the tidal damping in the inner WD, we proceed to estimate its tidal quantity parameter ($Q$) to be $\sim 10^6$, which was poorly constrained by observations. However, tidal damping may also happen during the preceding low-mass X-ray binary (LMXB) phase or hydrogen thermal nuclear flashes. But, in both cases, the inner companion fills its Roche lobe and probably suffers mass/angular momentum loss, which might cause $e_1$ to grow rather than decay.

Several pairs of solar system satellites occupy mean motion resonances (MMRs). We divide these into two groups according to their proximity to exact resonance. Proximity is measured by the existence of a separatrix in phase space. MMRs between Io-Europa, Europa-Ganymede and Enceladus-Dione are too distant from exact resonance for a separatrix to appear. A separatrix is present only in the phase spaces of the Mimas-Tethys and Titan-Hyperion MMRs and their resonant arguments are the only ones to exhibit substantial librations. When a separatrix is present, tidal damping of eccentricity or inclination excites overstable librations that can lead to passage through resonance on the damping timescale. However, after investigation, we conclude that the librations in the Mimas-Tethys and Titan-Hyperion MMRs are fossils and do not result from overstability.

Rubble piles are common in the solar system. Monolithic elements touch their neighbors in small localized areas. Voids occupy a significant fraction of the volume. In a fluid-free environment, heat cannot conduct through voids; only radiation can transfer energy across them. We model the effective thermal conductivity of a rubble pile and show that it is proportional the square root of the pressure, $P$, for $P\leq \epsy^3\mu$ where $\epsy$ is the material's yield strain and $\mu$ its shear modulus. Our model provides an excellent fit to the depth dependence of the thermal conductivity in the top $140\,\mathrm{cm}$ of the lunar regolith. It also offers an explanation for the low thermal inertias of rocky asteroids and icy satellites. Lastly, we discuss how rubble piles slow down the cooling of small bodies such as asteroids.

Electromagnetic (EM) follow-up observations of gravitational wave (GW) events will help shed light on the nature of the sources, and more can be learned if the EM follow-ups can start as soon as the GW event becomes observable. In this paper, we propose a computationally efficient time-domain algorithm capable of detecting gravitational waves (GWs) from coalescing binaries of compact objects with nearly zero time delay. In case when the signal is strong enough, our algorithm also has the flexibility to trigger EM observation {\it before} the merger. The key to the efficiency of our algorithm arises from the use of chains of so-called Infinite Impulse Response (IIR) filters, which filter time-series data recursively. Computational cost is further reduced by a template interpolation technique that requires filtering to be done only for a much coarser template bank than otherwise required to sufficiently recover optimal signal-to-noise ratio. Towards future detectors with sensitivity extending to lower frequencies, our algorithm's computational cost is shown to increase rather insignificantly compared to the conventional time-domain correlation method. Moreover, at latencies of less than hundreds to thousands of seconds, this method is expected to be computationally more efficient than the straightforward frequency-domain method.