Capon Estimator Research Articles

High resolution adaptive estimator of sinusoidal based on covariance matrix reconstruction

Exponentially decaying sinusoidal signal is often observed in engineering and research fields. In some practical applications, the frequencies of multiple signals are close to each other. It would face a challenging task to resolve these crowded signals processed by discrete Fourier transformation, especially in the case of the lack of sampling time and the fast-decaying signals. The Localised Capon estimator (LoCapE) has demonstrated its advantage in improving the resolution of exponentially decaying sinusoidal signals. However, the amplitude and the frequency of each component extracted by the LoCapE are not accurate due to the effect from interference components. Here, we improve the LoCapE method by reconstructing the interference-plus-noise covariance matrix instead of the signal covariance matrix. Experimental results on simulated and real exponentially decaying sinusoidal data show that the proposed method is more accurate in extracting parameters at the lack of sampling points.

High‐resolution angle estimation method in partly calibrated subarray‐based bistatic multiple‐input multiple‐output radar with unknown non‐uniform noise

Uniformly distributed white noise is often the desirable noise assumption for partly calibrated subarray-based radar systems. However, this assumption may practically not always be guaranteed. In this study, the angle estimation problem of the partly calibrated multi-subarray-based bistatic multiple-input multiple-output radar, where intersubarray calibration errors and non-uniform noise coexist, is considered. In addition, an inverse transformation-based enhanced Capon method is proposed to adapt to such a challenging situation. First, the noise-free covariance matrix of each subarray is obtained by formulating an imaginary selecting matrix following a covariance matrix invariance approach which performs an inverse transformation of the matrix. Thus, the noise identities at the different antennas of the subarray are estimated as nuisance parameters at the expense of the target signals. Second, assuming that the calibration within each subarray is well configured such that the steering vector of a subarray is known, knowledge is required of the intersubarray calibration mismatches. To this, a high-resolution Capon estimator which applies a combination of root and local searching orientation of the antenna arrays is implemented. The proposed approach resolves the intersubarray calibration errors and ultimately leads to obtaining the joint direction of departure and direction of arrival estimation of the target. Evidence of the proposed algorithm's validity and effectiveness over existing methods is demonstrated in numerical simulations under varying conditions.

An Estimator for Weather Radar Doppler Power Spectrum via Minimum Mean Square Error

This article proposes a method for estimating the Doppler power spectrum (DPS) of a weather radar via minimum mean square error (MMSE). In order to detect severe weather phenomena that mostly occur within the lowest few kilometers of the atmosphere, weather radars have to direct their beams at low elevation angles, and the received signals from such observations usually contain reflections from the ground and buildings, so-called ground clutter. The MMSE estimator, which is an adaptive spectral estimator, allows weather radar DPSs to be obtained with excellently reduced ground clutter contaminations. The MMSE estimator was examined by numerical simulations, which supposed various precipitation and ground clutter scenarios and DPS estimation parameter values. The MMSE estimator provided DPSs almost as accurate as those from the traditional Fourier and windowed Fourier estimators in simulations with no ground clutter and much better DPSs than those estimators in the presence of ground clutter. Furthermore, the MMSE estimator gave better suppression of ground clutter contamination than the Capon estimator, which is another adaptive spectral estimator. As a result of statistical evaluations, ground clutter signals with a strong clutter-to-noise ratio of 70 dB appeared only in a narrow velocity range of the MMSE DPS, from −2 to 2 m/s, and caused degradation of the mean and standard errors outside this velocity range by just 1 dB. The MMSE estimator was also applied to signals received by actual weather radar, and DPS estimation of precipitation signals with similarly low ground clutter contamination was demonstrated.

Parameter Estimation and Target Detection of Phased-MIMO Radar Using Capon Estimator

Phased-Multiple Input Multiple Output (PMIMO) radar is multi-antenna radar that combines the main advantages of the phased array (PA) and the MIMO radars. The advantage of the PA radar is that it has a high directional coherent gain making it suitable for detecting distant and small radar cross-section (RCS) targets. Meanwhile, the main advantage of the MIMO radar is its high waveform diversity gain which makes it suitable for detecting multiple targets. The combination of these advantages is manifested by the use of overlapping subarrays in the transmit (Tx) array to improve the performance of parameters such as angle resolution and detection accuracy at amplitude and phase proportional to the maximum number of detectable targets. This paper derives a parameter estimation formula with Capon's adaptive estimator and evaluates it for the performance of these parameters. Likewise, derivation for expressions of detection performance such as the probability of false alarm and the probability of detection is also given. The effectiveness and validation of its performance are compared to conventional estimator for other types of radars in terms of the effect of the number of target angles, the RCS of targets, and variations in the number of subarrays at Tx of this radar. Meanwhile, the detection performance is evaluated based on the effect of Signal to Noise Ratio (SNR) and the number of subarrays at Tx. The evaluation results of the estimator show that it is superior to the conventional estimator for estimating the parameters of this radar as well as the detection performance. Having no sidelobe makes this estimator strong against the influence of interference and jamming so that it is suitable and attractive for the design of radar systems. Root mean square error (RMSE) on magnitude detection from LS and Capon estimators were 0.033 and 0.062, respectively. Meanwhile, the detection performance for this radar has the probability of false alarm above 10-4 and the probability of detection of more than 99%.

The Capon Method for Mercury's Magnetic Field Analysis

Characterization of Mercury's internal and external magnetic field is one of the primary goals of the magnetometer experiment on board the BepiColombo MPO (Mercury Planetary Orbiter) spacecraft. A novel data analysis tool is developed to determine the Gauss coefficients in the multipole expansion using Capon's minimum variance projection method. The construction of the estimator is presented along with a test against the numerical simulation data of Mercury's magnetosphere and a comparison with the least square fitting method shows, that Capon's estimator is in better agreement with the coefficients, implemented in the simulation, than the least square fit estimator.

Differential SAR Tomography Reconstruction Robust to Temporal Decorrelation Effects

Temporal decorrelation is one of the major problems in synthetic aperture radar (SAR) tomography (TomoSAR) of a natural environment that leads to blurring and spreading in focused image space. In the context of spatiotemporal focusing using the multi-temporal multi-baseline (MB) SAR data, a model-based differential TomoSAR is employed. Along this and with the aim of temporal decorrelation-robust focusing, a differential tomography framework based on generalized Capon estimator is investigated. The method can cope with temporal decorrelation of the distributed environment by spatiotemporal focusing with optimal bandwidth of the distributed signal. In addition, the method employs an additional parameter for coherence channel balancing in the model of generalized Capon that benefits from it in characterizing the spatiotemporal backscattering by mitigating the inconsistency between channels. The analysis is performed with a realistic simulation of temporal decorrelation in the presence of different decorrelation sources and taking into account the dependence on the vertical structure of the forested area. Effectiveness of the proposed framework has been assessed on both simulated and real data sets by evaluation and characterization of the canopy and under foliage ground in terms of deviation between the estimated covariance matrix and one of the generalized TomoSAR models.

Estimation of Two-Dimensional Non-Symmetric Incoherently Distributed Source with L-Shape Arrays.

In the field of array signal processing, distributed sources can be regarded as an assembly of point sources within a spatial distribution. In this study, a two-dimensional (2D) non-symmetric incoherently distributed (ID) source model is proposed; we explore the estimation of a 2D non-symmetric ID source using L-shape arrays. The 2D non-symmetric ID source is established by modeling the angular power density function (APDF) as a Gaussian mixture model. Estimation of the non-symmetric distributed source is proposed based on the expectation maximization (EM) framework. The proposed EM iterative framework contains three steps in the process of each circle. Firstly, the nominal azimuth and nominal elevation of each Gaussian component are obtained from the phase parts of elements in sample covariance matrices. Then the angular spreads can be solved through a one-dimensional (1D) search by the original generalized Capon estimator. Finally, weights of each Gaussian component are obtained by solving the least-squares estimator. Simulations are conducted to verify the effectiveness of the estimation technique.

A Note on Capon's Minimum Variance Projection for Multi-Spacecraft Data Analysis

Capon’s minimum variance projection for the multi-point measurements is revisited using the method of likelihood function to derive the minimum variance projection and a simplified error estimate analytically. Theoretical construction of the minimum variance projection assumes a Gaussian form of the likelihood function and also regards the data covariance as a proxy of the noise covariance. The minimum variance projection is extended to the problem of two-spacecraft mode decomposition in the Mercury magnetosphere in which the magnetic field is a superposition of the constant field from the current sheet and the dipolar field from the planet. The extension of the Capon estimator (the data-variance projection) can identify the signal amplitudes of the different fields with a sufficient accuracy when the statistical averaging is properly done. The Capon estimator serves as a powerful analysis tool when the spatial resolution is limited to only a few points.

DOA Estimation for Underwater Target by Active Detection on Virtual Time Reversal Using a Uniform Linear Array.

Aiming at addressing the problem caused by multipath effects in direction of arrival (DOA) estimation for underwater targets, a method based on the active detection on virtual time reversal (ADVTR) Capon algorithm is proposed. Unlike the conventional passive target estimation method ignoring the multipath effects but only considering the direct wave, the proposed method is closer to the actual situation in that the multipath signal propagation model is fully taken into account; in addition, active detection (AD) and virtual time reversal (VTR) processes are added, which use active detection to estimate channels, and virtual time reversal to realize focusing in a computer after the source-receive array (SRA) receives the reflected signal of the target. The combination of the two methods can greatly improve the energy of SRA and the precision of target direction estimation. With the popular acoustic field simulation tool Bellhop, the model proposed in this paper is verified. Compared with the conventional Capon method without time reversal, the simulation results show that the ADVTR Capon estimation method is far better, in terms of resolution and suppressing the sidelobes. It is suitable for the target DOA estimation under low signal-to-noise ratio (SNR) conditions. Further, we also show the ADVTR Capon estimation method works well in a real tank experiment.

Quantitative Doppler Analysis Using Conventional Color Flow Imaging Acquisitions.

Interleaved acquisitions used in conventional triplex mode result in a tradeoff between the frame rate and the quality of velocity estimates. On the other hand, workflow becomes inefficient when the user has to switch between different modes, and measurement variability is increased. This paper investigates the use of power spectral Capon estimator in quantitative Doppler analysis using data acquired with conventional color flow imaging (CFI) schemes. To preserve the number of samples used for velocity estimation, only spatial averaging was utilized, and clutter rejection was performed after spectral estimation. The resulting velocity spectra were evaluated in terms of spectral width using a recently proposed spectral envelope estimator. The spectral envelopes were also used for Doppler index calculations using in vivo and string phantom acquisitions. In vivo results demonstrated that the Capon estimator can provide spectral estimates with sufficient quality for quantitative analysis using packet-based CFI acquisitions. The calculated Doppler indices were similar to the values calculated using spectrograms estimated on a commercial ultrasound scanner.

Computationally efficient direction of arrival estimation with unknown number of signals

In this paper, we investigate the problem of direction of arrival (DOA) estimation with unknown number of signals in the framework of beamforming. We show that the real part of the array output covariance matrix (R-AOCM) can be reformulated as an entire AOCM of a virtual array with available signal model for fast DOA estimate. By introducing an optimization problem to minimize the variance of the weighted output of this virtual array, DOA can be found by a novel real-valued real part Capon (R-Capon) estimator accordingly. Moreover, we prove that the rank of the R-AOCM is always no less than that of the entire AOCM, which suggests that R-Capon outperforms the standard Capon in scenarios with small numbers of snapshots. We also prove that the inverse of the R-AOCM can be equivalently jointed by those of two sub-matrices of about half sizes, and hence R-Capon has a significantly reduced computational complexity. These advantages as well as the theoretical analysis are finally verified by numerical simulations over a wide range of scenarios.

Improved Signal Reconstruction Algorithm for Multichannel SAR Based on the Doppler Spectrum Estimation

For high-resolution wide-swath synthetic aperture radar imaging algorithms, signal reconstruction is a key step. The steering vector plays an important role in signal reconstruction, which can be constructed by the ambiguity components. The information of the ambiguity components, e.g., number and index, is usually regarded to be constant and known. However, we find that the information of ambiguity components is always a piecewise function of the baseband frequency. This means that the steering vector cannot be preconstructed accurately and it will negatively affect the signal reconstruction. This paper presents an improved signal reconstruction method based on the Doppler spectrum estimation. The proposed method can estimate the variant ambiguity components to form the steering vector exactly by the Capon estimation. As a result, the method is able to restore the Doppler spectrum entirely and performs well on the noise reduction. Moreover, the baseband Doppler centroid and antenna pattern can be obtained in the proposed method. Simulated data and airborne raw data are processed to validate the algorithm.

Lateral resolution improvement of oversampled OCT images using Capon estimation of weighted subvolume contribution.

A novel technique for lateral resolution improvement in optical coherence tomography (OCT) is presented. The proposed method is based on lateral oversampling of the image. The locations and weights of multiple high spatial resolution sub-volumes are calculated using a Capon estimator assuming each contributes a weighted portion to the detected signal. This technique is independent of the delivery optics and the depth of field. Experimental results demonstrate that it is possible to achieve ~4x lateral resolution improvement which can be diagnostically valuable, especially in cases where the delivery optics are constrained to low numerical aperture (NA).

Localised high resolution spectral estimator for resolving superimposed peaks in NMR signals

Nuclear magnetic resonance (NMR) spectroscopy is the prime technique for studying molecular and biomolecular structure as well as dynamics. The time domain NMR signal can be ideally modelled as the sum of damped complex exponentials in additive Gaussian noise. The spectrum of the signal may contain regions having overlapping peaks. In order to understand the underlying chemical structure, these peaks need to be detected and resolved. In this paper, we propose the Localised Capon Estimator (LoCapE) for resolving closely spaced peaks in the NMR spectrum when the actual number of peaks is unknown. The novel method is able to efficiently retrieve the correct number of components and obtain high resolution spectral estimates in the selected regions of the spectrum. LoCapE is tested with simulated and actual proton NMR spectra to verify its performance.

Simultaneous Detection of Multiple Fiber-Optic Fabry–Perot Interferometry Sensors With Cepstrum-Division Multiplexing

Increasing the capacity of an array of sensors is a major challenge in fiber-optic sensing networks. Whereas fiber Bragg grating (FBG) sensors can be discriminated in the spectrum domain, through the popular wavelength-division-multiplexing (WDM) approach, Fabry–Perot interferometry (FPI) sensors have a broad spectrum and are not suitable for WDM. In this paper, a novel approach for the simultaneous detection of multiple FPI sensors is proposed. The key is decoding sensors in the optical cepstrum (spectrum of the optical spectrum) domain, using a Capon estimator. Subsequently, the small cepstrum shifts are accurately detected with a fast-phase correlation method. This approach is labeled cepstrum-division multiplexing (CDM). CDM has been tested on a simulated FPI array, with a coarse wavelength grid typical of most FBG interrogators. Unambiguous and simultaneous detection of 20–39 sensors is achieved, improving over previous methods limited to 1–3 sensors; 0.04-nm accuracy on cavity length estimation has been achieved. A preliminary experimental demonstration has been set up to validate the CDM approach.

Volume Height Estimation based on Fusion of Discrete Fourier Transform (DFT) and Least Square (LS) in a Tomographic SAR Application

Tomographic-SAR (Synthetic Aperture Radar) is a 3D Radar imaging technique, based on spectral estimation tools. This technique is used to estimate the distribution of the backscattering signal in the elevation axis, for each azimuth-range resolution cell of the SAR image. Spectral estimation algorithms belong to two families, non parametric estimation algorithms which include DFT (Discrete Fourier Transform), SVD (Single Value Decomposition), MUSIC (Multiple Signal Classification), CAPON and parametric estimation algorithms such as LS (Least Square) and ESPRIT (Estimation of signal parameters via rotation invariance techniques). In this paper we present an inversion algorithm based on the fusion of DFT and LS for the estimation of the reflectivity signal along the elevation axis. With an appropriate combination of these two algorithms and a realistic modeling of the signal distribution, we obtain a high resolution estimate of the reflectivity signal with medium computational effort. The inversion algorithm is tested on a forested area (Vasterbotten in northern Sweden), with multibaseline data set acquired in L-band (BioSAR-2008 project). Results are promising with the proposed algorithm. We used MUSIC and RVoG (Random Volume over Ground) inversions for comparison and LIDAR (Laser Imaging Detection And Ranging) image as datasets for validation of the results.

Scattering Centers Detection and Tracking in Refocused Spaceborne SAR Images for Infrastructure Monitoring

Infrastructure monitoring applications can require the tracking of slowly moving points of a certain structure. Given a certain point from a structure to be monitored, in the context of available spaceborne synthetic aperture radar (SAR) products, where the image is already focused in a slant range—azimuth grid, it is not obvious if this point is the scattering center, if it is in layover or if it is visible from the respective orbit. This paper proposes a scattering centers detection and tracking procedure based on refocusing a set of SAR images on a provided high-resolution grid of the structure. The refocusing procedure is designed for high-resolution spotlight and sliding spotlight SAR images and consists of an azimuth defocusing followed by a modified back-projection algorithm on the given set of points. The scattering centers of the refocused image are detected in the 4-D tomography framework by testing if the main response is at zero elevation in the local elevation–velocity spectral distribution obtained using the Capon estimator. The mean displacement velocity is estimated from the peak response on the zero elevation axis, whereas the displacements time series for detected single scatterers is obtained as phase difference of complex amplitudes. The algorithm is tested by simulations with an emphasis on its behavior for a low number of satellite passes and applied on real data acquired with the TerraSAR-X satellite over the Puylaurent dam. The relative displacements between scattering regions show very good agreement with in situ measurements.

Near-Field Beamforming Source Localization for Gear Transmission in a Semi-Anechoic Room

In this study, we discussed the sound sources localization and characterization for a spur gear transmission, which deals with a vibro-acoustic surveillance for predictive maintenance. The technique uses is based on the near-field Beam forming using Bartlett and Capon estimators. The measurements done on the gear transmission with a rectangular phased array of microphones were performed in acoustics and vibration laboratory (LVA) of the INSA of Lyon, in a semi-anechoic room. The gear transmission was considered sometimes open and sometimes closed. The results are presented as an image showing the acoustic field distribution radiated by the gear transmission at the most energetic frequencies. Both estimators have satisfactory results, but even better for the method with high resolution capon.

Damped CAPES 2D Spectral Estimation for Real-Valued Vibration Signals

We propose a 2D representation in the frequency-decay factor plane of an arbitrary real-world vibration signal. The signal is expressed as the sum of a decayed-attenuation sine term modulated by an amplitude function and a noise residue. We extend the combined approach of Capon estimation and amplitude and phase estimation (CAPES) to damped real vibration signals (DR-CAPES). In the proposed DR-CAPES method, the high-resolution amplitude and phase are estimated simultaneously for both angular frequency and decay factor grids. The performance of the proposed approach is tested numerically with noisy vibration data. Results show that the DR-CAPES method has an excellent frequency resolution, which helps to overcome difficulties in spectrum estimation when vibration modes are very close, and a small bias, which makes it suitable for obtaining accurate amplitude spectrums. The results also indicate that the proposed method can accurately estimate the amplitude spectrum with the use of averaging and denoising processes.

Insights Into MUSIC-Like Algorithm

In this work, we focus on a recent algorithm [Z. Ying and B. P. Ng, “MUSIC-like DOA Estaimation Without Estimating the Number of Sources,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1668-1676, 2010], which is remarked to have multiple signal classification (MUSIC)-like performance without requiring to segregate the signal and noise subspaces. The optimization problem solved by this algorithm in each look direction is analyzed to obtain insights into the working principle of the algorithm. Besides showing the similarity between this algorithm and the MUSIC algorithm, its distinction from the Capon's estimator is also highlighted. The bounds for the sole parameter embedded within the optimization problem is also discussed. Simulation results evaluate the performance of the technique in comparison with the MUSIC algorithm.