This paper presents an efficient method for computing Maximum Likelihood (ML) direction-of-arrival (DOA) estimates in scenarios in which the sensor noise powers are nonuniform and unknown. The method combines the Alternating Projection (AP) algorithm for coarsely locating additional DOAs and Newton iterations for finally obtaining the ML estimates. Compared with the existing approaches, the method reduces the computational burden significantly due to the small number of Newton iterations required and to the efficient computation of each iteration. Specifically, the iterations are computed in a small number of arithmetic operations thanks to the closed-form formulas for the gradient and Hessian of the ML cost functions presented in this paper. The method’s total computational burden is of just a few mega-flops (mega floating-point operations) in typical cases. We present the method for the deterministic and stochastic ML estimators. Then, an analysis of the deterministic ML cost function’s gradient reveals an unexpected drawback: its associated estimator is either degenerate or inconsistent. Finally, we assess the method’s root-mean-square (RMS) error and computational burden numerically and compare it with other relevant estimators in the literature.

In this paper, we present a method to estimate the signal subspace at all the frequencies in a given band, which is computed from the usual set of frequency-bin sample covariance matrices in wideband subspace estimation. Fundamentally, the method exploits the similarity between the signal subspace at any two near-by frequencies to produce an improved subspace estimate along the band. Its key idea consists of modeling the signal subspace by means of a projection matrix function which is approximated by a polynomial. The method provides two improvements: a reduced-size representation of the signal subspace along the frequency band, and a quality improvement in wideband direction-of-arrival (DOA) estimators such as Incoherent Multiple Signal Classification (IC-MUSIC) and Modified Test of Orthogonality of Projected Subspaces (MTOPS). The paper includes the derivation of asymptotic bounds for the bias and root-mean-square (RMS) error of the projection matrix estimate, and a numerical assessment of the method and its combination with the previous two DOA estimators.

Differential SAR Interferometry (DInSAR) is a powerful remote sensing technique employed to monitor surface displacements, such as ground subsidence or strong deformations caused by geological activity. The quality of the interferometric phase between two combined SAR images is essential for the estimation of the surface deformation. Multi-pIe decorrelation factors may degrade the quality of the measurements and, then, the development of filtering methods for noise suppression is mandatory. In this work, we propose a new strategy to improve noise reduction while preserving the original phase structure. The new method consists in an iterative filter in which noise reduction is achieved progressively. The original phase is filtered with adaptive kernels based on Chebyshev interpolation functions. The filter is especially useful for DInSAR geophysical applications, such as earthquakes or volcanic eruptions monitoring. The performance of the proposed method has been tested with both simulated data and recently acquired Sentinel-1 SAR data which mapped the August 2016 Central Italy earthquake.

The so‐called non‐uniform fast Fourier transform (NFFT) is a family of algorithms for efficiently computing the Fourier transform of finite‐length signals, whenever the time or frequency grid is non‐uniformly spaced. Among the five usual NFFT types, types 4 and 5 involve an inversion problem, and this makes them the most intensive computationally. The usual efficient methods for these last types are either based on a fast multipole (FM) or on an iterative conjugate gradient (CG) method. The purpose of this study is to present efficient methods for these type‐4 and type‐5 NFFTs in the one‐dimensional case that just require three NFFTs of types 1 or 2 plus some additional fast Fourier transforms (FFTs). Fundamentally, they are based on exploiting the Lagrange formula structure. The proposed methods roughly provide a factor‐ten improvement on the FM and CG alternatives in computational burden. The study includes several numerical examples in double precision, in which the proposed and the Gaussian elimination, CG and FM methods are compared, both in terms of round‐off error and computational burden.

Phase quality is a key element in the analysis of the deformation of the Earth's surface carried out with differential synthetic aperture radar interferometry. Various decorrelation sources may degrade the surface deformation estimates, and thus, phase filters are needed for this kind of application. The well-known Goldstein filter is the most widely used due to its simple implementation and computational efficiency. In the past years, improved filters have been proposed, which are based on this filter but introduce variations in the data processing. The effectiveness of these filters mostly depends on the size of the filtering window, the weight of the smoothed spectrum, and the kernel used to filter the spectrum. In this paper, we evaluate the performance of four of these filters and present a new method that outperforms all of them. The proposed filter is based on an iterative method in which the original phase is denoised progressively with adaptive filtering windows of different sizes. The effectiveness of the filter is controlled by the interferometric coherence, a direct indicator of the phase quality. Moreover, we introduce some modifications regarding the processing of the power spectrum. Specifically, we propose to smooth the original phase using a new filter which is based on a Chebyshev interpolation scheme. The performance of the new filter has been tested on both simulated and real interferograms, acquired by RADARSAT-2 and the Uninhabited Aerial Vehicle Synthetic Aperture Radar, which mapped two different geological events that caused surface deformation.

This letter presents a method to detect and estimate multiple frequencies based on the maximum-likelihood principle. The method addresses the three main difficulties in this kind of computation, which are the detection of the number of frequencies, the coarse localization of the cost function's global maximum, and the iterative refinement of an initial estimate. Fundamentally, it consists of first detecting and estimating single frequencies or frequency clusters using the periodogram, and then refining this last estimate through a Newton-type method. This second step is fast because its complexity is independent of the number of samples, once a single fast Fourier transform (FFT) has been computed. These two steps are iteratively repeated until no mode frequency is above a fixed detection threshold. The main advantage of the proposed method is its low complexity, given that its computational burden is just that of a few FFTs in typical scenarios. The method is assessed in a numerical example.

This letter presents a method to model propagation channels for estimation, in which the sampling scheme can be arbitrary. Additionally, the method yields accurate models, with a size that converges to the channel duration, measured in Nyquist periods. It can be viewed as an improvement on the usual discretization based on regular sampling at the Nyquist rate. The method is introduced in the context of multiple delay estimation using the MUSIC estimator, and is assessed through a numerical example.

This paper presents a novel method for the analytical mean time to lose lock (MTLL) computation of coherent second-order Langevin delay-locked loops (DLLs). Analytical MTLL computation is a key task for DLLs, since the computational complexity of numerical MTLL simulations is far too high in many operating ranges of the second-order Langevin DLLs. To obtain the crucial MTLL values analytically without simulations, we rewrite the Langevin stochastic differential equation (SDE) as a vector-valued Ornstein-Uhlenbeck (OU) SDE. It includes a Gaussian noise term, which yields as a solution of the vector-valued OU SDE a time-variant Gaussian distribution. Thus, the complementary error function yields the loss of lock probability and thereby the MTLL. If we replace the complementary error functions by suitable exponential approximations, we obtain a simple MTLL expression with an exponential function as dominant term. The simple exponential MTLL expression yields the optimum loop parameters corresponding to the maximum MTLL. Simulation results confirm that the optimum loop parameters corresponding to our analytical MTLL computation method and to the simplified exponential approximation coincide. Besides the crucial analytical MTLL results, the OU random processes yield additionally the likewise crucial analytical jitter results.

This paper presents an efficient method to compute the maximum likelihood (ML) estimation of the parameters of a complex 2-D sinusoidal, with the complexity order of the FFT. The method is based on an accurate barycentric formula for interpolating band limited signals, and on the fact that the ML cost function can be viewed as a signal of this type, if the time and frequency variables are switched. The method consists in first computing the DFT of the data samples, and then locating the maximum of the cost function by means of Newton's algorithm. The fact is that the complexity of the latter step is small and independent of the data size, since it makes use of the barycentric formula for obtaining the values of the cost function and its derivatives. Thus, the total complexity order is that of the FFT. The method is validated in a numerical example.

This paper presents an efficient implementation of the variable fractional-delay (VFD) filter, which is optimal in the time-domain L-infinity norm. The proposed filter has two stages. The first consist in computing the conventional Lagrange interpolator from the signal samples, but weighted using a set of fixed coefficients. And the second consists in multiplying the result of the previous step by a smooth function, which can be well approximated by a polynomial. The paper includes a numerical evaluation of this interpolator and a low-complexity, low-latency implementation based on multiplications.

An alternative interpolation technique for SAR image coregistration in interferometric processing is formulated and described. The proposed algorithm is based on the 1- D Farrow interpolator and, when combined with an adequate implementation, it involves a smaller computational burden than the conventional method and yields high accuracy. Basically, this technique enables to carry out the coregistration without resorting to a set of functional weights, and it only requires arithmetic operations and 2-D FFTs. This paper includes several results and comparisons that confirm its validity: the proposed technique is combined with two different polynomial interpolation procedures (Taylor and Chebyshev); bounds on its interpolation error in the 1-D and 2-D cases are derived; and, additionally, it is tested numerically with a synthetic image.

An efficient polynomial interpolation method is proposed to reduce the computational burden of the coregistration of synthetic aperture radar images in interferometric processing. The method can be viewed as an application of the Farrow interpolator technique and requires a series of 2-D fast Fourier transforms (FFTs). Mainly, it has two advantages relative to the usual coregistration procedure. First, it does not require to compute or store in memory any kernel sample. Second, it involves less floating-point operations than the conventional coregistration, with a clear advantage if the image needs to be oversampled. The efficiency of the method is based on the fact that the complexity of one 2-D FFT is much smaller than the complexity of one spatial convolution. These advantages are demonstrated by both analyzing the interpolation procedure itself and by defining an efficient code implementation.

Despite their good features, Newton-type methods are not usually employed in array processing due to the lack of appropriate formulas for the first- and second-order differentials. One specific property of most array processing models is that each column of the signal matrix depends only on the corresponding element of one or more parameter vectors. In this letter, we exploit this property to derive compact expressions of the gradient, Hessian, and Hessian approximation of common maximum-likelihood (ML) cost functions, using a proper symbolic technique. Specifically, we study the conditional ML, row-correlated ML, and asymptotic ML cost functions.

The time-of-arrival estimation of several short delayed replicas of a known signal in a single antenna receiver is an ill-conditioned problem, which improves if the amplitudes are not completely static. We present a signal model that includes slow amplitude variations using a Karhunen-Loeve expansion, and the corresponding maximum likelihood estimator. The results show that a clear performance improvement in the delay estimation can be achieved by including these variations in the model.

The time-of-arrival estimation error produced by multipath interference in a navigation receiver causes a strong degradation of the positioning accuracy. We present a synchronisation technique, operative in a navigation receiver under multipath interference, that takes into account its special conditions: low signal-to-noise ratio, DS-CDMA signals with long spreading codes, and very low data rates. The paper begins by introducing the signal model of a deterministic antenna array. In this context, the system-specific features translate into the assumption that all times of arrival must be inside a delay interval in which a truncated series approximation of the reference signal is valid. The introduction of this series in the signal model leads to an implementation based on a bank of correlators. Finally, the associated minimization problem is solved using three minimization algorithms: ESPRIT, IQML and a modification of Newton's method. The latter calculates the maximum likelihood estimator with a low computational burden.