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Editorial for coprime special issue
Digital Signal Processing 61 (2017) 1–2
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Editorial
Editorial for coprime special issue Historically speaking, the concept of coprimeness has had a place in classical signal processing starting from the late sixties, continuing on to middle seventies. An early occurrence of prime numbers and coprime pairs of integers in signal processing was in the development of fast Fourier transforms (FFT) and fast convolution using number theoretic techniques. For example, the Chinese Remainder Theorem has been used to develop fast transforms for the case where the data length can be expressed as a product of two coprime integers. In this case, the so-called twiddle-factors in the FFT algorithm could be avoided by using a very clever indexing scheme for the data and its transform. Furthermore, the residue number arithmetic system (RNS) has been employed in the early days of signal processing, to develop digital filtering algorithms that are free from roundoff and other quantization effects. In pulsed radar, the use of two pulse repetition rates, related by coprime integers, is known to increase the unambiguous target range that can be detected. Similar ideas have also been proposed for unambiguous identification of Doppler shifts due to moving targets, using Doppler filter banks. Thus the magic of coprime integers has long been well known to specialists in certain areas of signal processing. The recent upsurge of interest in coprime arrays in arrayprocessing arises for an altogether different reason. Simply put, these are sparse arrays with a large difference co-array containing a central uniform linear array (ULA). The ULA core has size O ( N 2 ) where N is the number of physical sensors. What this means is that, given a sufficient number of snapshots of array data, the covariance matrix can be estimated accurately at many more lags than the physical number of sensors. This allows one to identify many more source directions (DOAs) for uncorrelated sources than the number of physical sensors because, the number of identifiable DOAs depends only on the number of independent covariance lags that can be estimated. Coprime sampling has also been shown to be useful for time domain signals. It allows one to estimate samples of the correlation at Nyquist rate, from samples of the signal obtained at much lower sub-Nyquist rates using a pair of coprime samplers. In fact, sparse arrays with large uniform co-arrays were prevalent many decades before the advent of coprime arrays. A wellknown example is the class of minimum redundancy arrays (MRAs). However, the sensor locations in these arrays do not have simple expressions, unlike coprime arrays. For this reason, the MRA sensor locations have to be tabulated, and as the number of sensors increases, this becomes cumbersome. Furthermore, the MRA geometry cannot be easily extended to two dimensions. A natural question here is, “Are there other arrays besides coprime arrays, which have simple expressions for sensor locations while enjoying a good co-array?” Indeed, nested arrays, which were http://dx.doi.org/10.1016/j.dsp.2016.12.006 1051-2004/© 2016 Published by Elsevier Inc.
discovered before coprime arrays, offer an example, and have some advantages over coprime arrays, one of these being the fact that the entire co-array is a uniform linear array. In fact, it is the success of nested arrays that motivated the discovery of coprime arrays in the initial days of the development. Today there are many generalizations of these arrays proposed by many authors, the most recent ones being super nested arrays. These arrays enjoy all the properties of nested arrays, and at the same time have very few element pairs with minimum distance (half wavelength), thereby reducing mutual coupling effects. In short, the main properties of this new class of sparse arrays are that they have simple expressions for sensor locations, they enjoy a large co-array, and the co-array has a large ULA segment. Coprime arrays are only one of many such arrays, thus making the scope of this special issue broader than what its title suggests. All these arrays can identify many more sources than the number of physical sensors used. Furthermore, they have a larger aperture than the classical ULA (for fixed number of sensors), and therefore enjoy a lower Cramer–Rao bound (CRB) for source estimation, as well as better resolvability of closely spaced sources. In recent years, these coprime arrays and other sparse arrays have been the subject of considerable research activity worldwide. They have found applications in radar and sonar signal processing, Doppler estimation, channel identification, wideband spectrum sensing, and covariance compression. The basic ideas have also been extended to the world of sparse signal reconstruction, to significantly increase the allowed support size of an unknown sparse vector, for a fixed number of measurements. This special issue is a collection of papers, which report a number of new research directions in these areas. Here is a brief outline of the contents: The first three papers deal with nonideal situations that arise in realistic array processing scenarios. This includes mutual coupling between sensors, perturbation in sensor locations, and nonzero correlation or coherence between sources. In “Mutual coupling effect and compensation in non-uniform arrays for direction-ofarrival estimation”, BouDaher, Ahmad, Amin, and Hoorfar first evaluate the mutual coupling effects on the DOA estimation accuracy for three non-uniform array geometries, namely, the minimum redundancy array, the nested array, and the coprime array. The authors then propose methods that compensate for the mutual coupling effects to achieve accurate estimation of DOAs. In “Sparse source localization using perturbed arrays via biaffine modeling”, Koochakzadeh and Pal develop a novel selfcalibration approach for source localization using a sparse array with unknown perturbations on the sensor locations. Assuming small perturbations and a sparse grid-based DOA model, a bi-affine model with respect to perturbation and source power is derived
2
Editorial
from the covariance matrix. The redundancy in the co-array is then used to eliminate the perturbation variable and reduce the bi-affine problem into a linear sparse problem. In the paper “DOA estimation of mixed coherent and uncorrelated targets exploiting coprime MIMO radar”, the authors Qin, Zhang, and Amin, show how to handle coherent sources. For this they consider an active sensing framework (such as a MIMO radar), where the transmit and receive antennas form a pair of coprime arrays. By utilizing the sum co-array alongside the difference coarray, the paper shows that it is possible to successfully estimate the DOA of a mixture of coherent and uncorrelated sources. The next paper, “An iterative approach for sparse direction-ofarrival estimation in coprime arrays with off-grid targets”, by Sun, Wu, Sun, Ding, and Lan develops a sparsity-based DOA estimation approach for coprime arrays that accounts for targets with off-grid DOAs. Such DOA estimation problem is reformulated as a sparse recovery problem with an unknown grid offset vector. By introducing a convex function majorizing the given objective function, an iterative approach is developed to determine the offset vector and estimate the signal DOAs. The paper by Liu and Vaidyanathan, entitled “Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors”, marks the first contribution to statistical bounds on coprime arrays for a large number of sources that exceeds the number of physical sensors. The conditions for validity of the derived CRB expression are delineated and expressed in terms of the rank of an augmented coarray manifold matrix. The paper by Ramirez and Krolik, entitled “Synthetic aperture processing for passive co-prime linear sensor arrays”, introduces a novel synthetic aperture technique for estimating covariances using a moving thinned array. A thinned array can be visualized as a conventional ULA with some sensors removed, an example be-
ing a coprime array. Synthetic aperture processing allows one to fill holes in the difference co-array of the thinned array, thereby producing the effectiveness of a full ULA. The paper by Achanta et al., entitled “The spark of Fourier matrices: Connections to vanishing sums and coprimeness”, considers a certain base set of M rows selected from the N rows of the discrete Fourier Transform (DFT) matrix in a specific manner, and derives necessary and sufficient conditions under which full spark is retained for this choice of the base set. The conditions rely upon recent characterizations of vanishing sums of Nth roots of unity. Finally, in the paper “Through-the-wall radar imaging exploiting Pythagorean apertures with sparse reconstruction”, Muqaibel, Abdalla, Alkhodary, and Alawsh deal with mitigation of multipath arising from interior walls and other indoor reflectors using coprime subarray processing and utilizing aspect-dependent features. The sensing matrices associated with the proposed subarrays are developed and analyzed. We hope that the reader finds this special issue informative with the proper coverage of the theory and applications of coprime and sparse array signal processing. Finally, many of the papers in this special issue cover research findings which are sponsored by the US Office of Naval Research. The Guest Editors would like to thank the ONR Program Manager, Dr. John Tague, for his support for co-prime research over the past 4 years.
Moeness G. Amin, Palghat P. Vaidyanathan, Yimin D. Zhang, Piya Pal
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Editorial
Editorial for coprime special issue Historically speaking, the concept of coprimeness has had a place in classical signal processing starting from the late sixties, continuing on to middle seventies. An early occurrence of prime numbers and coprime pairs of integers in signal processing was in the development of fast Fourier transforms (FFT) and fast convolution using number theoretic techniques. For example, the Chinese Remainder Theorem has been used to develop fast transforms for the case where the data length can be expressed as a product of two coprime integers. In this case, the so-called twiddle-factors in the FFT algorithm could be avoided by using a very clever indexing scheme for the data and its transform. Furthermore, the residue number arithmetic system (RNS) has been employed in the early days of signal processing, to develop digital filtering algorithms that are free from roundoff and other quantization effects. In pulsed radar, the use of two pulse repetition rates, related by coprime integers, is known to increase the unambiguous target range that can be detected. Similar ideas have also been proposed for unambiguous identification of Doppler shifts due to moving targets, using Doppler filter banks. Thus the magic of coprime integers has long been well known to specialists in certain areas of signal processing. The recent upsurge of interest in coprime arrays in arrayprocessing arises for an altogether different reason. Simply put, these are sparse arrays with a large difference co-array containing a central uniform linear array (ULA). The ULA core has size O ( N 2 ) where N is the number of physical sensors. What this means is that, given a sufficient number of snapshots of array data, the covariance matrix can be estimated accurately at many more lags than the physical number of sensors. This allows one to identify many more source directions (DOAs) for uncorrelated sources than the number of physical sensors because, the number of identifiable DOAs depends only on the number of independent covariance lags that can be estimated. Coprime sampling has also been shown to be useful for time domain signals. It allows one to estimate samples of the correlation at Nyquist rate, from samples of the signal obtained at much lower sub-Nyquist rates using a pair of coprime samplers. In fact, sparse arrays with large uniform co-arrays were prevalent many decades before the advent of coprime arrays. A wellknown example is the class of minimum redundancy arrays (MRAs). However, the sensor locations in these arrays do not have simple expressions, unlike coprime arrays. For this reason, the MRA sensor locations have to be tabulated, and as the number of sensors increases, this becomes cumbersome. Furthermore, the MRA geometry cannot be easily extended to two dimensions. A natural question here is, “Are there other arrays besides coprime arrays, which have simple expressions for sensor locations while enjoying a good co-array?” Indeed, nested arrays, which were http://dx.doi.org/10.1016/j.dsp.2016.12.006 1051-2004/© 2016 Published by Elsevier Inc.
discovered before coprime arrays, offer an example, and have some advantages over coprime arrays, one of these being the fact that the entire co-array is a uniform linear array. In fact, it is the success of nested arrays that motivated the discovery of coprime arrays in the initial days of the development. Today there are many generalizations of these arrays proposed by many authors, the most recent ones being super nested arrays. These arrays enjoy all the properties of nested arrays, and at the same time have very few element pairs with minimum distance (half wavelength), thereby reducing mutual coupling effects. In short, the main properties of this new class of sparse arrays are that they have simple expressions for sensor locations, they enjoy a large co-array, and the co-array has a large ULA segment. Coprime arrays are only one of many such arrays, thus making the scope of this special issue broader than what its title suggests. All these arrays can identify many more sources than the number of physical sensors used. Furthermore, they have a larger aperture than the classical ULA (for fixed number of sensors), and therefore enjoy a lower Cramer–Rao bound (CRB) for source estimation, as well as better resolvability of closely spaced sources. In recent years, these coprime arrays and other sparse arrays have been the subject of considerable research activity worldwide. They have found applications in radar and sonar signal processing, Doppler estimation, channel identification, wideband spectrum sensing, and covariance compression. The basic ideas have also been extended to the world of sparse signal reconstruction, to significantly increase the allowed support size of an unknown sparse vector, for a fixed number of measurements. This special issue is a collection of papers, which report a number of new research directions in these areas. Here is a brief outline of the contents: The first three papers deal with nonideal situations that arise in realistic array processing scenarios. This includes mutual coupling between sensors, perturbation in sensor locations, and nonzero correlation or coherence between sources. In “Mutual coupling effect and compensation in non-uniform arrays for direction-ofarrival estimation”, BouDaher, Ahmad, Amin, and Hoorfar first evaluate the mutual coupling effects on the DOA estimation accuracy for three non-uniform array geometries, namely, the minimum redundancy array, the nested array, and the coprime array. The authors then propose methods that compensate for the mutual coupling effects to achieve accurate estimation of DOAs. In “Sparse source localization using perturbed arrays via biaffine modeling”, Koochakzadeh and Pal develop a novel selfcalibration approach for source localization using a sparse array with unknown perturbations on the sensor locations. Assuming small perturbations and a sparse grid-based DOA model, a bi-affine model with respect to perturbation and source power is derived
2
Editorial
from the covariance matrix. The redundancy in the co-array is then used to eliminate the perturbation variable and reduce the bi-affine problem into a linear sparse problem. In the paper “DOA estimation of mixed coherent and uncorrelated targets exploiting coprime MIMO radar”, the authors Qin, Zhang, and Amin, show how to handle coherent sources. For this they consider an active sensing framework (such as a MIMO radar), where the transmit and receive antennas form a pair of coprime arrays. By utilizing the sum co-array alongside the difference coarray, the paper shows that it is possible to successfully estimate the DOA of a mixture of coherent and uncorrelated sources. The next paper, “An iterative approach for sparse direction-ofarrival estimation in coprime arrays with off-grid targets”, by Sun, Wu, Sun, Ding, and Lan develops a sparsity-based DOA estimation approach for coprime arrays that accounts for targets with off-grid DOAs. Such DOA estimation problem is reformulated as a sparse recovery problem with an unknown grid offset vector. By introducing a convex function majorizing the given objective function, an iterative approach is developed to determine the offset vector and estimate the signal DOAs. The paper by Liu and Vaidyanathan, entitled “Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors”, marks the first contribution to statistical bounds on coprime arrays for a large number of sources that exceeds the number of physical sensors. The conditions for validity of the derived CRB expression are delineated and expressed in terms of the rank of an augmented coarray manifold matrix. The paper by Ramirez and Krolik, entitled “Synthetic aperture processing for passive co-prime linear sensor arrays”, introduces a novel synthetic aperture technique for estimating covariances using a moving thinned array. A thinned array can be visualized as a conventional ULA with some sensors removed, an example be-
ing a coprime array. Synthetic aperture processing allows one to fill holes in the difference co-array of the thinned array, thereby producing the effectiveness of a full ULA. The paper by Achanta et al., entitled “The spark of Fourier matrices: Connections to vanishing sums and coprimeness”, considers a certain base set of M rows selected from the N rows of the discrete Fourier Transform (DFT) matrix in a specific manner, and derives necessary and sufficient conditions under which full spark is retained for this choice of the base set. The conditions rely upon recent characterizations of vanishing sums of Nth roots of unity. Finally, in the paper “Through-the-wall radar imaging exploiting Pythagorean apertures with sparse reconstruction”, Muqaibel, Abdalla, Alkhodary, and Alawsh deal with mitigation of multipath arising from interior walls and other indoor reflectors using coprime subarray processing and utilizing aspect-dependent features. The sensing matrices associated with the proposed subarrays are developed and analyzed. We hope that the reader finds this special issue informative with the proper coverage of the theory and applications of coprime and sparse array signal processing. Finally, many of the papers in this special issue cover research findings which are sponsored by the US Office of Naval Research. The Guest Editors would like to thank the ONR Program Manager, Dr. John Tague, for his support for co-prime research over the past 4 years.
Moeness G. Amin, Palghat P. Vaidyanathan, Yimin D. Zhang, Piya Pal