This paper considers the sampling of temporal or spatial wide sense stationary (WSS) signals using a co-prime pair of sparse samplers. Several properties and applications of co-prime samplers are developed. First, for uniform spatial sampling with M and N sensors where M and N are co-prime with appropriate interelement spacings, the difference co-array has O(MN) freedoms which can be exploited in beamforming and in direction of arrival estimation. An M -point DFT filter bank and an N-point DFT filter bank can be used at the outputs of the two sensor arrays and their outputs combined in such a way that there are effectively MN bands (i.e., MN narrow beams with beamwidths proportional to 1/MN), a result following from co-primality. The ideas are applicable to both active and passive sensing, though the details and tradeoffs are different. Time domain sparse co-prime samplers also generate a time domain co-array with O(MN) freedoms, which can be used to estimate the autocorrelation at much finer lags than the sample spacings. This allows estimation of power spectrum of an arbitrary signal with a frequency resolution proportional to 2π/(MNT) even though the pairs of sampled sequences xc(NTn) and xc(MTn) in the time domain can be arbitrarily sparse - in fact from the sparse set of samples xc(NTn) and xc(MTn) one can estimate O(MN) frequencies in the range |ω| <; π/T. It will be shown that the co-array based method for estimating sinusoids in noise offers many advantages over methods based on the use of Chinese remainder theorem and its extensions. Examples are presented throughout to illustrate the various concepts.

Sparse Sensing With Co-Prime Samplers and Arrays

The paper focuses on continuous models derived from a discrete microstructure Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed

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Improved Continuous Models for Discrete Media

The standard technique used by commercial medical ultrasound systems to form B-mode images is delay and sum (DAS) beamforming. However, DAS often results in limited image resolution and contrast that are governed by the center frequency and the aperture size of the ultrasound transducer. A large number of elements lead to improved resolution but at the same time increase the data size and the system cost due to the receive electronics required for each element. Therefore, reducing the number of receiving channels while producing high-quality images is of great importance. In this paper, we introduce a nonlinear beamformer called COnvolutional Beamforming Algorithm (COBA), which achieves significant improvement of lateral resolution and contrast. In addition, it can be implemented efficiently using the fast Fourier transform. Based on the COBA concept, we next present two sparse beamformers with closed-form expressions for the sensor locations, which result in the same beam pattern as DAS and COBA while using far fewer array elements. Optimization of the number of elements shows that they require a minimal number of elements that are on the order of the square root of the number used by DAS. The performance of the proposed methods is tested and validated using simulated data, phantom scans, and in vivo cardiac data. The results demonstrate that COBA outperforms DAS in terms of resolution and contrast and that the suggested beamformers offer a sizable element reduction while generating images with an equivalent or improved quality in comparison with DAS.

Sparse Convolutional Beamforming for Ultrasound Imaging

Millimeter-wave (mmWave) wireless communication system allows a compact antenna array with a large number of colocated elements, which enables flexible array design and beampattern synthesis. However, security is also particularly important but challenging for the fifth-generation (5G) wireless communications, as the broadcasting nature allows any receivers to have a copy of the transmitted signals. In this paper, we propose a hybrid multiple-input multiple-output (MIMO) phased-array time-modulated directional modulation (DM) scheme for physical layer security in mmWave wireless communications. The hybrid MIMO phased-array enjoys the advantages of MIMO in spatial diversity without sacrificing the main advantage of phased-array in coherent directional gain. The essence is to divide the transmit array into multiple subarrays. Each subarray can be used to form a directional beam, while all subarrays are jointly combined to work as an MIMO for higher angular resolution capabilities. More importantly, a time-modulated DM scheme is applied for the phased-MIMO to achieve physical layer security for mmWave wireless communications, even without the knowledge of the eavesdropper’s position information. The feasibility of the proposed scheme is verified by numerical results.

Hybrid MIMO and Phased-Array Directional Modulation for Physical Layer Security in mmWave Wireless Communications

We have developed several methods of designing sparse periodic arrays based upon the polynomial factorization method. In these methods, transmit and receive aperture polynomials are selected such that their product results in a polynomial representing the desired combined transmit/receive (T/R) effective aperture function. A desired combined T/R effective aperture is simply an aperture with an appropriate width exhibiting a spectrum that corresponds to the desired two-way radiation pattern. At least one of the two aperture functions that constitute the combined T/R effective aperture function will be a sparse polynomial. A measure of sparsity of the designed array is defined in terms of the element reduction factor. We show that elements of a linear array can be reduced with varying degrees of beam mainlobe width to sidelobe reduction properties.

General polynomial factorization-based design of sparse periodic linear arrays

For arbitrary matrix dilation M whose determinant is odd or equal to ±2, we describe all symmetric interpolatory masks generating dual compactly supported wavelet systems with vanishing moments up to arbitrary order n. For each such mask, we give explicit formulas for a dual refinable mask and for wavelet masks such that the corresponding wavelet functions are real and symmetric/antisymmetric. We proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0. For matrix dilations M with |det M| = 2, we also give an explicit method for construction of masks (non-interpolatory) m0 symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n. It is proved that for some matrix dilations (in particular, for the quincunx matrix) such a mask does not have a dual mask. Some of the constructed masks were successfully applied for signal processes.

Symmetric multivariate wavelets

In ultrasound scanners, a two-way radiation pattern is generated by a transmit array and a receive array The design of such arrays is simplified by treating the problem as the design of an “effective aperture function” w eff (n) which is given by the convolution of the transmit aperture function w T (n) and the receive aperture function w R (n) In this paper we consider the design of one-dimensional sparse transmit and receive arrays with an apodized effective aperture function The design approach is based on the factorization of a polynomial representation of the effective aperture function

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On the design of one-dimensional sparse arrays with apodized end elements

This paper outlines a general method for the factorization of a polynomial with unity coefficients into a product of sparse polynomials with fewer nonzero coefficients with unity values. The factorization method is then used to design sparse antenna arrays with uniform and linearly tapered effective aperture functions.

A general method for designing sparse antenna arrays

In ultrasound scanners, a two-way radiation pattern is generated by a transmit array and a receive array. The design of such arrays is simplified by treating the problem as the design of an effective aperture function w/sub eff/(n) which is given by the convolution of the transmit aperture function w/sub T/(n) and the receive aperture function w/sub R/(n). In this paper we consider the design of one-dimensional sparse transmit and receive arrays with a uniform effective aperture function. The design approach is based on the factorization of a polynomial representation of the effective aperture function.

M. K. Tchobanou

Papers

General polynomial factorization-based design of sparse periodic linear arrays

Symmetric multivariate wavelets

On the design of one-dimensional sparse arrays with apodized end elements

A general method for designing sparse antenna arrays

A simple approach to the design of one-dimensional sparse arrays