The Toom-Cook multiplier (TCM) is the most suitable method for multiplying large integers with key widths typically employed in ECC or RSA due to its computational efficiency. In this brief, we improve the accuracy of hardware TCM algorithm, and propose a TCM-based Barrett modular multiplication algorithm. We provide detailed mathematical proofs for the determinantion of parameters, error ranges, and input-output bounds, which serve to validate the robustness and reliability of our proposed algorithm. Further, the hardware architecture and the ASIC implementation results under 40nm, 65nm, 90nm process are presented. Compared with existing works, the proposed modular multiplier exhibits superior performance in terms of Area-Time-Product.

In this paper, vertical motions of sparse linear arrays (SLAs) are utilized to generate equivalent synthetic planar arrays for two-dimensional (2D) direction-of-arrival (DOA) estimation. The proposed array geometry named vertical synthetic planar array (VSPA) consists of an arbitrary SLA on the x-axis and a series of its time-shifted arrays in the vertical orientation. With the original linear array and the moving trail known, the difference coarray of VSPA can be easily obtained. By utilizing both the synthetic aperture processing and the coarray technique, VSPA has the ability to construct a synthetic planar array with only an SLA used. Compared with the traditional sparse planar arrays (SPAs), such as 2D nested array and 2D coprime array, VSPA can achieve higher degree-of-freedom and improved 2D DOA estimation performance with the same number of sensors. Moreover, as different original linear arrays and moving trails can be chosen for different applications, the construction of VSPA is of high flexibility. Numerical simulations are presented to verify the superiority of the proposed VSPA geometry over other typical SPAs.

Maximum likelihood (ML) estimation algorithm is a special case of Bayesian estimation method in the field of direction of arrival (DOA) estimation. The ML algorithm exhibits robust capabilities with small estimation errors, making it applicable to challenging conditions, such as dealing with correlated signals and limited snapshots. However, a drawback of the ML algorithm is its multidimensional iterative nature, which results in huge computation and time consumption. To address these challenges, this article proposes a hardware implementation of the ML algorithm on a field-programmable gate array (FPGA). The design scheme presented in this article uses alternating projection (AP) to transform multidimensional operations into 1-D operations, effectively reducing the computational load. On the other hand, combined with hardware characteristics, pipeline design is used to save computing time. This design achieves a time reduction of approximately 60% in contrast to the design without the pipeline. In addition, when the number of sources is 1, the number of snapshots is 1000, the signal-to-noise ratio (SNR) is 10 dB, and the number of elements is 10, the root mean square error (RMSE) can be close to 0.02°. This exemplary performance serves as evidence that the hardware implementation in this article ensures accuracy.

Recently, the difference and sum (diff-sum) coarray has attracted much attention in one-dimensional direction-of-arrival estimation for its high degrees-of-freedom (DOFs). In this paper, we utilize both the spatial information and the temporal information to construct the diff-sum coarray for planar sparse arrays. The diff-sum coarray contains both the difference coarray and the sum coarray, which provides much higher DOFs than the difference coarray alone. We take a planar coprime array consisting of two uniform square subarrays as the array model. To fully use the aperture-extending ability of the diff-sum coarray, we propose two novel configurations to improve the planar coprime array. The first configuration compresses the inter-element spacing of one subarray and results in a larger consecutive area in the coarray. The second configuration rearranges the two subarrays and introduces a proper separation between them, which can significantly reduce the redundancy of the diff-sum coarray and increase the DOFs. Besides, we derive the closed-form expressions of the central consecutive ranges in the coarrays of the proposed array configurations. Simulations verify the superiority of the proposed array configurations.

In this paper, we propose a new strategy with increased element spacings placed on two-side subarrays and increase the physical sparse sensors array aperture. In order to ensure the continuity of difference co-array, four conditions are proposed. Based on these conditions, a new array structure named TS2-ENA is derived. Compared with other nested-like arrays with a fixed number of elements, our proposed structure possesses a higher degree of freedom, closed-form expression and outstanding performance. Simulation results show the superiority of the new configuration with the spacial smoothing MUSIC algorithm conducted.