Multiplier-free DCT approximations for RF multi-beam digital aperture-array space imaging and directional sensing

Abstract

Multi-beamforming is an important requirement for broadband space imaging applications based on dense aperture arrays (AAs). Usually, the discrete Fourier transform is the transform of choice for AA electromagnetic imaging. Here, the discrete cosine transform (DCT) is proposed as an alternative, enabling the use of emerging fast algorithms that offer greatly reduced complexity in digital arithmetic circuits. We propose two novel high-speed digital architectures for recently proposed fast algorithms (Bouguezel, Ahmad and Swamy 2008 Electron. Lett. 44 1249–50) (BAS-2008) and (Cintra and Bayer 2011 IEEE Signal Process. Lett. 18 579–82) (CB-2011) that provide good approximations to the DCT at zero multiplicative complexity. Further, we propose a novel DCT approximation having zero multiplicative complexity that is shown to be better for multi-beamforming AAs when compared to BAS-2008 and CB-2011. The far-field array pattern of ideal DCT, BAS-2008, CB-2011 and proposed approximation are investigated with error analysis. Extensive hardware realizations, implementation details and performance metrics are provided for synchronous field programmable gate array (FPGA) technology from Xilinx. The resource consumption and speed metrics of BAS-2008, CB-2011 and the proposed approximation are investigated as functions of system word size. The 8-bit versions are mapped to emerging asynchronous FPGAs leading to significantly increased real-time throughput with clock rates at up to 925.6 MHz implying the fastest DCT approximations using reconfigurable logic devices in the literature.

Area and Power Efficient VLSI Architecture for Two Dimensional 16-point Modified Gate Diffusion Input Discrete Cosine Transform

“…In [9], a DCT approximation tailored for a particular radio-frequency (RF) application was obtained in accordance with an complete computational hunt. The transformation matrix C 1 =D 1 .T 1 where D 1 =1/2.diag (1/√2, 1/√3, 1/√5, 1/√3, 1/√2, 1/√3, 1/√5, 1/√3}.…”

“…The proposed digital architectures are custom designed for the real-time implementation of the approximate algorithms [5][6][7][8][9][10][11].The proposed architecture employs two parallel realizations of DCT approximation blocks, as shown in Fig. 5.…”

Multi-beam receiver apertures using multiplierless 8-point approximate DFT

“…To derive a low complexity algorithm, each f i,k approximated by elements defined over a set of small integer values, usually P = {0, ±1/2, ±1, ±2} [17], [18]. Because the DFT operates over complex numbers, the following extended set based on P is considered:…”

“…Then, employing the parametric-based optimization method described in [14], [17], we can derive an approximation for F 8 . The two constraints that were imposed on the proposed approximation are: (i) nearorthogonality and (ii) low-complexity.…”

Multiplierless 16-point DCT approximation for low-complexity image and video coding

“…Such approximations do not compute the exact DCT, but are capable of providing energy compaction [21,22] at a very low computational cost. In particular, the 8-point DCT was given a number of approximations: the signed DCT [17], the level 1 approximation [16], the Bouguezel-Ahmad-Swamy (BAS) transforms [21,[23][24][25][26], the rounded DCT (RDCT) [27], the modified RDCT [28], the approximation in [29], and the improved DCT approximation introduced in [30]. These methods furnish meaningful DCT approximations using only addition and bit-shifting operations, whilst offering sufficient computational accuracy for image and video processing [31].…”