An 8-point discrete cosine transform (DCT) fast algorithm based on the Loeffler DCT factorisation and algebraic integer (AI) representation is proposed. The proposed algorithm is an error-free implementation of the Loeffler algorithm and it is capable of computing the 8-point DCT multiplierlessly. Decoding architectures are also proposed for mapping AI encoded quantities back to usual fixed point arithmetic using canonical signed digit representation and the expansion factor method. The proposed algorithm is mapped into systolic-array digital architectures and physically realised as digital prototype circuits using field-programmable gate array technology on a Reconfigurable Open Architecture Computing Hardware board and mapped to 0.18 μm complementary metal-oxide-semiconductor technology using AMS Encounter Digital Implementation libraries at 1.8 V supply.
This paper introduced a matrix parametrization method based on the Loeffler discrete cosine transform (DCT) algorithm. As a result, a new class of 8-point DCT approximations was proposed, capable of unifying the mathematical formalism of several 8-point DCT approximations archived in the literature. Pareto-efficient DCT approximations are obtained through multicriteria optimization, where computational complexity, proximity, and coding performance are considered. Efficient approximations and their scaled 16- and 32-point versions are embedded into image and video encoders, including a JPEG-like codec and H.264/AVC and H.265/HEVC standards. Results are compared to the unmodified standard codecs. Efficient approximations are mapped and implemented on a Xilinx VLX240T FPGA and evaluated for area, speed, and power consumption.
Emerging millimeter-wave (mmW) wireless systems require beamforming and multiple-input multipleoutput (MIMO) approaches in order to mitigate path loss, obstructions, and attenuation of the communication channel. Sharp mmW beams are essential for this purpose and must support baseband bandwidths of at least 1 GHz to facilitate higher system capacity. This paper explores a baseband multibeamforming method based on the spatial Fourier transform. Approximate computing techniques are used to propose a lowcomplexity fast algorithm with sparse factorizations that neatly map to integer W/L ratios in CMOS current mirrors. The resulting approximate fast Fourier transform (FFT) can thus be efficiently realized using CMOS analog integrated circuits to generate multiple, parallel mmW beams in both transmit and receive modes. The paper proposes both 8-and 16-point approximate-FFT algorithms together with circuit theory and design information for 65-nm CMOS implementations. Postlayout simulations of the 8-point circuit in Cadence Spectre provide well-defined mmW beam shapes, a baseband bandwidth of 2.7 GHz, a power consumption of 70 mW, and a dynamic range >42.2 dB. Preliminary experimental results confirm the basic functionality of the 8-beam circuit. Schematic-level analysis of the 16-beam I/Q version show worst-case and average side lobe levels of −10.2 dB and −12.2 dB at 1 GHz bandwidth, and −9.1 dB and −11.3 dB at 1.5 GHz bandwidth. The proposed multibeam architectures have the potential to reduce circuit area and power requirements while meeting the bandwidth requirements of emerging 5G baseband systems.
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