A Real/Complex Logarithmic Number System ALU

• Published in: IEEE transactions on computers 2011-02, Vol.60 (2), p.202-213
• Main Authors: Arnold, Mark G., Collange, Caroline
• Format: Article
• Language: English
• Subjects: Complex arithmetic; Cost engineering; Economics; Errors; fast Fourier transform; Field programmable gate arrays; FPGA; Hardware; hardware function evaluation; Interpolation; logarithmic number system; Memory management; Number systems; Quantization; Representations; Singularities; Synthesis; Table lookup; Tables (data); VHDL
• ISSN: 0018-9340; 1557-9956
• DOI: 10.1109/TC.2010.154

The real Logarithmic Number System (LNS) offers fast multiplication but uses more expensive addition. Cotransformation and higher order table methods allow real LNS ALUs with reasonable precision on Field-Programmable Gate Arrays (FPGAs). The Complex LNS (CLNS) is a generalization of LNS, which represents complex values in log-polar form. CLNS is a more compact representation than traditional rectangular methods, reducing bus and memory cost in the FFT; however, prior CLNS implementations were either slow CORDIC-based or expensive 2D-table-based approaches. Instead, we reuse real LNS hardware for CLNS, with specialized hardware (including a novel \log \sin that overcomes singularity problems) that is smaller than the real-valued LNS ALU to which it is attached. All units were derived from the Floating-Point-Cores (FloPoCo) library. FPGA synthesis shows our CLNS ALU is smaller than prior fast CLNS units. We also compare the accuracy of prior and proposed CLNS implementations. The most accurate of the proposed methods increases the error in radix-two FFTs by less than half a bit, and a more economical FloPoCo-based implementation increases the error by only one bit.