{"title":"On Finite Wordlength Properties of Block-Floating-Point Arithmetic","authors":"Abhijit Mitra","volume":20,"journal":"International Journal of Electronics and Communication Engineering","pagesStart":1709,"pagesEnd":1715,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/8380","abstract":"A special case of floating point data representation is block\r\nfloating point format where a block of operands are forced to have a joint\r\nexponent term. This paper deals with the finite wordlength properties of\r\nthis data format. The theoretical errors associated with the error model for\r\nblock floating point quantization process is investigated with the help of error\r\ndistribution functions. A fast and easy approximation formula for calculating\r\nsignal-to-noise ratio in quantization to block floating point format is derived.\r\nThis representation is found to be a useful compromise between fixed point\r\nand floating point format due to its acceptable numerical error properties over\r\na wide dynamic range.","references":"[1] K. R. Ralev and P. H. Bauer, \"Realization of Block Floating Point\r\nDigital Filters and Application to Block Implementations,\" IEEE Trans.\r\nSignal Processing, vol. 47, no. 4, pp. 1076-1086, April 1999.\r\n[2] K. Kallioja\u252c\u00bfrvi and J. Astola, \"Roundoff Errors in Block-Floating-Point\r\nSystems,\" IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 783-790,\r\nApril 1996.\r\n[3] J. Kontro, K. Kallioja\u252c\u00bfrvi and Y. Neuvo, \"Floating-point arithmetic in\r\nsignal processing,\" in Proc. 1992 IEEE Int. Symp. Circuits, Syst., San\r\nDiego, CA, May 10-13, 1992, pp. 1784-1791.\r\n[4] S. Sridharan and G. Dickman, \"Block floating point implementation of\r\ndigital filters using the DSP56000,\" Microprocess. Microsyst., vol. 12,\r\nno. 6, pp. 299-308, July-Aug. 1988.\r\n[5] P. H. Bauer, \"Absolute Error Bounds for Block-Floating-Point Direct-\r\nForm Digital Filters,\" IEEE Trans. Signal Processing, vol. 43, no. 8,\r\npp. 1994-1996, Aug. 1995.\r\n[6] A. V. Oppenheim, \"Realization of digital filters using block floating\r\npoint arithmetic,\" IEEE Trans. Audio Electroaccoust., vol. AE-18, no.\r\n2, pp. 130-136, June 1970.\r\n[7] K. Kallioja\u252c\u00bfrvi, \"Analysis of Block-Floating-Point Quantization Error,\"\r\nin Proc. 11th Euro. Conf. Circuit Theo., Design, Davos, Switzerland,\r\nAug. 30- Sep. 3, 1993, pp. 791-796.\r\n[8] A. Mitra, \"A New Block-based NLMS Algorithm and Its Realization\r\nin Block Floating Point Format,\" Int. J. Info. Tech., vol. 1, no. 4, pp.\r\n244-248, 2004.\r\n[9] A. Mitra, \"Efficient Realization of Gradient Based Adaptive Filters using\r\nBlock Floating Point Arithmetic,\" Ph.D. Dissertation, Indian Institute\r\nof Technology Kharagpur, India, Jan. 2004.\r\n[10] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing,\r\nEnglewood Cliffs, NJ: Prentice-Hall, 1989.\r\n[11] A. Fettweis, \"On Properties of Floating-Point Roundoff Noise,\" IEEE\r\nTrans. Accoust. Speech Signal Processing, pp. 149-151, April 1974.\r\n[12] A. Papoulis, Probability, Random Variables and Stochastic Processes,\r\nNew York: McGraw-Hill, 1965.\r\n[13] T. Kaneko and B. Liu, \"On local roundoff errors in floating-point arithmetic,\"\r\nJournal Ass. Comp. Mach., vol. 20, pp. 391-398, July 1973.\r\n[14] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing, Englewood\r\nCliffs, NJ: Prentice-Hall, 1975.\r\n[15] B. Liu, \"Effect of finite wordlength on the accuracy of digital filters-\r\nA review,\" IEEE Trans. Circuit Theory, vol. CT-18, pp. 670-677, Nov \r\n1971.\r\n[16] J. H. Wilkinson, Rounding Errors in Algebraic Processes, Englewood\r\nCliffs, NJ: Prentice-Hall, 1963.\r\n[17] A. B. Sripad and D. L. Snyder, \"Quantization Errors in Floating-\r\nPoint Arithmetic,\" IEEE Trans, Accoust, Speech Signal Processing, vol.\r\nASSP-26,pp. 149-151, Oct 1978.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 20, 2008"}