H. Chibl`∗ and A. Ghandour∗ e
[1] G. Han & E.S. Sinencio, CMOS transconductance multipliers:A tutorial, IEEE Trans. on Circuit and Systems: Analog andDigital Signal Processing, 45(12), 1998, 1550–1563. [2] V.I. Prodanov & M.M. Green, Bipolar/CMOS (WI) rail-to-rail constant-gm input stage, Electronics Letters, 33(5), 1997,386–387. [3] N. Saxena & J.J. Clark, A four-quadrant CMOS analog multi-plier for analog neural networks, IEEE Journal of Solid StateCircuits, 29(6), 1994, 746–749. [4] A.J. Annema, Feed forward neural networks (Norwell, MA:Kluwer Academic Publishers, 1995). [5] G.M. Bo, D.D. Caviglia, H. Chibl`e, & M. Valle, A circuitarchitecture for on-chip learning, Analog Integrated Circuitsand Signals Processing, 18, 1999, 163–173. [6] H. Chibl`e, Analysis and design of analog microelectronic neuralnetwork architectures with on-chip supervised learning, Doc-toral Dissertation, University of Genoa, Genoa-Italy, 1997. [7] H. Chible, Four-quadrant multiplier for analog VLSI neuralnetworks, Lebanese Science Journal, 1 (2), 2000, 51–62. [8] M. Valle, D.D. Caviglia, & G.M. Bisio, An experimental analogVLSI neural network with on-chip back-propagation learning,Appendix A: A Summary of All Equations Discussed in This PaperExplanation of Each Equation1. It is a static quadratic function between output current and input voltage that varies in the range “V0 < Vw < Vdd”.2. It is a static quadratic function between output current and input voltage that varies in the range “0 < Vw < V0.3. It is the combination of (1) and (2). It gives a static quadratic function (positive values only) between outputcurrent and input voltage that varies in the range “0 < Vw < Vdd”.4. It is the difference between (1) and (2). It gives a static quadratic function (positive and negative values) betweenoutput current and input voltage that varies in the range “0 < Vw < Vdd”.5. It is a static square-root function with respect to Ib, which varies between 0 and Ibmax.6. It is a static linear function with respect to Ib, which varies between 0 and Ibmax.7. It is a combination of (5) and (6). It gives a static general equation in SI and WI.8. It is a linear function “SI” or square-root function “WI”. It is a static function.9. It presents a static difference linear function.10. It presents a two-quadrant multiplier that multiplies Vin by Vw, where Vin can be positive or negative and Vw is onlypositive. It can be viewed in two modes as in (11) and (12).11. It shows a wide-range linear function “SI” and wide-range quadratic function “WI”. It is a dynamic function.The parameter a1 depends on Vin, which controls the linear or quadratic function.12. It shows a linear function in SI or WI. The wideness of the linear function in SI is larger than in WI. It is a dynamicfunction. The parameter b1 depends on Vw, which controls the linear function slope.13. It can be considered four-quadrant multiplier, which multiplies Vin by Vw, where Vin and Vw can be positiveor negative.Analog Integrated Circuits and Signals Processing, 9, 1996,231–245. [9] E.A. Vittoz, Analog VLSI signal processing: Why, Where andhow? Journal of VLSI Signal Processing, 8, 1994, 27–44. [10] C.A. Mead, Analog VLSI and neural systems (Reading, Boston,MA: Addison-Wesley, 1989). [11] H. Chibl`e, Experimental results of an analog VLSI multi-plier/tranconductance circuit, Lebanese Science Journal, 4(2),2003, 73–85. [12] H. Chibl`e, Experimental results of an analog VLSI multi-plier/synapse/transconductance circuit, International Journalof Modelling and Simulation, 24(4), 2004, 224–230. [13] J. Hertz, A. Kough, & R.G. Palmer, Introduction to thetheory of neural computation (Reading, Redwood City, Calif.:Addison-Wesley, 1991). [14] R.P. Lippmann, An introduction to computing with neuralnets, IEEE ASSP Magazine, 4(2), April 1987, 4–22. [15] D.E. Rumelhart & J.L. McClelland, Parallel distributed pro-cessing (Cambridge, Massachusetts: MIT Press, 1986). [16] H. Chible, M. Valle, & D.D. Caviglia, A modified vogl-basedback propagation algorithm for VLSI hardware implementa-tion, Proc. of the fifteenth IASTED International Conf. onApplied Informatics, Austria, 1997, 83–86. [17] M. Valle, D.D. Caviglia, G. Donzellini, A. Mussi, F. Oddone,& G.M. Bisio, Neural computer based on an analog VLSIneural network, Proc. of ICANN’94, Sorrento, Italy, 2, 1994,1339–1342. [18] T.P. Vogl, J.K. Mangis, A.K. Rigler, W.T. Zink, & D.L. Alkon,Accelerating the convergence of the back-propagation method,Biological Cybernetics, V59(4), 1988, 257–263.236
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