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MASON ECE 645 - Lecture 1: Introduction and Number Representations

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ECE 645 – Computer ArithmeticLecture 1:Introduction and Number RepresentationsECE 645—Computer Arithmetic1/22/082Lecture Roadmap• Syllabus and Course Objectives• ECE 645 CAD Tools• Computer Arithmetic: Introduction• Fixed-Point Number System Representations• Fixed-Radix Unsigned Representations• Fixed-Radix Signed Representations• Signed-Magnitude• Biased• Digit Complement (One's Complement)• Radix Complement (Two's Complement)3Required Reading• B. Parhami, "Computer Arithmetic: Algorithms and Hardware Design"• Chapter 1, Numbers and Arithmetic (entire chapter)• Chapter 2, Representing Signed Numbers (entire chapter)• Note errata at:• http://www.ece.ucsb.edu/~parhami/text_comp_arit.htm#errorsSyllabus and Course ObjectivesECE 645 – Computer Arithmetic5About the Instructor• Dr. David Hwang• PhD in Electrical Engineering, UCLA 2005• Thesis: System Architectures and VLSI Implementations of Secure Embedded Systems• Worked in industry designing VLSI signal processing algorithms and circuits• Research Interests• Secure embedded systems• Cryptographic hardware and circuits• VLSI digital signal processing• VLSI systems and circuits6Course Objectives• At the end of this course you should be able to:• Understand mathematical and gate-level algorithms for computer addition, multiplication, and division• Understand tradeoffs involved with different arithmetic architectures between performance, area, latency, etc.• Understand sources of error in computer arithmetic and error analysis• Be comfortable with different number systems, and have familiarity with Galois field and finite field arithmetic for future study• Synthesize and implement computer arithmetic blocks on FPGAs• This knowledge will come about through homework, exams, and projectsECD 645 CAD ToolsECE 645 – Computer Arithmetic8FPGA CAD Tools• This class assumes proficiency with the FPGA CAD tools from ECE 545• As a refresher, go to last semester's ECE 545 and run through the hands-on sessions• You are expected to be proficient with:• Synthesizable VHDL coding• Advanced VHDL testbenches, including file input/output• Xilinx FPGA synthesis and post-synthesis simulation• Xilinx FPGA place-and-route and post-place and route simulation• Reading and interpreting all synthesis and implementation reports9ECE 645 CAD Tool Flows"Xilinx XST 9.1 SP3"""Xilinx XST 9.1 SP3""Xilinx ISE Foundation 9.1 SP3Synplicity Synplify Pro 8.6.2Mentor Graphics Modelsim SE 6.3aXilinx ISE Foundation 9.1 SP3Xilinx ISE Foundation 9.1 SP3Synplicity Synplify Pro 8.6.2Aldec Active-HDL 7.2 SP2Aldec Active-HDL 7.2 SP2ImplementationSynthesisSimulationEnvironment• The above four design flows are all installed on the lab computers in ST2 203 and ST2 265• The two design flows using XST can also be emulated on your laptop or home computer using the techniques shown on the web site:Computer Arithmetic:IntroductionECE 645 – Computer Arithmetic11Proceedings of conferencesARITH - International Symposium on Computer ArithmeticASIL - Asilomar Conference on Signals, Systems, and Computers ICCD - International Conference on Computer DesignCHES - Workshop on Cryptographic Hardware and Embedded SystemsJournals and periodicalsIEEE Transactions on Computers,in particular special issues on computer arithmetic: 8/70, 6/73, 7/77, 4/83, 8/90, 8/92, 8/94, 7/00, 3/05. IEEE Transactions on Circuits and Systems IEEE Transactions on Very Large Scale Integration IEE Proceedings: Computer and Digital Techniques Journal of VLSI Signal Processing Computer Arithmetic Advances12What is Computer Arithmetic?• From Parhami's slidesHardware (our focus in this class) Software––––––––––––––––––––––––––––––––––––––––––––––––– ––––––––––––––––––––––––––––––––––––Design of efficient digital circuits for Numerical methods for solvingprimitive and other arithmetic operations systems of linear equations,such as +, –, ×, ÷, √, log, sin, cos partial differential equations, etc.Issues: Algorithms Issues: AlgorithmsError analysis Error analysisSpeed/cost trade-offs Computational complexityHardware implementation ProgrammingTesting, verification Testing, verificationGeneral-purpose Special-purpose–––––––––––––––––––––– –––––––––––––––––––––––Flexible data paths Tailored toFast primitive applications like:operations like Digital filtering+, –, ×, ÷, √ Image processingBenchmarking Radar tracking13• In digital arithmetic one has to come to grips with approximation and questions like:• When is approximation good enough • What margin of error is acceptable• Be aware of the applications you are designing the arithmetic circuit or program for• Analyze the implications of your approximationApproximations and Error142.....u =10 timesv = 21/1024 = 1.000 677 131= 1.000 677 131x = (((u2)2)…)2= 1.999 999 96310 timesx’ = u1024 = 1.999 999 973y = (((v2)2)…)2= 1.999 999 98310 timesy’ = v1024 = 1.999 999 994Hidden digits in the internal representation of numbersDifferent algorithms give slightly different resultsVery good accuracyCalculators15Example: Failure of Patriot Missile (1991 Feb. 25)Source http://www.math.psu.edu/dna/455.f96/disasters.htmlAmerican Patriot Missile battery in Dharan, Saudi Arabia, failed to intercept incoming Iraqi Scud missile The Scud struck an American Army barracks, killing 28Cause, per GAO/IMTEC-92-26 report: “software problem” (inaccurate calculation of the time since boot) Specifics of the problem: time in tenths of second as measured by the system’s internal clock was multiplied by 1/10 to get the time in seconds. Internal registers were 24 bits wide 1/10 = 0.0001 1001 1001 1001 1001 100 (chopped to 24 b) Error ≅ 0.1100 1100 × 2–23≅ 9.5 × 10–8Error in 100-hr operation period≅ 9.5 × 10–8× 100 × 60 × 60 × 10 = 0.34 sDistance traveled by Scud = (0.34 s) × (1676 m/s) ≅ 570 mThis put the Scud outside the Patriot’s “range gate” Ironically, the fact that the bad time calculation had been improved in some (but not all) code parts contributed to the problem, since it meant that inaccuracies did not cancel outConsequences of Bad Approximations16Example: Explosion of Ariane


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