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Behrooz Parhami's ECE 252B Course Page for Spring 2021

Digits and basic arithmetic operations

Computer Arithmetic

Page last updated on 2021 June 15

Enrollment code: 12534
Prerequisite: ECE 154A (in lieu of ECE 152A and 152B)
Class meetings: MW 12:00-1:30, asynch. recorded lectures
Instructor: Professor Behrooz Parhami
Open office hours: MW 12:00-1:00 (via Zoom)
Course forum on GauchoSpace: For messaging/discussion only
Course announcements: Listed in reverse chronological order
Course calendar: Schedule of lectures & other activities
Homework assignments: Four assignments, worth 40% in all
Exams: None for spring 2021
Research paper: Worth 60% of course grade
Research paper guidlines: Brief guide to format and contents
Poster presentation tips: Brief guide to format and structure
Policy on academic integrity: Please read very carefully
Grade stats: Grade range, mean, etc. for each activity
References: Textbook and other information sources
Lecture slides: Available on the textbook's Web page
Miscellaneous information: Motivation, catalog entry, history

Course Announcements

Megaphone 2021/06/15: The spring 2021 offering of ECE 252B is officially over and grades have been reported to the Registrar. You should have received an individual e-mail message from me with feedback on your research paper, along with grades for the paper, homework assignments, and the course. Looking forward to having some of you in my other graduate courses next year. Wishing you a relaxing summer in preparation for return to campus in fall 2021!
2021/06/04: With HW4 already submitted by all students (grading is in progress), only submission of your research paper remains for you to complete the course requirements. Do watch Lecture 19 about CORDIC, if you haven't done so already. It covers one of the most interesting ideas in computer arithmetic, a scheme used extensively in building scientific calculators that need to evaluate functions with low cost and energy. Your completed research paper will be due by W 6/09 (any time). Because I will need a few days before the grade-submission deadline of 6/16 to read and evaluate your papers, no time extension can be granted.
2021/05/22: This coming week is the last full week of classes, with only one lecture scheduled during week 10. Please watch Lectures 17 & 18 (FLP errors/precision and square-rooting) during the week ahead, to keep synchronized with our progress and the material needed for homework assignments. HW4, the last one for the course, has been posted and will be due by 12:00 noon on W 6/02. Your completed research paper will be due by W 6/09 (any time). Because I will need a few days before the grade-submission deadline of 6/16 to read and evaluate your papers, no time extension can be granted. Please do not forget to complete the instructor and course evaluation surveys. I will very much appreciate your feedback that helps me improve the course in future offerings.
2021/05/16: We have 2.5 weeks to the end of classes. Please watch Lectures 15 & 16 (FLP number representation & operations) during the week ahead, to keep synchronized with our progress and the material needed for homework assignments. HW3 will be due by noon on W 5/19. HW4, the last one for the course, will be posted by the May 22-23 weekend. Your final list of references and provisional abstract are due by M 5/17 (any time). Of course, an abstract is usually written after the completion of research. A "provisional" abstract is your view/hope at this point on what the paper will contain or accomplish. It can be changed/refined once you make further progress. A final note: Instructor and course evaluation surveys will be coming your way soon! Please complete the surveys: Your input will help me develop and improve this course for future students. Besides questions that ask for numerical ratings, you can also write comments in free form. Be constructive: Tell me what worked and what didn't.
2021/05/09: Please watch Lectures 13 & 14 during the week ahead, to finish our discussion on division and to keep synchronized with our progress and the material needed for homework assignments. HW3 has been posted. Hope you are making good progress on your research. I have already examined your preliminary reference lists and provided feedback where needed. Your next research milestone is to finalize your reference list and add a provisional abstract by the M 5/17 (any time) deadline. Of course, an abstract is usually written after the completion of research. A "provisional" abstract is your view/hope at this point on what the paper will contain or accomplish. It can be changed/refined once you make further progress.
2021/05/03: Please watch Lectures 11 & 12 (basic and high-radix division) during the week ahead, to keep synchronized with our progress and the material needed for homework assignments. HW3 will be posted by the coming weekend, at least a couple of days ahead of schedule. Hope you are making good progress on your research. Your preliminary list of references will be due on W 5/05 (any time). The submitted list should convey that you have been able to locate key resources for doing your research. You can always augment the list later, as you discover new sources, or delete some items that turn out to be less relevant or helpful.
2021/04/25: Please watch Lectures 9 & 10 (more about multipliers) during the coming week, to keep synchronized with our progress and the material needed for homework assignments. HW2 will be due by 12:00 noon on W 4/28. HW3 will be posted by M 5/10 (probably a couple of days earlier), so you will have ~ 10 days to focus on your research project, sending me your list of preliminary references by W 5/05 and getting ready for submitting your final references, along with an abstract, by M 5/17.
2021/04/18: HW2, consisting of 7 problems of various complexities, has been posted to the homework area below. All but one of the class students now have their research topic assignments finalized, recorded in the research area of this page, and communicated via e-mail. Please watch Lectures 7 & 8 over the coming week to keep synchronized with our progress and the material needed for homework assignments.
2021/04/15: I have assigned research topics to six students who submitted their topic preferences by the 4/14 due date. The remaining five students should send me their preferences no later than Friday, 4/16. Please note that the assigned topics are no loger available for selection.
2021/04/13: Hope you have settled into the spring 2021 quarter, now that we are in the middle of its third week. HW1 will be due on W 4/14, 12:00 noon. HW2 will be posted over the coming weekend. Please watch Lectures 5 & 6 during this week and send me your top 3 research-topics choices, in order of your preference, by the W 4/14 deadline (topics available for this quarter are so designated in the long list of research topics). I will begin making topic assignments before the W 4/21 target date to give you more time to do your research. So, if you miss the 4/14 deadline, it will be less likely for you to get one of your top choices. In such a case, I will ask you to submit additional choices.
2021/04/04: HW1, consisting of 7 problems of various complexities, has been posted to the homework area below. Please start working on it right away, because no time extension will be granted. Please watch Lectures 3 & 4 in the coming week and begin thinking about your research topic preferences (topics available for this quarter are so designated in the long list of research topics below). You should send me by W 4/14 your top three choices of topics, in order of your preference.
2021/03/24: Welcome to the Web page for the graduate course ECE 252B in spring 2021. Information on the spring 2020 and earlier offerings of the course is available under the "History" section at the end of this page. Throughout the current quarter, this "Course Announcements" section will alert you to significant additions or changes to this Web page. Please visit regularly.
Important special announcement about the spring 2021 offering of ECE 252B:
a. About your instructor: I have been at UCSB since 1988 (almost 33 years) and, before that, taught at other institutions for 15 years. I got my PhD at UCLA in 1973. I am looking forward to celebrating my 50th year as a professor in March 2023!
b. On coronavirus and COVID-19: We are experiencing enormous difficulties, as we adhere to social-distancing and other restrictions and worry about our own health and the health of our loved ones. With vaccinations already begun, there is light at the end of the tunnel. It appears that we may be able to return to normal instruction in fall 2021. Meanwhile, I will try to be as flexible as possible to accommodate any special needs.
c. Your photo: Those of you who don't have a photo on GauchoSpace, please consider adding one, to make it easier for us to connect. Alternatively, you can e-mail me a recent photo before the first week of classes.
d. Enrollment: Course enrollment stands at 11 as of today, but it may increase, given the enrollment/add petitions we normally receive just before start of classes.
e. Lectures: I will use recorded lectures from spring 2020, available as YouTube videos, so there will be no live questioning or discussion. I will accept your questions via e-mail (for quick questions needing brief answers) or during my office hours (for lengthier ones). Links to lecture videos are provided under Course Calendar. Please ignore any mention of dates in the various videos, which were recorded in spring 2020. For example, the first lecture instead of being on 3/30 will be on 3/29
f. Introductory video: Please watch the 11-minute introductory video, describing the course and its requirements.
g. First lecture: Please watch the 81-minute video covering Chapter 3. You are responsible for reading Chapters 1-2, which will not be covered in the lectures.
h. The remaining lectures: The remaining lectures are also available as YouTube videos, so there will be no live questioning or discussion. I will accept your questions via e-mail (for quick questions needing a brief answer) or during my Zoom office hours (for lengthier ones). Links to lecture videos are provided under Course Calendar
i. Submissions: You will submit solutions to homework assignments and various research reports as PDF attachments to e-mails. Please use a descriptive subject line to let me quickly identify that your e-mail is related to ECE 252B (I will be receiving many e-mails this quarter, given my ECE 1B freshman seminar with an enrollment of ~100).
j. Office hours: I have converted the first hour of our scheduled class time (MW 12:00-1:00 PM) to Zoom office hours specific to this course. These hours should not conflict with your other course commitments, given how courses are scheduled. So, we are using what is known as the "flipped classroom model": Lectures watched ahead of time, with questions/discussion, if any, during the scheduled lecture time. Zoom meeting details will be sent to you via GauchoSpace.

Course Calendar


Course lectures, homework assignments, and research milestones have been scheduled as follows. This schedule will be strictly observed. Please review the first two chapters in the textbook (before the first class, if possible). These chapters contain material that you should already know. PowerPoint and pdf files of course lectures, including skipped material in Chapters 1-2, can be found on the textbook's web page. [Introductory video]

Day & Date (book chapters) Lecture/discussion topic [Homework posted/due] {Special notes}
M 03/29 (ch. 3) Redundant number representation {Lecture 1 (81 min.)}
W 03/31 (ch. 4) Residue number systems {Lecture 2 (74 min.)}

M 04/05 (Ch. 5) Basic addition and counting [HW1 posted] {Lecture 3 (69 min.)}
W 04/07 (ch. 6) Carry-lookahead adders {Lecture 4 (86 min.)}

M 04/12 (ch. 7) Variations in fast adders {Lecture 5 (91 min.)}
W 04/14 (ch. 8) Multioperand addition [HW1 due] {Lecture 6 (85 min.)} {Research topic preferences}

M 04/19 (ch. 9) Basic multiplication schemes [HW2 posted] {Lecture 7 (66 min.)}
W 04/21 (ch. 10) High-radix multipliers {Lecture 8 (79 min.)} {Research topics assigned}

M 04/26 (ch. 11) Tree and array multipliers {Lecture 9 (85 min.)}
W 04/28 (ch. 12) Variations in multipliers [HW2 due] {Lecture 10 (74 min.)}

M 05/03 (ch. 13) Basic division and some speedup methods {Lecture 11 (92 min.)}
W 05/05 (ch. 14) High-radix division {Lecture 12 (84 min.)} {Preliminary references due}

M 05/10 (ch. 15) Variations in dividers [HW3 posted] {Lecture 13 (65 min.)}
W 05/12 (ch. 16) Division by convergence {Lecture 14 (79 min.)}

M 05/17 (ch. 17) Floating-point number representation {Lecture 15 (86 min.)} {Abstract & ref's due}
W 05/19 (ch. 18) Floating-point operations [HW3 due] {Lecture 16 (75 min.)}

M 05/24 (ch. 19-20) Errors, precision, and certifiability [HW4 posted] {Lecture 17 (82 min.)}
W 05/26 (ch. 21) Square-rooting methods {Lecture 18 (89 min.)}

M 05/31 (No lecture) Memorial Day observance
W 06/02 (ch. 22-23) CORDIC algorithms and function evaluation [HW4 due] {Lecture 19 (84 min.)}

W 06/09 {Research paper due by midnight}
W 06/16 {Course grades due by midnight}

Homework Assignments

Homework image

-Turn in your solutions as a PDF file attached to an e-mail sent by the due date/time.
-Because solutions will be handed out on the due date, no extension can be granted.
-Include your name, course name, and assignment number at the top of the first page.
-If homework is handwritten and scanned, make sure that the PDF is clean and legible.
-Although some cooperation is permitted, direct copying will have severe consequences

Homework 1: Number systems, addition/subtraction (ch. 1-7, due W 2021/04/14, 12:00 noon)
Do the following problems from the textbook: 1.2, 2.10, 3.2, 4.19, 5.5ckm, 6.15, 7.5

Homework 2: Multioperand addition, multiplication (ch. 8-11, due W 2021/04/28, 12:00 noon)
Do the following problems from the textbook: 8.4, 8.26, 9.2b, 9.14fg, 10.13, 11.9, 11.19

Homework 3: Variations in multiplication, division (ch. 12-16, due W 2021/05/19, 12:00 noon)
Do the following problems from the textbook: 12.13, 13.6, 13.21, 14.10, 15.6, 16.4, 16.6

Homework 4: Floating-point arithmetic, square-rooting (ch. 17-21, due W 2021/06/02, 12:00 noon)
Do the following problems from the textbook: 17.2, 17.11, 18.6bc, 18.19, 19.23, 20.25, 21.2a

Sample Exams and Study Guide [Not applicable to spring 2021]

Answer sheet

The following sample exams (from spring 2007 and 2008) are meant to indicate the types and levels of problems, rather than the coverage (which is outlined in the course calendar). Students are responsible for all sections and topics (in the textbook and class handouts) that are not explicitly excluded in the study guides that follow the sample exams, even if the material was not covered in class lectures.

Sample Midterm Exam (105 minutes)
Problem 1 [15 points] Defining concepts and terms. Define each of the following concepts/terms precisely and concisely within the space provided (about 1.5 inches per term) [3 points each]: Manchester carry chain; Multiplier recoding; ulp; Conditional-sum adder; Parallel prefix graph
Problem 2 [10 points] Number representation. Show that flipping (complementing) the sign bit of k-bit numbers in 2's-complement format results in biased representation and determine the bias amount that characterizes this new representation.
Problem 3 [20 points] Basic design concepts. Draw diagrams showing each of the following. No explanation is necessary; the diagrams should be self-explanatory.
a. How an ordinary binary adder can be augmented to perform addition or subtraction of 2's-complement numbers under the control of an add'/sub signal (0 means "add", 1 means "subtract").
b. How 2-bits-at-a-time or radix-4 sequential multiplication might be performed at high speed without Booth's recoding and without precomputing 3 times the multiplicand.
Problem 4 [15 points] Carry-skip addition.
a. Show that the optimal block width b in a fixed-block carry-skip adder is proportional to the square root of the word width k. [10 points]
b. Briefly discuss why carry-skip adders are of interest at all, given that faster logarithmic-time adders are available. [5 points]
Problem 5 [15 points] Multioperand addition. The following describes a multioperand addition process in tabular form:
0 0 8 8 8 8 8 8 8 8
0 2 6 6 6 6 6 6 6 4
0 4 4 4 4 4 4 4 3 2
1 3 3 3 3 3 3 3 2 1
2 2 2 2 2 2 2 2 1 1
a. Explain the process described by this table. [5 points]
b. In the hardware implementation implied by the table, what component types are used and how many of each? Be as precise as possible in specifying the components used. [10 points]
Problem 6 [25 points] Two's-complement multiplication.
a. Represent x = 3, y = −3, and z = 5 as 4-bit 2's-complement numbers. [5 points]
b. Using the right-shift algorithm, perform x times z, using the representations of part a, to get the 8-bit product p = 15. [10 points]
c. Using the left-shift algorithm, perform y times z, to get the 8-bit product p' = −15. [10 points]

Sample Final Exam (2.5 hours)
Do Problems 1-2, plus 5 of the remaining 6. If all optional problems are answered, the first 5 will be graded.
Problem 1 [15 points]
a. The standard 2-way carry operator has two pairs of inputs and a pair of outputs. Present a suitable generalization to an h-way operator with h pairs of inputs.
b. Name and justify one, and only one, advantage of each of the following dividers over the other two: high-radix, array, convergence.
c. Explain why square-rooting cannot be viewed as a special case of division, in the same way that squaring is a special case of multiplication.
Problem 2 [10 points] Problem 5.18c from the textbook.
Problem 3 [15 points] Problem 7.28 from the textbook.
Problem 4 [15 points] Problem 11.22 from the textbook.
Problem 5 [15 points] Problem 15.6 from the textbook.
Problem 6 [15 points] Problem 19.2 from the textbook.
Problem 7 [15 points] Problem 21.18 from the textbook.
Problem 8 [15 points] Problem 22.20abc from the textbook.

Midterm Exam Study Guide
The following textbook sections are excluded from the midterm exam: 3.4-3.6, 4.4-4.6, 6.3, 7.2, 10.5

Final Exam Study Guide
In addition to the midterm exclusions, the following textbook sections are excluded from the final exam covering Chapters 3-21: 15.4, 15.6, 19.4, 19.5, 19.6, 20.2, 21.4, 21.6

Research Paper and Presentation

Colored marbles

Each student will review a subfield of computer arithmetic or do original research on a selected and approved topic. A list of research topics is provided below ("TBD" designates topics that are not available for the current quarter); however, students should feel free to propose their own topics for approval (include a brief description and a couple of references). A publishable report earns an "A" for the course, regardless of homework grades. See the course calendar for research milestones and due dates. Consult Research Paper Guidlines for formatting tips.

Topics for Part I of the Textbook: Number Representation

01. Implementation of Arithmetic Operations in Mechanical Calculators (Assigned to: TBD)

02. The Need for, and Practicality of, Decimal Computer Arithmetic in Hardware (Assigned to: TBD)

03. Practical Implementations of Ternary Computer Arithmetic (Assigned to: TBD)

04. A Comparison of Carry-Save and Borrow-Save Number Systems and Arithmetic (Assigned to: TBD)

05. Modulo-(2a+1) Number Representations and Arithmetic (Assigned to: Available for S'21)
H. T. Vergos and C. Efstathiou, "Efficient Modulo 2n + 1 Adder Architectures," Integration, the VLSI J., Vol. 42, pp. 149-157, 2009.
G. Jaberipur and B. Parhami, "Unified Approach to the Design of Modulo-(2n±1) Adders Based on Signed-LSB Representation of Residues," Proc. 19th IEEE Int'l Symp. Computer Arithmetic, 8-10 June 2009, to appear. [Preprint available via B. Parhami's publications Web page.]

05a. Robust Residue Number Systems and Arithmetic (Assigned to: Pengfei Xu)
B. Parhami, "Digital Arithmetic in Nature: Continuous-Digit RNS," The Computer J., Vol. 58, No. 5, pp. 1214-1223, May 2015.
L. Xiao, X.-G. Xia, and H. Huo, "Toward Robustness in Residue Number Systems," IEEE Trans. Signal Processing, Vol. 65, No. 6, pp. 1497-1510, March 2017.

06. Number Representation with Discrete Logarithms (Assigned to: TBD)
A. Fit-Florea, L. Li, M. A. Thornton, and D. W. Matula, "A Discrete Logarithm Number System for Integer Arithmetic Modulo 2k: Algorithms and Lookup Structures," IEEE Trans. Computers, Vol. 58, No. 2, pp. 163-174, February 2009.

06a. Arithmetic in the Human Brain (Assigned to: TBD)
S. Cordes, C. R. Gallistel, R. Gelman, and P. Latham, P., "Nonverbal Arithmetic in Humans: Light from Noise," Perception & Psychophysics, Vol. 69, No. 7, pp. 1185-1203, 2007.
S. Dehaene, "The Neural Basis of the Weber-Fechner Law: A Logarithmic Mental Number Line," Trends in Cognitive Sciences, Vol. 7, No. 4, pp. 145-147, 2003.
A. Nieder and E. K. Miller, "Coding of Cognitive Magnitude: Compressed Scaling of Numerical Information in the Primate Prefrontal Cortex," Neuron, Vol. 37, pp. 149-157, 2003. {The mental number line seems to be logarithmic rather than linear.}
M. Piazza and S. Dehaene, "From Number Neurons to Mental Arithmetic: The Cognitive Neuroscience of Number Sense," The Cognitive Neurosciences, ed. by M. S. Gazzaniga, et al., pp. 865-77, 3rd ed., 2004.
R. Sarpeshkar, "Analog Versus Digital: Extrapolating from Electronics to Neurobiology," Neural Computation, Vol. 10, pp. 1601-1638, 1998.

Topics for Part II of the Textbook: Addition/Subtraction

07. Variable-Block Carry-Lookahead Adders (Assigned to: TBD)
V. Kantabutra, "A Recursive Carry-Lookahead/Carry-Select Hybrid Adder," IEEE Trans. Computers, Vol. 43, No. 12, pp. 1495-1499, December 1993.

08. Parallel-Prefix Ling Adders (Assigned to: TBD)
N. Burgess, "Implementation of Recursive Ling Adders in CMOS VLSI," Proc. 43rd Asilomar Conf. Signals, Systems, and Computers, November 2009, pp. 1777-1781.

09. Design of Optimal Adders with Input Timing Profile (Assigned to: TBD)

09a. Design of Approximate Adders for Speed and Energy Efficiency (Assigned to: TBD)
[Path18] R. Pathak, "A Review of Approximate Adders for Energy-Efficient Digital Signal Processing," Int'l Research J. Engineering and Technology, Vol. 5, No. 10, pp. 1895-1900, October 2018.

10. Saturating Two-Operand and Multioperand Adders (Assigned to: TBD)

11. Saturating Parallel Counters and Compressors (Assigned to: TBD)

12. Nonbinary Parallel Counters: The Ternary Example (Assigned to: Kerr Ding)
M. De and B. P. Sinha, "Fast Parallel Algorithm for Ternary Multiplication Using Multivalued I2L Technology," IEEE Trans. Computers, Vol. 43, No. 5, pp. 603-607, May 1994.

13. Implementation of Parallel Counters by Means of Sorting Networks (Assigned to: TBD)

13a. Multiplexer-Based Designs for Parallel Counters and Compressors (Assigned to: TBD)
W. Hong, R. Modugu, and M. Choi, "Efficient Online Self-Checking Modulo 2^n + 1 Multiplier Design," IEEE Trans. Computers, Vol. 60, No. 9, pp. 1354-1365, September 2011.

14. Counting Networks: Design Methods and Applications (Assigned to: TBD)

Topics for Part III of the Textbook: Multiplication

15. Trade-offs in Compensation Methods for Truncated Multipliers (Assigned to: Ziming Qi)
N. Petra, D. De Caro, V. Garofalo, E. Napoli, and A. G. M. Strollo, "Truncated Binary Multipliers with Variable Correction and Minimum Mean Square Error," IEEE Trans. Circuits and Sustems I, Vol. 57, No. 6, pp. 1312-1325, June 2010.

16. Truncated Squarers and Cubers (Assigned to: Available for S'21)
E. G. Walters, M. J. Schulte, and M. G. Arnold, "Truncated Squarers with Constant and Variable Correction," Advanced Signal Processing Algorithms, Architectures, and Implemenatations XIV (Proc. SPIE Conf. 5559), 2004, pp. 40-50.

17. Multimode Multiplication and Squaring Circuits (Assigned to: TBD)
K. E. Wires, M. J. Schulte, L. P. Marquette, and P. I. Balzola, "Combined Unsiged and Two's Complement Squarers," Proc. 33rd Asilomar Conf. Signals Systems and Computers, 1999, pp. 1215-1219.

18. Design of Cubing Circuits (Assigned to: TBD)

19. Generalized Recursive Multipliers Built of Possibly Nonsquare Modules (Assigned to: Zexi Liu)
B. Parhami, "A Theoretical Analysis of Square versus Rectangular Component Multipliers in Recursive Multiplication," Proc. 50th Asilomar Conf. Signals, Systems, and Computers, 2016, pp. 157-161.

20. Merged Arithmetic: The Case of Add-Multiply-Add Circuits (Assigned to: TBD)
E. Hakkennes and S. Vassiliadis, "Multimedia Execution Hardware Accelerator," J. VLSI Signal Processing, Vol. 28, No. 3, pp. 221-234, July 2001.

21. A Comparison of Hardware Multipliers Used in Microprocessors (Assigned to: Zahra Fahimi)
G. Colon-Bonet and P. Winterrowd, "Multiplier Evolution: A Family of Multiplier VLSI Implementations," Computer J., Vol. 51, No. 5, pp. 585-594, 2008.

22. A Survey of Multiplier Circuits in Digital Signal Processors (Assigned to: TBD)
J. M. Jou, S. R. Kuang, and R. D. Chen, "Design of Low-Error Fixed-Width Multipliers for DSP Applications," IEEE Trans. Circuits and Systems II, Vol. 46, pp. 836-842, June 1999.
S. J. Jou, M.-H. Tsai, and Y.-L. Tsao, "Low-Error Reduced-Width Booth Multipliers for DSP Applications," IEEE Trans. Circuits and Systeme I, Vol. 50, No. 11, pp. 1470-1474, November 2003.

Topics for Part IV of the Textbook: Division

23. Radix-16 SRT Division: Algorithm and Implementations (Assigned to: TBD)
[Intel's] New Radix-16 Divider

24. A Survey of the Applications of Reciprocation and Square-Rooting (Assigned to: TBD)

25. Combinational Circuits for Fast Approximate Reciprocation (Assigned to: TBD)
P.-M. Seidel, "High-Speed Redundant Reciprocal Approximation," Integration, The VLSI Journal, Vol. 28, No. 1, pp. 1-12, September 1999.

26. On-the-Fly Conversion of Redundant Quotients into Nonredundant Form (Assigned to: TBD)

27. Practical Hardware Implementation of Montgomery Modular Multiplication (Assigned to: Available for S'20)
S. Fatemi, M. Zare, A. F. Khavari, and M. Maymandi-Nejad, "Efficient Implementation of Digit-Serial Montgomery Modular Multiplier Architecture," IET Circuits, Devices & Systems, Vol. 13, No. 7, pp. 942-949, 2019.

27a. Convergence Division with Faster-than-Quadratic Convergence (Assigned: TBD)
I. Kong and E. E. Swartzlander, "A Goldschmidt Division Method with Faster than Quadratic Convergence," IEEE Trans. VLSI Systems, Vol. 19, No. 4, pp. 696-700, April 2011.

Topics for Part V of the Textbook: Real Arithmetic

28. History of Floating-Point Number Representation Formats and Associated Standards (Assigned to: TBD)

29. Level-Index Number Rerpesentation and Arithmetic (Assigned to: TBD)

30. Sign/Logarithmic Arithmetic and the European Logarithmic Microprocessor (Assigned to: TBD)
J. N. Coleman, et al., "The European Logarithmic Microprocessor," IEEE Trans. Computers, Vol. 57, No. 4, pp. 532-546, April 2008.
J. N. Coleman, E. I. Chester, C. I. Softley, and J. Kadlec, "Arithmetic on the European Logarithmic Microprocessor," IEEE Trans. Computers, Vol. 49, No. 7, pp. 702-715, July 2000.

31. Residue Logarithmic Number Representation and Arithmetic (Assigned to: TBD)

32. Accurate Summation of Sets of Floating-Point Numbers (Assigned to: TBD)
A. Eisinberg and G. Fedele, "Accurate Floating-Point Summation: A New Approach," Applied Mathematics and Computation, Vol. 189, pp. 410-424, 2007.
T. Ogita, S. M. Rump, and S. Oishi, "Accurate Sum and Dot Product," SIAM J. Scientific Computing, Vol. 26 , No. 6, pp. 1955-1988, 2005.

33. Multiprecision Arithmetic on Media Processors (Assigned to: TBD)

33a. Implementing Arithmetic Functions with Interval Arithmetic (Assigned to: Available for S'21)
U. Kulisch, "Mathematics and Speed for Interval Arithmetic: A Complement to IEEE 1788," ACM TOMS, Vol. 45, No. 1, March 2019, pp. 5:1-5:22. []

Topics for Part VI of the Textbook: Function Evaluation

34. Combinational Circuits for Fast Approximate Square-Rooting (Assigned to: TBD)

35. Cube Roots: Hardware Algorithms and Applications (Assigned to: Luyao Han)
A. Pineiro, J. D. Bruguera, F. Lamberti, and P. Montuschi, "A Radix-2 Digit-by-Digit Architecture for Cube Root," IEEE Trans. Computers, Vol. 57, No. 4, pp. 562-566, April 2008.
Cube-Roots via Newton-Raphson Method

36. Argument Reduction for Faster, More Accurate Function Evaluation (Assigned to: Available for S'21)
S. Boldo, M. Daumas, and R.-C. Li, "Formally Verified Argument Reduction with a Fused Multiply-Add," IEEE Trans. Computers I, Vol. 58, No. 8, pp. 1139-1145, August 2009.
N. Brisebarre, et al., "A New Range-Reduction Algorithm," IEEE Trans. Computers, Vol. 54, No. 3, pp. 331-339, March 2005.

37. Function Evaluation by Piecewise Linear Approximation (Assigned to: TBD)
N. Takagi, "Powering by a Table Look-Up and A Multiplication with Operand Modification," IEEE Trans. Computers, Vol. 47, No. 11, pp. 1216-1222, Nov. 1998.
O. Gustafsson and K. Johanson, "Multiplierless Piecewise Linear Approximation of Elementary Functions," Proc. 40th Asilomar Conf. Signals, Systems, and Computers, October 2006.

38. Smaller Lookup Tables by Exploiting Symmetry and Nonuniform Segmentation (Assigned to: TBD)
D.-U Lee, R. C. C. Cheung, W. Luk, and J. D. Villasenor, "Hierarchical Segmentation for Hardware Function Evaluation," IEEE Trans. VLSI Systems, Vol. 17, No. 1, pp. 103-116, January 2009.
T. Sasao, S. Nagayama, and J. T. Butler, "Numerical Function Generators Using LUT Cascades," IEEE Trans. Computers, Vol. 56, No. 6, pp. 826-838, June 2007.

Topics for Part VII of the Textbook: Implementation Topics

39. Pipelined Arithmetic in Vector Supercomputers (Assigned to: TBD)

40. Online or Digit-Pipelined Arithmetic with Carry-Save Operands (Assigned to: TBD)

40a. On-the-Fly Arithmetic Converters as Finite Automata (Assigned to: Available for S'21)
N. Pippenger, "On-the-Fly Algorithms and Sequential Machines," IEEE Trans. Computers, Vol. 60, No. 9, pp. 1372-1375, September 2011.

41. Low-Power Full-Adder Cells and Their Applications (Assigned to: TBD)
K. Navi, et al., "A Novel Low-Power Full-Adder Cell for Low Voltage," Integration, the VLSI J., Vol. 42, No. 4, pp. 457-467, September 2009.
M. Aguirre-Hernandez and M. Linares-Aranda, "CMOS Full-Adders for Energy-Efficient Arithmetic Applications," IEEE Trans. VLSI Systems, Vol. 19, No. 4, pp. 718-721, April 2011.

42. Low-Power Design Techniques for Multipliers (Assigned to: TBD)
I. S. Abu-Khater, A. Bellaouar, and M. I. Elmasry, "Circuit Techniques for CMOS Low-Power High-Performance Multipliers," IEEE J. Solid-State Circuits, Vol. 31, pp. 1535-1546, October 1996.
S. S. Mahant-Shetti, P. T. Balsara, and C. Lemonds, "High Performance Low Power Array Multiplier Using Temporal Tiling," IEEE Trans. VLSI Systems, Vol. 7, No. 1, pp. 121-124, March 1999.

43. Dedicated Hardware Multipliers on FPGA Chips (Assigned to: TBD)
Using Embedded Multipliers in Spartan-3 FPGAs

44. Design Methods for Implementing Wider Multipliers Using Embedded FPGA Multipliers (Assigned to: TBD)
S. Gao, D. Al-Khalili, and N. Chabini, "Efficient Realization of Large Size Two's Complement Multipliers Using Embedded Blocks in FPGAs," Circuits, Systems, and Signal Processing, Vol. 27, No. 5, pp. 713-731, October 2008.
J.-L. Beuchat and A. Tisserand, "Small Multiplier Based Multiplication and Division Operators for Virtex-II Devices," Proc. 12th Int'l Conf. Field-Programmable Logic and Applications, 2002, pp. 513-522.

45. Augmenting FPGAs for Faster Arithmetic Operations (Assigned to: TBD)
H. Parandeh-Afshar, A. K. Verma, P. Brisk, and P. Ienne, "Improving FPGA Performance for Carry-Save Arithmetic," IEEE Trans. VLSI Systems, Vol. 18, No. 4, pp. 578-590, April 2010.

General Research Topics Spanning Multiple Parts

46. A Survey of Arithmetic Circuits in Electronic Scientific Calculators (Assigned to: Jackie Burd)
G. R. Rising, Inside Your Calculator: From Simple Programs to Significant Insights, Wiley, 2007.
H.-J. Bohm, "Small-Data Computing: Correct Calculator Arithmetic," Communications of the ACM, Vol. 60, No. 8, pp. 44-49, August 2017.

47. Arithmetic in Early Supercomputers: IBM System/360 Model 91 and CDC 6600 (Assigned to: TBD)

48. Arithmetic and Energy Economy Provisions in IBM Blue Gene/L Parallel Supercomputer (Assigned to: TBD)
J. Lorenz, S. Kral, F. Franchetti, and C.W. Ueberhuber, "Vectorization Techniques for the Blue Gene/L Double FPU," IBM J. Research and Development, Vol. 49, Nos. 2/3, pp. 437-446, March/May 2005.
S. Chatterjee, et al., "Design and Exploitation of a High-Performance SIMD Floating-Point Unit for Blue Gene/L," IBM J. Research and Development, Vol. 49, Nos. 2/3, pp. 377-391, March/May 2005.

49. Implementation of Arithmetic Operations in Graphics Processors (Assigned to: Justin Hemphill)
D. De Caro, N. Petra, and A. G. M. Strollo, "High-Performance Special Function Unit for Programmable 3-D Graphic Processors," IEEE Trans. Circuits and Systems I, Vol. 56, No. 9, pp. 1968-1978, September 2009.
D. Blythe, "Rise of the Graphics Processor," Proc. IEEE, Vol. 96, No. 5, pp. 761-778, May 2008.

49a. Arithmetic in a Near-Pixel SIMD Architecture for 3D Stacked Chips (Assigned to: TBD)
B. Pfundt, M. Reichenbach, C. Soll, and D. Fey, "Novel Image Processing Architecture for 3D Integrated Circuits," PARS-Mitteilungen, Vol. 32, No. 1, 2015.
S. Chevobbe, M. Lepecq, K. Benchehida, M. Darouich, T. Dombek, F. Guellec, and L. Millet, "A Versatile 3D Stacked Vision Chip with Massively Parallel Processing Enabling Low Latency Image Analysis," Proc. Int'l Image Sensor Workshop, June 2019.

49b. Implementation of Arithmetic Operations in Tensor Processing Units (Assigned to: Available for S'21)
N. Jouppi, C. Young, N. Patil, and D. Patterson, D., "Motivation for and Evaluation of the First Tensor Processing Unit, IEEE Micro, Vol. 38, No. 3, 2018, pp. 10-19.

50. Number Crunching for Computer Games: History and Techniques (Assigned to: TBD)

51. Arithmetic Operations for Elliptic Curve Cryptography (Assigned to: TBD)
C. K. Koc (ed.), Cryptographic Engineering, Springer, 2009.
S. Kumar, Elliptic Curve Cryptography for Constrained Devices, VDM Verlag, 2008.
J. Solinas, "Generalized Mersenne numbers," Tech. Report CORR 99-39, Dept. C&O, U. Waterloo, 1999.

52. Implementation of Ultrahigh-Precision Arithmetic on Parallel Computers (Assigned to: TBD)
D. Takahashi, "Parallel Implementation of Multiple-Precision Arithmetic and 2,576,980,370,000 Decimal Digits of Pi Calculation," Parallel Computing, Vol. 36, No. 8, pp. 439-448, August 2010.

53. A Comparison of Synchronous and Asynchronous Arithmetic Circuits (Assigned to: TBD)

54. Implementing Arithmetic Operations with Neuronlike Hardware Elements (Assigned to: TBD)

54a. Number Storage and Arithmetic with DNA (Assigned to: Min Jian Yang)
A. K. George, I. O. Kunnummal, L. Alazzawi, and H. Singh, "Design of DNA Digital Circuits," IEEE Potentials, March/April 2020, pp. 35-40.

55. Computer Arithmetic with Emerging Technologies (Assigned to: TBD)
G. Bourianoff, "The Future of Nanocomputing," IEEE Computer, Vol. 36, No. 8, pp. 44-53, August 2003.
Y. Brun, "Arithmetic Computation in the Tile Assembly Model: Addition and Multiplication," Theoretical Computer Science, Vol. 378, No. 1, pp. 17-31, June 2007.
G. Jaberipur, B. Parhami, and D. Abedi, "Adapting Computer Arithmetic Structures to Sustainable Supercomputing in Low-Power, Majority-Logic Nanotechnologies," IEEE Trans. Sustainable Computing, Vol. 3, No. 4, pp. 262-273, October-December 2018.

Poster Presentation Tips

Poster format

Here are some guidelines for preparing your research poster. The idea of the poster is to present your research results and conclusions thus far, get oral feedback during the session from the instructor and your peers, and to provide the instructor with something to comment on before your final report is due. Please send a PDF copy of the poster via e-mail by midnight on the poster presentation day.

Posters prepared for conferences must be colorful and eye-catching, as they are typically competing with dozens of other posters for the attendees' attention. Here is an example of a conference poster. Such posters are often mounted on a colored cardboard base, even if the pages themselves are standard PowerPoint slides. In our case, you should aim for a "plain" poster (loose sheets, to be taped to the wall in our classroom) that conveys your message in a simple and direct way. Eight to 10 pages, each resembling a PowerPoint slide, would be an appropriate goal. You can organize the pages into 2 x 4 (2 columns, 4 rows), 2 x 5, or 3 x 3 array on the wall. The top two of these might contain the project title, your name, course name and number, and a very short (50-word) abstract. The final two can perhaps contain your conclusions and directions for further work (including work that does not appear in the poster, but will be included in your research report). The rest will contain brief description of ideas, with emphasis on diagrams, graphs, tables, and the like, rather than text which is very difficult to absorb for a visitor in a very limited time span.

Grade Statistics


Grades are in [F, A+], unless otherwise noted..
HW1 grades: Range = [B, A+], Mean = 3.58, Median = B+/A–
HW2 grades: Range = [B, A+], Mean = 3.41, Median = B+
HW3 grades: Range = [B, A+], Mean = 3.36, Median = B+
HW4 grades: Range = [B–, A+], Mean = 3.47, Median = B+
Research paper grade (percent): Range = [55, 90], Mean = 72, Median = 70
Course letter grades: Range = [B, A+], Mean = 3.40, Median = B+


Image of a reference book

Primary textbook (required):
Parhami, Computer Arithmetic: Algorithms and Hardware Designs, Oxford, 2nd ed., 2010.

Verilog descriptions of arithmetic circuits (recommended):
This course does not involve a lab component or implementation projects. For those interested in pursuing practical circuit implementations, the following book may be useful:
Cavanagh, Computer Arithmetic and Verilog HDL Fundamentals, CRC Press, 2010.

Other useful books (not required):
Brent/Zimmermann, Modern Computer Arithmetic, Cambridge, 2011
Deschamps/Bioul/Sutter, Synthesis of Arithmetic Circuits: ... , Wiley, 2006 (TK7895.A65D47)
Ercegovac/Lang, Digital Arithmetic, Morgan Kaufmann, 2004 (QA76.9.C62E73)
Ercegovac/Lang, Division and Square Root: ... , Kluwer, 1994 (QA76.9.C62E73)
Knuth, The Art of Computer Programming: Seminumerical Algorithms, Wiley, 1981 (QA76.6.K64 vol 2)
Kulisch/Miranker, Computer Arithmetic in Theory and Practice, Academic Press, 1981 (QA162.K84)
Koren, Computer Arithmetic Algorithms, 2nd ed., A K Peters, 2002 (QA76.9.C62K67)
Muller, Elementary Functions: Algorithms and Implementation, Birkhauser, 2006 (QA331.M866)
Muller et al., Handobook of Floating-Point Arithmetic, Birkhauser, 2010
Oklobdzija, High-Performance System Design, IEEE Press, 1999 (TK7871.99.M44037)
Omondi, Computer Arithmetic Systems: ... , Prentice-Hall, 1994 (QA76.9.C62O46)
Omondi, Computer-Hardware Evaluation of Mathematical Functions, Imperial College Press, 2015
Swartzlander, Computer Arithmetic, vols. 1-2, IEEE Computer Society Press, 1990 (QA76.6.C633)
Waser/Flynn, Intro. Arithmetic for Digital Systems Designers, HR&W, 1982 (TK7895.A65W37.1982)

Research resources:
Proc. IEEE Symp. Computer Arithmetic, 1969, 72, 75 78, 81 and subsequent odd years; beginning with ARITH-23 in July 2016, the conference became an annual event.
On-line proceedings for IEEE Symp. Computer Arithmetic, 1969-2019
IEEE Trans. Computers, particularly special issues/sections on computer arithmetic: 8/1970, 6/1973, 7/1977, 4/1983, 8/1990, 8/1992, 8/1994, 7/1998, 7/2000, 3/2005, 2/2009, 2/2011, 8/2012, 8/2014, 12/2017, 7/2019
UCSB library's electronic journals, collections, and other resources

Miscellaneous Information

Motivation: Computer arithmetic is a subfield of digital computer organization. It deals with the hardware realization of arithmetic functions to support various computer architectures as well as with arithmetic algorithms for firmware/software implementation. A major thrust of digital computer arithmetic is the design of hardware algorithms and circuits to enhance the speed of various numeric operations. Thus much of what is presented in this course complements the architectural and algorithmic speedup techniques covered as part of the advanced computer architecture (ECE 254A/B/C) sequence.

Catalog entry: 252B. Computer Arithmetic. (4) PARHAMI. Prerequisites: ECE 152A-B (Changing to ECE 154A). Lecture, 4 hours. Standard and unconventional number representations. Design of fast two-operand and multioperand adders. High-speed multiplication and division algorithms. Floating-point numbers, algorithms, and errors. Hardware algorithms for function evaluation. Pipelined, digit-serial and fault-tolerant arithmetic processors.

History: Professor Parhami took over the teaching of ECE 252B from the late Dr. James Howard in the winter quarter of 1989. By covering sequential machines, computer arithmetic, and advanced microprocessor-based design, the graduate course sequence ECE 252A/B/C was meant to provide a firm foundation in the theories and techniques of advanced digital design. During the first few offerings of ECE 252B, Professor Parhami gradually modified the content to increase both its coverage and research orientation (the now-discontinued ECE 252A & 252C underwent similar transformations by Professor Kwang-Ting Cheng and Professor Parhami, respectively). In 2000, based on a decade of experience in teaching computer arithmetic, Professor Parhami published a reference volume and graduate-level textbook, Computer Arithmetic: Algorithms and Hardware Designs (Oxford Univ. Press), which is being used at many universities worldwide. The 2nd edition of this textbook appeared in 2010.
Offering of ECE 252B in spring 2021
Offering of ECE 252B in spring 2020
Offering of ECE 252B in spring 2019
Offering of ECE 252B in spring 2018
Offering of ECE 252B in spring 2017
Offering of ECE 252B in spring 2016 (PDF file)
Offering of ECE 252B in spring 2015 (PDF file)
Offering of ECE 252B in spring 2014 (PDF file)
Offering of ECE 252B in spring 2013 (PDF file)
Offering of ECE 252B in spring 2012 (PDF file)
Offering of ECE 252B in spring 2011 (PDF file)
Offering of ECE 252B in spring 2010 (PDF file)
Offering of ECE 252B in spring 2009 (PDF file)
Offerings of ECE 252B from 2000 to 2008 (PDF file)