Digital Arithmetic. Slides. Errata. Solutions

555 Pages·2016·2 MB·English

by  Ercegovac M.D.

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DIGITAL ARITHMETIC
Milo(cid:20)s D. Ercegovac
Computer Science Department
University of California Los Angeles
and
Tom(cid:19)as Lang
Department of Electrical and Computer Engineering
University of California at Irvine
Viewgraphs Copyright c 2003 by the authors
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Copyright c 2004 by Elsevier Science (USA)
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Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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ABOUT THE BOOK
1. Overview [Chapter 1]
2. Two-Operand Addition [Chapter 2]
3. Multi-operand Addition [Chapter 3]
4. Multiplication [Chapter 4]
5. Division by Digit Recurrence [Chapter 5]
6. Square Root by Digit Recurrence [Chapter 6]
7. Reciprocal, Division, Reciprocal Square RToot, and Inverse Square Root by
Iterative Approximation [Chapter 7]
8. Floating-point Representation, Algorithms, and Implementations [Chapter 8]
9. Digit-Serial Arithmetic [Chapter 9]
10. Function Evaluation [Chapter 10]
11. CORDIC Algorithm and Implementations [Chapter 11]
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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Chapter 1: Review of the Basic Number Representations and Arithmetic
Algorithms
General-purpose processors
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3
Main use: numerical computations
3
Address calculations
- basic operations
- (cid:12)xed point and (cid:13)oating point
- IEEE standard
- vector processors
Special-purpose (application-speci(cid:12)c) processors
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- for numerically intensive applications
- single computation or classes of computations
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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Application-speci(cid:12)c processor
Areas of application:
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- signal processing
- embedded systems
- matrix computations
- graphics, vision, multi-media
- cryptography and security
- robotics, instrumentation; others?
Features:
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- better use of technology
- improvement in speed, area, power
- (cid:13)exibility in
* implementation; decomposition into modules
* number systems and data formats; algorithms
Need good design tools; di(cid:14)cult to change; FPGAs?
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Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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GENERAL-PURPOSE VS. APPLICATION-SPECIFIC
Flexibility
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Matching speci(cid:12)c applications
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Use of VLSI and special technology
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Use of hardware-level parallel processing
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Lower software overhead
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Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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ARITHMETIC PROCESSORS: USER’S VIEW
AP = (operands, operation, results, conditions, singularities)
Numerical operands and results speci(cid:12)ed by
(cid:15)
{
Set of numerical values x N ((cid:12)nite and ordered set)
2
{
Range V x V
min max
(cid:20) (cid:20)
{
Precision
{
Number representation system (NRS)
Set of operations: addition, subtraction, multiplication, division, ...
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Conditions: values of the results { zero, negative, etc.
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Singularities: Illegal results { over(cid:13)ow, under(cid:13)ow, Nan, etc.
(cid:15)
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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LEVELS OF DESCRIPTION AND IMPLEMENTATION
Numerical computations (applications)
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Algorithms
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Arithmetic operations
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Operations Operands-Result Algorithm
Numerical function Numbers Numerical Algorithm
# #
Number System Arithmetic design
#
# #
Digit-vector function Digit vector Digit-vector Algorithm
# #
Digit Coding RTL design
#
# #
Bit-vector function Bit vector Bit-vector Algorithm
#
Logic Design
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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NUMBER REPRESENTATION SYSTEMS
value: x N = NRS = X V : digit vector
2 ) ) 2
Digit Vector
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X = (x ; : : : ; x ; : : : ; x )
a i b
{
indexing:
Leftward Zero Origin (LZ) (integers)
X = (x ; x ; : : : ; x )
n 1 n 2 0
(cid:0) (cid:0)
Rightward One Origin (RO) (fractions)
X = (x ; x ; : : : ; x )
1 2 n
{
Digit set D - set of values for digit x (usually consecutive integers)
i i
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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NUMBER REPRESENTATION (cont.)
Number of (unambiguously) representable numbers
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N (cid:5) D
i
j j (cid:20) j j
Number representation system
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F : N V
!
Choose NRS to
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{
allow e(cid:14)cient computation(s)
{
suitable interface with other systems
Di(cid:11)erent implementation-performance constraints
(cid:15)
= variety of NRS
)
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction
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SOME CHARACTERISTICS OF NRS
a) Range: (cid:12)nite set of digit-vector values
b) Unambiguity: two numbers should not have same representation
If x N; y N; x = y then F(x) = F(y)
2 2 6 6
c) Nonredundant/redundant
Redundant: F 1(X) = F 1(Y )
(cid:0) (cid:0)
Digital Arithmetic - Ercegovac/Lang 2003 1 { Introduction