The Social Digitization Workshop of the Silesian Digital Library at the Silesian Library
Silesian University of Technology as Centre of Modern...
Transcript of Silesian University of Technology as Centre of Modern...
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Course title: Arithmetic of Digital Systems (ADS)
Faculty of Automatic Control, Electronics and Computer Science,
Institute of Informatics
Field of study: Informatics
Stationary first degree studies
Silesian University of Technology as Centre of Modern Education
Based on Research and Innovations
POWR.03.05.00-IP.08-00-PZ1/17Project co-financed by the European Union under the European Social Fund
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LECTURE 1
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Introduction: Course Description
• Teaching modes and hours• Semester 1: lecture 15, classes 15
• Method of assessment: tests• References
[1] Stańczyk U., Cyran K., Pochopień B. Theory of logic circuits volume 1 Fundamental issues, Publishers of the Silesian University of Technology, Gliwice 2007[2] Pochopień B. Arytmetyka komputerowa. Akademicka Oficyna Wydawnicza EXIT, Warszawa 2012[3] Pochopień B., Stańczyk U., Wróbel E.: Arytmetyka systemów cyfrowych w teorii i praktyce. Wydanie II poprawione i uzupełnione. Wydawnictwo Politechniki Śląskiej, Gliwice 2012
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Important Practical Information
• Course instructors• PhD Eng. Urszula Stańczyk• PhD DSc Eng. Bartłomiej Zieliński
• Lecturer• PhD Eng. Urszula Stańczyk• Office hours: room 315• E-mail: [email protected]
• Database• zmitac.aei.polsl.pl• user accounts• access to courses and grades• more detailed information to follow in classes
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Course Objectives
Getting acquainted with the theory and gaining practical skills in the scope of: principles of the implementation of basic arithmetic operations and methods of arithmetic operations in fixed-point and floating-point arithmetic and their selection, evaluation and application.
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Discussed topics (1)• Number systems
• Arithmetic operations on single digits in a system with radix R
• Complements in positional number system with radix R
• Representation of numbers with sign
• Representation of numbers in digital systems
• Codes
• BCD numbers – Representation
– Complements
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Discussed topics (2)• Conversions between positional number systems
with different radixes
• Arithmetic of fixed-point numbers – Binary addition and subtraction
– Binary multiplication and division
– Addition and subtraction for BCD numbers
– Multiplication and division for BCD numbers
• Floating point arithmetic– Addition and subtraction
– Multiplication and division
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Discussed topics (3)• Fundamental arithmetic circuits
– Adder
– Subtractor
– Comparator
• Parallel and serial circuits
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Number system• A number – an abstract entity that represents
a count or measurement
• Components of a number system
– Set of arbitrarily established symbols for representing numbers
– Set of rules dictating representation of any number by these symbols
– Set of rules for performing arithmetic operations on numbers
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Number systems• Symbolic non-positional systems
– The numerical value of a digit is independent on its position within a number
– Example: Roman number system
I, II, III, IV, IX,…
• Weighted positional number systems– The numerical value of a digit is indicated by its
position, as to all specific weights are assigned
– Example: Arabic number system
1, 11, 111, 12, 212,…
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Positional number system• In a weighted positional number system
(N+M)-positional non-negative number
is represented as:
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Characteristics of a positional number
system• The maximal value of number A that can be
represented
• The minimal value of non-zero A number
• The number of all different numbers that can be represented in the system
• Absolute error of representation of A
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Positional systems with positive radix
• The most widely used systems:
– binary
– octal
– decimal
– hexadecimal
– binary-coded decimal (BCD)
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Representation of integers in systems
with various radixes
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Arithmetic operations in a number
system with radix R
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• Four basic operations:
• Addition, subtraction and multiplication on two (N+M)-positional nonnegative numbers
in a number system with radix R can be reduced to these operations on single digits
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Arithmetic operations on single digits
in a system with radix R
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Results of basic operations in binary
system
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LECTURE 2
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Conversion of numbers• Converting a number X represented in a number system
with radix R
into its equivalent form in a number system with radix Smeans finding
• Methods convenient when– R=10 and S≠10– R ≠10 and S=10
• Fraction that is finite in one system can become infinite when we change radix, then we obtain rounding off error
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Quotient-product method• Two parts of the conversion process
– digits of integer part of the number are found as numerators of fractional remainders obtained from division by S in the system with radix R
• Firstly we divide the integer part , then resulting quotients.
• The division stops when we reach the quotient equal 0.
– digits of fractional part of the number are found as carry digits shifted to the integer part when multiplying by S in the system with radix R
• Firstly we divide the fractional part , then resulting fractions.
• The multiplication stops when we reach the fraction equal 0, or when we find the required number of digits
• Most convenient when R=10 and S≠10
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Example
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Direct method• Digits of a number and radix R are
expressed by their equivalents in a number system with radix S as
• Representation of the number is found by performing operations in system with radix S
• Most convenient when R ≠10 and S=10
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Example
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For binary number find its decimal equivalent
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Tabular version of direct method
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2nd version of direct method• Repetitive multiplication by R i and R -i
computationally expensive
• Instead nested calculations can be employed
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Example
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For binary number find its decimal equivalent(�)�= 1011.101
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Differential method• Conversion by subtracting multiples of powers
of a radix – Firstly from the converted number we subtract the
highest multiple of the highest power of a radix that is not grater than the converted number
– Next we subtract from the obtained difference, and the powers gradually decrease
– The process stops when we reach zero or required accuracy
• Most convenient when R = 10 and S≠ 10
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Example
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Conversion between systems with
radix BK
• It is fairly easy to convert numbers between systems for which radixes are equal to powers of the same base
• The simplest case: powers of 2 – binary – 20
– octal – 23
– hexadecimal 24
• Conversions
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Example
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Conversion accuracy• To maintain accuracy through conversion we need
to find the required number of digits in the fractional part K
• Generally an absolute error of representation of A
is
• For systems with radixes R and S
• It is common to use
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ExampleFor a decimal number with 2 fractional digits (soK10=2) find numbers of fractional digits required to maintain accuracy for the conversions:
decimal-binary, decimal-octal, decimal-hexadecimal
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