FYSE410 DIGITAL ELECTRONICS [1] [2] [3] [4] [5] A number system consists of an ordered set of symbols (digits).

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1 FYSE4 DIGITAL ELECTRONICS Litterature: LECTURE [] [] [4] [5] DIGITAL LOGIC CIRCUIT ANALYSIS & DESIGN Victor P. Nelson, H. Troy Nagle J. David Irwin, ill D. Carroll ISN DIGITAL DESIGN M. Morris Mano Michael D. Ciletti ISN Introduction ToDigital Techniques Dan I. Porat Arpad arna ISN Digital Logic Design A. P. Godse D. A. Godse Digital Design Principles and Practices John F. Wakerly ISN Number Systems Positional Notation N ( an an aa. a a a m ) r n integer digits m fractional digits A number system consists of an ordered set of symbols (digits). The total number of digits allowed in the number system is called base or radix (r) an a m Most significant digit Least significant digit Addition + Subraction Multiplication Division

2 Number Systems Polynomial Notation N (.5 ) N n i m a i r i For desimal number : r anda i {,,,,4,5,67,,,9 } Number Systems Commonly Used Number Systems Desimal number : r andai {,,,,4,5,67,,,9 } inary number : r andai {,} Octal number : r anda i {,,,,4,5,67, } Hexadecimal number : r 6anda i {,,,,4,5,67,,,9, A,,C,D,E, F} 4

3 Number Systems Commonly Used Number Systems Decimal inary Octal Hexadecimal A C D E F First seventeen positive integers yte bit MS bits Kb MS Word yte 6bit 4Kb 4 K bits 7 One bit it or yte LS Word For example the size of memory : the number of one bit memory locations. LS 5 inary Addition + inary addition table Carry Example ( ) ( ) + + ( + ) + ( ) + ( ) ( ) + Carry Augend Addend Sum 6

4 inary Addition The computation is easily performed by adding the numbers in pairs. Example ( ) + ( ) + ( ) + ( ) Sum 7 inary Addition We can also perform the addition directly to avoid the intermediate steps. Example ( ) + ( ) + ( ) + ( ) ( ) + ( + ) + ( + ) ( ) + ( ) ( ) + ( ) + ( + ) + ( + ) ( ) + ( ) ( ) + ( ) + ( + ) + ( + ) ( ) + ( ) ( ) ( + ) + ( + ) ( ) + ( ) + ( ) + ( ) ( + ) + ( + ) ( ) + ( ) ( )

5 inary Subtraction Example ( ) ( ) Column withaborrowofor orrows orrows Minuend Subtrahend Difference 9 inary Example ( ) ( ) + Multiplication x inary Multiplication table ( ) x x We don't need to list an all-zero partial product for a multiplier bit of.

6 inary Division Example : Divide ( ) 65 by ( ) Example : Divide ( ) 9 by ( ) 9 5 Divisor Quotient Dividend Reminder Octal Addition 46 Example ( 46 ) + ( 75) Octal addition table Carries Augend Addend Sum

7 Octal Subtraction Example ( 64 ) ( 57) orrows Minuend Subtrahend Difference Octal Multiplication x Multiplicand Multiplier Partial products Product Example ( 467 ) ( 5 ) Octal multiplication table

8 Octal Division Example : Divide ( 46 ) 6 by ( 5 ) 47 Quotient 46 Dividend Reminder Hexadecimal Addition A D Example ( ) ( ) 6 A5 +7 D 9C Carries Augend Addend Sum Hexadecimal addition table 6

9 Hexadecimal Subtraction 9 F + 6 4A6 Example ( ) ( ) 6 E 9F - 4A6 54 E5 orrows Minuend Subtrahend Difference 7 Hexadecimal Multiplication 5 CA 6 7D Example ( ) ( ) 6 + 5C A x7 D 5C E 4A A 56 F96C Multiplicand Multiplier Partial products Product Hexadecimal Multiplication table

10 Hexadecimal Division Example : Divide ( 7 FCA) 6 by ( E) 6 A5 7 FCA 6C C 6 6 A E C E 6 6 Quotient Dividend Reminder 9 ase Conversion Conversion : NA N A < k We have numbers N A and N Assumption: A, k {,,,... } Group the digits of N in groups of k digits in both directions from the radix point and then replace each group with the equivalent digit in base. Examples : inary to Octal A and A

11 ase Conversion Conversion : NA N A < k Assumption: A, k {,,,... } Replace each base digit in N with equivalent k digits in base A. Example : Hexadecimal to inary 4 4 A and 6 A k 4 AF.6C6. Replace each base 6 digit in N 6 with equivalent 4 digits in base. Example : Octal to inary A and A k Replace each base digit in N with equivalent digits in base. ase Conversion Conversion : NA N Hexadecimal to Octal Assumption: 4 anda Example : Convert AF.6C 6 to base AF.6C A F 6 C

12 ase Conversion Conversion : Radix Divide Method N N A For integers n ( N ) b b I A n Integer in base A The b i 's represent the digits of (N I ) in base A ( NI ) ( ) b A A n b n n n b +...b + b + b Quotient Q Reminder R Quotient Q Q ( ) A Reminder R Quotient Q Q ( ) A Reminder R Stop when Quotient is zero Example : Convert b 9 4 to base 4 5 b 4 5 b Conversion : Radix Divide Method ase Conversion For integers Example : Convert4 to base A ( ) 6 b 6 4 ( E ) 6 b 4 4 EA 6 4

13 Conversion : Radix Multiply Method ase Conversion For fractoins The fraction can be written in series form. Fraction in base A m ( N ) b + b... b F A + m The b i 's represent the digits of (N F ) in base A A b m ( N ) ( b + b... + b ) F + b A A... + b m ( m ) m 5 Conversion : Radix Multiply Method ase Conversion For fractoins Example : Convert.5 to base b b b b 4 b 5 b 6 b 7 b (.654 )

14 Conversion : Radix Multiply Method ase Conversion For fractoins Example : Convert.5 to base MSD LSD Signed Number Representation Sign and Magnitude N ( sa n a.a a m ) rsm wheres if N positive s r if N negative Examples : ( ) ( ) (, ) sm ( ) ( 9, ) sm N N Sign r-

15 Radix Complement [ N ] r Signed Number Representation Complements are used in digital computers to simplify the subtraction operation. (simpler and less expensive circuits) In general, for base-r system : (r's complement ) n [ N] r r ( N) r n numberof digits in Largest positive number is r n r r Most negative number is n ( N ) r n 5 r Two's Complement : [ ] ( ) Example of Two's Complement : [ N] [ ] ( ) ( ) ( ) ( ) "Invert bits and add one", [] N n N 9 Signed Number Representation Diminished Radix Complement In general, diminished radix complement [N] r- of number (N) r is: ((r-)'s complement ) n [ N] r ( N) ( ) r r r n numberof digits in N Ones's Complement : r n [ ] r ( N ) N Example of One's Complement : "Invert bits" [ N] [ ] ( ) ( ) ( ) ( ) ( )

16 Signed Number Representation Subtraction with Radix Complement n ( M ) ( N) ( M) + [ N] ( M) ( N) + r r r r r r r If ( M ) r ( N ) r The sum will generate an end carry n r Discard the end carry The result is M-N If ( M ) r ( N ) r The sum does not produce an end carry The result is [ N M ] r Examples of 's Complement : ( ) ( N ) 75 5 M 's complement of N is :9675 Discard end carry (r's complement of (N-M) ) ( ) ( N ) 5 75 M 's complement of N is : 746 's complement of is : [ 7 ] 69 Numeric Codes Computer Codes Fixed-point Numbers Fixed-point integer Sign bit Fixed-point fraction n n n n n Implied binary point n Implied binary point Magnitude representation Excess or iased Representations Excess EXCESS- code Excess-K representation of a code C is C+K. Excess- n numbers are two's complement numbers with the sign bit reversed.

17 Numeric Codes Computer Codes Floating-point Numbers N M r E where M mantissa E exponent Mantissa M is often coded in sign magnitude, usually as a fraction. M ( SM.an a m) rsm S M positivenumber Numeric Codes Computer Codes Floating-point Numbers Exponent E is most often coded in excess-k two's complement. ias K is added to the 's complement integer value of the exponent. For binary floating-point numbers, K is usually selected to be e where e is the number of bits in the exponent. N M r N M r E E E+ ( M r) r E ( M r) r Example : M + (.) 4 (.) 5 (.) 6 (.) 4

18 Character and Other Codes Computer Codes inary Coded Decimal (CD) (4-code) The CD code is weighted code: Each bit position in the code has a fixed weight associated with it. Example ( 975) ( ) CD N : : : : 4: 5: 6: 7: : 9: DC Codes 5 Character and Other Codes Computer Codes Gray Code (unit distance code) 6

19 Computer Codes Character and Other Codes ASCII (American Standard Code for Information Interchange) 7 Error Detection - and Correction Codes Computer Codes Parity Codes Parity it P Information bits Even Parity P Information bits : the number of is odd P Information bits : the number of is even [] Odd Parity P Information bits : the number of is odd P Information bits : the number of is even

20 Computer Codes Classification of inary Codes Weighted Non-weighted Reflective Sequential Alphanumeric Error Detecting and Correcting inary Excess- Gray 4 5 Excess- 4 ASCII ECDIC [4] CD Excess- Hollerith Parity Hamming Odd Even [] 9 The End 4