CSI-2111 Computer Architecture Ipage Revision  Objective : To examine basic concepts of: –2.1 Numbering Systems –2.2 Binary Numbers –2.3 Boolean Algebra –2.4 Logic Gates –2.5 Adders –2.6 Timing Diagram

CSI-2111 Computer Architecture Ipage Number System (base B)  Number is represented in terms of Positional Weighting:  ( N ) B = d n-1 B n-1 + d n-2 B n d 1 B 1 + d 0 B 0 · d -1 B d -m B -m Integral Part. Fractional Part B = base d k = digit in position k, -m ≤ k ≤ n-1 B k = weight of position k, -m ≤ k ≤ n-1 n = number of integral digits in N m = number of fractional digits in N

CSI-2111 Computer Architecture Ipage 2-3 The most known systems  B = 10 (Decimal) digits : (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)  B = 2(Binary) digits: (0, 1)  B = 8(Octal) digits: (0, 1, 2, 3, 4, 5, 6, 7)  B = 16(Hexadecimal) digits: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)

CSI-2111 Computer Architecture Ipage 2-4 Conversions (I)  How to convert from one number system to the other (change of base B )?  Algorithm 1: –favorable for conversions to the decimal system (N 1 ) A  (N 2 ) 10

CSI-2111 Computer Architecture Ipage 2-5 Conversions (II)  Algorithm 2: Use of successive divisions and multiplications –favorable for conversions from the decimal system (N 1 ) 10  (N 2 ) B (34.625) 10  (?) 2

CSI-2111 Computer Architecture Ipage 2-6 Conversions (II) - Example (34.625) 10  ( ) 2 Integral Part: Fractional Part: 34 ÷ 2 = 17r x 2 = ÷ 2 = 8r x 2 =0.5 8 ÷ 2 = 4r 00.5 x 2 =1 4 ÷ 2 = 2r 0 2 ÷ 2 = 1r 0 1 ÷ 2 = 0r 1

CSI-2111 Computer Architecture Ipage 2-7 Binary  Hexadecimal  Binary  Hexadecimal –Form groups of 4 bits starting at binary point. –Each group of 4 bits represents a hexadecimal digit.  Hexadecimal  Binary –Convert each hexadecimal digit to its binary equivalent (4 bits).

CSI-2111 Computer Architecture Ipage 2-8 Binary  Octal  Binary  Octal –Form groups of 3 bits starting at binary point. –Each group of 3 bits represents an octal digit.  Octal  Binary –Convert each octal digit to its binary equivalent (3 bits).

CSI-2111 Computer Architecture Ipage Binary Numbers  Fixed Point Representation. –(N) 2 has an implicit binary point in a fixed position. –Notion of complement : Complement(d) = (Base - 1) - d

CSI-2111 Computer Architecture Ipage 2-10 Complements of Binary Numbers  1’s Complement : (1CF)  2’s Complement : (2CF)  2’s Complement of = ?

CSI-2111 Computer Architecture Ipage 2-11 Unsigned Binary Integers (UBI)  N is represented in terms of positional weighting: 0 ≤ N ≤ 2 n – 1 d n-1 d n-2 d n-3... d 2 d 1 d 0  No sign  N = ( ) 2(7.4) = (?) 10 –Word of 11 digits –Fixed point representation, with 4 digits for the fractional part

CSI-2111 Computer Architecture Ipage 2-12 Signed Binary Integers  3 ways of representing ± N with n bits: –Sign-Magnitude Form (SMF) –1’s Complement Form(1CF) –2’s Complement Form(2CF)

CSI-2111 Computer Architecture Ipage 2-13 Comparison of SMF, 1CF, and 2CF Sign Magnitude / 2’s Complement / 1’s Complement (4 bits word length) BinarySMF2CF1CF

CSI-2111 Computer Architecture Ipage 2-14 Binary Addition (by complement)  The bit sign is treated like any other bit (they are added!)  The subtraction is performed by addition; the negative numbers are treated like numbers to add.  Addition by 1CF  Addition by 2CF

CSI-2111 Computer Architecture Ipage 2-15 Overflow  An overflow occurs when the operands have the same sign and the result has a sign different from that of the operands.  Ex ; 4 bits word; SMF (-2)1 010

CSI-2111 Computer Architecture Ipage Boolean Algebra  Two elements: 0 and 1  Elementary operators: {AND, OR, NOT}  Representation and algebraic simplification of Boolean functions and their realization using logic gates will be studied.

CSI-2111 Computer Architecture Ipage 2-17 Boolean Functions & Truth Tables  Any Boolean function defined over n variables, each taking a Boolean constant value (0 or 1).  The Truth Table represents the function f, with all 2 n combinations of 1’s and 0’s of its variables.  Each Boolean function is defined by its truth table and is represented by inter- connected logic gates.

CSI-2111 Computer Architecture Ipage 2-18 Functions and Logic Gates

CSI-2111 Computer Architecture Ipage 2-19 Laws of Boolean Algebra  Together with –Postulates (or axioms) –Theorems  Manipulations (proof) –Algebraic –Tabular  Simplifications –Algebraic

CSI-2111 Computer Architecture Ipage 2-20 Proofs and Simplifications  Algebraic Proofs  Tabular Proofs  Algebraic Simplifications

CSI-2111 Computer Architecture Ipage 2-21 Examples *  Algebraic Proof: (A B’) + B = A + B = B + (A B’) ( Commutative, OR ) = (B + A) (B + B’) (Distributive, OR) = (B + A) (1) (Complementation, OR) = (B + A) ( Identity Element, AND ) = A + B( Commutative, OR )

CSI-2111 Computer Architecture Ipage 2-22 Examples *  Proof by Truth Table: (A B’) + B = A + B A B (A B’) + B A + B

CSI-2111 Computer Architecture Ipage 2-23 Examples *  Algebraic Simplification Example: f(ABC) = AC’ + A’B + A’B’C’ = ? = AC’ + A’B + A’BC’ + A’B’C’(Absorption, OR) = AC’ + A’B + A’C’ (B + B’)(Distributive, AND) = AC’ + A’B + A’C’ (1) (Complementation, OR) = AC’ + A’B + A’C’ (Identity Element, AND) = A’B + AC’ + A’C’ (Commutative, OR) = A’B + C’A + C’A’ (Commutative, AND (2 times)) = A’B + C’ (A+A’) (Distributive, AND) = A’B + C’ (1) (Complimentation, OR) = A’B + C’ (Identity Element, AND)

CSI-2111 Computer Architecture Ipage 2-24 Representation: Canonical Forms  Canonical Sum of Products Form (CSOP or  m) –Sum of mintermes –Ex.: f (A, B) = (A’B) + (AB) =  m(1, 3)  Canonical Product of Sums Form (CPOS or  M) –Product of maxtermes –Ex.: f (A, B) = (A+B’) (A’+B) =  M(1, 2)

CSI-2111 Computer Architecture Ipage 2-25 Examples of Canonical Forms  According to De Morgan’s law: M i ' = m i and m i ' = M i  Say f(A, B, C) defined by: ABCf

CSI-2111 Computer Architecture Ipage 2-26 Examples of Canonical Forms *  f(A, B, C) defined by: A BCf = A’B’C + AB’C’ + AB’C + ABC’ + ABC =  m (1, 4, 5, 6, 7) is the CSOP form of f

CSI-2111 Computer Architecture Ipage 2-27 Examples of Canonical Forms *  f(A, B, C) defined by: ABCf ABCf = (A+B+C) (A+B’+C) (A+B’+C’) =  M (0, 2, 3) is the CPOS form of f.

CSI-2111 Computer Architecture Ipage 2-28 Equivalence of Canonical Forms* f(A, B, C) =  m (0, 4, 5, 7) =  M (?)  M (1, 2, 3, 6) f(A, B, C, D) =  M (2, 3, 5, 6, 7) =  m (?)  m (0, 1, 4) ???  m (0, 1, 4, 8, 9, 10, 11, 12, 13, 14, 15) !!!

CSI-2111 Computer Architecture Ipage 2-29 Canonical Forms (encore!)  A function f is not necessarily represented in a canonical form. f(A, B, C) = A’B’C + AB’+ BC’  How to obtain the canonical forms of such functions? –Algebraic method

CSI-2111 Computer Architecture Ipage 2-30 Algebraic Method* f(A, B, C) = A’B + C’ + ABC =A’B(C+C’) + C’(A+A’)(B+B’) +ABC =A’BC + A’BC’ + ABC’ + AB’C’ + A’BC’ + A’B’C’ + ABC =A’BC + A’BC’ + ABC’ + AB’C’ + A’B’C’ + ABC =  m(3, 2, 6, 4, 0, 7)

CSI-2111 Computer Architecture Ipage 2-31 Remarks on Boolean Functions  Single representation ( POS or SOP ).  Logical equivalence.  How many possible functions for N Boolean variable?  Functions with 2 Boolean variables

CSI-2111 Computer Architecture Ipage 2-32 Functionally Complete Sets  Together of operators being able to represent all the functions  {AND, NOT, OR}, {NOR}, {NAND}  POS form with {NOR}  SOP form with {NAND}

CSI-2111 Computer Architecture Ipage Logic Gates   The logic gates implement the switching functions  A gate with N inputs represents a function with N Boolean variables  One comes across OR, AND, NOR, and NAND gates with N inputs

CSI-2111 Computer Architecture Ipage 2-34 Synthesis with Logic Gates  To implement a switching function with logic gates f (A, B, C) = (A + (BC)')'  B C A f

CSI-2111 Computer Architecture Ipage 2-35 Analysis with Logic Gates  To find the functionality of the circuit made up of logic gates  f (A, B, C) = (A + (BC)')' B C A f

CSI-2111 Computer Architecture Ipage 2-36 Synthesis with Single Gates  More economical than {AND, OR, NOT}  SOP form with {NAND}  POS form with {NOR} –Similar to SOP. A B A+B A 1 B 1 

CSI-2111 Computer Architecture Ipage 2-37 Return to NOR-NAND *  f (A, B, C, D) = AB’ +A’C + D  To implement with NAND (  ) only: = ((AB’ +A’C + D)’)’ = ((AB’)’. (A’C)’. (D)’)’ = (AB’)’  (A’C)’  (D)’ = (A  B’)  (A’  C)  (D)’

CSI-2111 Computer Architecture Ipage 2-38 Return to NOR-NAND *  f (A, B, C, D) = AB’ +A’C + D  To implement with NOR (  only: = ((AB’)’)’ + ((A’C)’)’ + D = (A’+B)’ + (A+C’)’ + D = (A’  B) +(A  C’) + D = (((A’  B) +(A  C’) + D)’)’ = ((A’  B)  (A  C’)  D)’ = ((A’  B)  (A  C’)  D)  0

CSI-2111 Computer Architecture Ipage 2-39 Return to NOR-NAND *  g (A, B, C, D) = (A+B’).(A’+C). D  Implement with NAND (  ) only: = ((A+B’)’)’. ((A’+C)’)’. D = (A’B)’. (AC’)’. D = (A’  B). (A  C’). D = (((A’  B). (A  C’). D)’)’ = ((A’  B)  (A  C’)  D)’ = ((A’  B)  (A  C’)  D) 

CSI-2111 Computer Architecture Ipage 2-40 Return to NOR-NAND *  g (A, B, C, D) = (A+B’).(A’+C). D  Implement with NOR (  only: = (((A+B’).(A’+C).D)’)’ = ((A+B’)’ + (A’+C)’ + D’)’ = (A+B’)’  (A’+C)’  D’ = (A  B’)  (A’  C)  D’

CSI-2111 Computer Architecture Ipage Adders  Classic combinational circuits  Various common circuits –Half-Adders –Elementary adder –Parallel full-adder –Elementary subtracter –Adder-substracter

CSI-2111 Computer Architecture Ipage 2-42 Adding words of several bits?  Parallel full-adder of 4 bits  EA = Elementary Adder EA B4B4 A4A4 C4C4 S4S4 B3B3 A3A3 C3C3 S3S3 B2B2 A2A2 C2C2 S2S2 B1B1 A1A1 C1C1 S1S1 R5R5

CSI-2111 Computer Architecture Ipage Logic Timing Diagram

CSI-2111 Computer Architecture Ipage 2-44 Complementary Reading  In Mano and Kime –Sections 1.2 and 1.3 Numbers and binary arithmetic –Sections 2.1, 2.2, 2.3, 2.6 and 2.7 Boolean algebra, logic gates, canonical forms –Section 3.8, except « Carry Lookahead Adder » Adders –Sections 3.9 and 3.10 Subtracters