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Review

Review. D: n bit binary number D = (d n-1 ∙ ∙ ∙ d 1 d 0 ) 2 If D is an unsigned binary number D = ( 2 n-1 d n-1 + ∙ ∙ ∙ 2 1 d 1 + 2 0 d 0 ) 10 If D is a sign-magnitude binary number D = + ( 2 n-2 d n-2 + ∙ ∙ ∙ 2 1 d 1 + 2 0 d 0 ) 10 if d n-1 =0

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Review

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  1. Review • D: n bit binary number D = (dn-1 ∙ ∙ ∙ d1 d0)2 • If D is an unsigned binary number D = (2n-1 dn-1+∙ ∙ ∙21 d1 + 20 d0)10 • If D is a sign-magnitude binary number D = + (2n-2 dn-2+∙ ∙ ∙21 d1 + 20 d0 )10 if dn-1=0 = – (2n-2 dn-2+∙ ∙ ∙21 d1 + 20 d0 )10 if dn-1=1 (–D)= (d’n-1 dn-2 ∙ ∙ ∙ d1 d0)2 • If D is in two`s complement system D = (-2n-1 dn-1+ 2n-2 dn-2 + ∙ ∙ ∙21 d1 + 20 d0)10 (–D) = 2n – D = (2n-1) – D + 1 = (d’n-1 d’n-2 ∙ ∙ ∙ d’1 d’0)2 + 1 Digital Systems Design

  2. Review • Two’s complement multiplication • Shift and two’s complement addition except for the last step. Remember MSB represent (-2n-1) -5 1011 x –3 1101 0000 initial partial product, which is zero. 1011 11011 partial product 0000 111011 partial product 1011 11100111 0101 shifted-and-negated 1 00001111 Digital Systems Design

  3. Review BCD: Binary-Coded Decimal • 0-9 encoded with their 4-bit unsigned binary representation (0000 – 1001). The codewords (1010 – 1111) are not used. • 8-bit byte represent values from 0 to 99. • BCD Addition: Carry 11 448 0100 0100 1000 + 489 0100 1000 1001 937 Sum 1001 1101 10001 Add 6 + 0110+ 0110 BCD sum 1 0011 1 0111 BCD result 1001 0011 0111 Digital Systems Design

  4. 2. Combinational Logic Circuits • Boolean Algebra • switching algebra • deals with Boolean values --- 0, 1 • Positive-logic convention • analog voltages LOW, HIGH  0, 1 • Negative logic --- seldom used • Signal values denoted by variables(X, Y, FRED, etc.) Digital Systems Design

  5. Boolean operators • Complement: X¢ (opposite of X) • AND: X × Y • OR: X + Y Digital Systems Design

  6. Literal: a variable or its complement • X, X¢, FRED¢, CS_L • Expression: literals combined by AND, OR, parentheses, complementation • X+Y • P × Q × R • A + B × C • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ • Equation: Variable = expression • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ Digital Systems Design

  7. Basic Logic Gates Digital Systems Design

  8. Theorems Digital Systems Design

  9. More Theorems Digital Systems Design

  10. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Why? • Each axiom (T1-T5) has a dual (T1¢-T5¢) • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual) X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8) Digital Systems Design

  11. N-variable Theorems Digital Systems Design

  12. DeMorgan Symbol Equivalence Digital Systems Design

  13. Similar for OR Digital Systems Design

  14. Complement of a function • F1 = XYZ’ + X’Y’Z • F1’ = (XYZ’ + X’Y’Z)’ = (XYZ’)’× (X’Y’Z)’ = (X’+Y’+Z) × (X+Y+Z’) • Complement = take dual +complement each literal • Dual of F1 = (X+Y+Z’) × (X’+Y’+Z) • F1’ = (X’+Y’+Z) × (X+Y+Z’) Digital Systems Design

  15. Standard Forms: • Product and sum terms • Minterm: A product term in which all variables appear exactly once, either complemented or not (2n minterms) • For a two variable function, minterms are • X’Y’, X’Y, XY’, XY • m0 , m1 , m2 , m3 • Maxterms: A sum term that contains all variables in complemented or uncomplemented form • X+Y, X+Y’, X’+Y, X’+Y’ • M0 , M1 , M2 , M3 Digital Systems Design

  16. Digital Systems Design

  17. Alternative representations • F(X,Y,Z) = X’Y’Z’ + X’YZ’ +XY’Z + XYZ = m0 + m2 + m5 + m7 = • F’(X,Y,Z) = X’Y’Z + X’YZ + XY’Z’ + XYZ’ = m1 + m3 + m4 + m6 = • F(X,Y,Z) = (m1 + m3 + m4 + m6)’ = m1’ m3’ m4’ m6’ = M1 M3 M4 M6 = Digital Systems Design

  18. Maxterms are seldom used, we’ll use minterms rather. • Properties of minterms: • There are 2n minterms. 1-1 with binary numbers 0-(2n-1) • Every Boolean function can be expressed as sum of minterms. • Absent minterms belong to complement function • A function that include all minterms is equal to logic 1. Digital Systems Design

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