l5_15a_6

l5_15a_6 - Last time ECE 15A Fundamentals of Logic Design...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 ECE 15A Fundamentals of Logic Design Lecture 5 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Last time ± Minterm and maxterm representations of Boolean functions ± Any Boolean function can be expressed as a sum of 1- minterms (dnf) = ) , , ( 2 1 n x x x F L (1_minterms) Example x y z minterms notation F 0 0 0 x’y’z’ m0 0 0 0 1 x’y’z m1 0 0 1 0 x’yz’ m2 0 0 1 1 x’yz m3 1 1 0 0 x y’z’ m4 0 1 0 1 xy’z m5 1 1 1 0 xyz’ m6 1 1 1 1 xyz m7 1 = ) , ( y x F (m3,m5,m6,m7) 3 Last time ± Any Boolean function can be expressed as a product of its 0-maxterms (cnf) = ) , , ( 2 1 n x x x F L (0_maxterms) Example x y z maxterms notation F 0 0 0 x+y+z’ M0 0 0 0 1 x+y+z’ M1 0 0 1 0 x+y’+z M2 0 0 1 1 x’+y+z M3 1 1 0 0 x’+y+z M4 0 1 0 1 x’+y+z’ M5 1 1 1 0 x’+y’+z M6 1 1 1 1 x’+y’+z’ M7 1 = ) , ( y x F (M0,M1,M2,M4) 4 Canonical representations of Boolean functions ± Canonical forms are unique representations of Boolean functions. ± dnf and cnf are canonical representations of Boolean functions ± To convert between dnf and cnf, we interchange the sum and product symbols and list those numbers which were absent in the original form. = ) , ( y x F (m3,m5,m6,m7) = ) , ( y x F (M0,M1,M2,M4) Example: 5 Boolean functions in standard forms ± Sum-of-products (SOP) ² Boolean expression that contains AND terms (called products or implicants) which are ORed together ² Example: F1(x,y,z,w)=xy+z’+x’w’w’ ± Product of sums (POS) ² Boolean expression that contains OR terms (called sum terms or implicants) which are ANDed together ² Example: F2=(x+z’+w)(x’+y+w’) ± These normal forms are not unique: the same functions can be expressed in many different ways ± Standard forms usually contain fewer terms and literals than the corresponding dnfs or cnfs. 6 XOR operation 0 1 0 0 1 1 1 0 A B = B A (commutative) (A B) C = A (B C) (associative) (AB) (AC) = A (B C) multiplication distributive over L = (A B)(A C) = A (BC)= R A B C A B A C L BC R 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 A C = B B C = A A B C = 0 If A B = C, then
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 7 Other logic operations ± NOR f(a,b) = (a+b)’ ± NAND f (a,b) = (ab)’ ± XNOR (Equivalence) f (a,b)= ab + a’b’ NOR 0 1 0 1 0 1 0 0 NAND 0 1 0 1 1 1 1 0 XNOR 0 1 0 1 0 1 0 1 8 Functionally complete operations ± A set of operations is functionally complete (universal) if and only if every Boolean function can be expressed entirely by means of operations from this set.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/20/2011 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

Page1 / 6

l5_15a_6 - Last time ECE 15A Fundamentals of Logic Design...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online