{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

l5_15a_6

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

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

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

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

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online