# Gate-Level Minimization

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1 Chapter 3 Gate-Level Minimization 3- Outline! Karnaugh Map Method! NAND and NOR Implementations! Other Two-Level Implementations! Exclusive-OR Function! Hardware Description Language 3-2

2 Why Logic Minimization?! Minimize the number of gates used! Reduce gate count = reduce cost! Minimize total delay (critical path delay)! Reduce delay = improve performance! Satisfy design constrains! Maximum fanins and fanouts,! Remove undesired circuit behavior! Hazard, race, 3-3 The Map Method! The map method is also known as the Karnaugh map or K-map! Provide a straightforward procedure for minimizing Boolean functions! The simplified expressions are always in one of the two standard forms:! Sum of Products (SOP)! Product of Sums (POS) 3-4 2

3 Two-Variable Map (/2)! Two-variable function has four minterms! Four squares in the map for those minterms! The corresponding minterm of each square is determined by the bit status shown outside 3-5 Two-Variable Map (2/2) y x x + y = x(y+y ) + y(x+x ) = xy (m 3 ) + xy (m 2 ) + x y (m ) = m + m 2 + m

4 Three-Variable Map (/2) Gray Code 3-7 Three-Variable Map (2/2) F( x, y, z) = (3,4,6,7) = yz + xz' F ( x, y, z) = (,2,4,5,6) = z' + xy ' xz (all y covered) yz (all x covered) z (x, y covered) xy (all z covered) 3-8 4

5 Four-Variable Map (/2) Gray Code Gray Code 3-9 Four-Variable Map (2/2) F = A B C + B CD + A BCD + AB C = = B D + B C + A CD (,,2,6,8,9,) A CD B D B C 3-5

6 Five-Variable Map (/2) * Maps with six or more variables need too many squares and are impractical to use. 3- Five-Variable Map (2/2) F(A,B,C,D,E) = (,2,4,6,9,3,2,23,25,29,3) = A B E + BD E + ACE BD E (same position) 3-2 6

7 Prime Implicants! Implicant (cube) : A group of minterms that form a cube! Prime implicant : Combine maximum possible number of adjacent squares in the map cd a'b'c'd' ab a'b'c prime implicant prime implicant ac' ab'c' abc' a'cd' prime implicant 3-3 Essential Prime Implicants! If a minterm is covered by only one prime implicant, that prime implicant is essential and must be included ab cd a'c' acd Note: 's in red color are covered by only one prime implicant. All other 's are covered by at least two prime implicants a'b'd' 3-4 7

8 Systematic Simplification! Identify all prime implicants on the k-map! Select all essential prime implicants! Select a minimum subset of the remaining prime implicants that cover all s! Ex: F( A, B, C, D) = (,2,3,5,7,8,9,,,3,5) F = BD + B' D' CD + AD CD + AB' + B' C + AD B' C + AB' 3-5 Product of Sums Simplification! The complement of a function is represented in the map by the squares not marked by s! Choose " sum of products (minterms)! Choose " product of sums (Maxterms) F( A, B, C, D) = (,,2,5,8,9,) =B D +B C +A C D =(A +B )(C +D )(B +D) 3-6 8

9 Two Gate Implementations! Sometimes product-of-sums representations may have smaller implementations 7 literals, 4 gates 6 literals, 4 gates 3-7 Don t Care Conditions! X = don't care (can be or )! Don t cares can be included to form a larger cube, but not necessary to be completely covered! Ex: F( w, x, y, z) = (,3,7,,5) d( w, x, y, z) = (,2,5) larger cube with don t cares 3-8 9

10 Outline! Karnaugh Map Method! NAND and NOR Implementations! Other Two-Level Implementations! Exclusive-OR Function! Hardware Description Language 3-9 NAND and NOR Implementation! Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gate! NAND and NOR gates are much easier to fabricate! NAND or NOR gates are both universal gates! Any digital system can be implemented with only NAND gates or NOR gates 3-2

11 Alternative Graphic Symbols! To facilitate the conversion to NAND or NOR logic, it is convenient to define alternative graphic symbols! Bubble means complement obtained by DeMorgan s theorem 3-2 Two-Level Implementation (NAND)! It s easy to implement a Boolean function with only NAND gates if converted from a sum of products form! Ex: F = AB+CD = ((AB) (CD) ). add two bubbles at the ends 3. all-nand circuit has produced 2. convert to NAND gate using DeMorgan s theorem 3-22

12 Procedures: Example 3-. Simplify the function in sum of products 2. Draw NAND gates for the first level 3. Draw a single AND-invert or invert-or in the second level 4. Add an inverter at the first level for the term with a single literal F( x, y, z) = (,2,3,4,5,7 ) 3-23 Two-Level Implementation (NOR)! It s easy to implement a Boolean function with only NOR gates if converted from a product of sums form! Ex: F=(A+B)(C+D)E. add two bubbles at the ends 2. complement this input to add the third bubble (if required) 3. convert to NOR gate using DeMorgan s theorem

13 Multilevel NAND Circuits F=A(CD+B)+BC! Procedures:. Convert all AND gates to NAND gates with AND-invert symbols 2. Convert all OR gates to NAND gates with invert-or symbols 3. Check all bubbles and insert an inverter for the bubble that are not compensated by another bubble 3-25 Multilevel NOR Circuits! For NOR gates, AND " invert-and, OR " OR-invert! Other procedures are the same as those for NAND F=(AB +A B)(C+D ) complemented complemented

14 Outline! Karnaugh Map Method! NAND and NOR Implementations! Other Two-Level Implementations! Exclusive-OR Function! Hardware Description Language 3-27 Wired Logic! Wired logic: direct wire connection that results in a specific logic function! Wired-AND! Wired-OR! Some NAND and NOR implementations (not all) have such a property

15 Nondegenerate Forms! There are 6 possible combinations of two-level forms! Eight of these combinations will degenerate to a single operation! The other eight nondegenerate forms produce an implementation in SOP or POS AND-OR NAND-NAND NOR-OR OR-NAND The two forms at the same line are dual to each other OR-AND NOR-NOR NAND-AND AND-NOR 3-29 AND-OR-INVERT Implementation! NAND-AND and AND-NOR are equivalent and both perform the AND-OR-INVERT (AOI ) function! Require sum-of-products form in nature! When starting from product-of-sums form, complement it using DeMorgan s theorem to obtain sum-of-products form! Ex: F = (AB + CD + E) 3-3 5

16 OR-AND-INVERT Implementation! OR-NAND and NOR-OR are equivalent and both perform the OR-AND-INVERT (OAI ) function! Require product-of-sums form in nature! When starting from sum-of-products form, complement it using DeMorgan s theorem to obtain product-of-sums form! Ex: F = [(A+B)(C+D)E] 3-3 Implement with Two-Level Forms

17 Example 3- choose! F = x y z + xyz (choose in K-map) = (x y + xy + z) (choose in K-map) choose 3-33 Outline! Karnaugh Map Method! NAND and NOR Implementations! Other Two-Level Implementations! Exclusive-OR Function! Hardware Description Language

18 Exclusive-OR (XOR) Function! XOR is often denoted by the symbol! Logic operation of XOR! X Y = XY + X Y! Equal to if only x is equal to or if only y is equal to, but not when both are equal to! It s complement, exclusive-nor (XNOR), is often denoted by the symbol! Logic operation! X Y = XY + X Y! It is equal to if both x and y are equal to or if both are equal to! Seldom used in general Boolean functions! Particularly useful in arithmetic operations and error detection and correction circuits 3-35 Exclusive-OR Implementations

19 Odd Function! The multiple-variable XOR operation is defined as an odd function! TRUE when no. of in inputs is odd F C B A 3-38 Even Function! The multiple-variable XNOR operation is defined as an even function! TRUE when no. of in inputs is even F C B A

20 Four-Variable XOR Function 3-39 Parity Generation and Checking! An extra parity bit is often added and checked at the receiving end for error! The circuit that generates the parity bit in the transmitter is called a parity generator! The circuit that checks the parity in the receiver is called a parity checker! Exclusive-OR functions are very useful to construct such circuits 3-4 2

21 Parity Generator! For even parity:! The total number of (including P) is even! The number of at inputs is odd! Generated with an XOR gate (odd function)! P = x y z (for 3-bit message)! Similarly, odd parity can be generated with an XNOR gate 3-4 Parity Checker! For even parity, the total number of in the message is even! An error occurs when the received number of is odd! An XOR gate (odd function) can detect such an error! Has n+ inputs

22 Outline! Karnaugh Map Method! NAND and NOR Implementations! Other Two-Level Implementations! Exclusive-OR Function! Hardware Description Language 3-43 Hardware Description Language! Have high-level language constructs to describe the functionality and connectivity of the circuit! Can describe a design at some levels of abstraction! Behavioral, RTL, Gate-level, Switch! Can describe functionality as well as timing! Can model the concurrent actions in real hardware! Can be used to document the complete system design tasks! testing, simulation related activities! Comprehensive and easy to learn! Two popular languages: Verilog & VHDL! Will be taught in another course

23 Why Use an HDL?! Hard to design directly for complex systems! Formal description using HDL! Verify the specification through simulation or verification! Easy to change! Enable automatic synthesis! Allow architectural tradeoffs with short turnaround! Reduce time for design capture! Encourage focus on functionality! Shorten the design verification loop *HDL = Hardware Description Language

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