Combinational Circuit Minimization
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1 Combinational Circuit Minimization Canonical sum and product logic expressions do not provide a circuit realization with the minimum number of gates. Minimization methods reduce the cost of two level AND-OR, NAND-NAND, OR-AND, NOR-NOR circuits in three ways: By minimizing the number of first level gates By minimizing the number of inputs of each first-level gate. Minimizing the inputs of the second level gate Most minimization methods are based on the combining theorems T, T : given product term. + given product term. = given product term (given sum term+).(given sum term + ) = given sum term EECC - Shaaban # Lec # inter --
2 Karnaugh Maps A Karnaugh Map or ( for short) is a graphical representation of the truth table of a logic function. The for an n-input logic function is an array with n cells or squares, one for each input combination or minterm. The rows and columns are labeled so that the input combination for any cell is determined from the row and column headings. The row and columns of the map are ordered in such a way that each cell differs from an adjacent cell in only one input variable: Thus for an n-variable, each cell has n adjacent cells. The for a function is filled by putting: a in the square corresponding to a minterm a otherwise (maybe omitted) EECC - Shaaban # Lec # inter --
3 -Variable For a -variable logic function F(,): Truth Table: Row F Minterm F(,). F(,). F(,). F(,). Example: For the function F(,) = S, (,,) Truth Table: Row F EECC - Shaaban # Lec # inter --
4 -Variable For a -variable logic function F(,,): Truth Table: Row F Minterm F(,,).. F(,,).. F(,,).. F(,,).. F(,,).. F(,,).. F(,,).. F(,,).. Example: For the function F(,,) = S,, (,,,) Truth Table: Row F EECC - Shaaban # Lec # inter --
5 -Variable (continued) There is a horizontal adjacency wrap-around in the -variable : For example: Cell (minterm =.. ) is adjacent to: cell (minterm, =.. ) by wrap-around. in addition to being adjacent to cells, (minterm =.. minterm, =.. ) Cell (minterm,..) is adjacent to: cell (minterm,..) by wrap-around. in addition to being adjacent to cells, (minterm =.. minterm =..) EECC - Shaaban # Lec # inter --
6 -Variable For a -variable logic function F(,,,): Truth Table: Row F Minterm F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... 8 F(,,,)... 9 F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,)... F(,,,) EECC - Shaaban # Lec # inter --
7 -Variable (continued) There are adjacency wrap-arounds in the -variable : a horizontal wrap-around and a vertical wrap-around. Every cell thus has neighbours. For example, cell corresponding to minterm is adjacent to: cells,,, EECC - Shaaban # Lec # inter --
8 -Variable Example For the function F(,,,) = S,,, (,,,,,) Truth Table: Row F EECC - Shaaban #8 Lec # inter --
9 Minimizing Sum of Products using s Each input combination with in a Karnaugh map or truth table correspond to a minterm in the function s canonical sum representation. Pairs of adjacent cells in the Karnaugh map indicate minterms that differ in only one variable. Using the generalization of T, such adjacent minterm pairs can be combined into a single product term. In general, one can simplify a logic function by combining pairs of adjacent -cell minterms and writing a sum of products expression to cover all of the -cells. EECC - Shaaban #9 Lec # inter --
10 K-Map Minimization Rules and Definitions A minimal sum of a logic function F(,,.. n ) is a sum-ofproducts expression for F such that no other similar expression for F has fewer product terms, and other expressions with the same number of product terms have at least the same number of literals. A set of i -cells are combined into a single square or rectangle if i variables take all i possible combinations within the set while the remaining variables have the same value. The corresponding product term for the combined cells has n-i literals. Only the variables that have the same value appear in the resulting product term: A variable in the resulting product term is complemented if it appears as in all the -cells, and uncomplemented if it appears as. EECC - Shaaban # Lec # inter --
11 Minimization Using s Group or combine as many adjacent -cells as possible: The larger the group is, the fewer the number of literals in the resulting product term. Each group of combined adjacent -cells must have a number of cells equal to powers of two:,,, 8, Grouping adjacent -cells eliminates variable, grouping - cells eliminates variables, grouping 8 -cells eliminates variables, and so on. In general, grouping n squares eliminates n variables. Select as few groups as possible to cover all the -cells (minterms) of the function: The fewer the groups, the fewer the number of product terms in the minimized function. EECC - Shaaban # Lec # inter --
12 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,) = S,, (,,,) Truth Table Row F EECC - Shaaban # Lec # inter --
13 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,) = S,, (,,,)... Truth Table Row F. Minimum SOP for F = EECC - Shaaban # Lec # inter --
14 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,) = S,, (,,,, ) Truth Table Row F EECC - Shaaban # Lec # inter --
15 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,) = S,, (,,,, ). Truth Table Row F Minimum SOP for F = +. EECC - Shaaban # Lec # inter --
16 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(N,N,N,N) = S N,N,N,N (,,,,,,) Truth Table: Row F 8 9 N N N N N N 8 N 9 N EECC - Shaaban # Lec # inter --
17 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(N,N,N,N) = S N,N,N,N (,,,,,,) Truth Table: Row F 8 9 N. N N N N N.N.N N N N 8 N 9 N. N. N N N. N. N Minimum SOP for F = N. N + N. N. N + N. N. N + N. N.N EECC - Shaaban # Lec # inter --
18 K-Map Minimization Rules and Definitions A logic function P(,,.. n ) implies a logic function F(,, n ) if for every input combination such that P=, then F= (F includes P, or F covers P). A prime implicant of a logic function F(,.. n ) is a normal product term P(,.. n ) that implies F, such that if any variable is removed from P, the the resulting product term does not imply F. A minimal sum is a sum of prime implicants (not necessarily all of them). A distinguished -cell of a logic function is an input combination that is covered by only one prime implicant. An essential prime implicant of a logic function is a prime implicant that covers one or more distinguished -cells and must be included every minimal sum expression for the function. EECC - Shaaban #8 Lec # inter --
19 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,,) = S,,, (,,,,,,,,) Also identify all prime implicants, distinguished -cells and the corresponding essential prime implicants that cover them. 8 9 EECC - Shaaban #9 Lec # inter --
20 -Variable Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(,,,) = S,,, (,,,,,,,,) Also identify all prime implicants, distinguished -cells and the corresponding essential prime implicants that cover them. From : Prime Implicants:.... Distinguished -cells: Cell covered by. Cell covered by. Cell covered by. Cell covered by. Here all prime implicants are essential prime implicants and all of them must be included in minimum SOP expression: F = EECC - Shaaban # Lec # inter --.
21 Minimization with Don t care Input Combinations In some cases, the output of a combinational circuit doesn t matter for certain input combinations. Such combinations are called don t cares and the output is represented in the truth table and s as d. hen using s to minimize such functions: Allow d s to be included when circling sets of s to make the sets as large as possible. Do not circle any set that only contains d s. EECC - Shaaban # Lec # inter --
22 -Variable Minimization Example ith Don t cares Using, find a minimal sum of products (SOP) expression for prime BCDdigit detector which gives when the input BCD digit is prime, Since the values - do not occur in a BCD digit minterms - are treated as don t cares giving the expression: F(N,N,N,N) = S N,N,N,N (,,,,) + d(,,,,,) From : Prime Implicants: N. N N. N N. N Distinguished -cells: Cell covered by N. N Cell covered by N. N Here not all prime implicants are essential prime implicants that must be included minimum SOP expression: N N N N N N N. N d d d d 8 9 d d N. N N F = N. N + N. N N N. N EECC - Shaaban # Lec # inter --
23 Minimization of Product of Sums Similar to minimization of sum of products by using duality and looking at -cells instead of -cells. A set of i -cells may be combined if i variables take all i possible combinations within the set while the remaining variables have the same value. In the resulting n-i literals sum term, a variable is complemented if it appears as in all the -cells, and uncomplemented if it appears as. A prime implicate of a logic function F(,.. n ), is a normal sum term S(,.. n ) implied by F, such as if any variable is removed from S, then the resulting sum term is not implied by F. A minimal product is a product of prime implicates. EECC - Shaaban # Lec # inter --
24 Product of Sums Minimization Example Using, find a minimal product of sums (POS) expression expression for the function: F(,,) = P,, (,,,) Truth Table Row F EECC - Shaaban # Lec # inter --
25 Product of Sums Minimization Example Using, find a minimal product of sums (POS) expression expression for the function: F(,,) = P,, (,,,) ( + ) Truth Table Row F ( + ) Minimum POS for F = ( + ). ( + ) EECC - Shaaban # Lec # inter --
26 Product of Sums Minimization Example Using, find a minimal product of sums (POS) expression expression for the function: F(,,,) = P,,, (,,8,,,,,) 8 9 EECC - Shaaban # Lec # inter --
27 Product of Sums Minimization Example Using, find a minimal product of sums (POS) expression expression for the function: F(,,,) = P,,, (,,8,,,,,) 8 ( + ) ( + + ) 9 ( + ) Minimum POS for F = ( + + ). ( + ). ( + ) EECC - Shaaban # Lec # inter --
28 -variable s The for a -variable logic function F(V,,,,) is organized as two -variable s: V = V = Corresponding squares of each map are adjacent. Can be visualised as being one -variable map on top of another -variable map. EECC - Shaaban #8 Lec # inter --
29 -Variable SOP Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(V,,,,) = S V,,,, (,,,,9,,,,,,9,) V = V = 8 9 EECC - Shaaban #9 Lec # inter --
30 -Variable SOP Minimization Example Using, find a minimal sum of products (SOP) expression expression for the function: F(V,,,,) = S V,,,, (,,,,9,,,,,,9,) V V = V = 8 9 Minimum SOP for F = V.. +. EECC - Shaaban # Lec # inter --
31 -variable s for a -variable logic function F(U,V,,,,) is organized as two -variable s: U,V =, U,V =, U,V =, U,V =, EECC - Shaaban # Lec # inter --
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