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Circuit analysis summary

Circuit analysis summary. After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert between expressions and truth tables. Find the circuit’s inputs and outputs. Find a Boolean expression

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Circuit analysis summary

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  1. Circuit analysis summary • After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. • You can easily convert between expressions and truth tables. Find the circuit’s inputs and outputs Find a Boolean expression for the circuit Find a truth table for the circuit CS231 Boolean Algebra

  2. Boolean operations summary • We can interpret high or low voltage as representing true or false. • A variable whose value can be either 1 or 0 is called a Boolean variable. • AND, OR, and NOT are the basic Boolean operations. • We can express Boolean functions with either an expression or a truth table. • Every Boolean expression can be converted to a circuit. • Nex, we’ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. CS231 Boolean Algebra

  3. Boolean algebra • Last time we talked about Boolean functions, Boolean expressions, and truth tables. • Today we’ll learn how to how use Boolean algebra to simplify Booleans expressions. • Last time, we saw this expression and converted it to a circuit: (x + y’)z + x’ • Can we make this circuit “better”? • Cheaper: fewer gates • Faster: fewer delays from inputs to outputs ©2000-2002 Howard Huang

  4. Expression simplification • Normal mathematical expressions can be simplified using the laws of algebra • For binary systems, we can use Boolean algebra, which is superficially similar to regular algebra • There are many differences, due to • having only two values (0 and 1) to work with • having a complement operation • the OR operation is not the same as addition CS231 Boolean Algebra

  5. Formal definition of Boolean algebra • A Boolean algebra requires • A set of elements B, which needs at least two elements (0 and 1) • Two binary (two-argument) operations OR and AND • A unary (one-argument) operation NOT • The axioms below must always be true (textbook, p. 33) • The magentaaxioms deal with the complement operation • Blueaxioms (especially 15) are different from regular algebra CS231 Boolean Algebra

  6. Comments on the axioms • The associative laws show that there is no ambiguity about a term such as x + y + z or xyz, so we can introduce multiple-input primitive gates: • The left and right columns of axioms are duals • exchange all ANDs with ORs, and 0s with 1s • The dual of any equation is always true CS231 Boolean Algebra

  7. Are these axioms for real? • We can show that these axioms are valid, given the definitions of AND, OR and NOT • The first 11 axioms are easy to see from these truth tables alone. For example, x + x’ = 1 because of the middle two lines below (where y = x’) CS231 Boolean Algebra

  8. Proving the rest of the axioms • We can make up truth tables to prove (both parts of) DeMorgan’s law • For (x + y)’ = x’y’, we can make truth tables for (x + y)’ and for x’y’ • In each table, the columns on the left (x and y) are the inputs. The columns on the right are outputs. • In this case, we only care about the columns in blue. The other “outputs” are just to help us find the blue columns. • Since both of the columns in blue are the same, this shows that (x + y)’ and x’y’ are equivalent CS231 Boolean Algebra

  9. Simplification with axioms • We can now start doing some simplifications x’y’ + xyz + x’y = x’(y’ + y) + xyz [ Distributive; x’y’ + x’y = x’(y’ + y) ] = x’1 + xyz [ Axiom 7; y’ + y = 1 ] = x’ + xyz [ Axiom 2; x’1 = x’ ] = (x’ + x)(x’ + yz) [ Distributive ] = 1  (x’ + yz) [ Axiom 7; x’ + x = 1 ] = x’ + yz [ Axiom 2 ] CS231 Boolean Algebra

  10. Let’s compare the resulting circuits • Here are two different but equivalent circuits. • In general the one with fewer gates is “better”: • It costs less to build • It requires less power • But we had to do some work to find the second form CS231 Boolean Algebra

  11. Some more laws • Here are some more useful laws (p. 37). Notice the duals again! • We can prove these laws by either • Making truth tables: • Using the axioms: x + x’y = (x + x’)(x + y) [ Distributive ] = 1  (x + y) [ x + x’ = 1 ] = x + y [ Axiom 2 ] CS231 Boolean Algebra

  12. The complement of a function • The complement of a function always outputs 0 where the original function outputted 1, and 1 where the original produced 0. • In a truth table, we can just exchange 0s and 1s in the output column(s) f(x,y,z) = x(y’z’ + yz) CS231 Boolean Algebra

  13. Complementing a function algebraically • You can use DeMorgan’s law to keep “pushing” the complements inwards • You can also take the dual of the function, and then complement each literal • If f(x,y,z) = x(y’z’ + yz)… • …the dual of f is x + (y’ + z’)(y + z)… • …then complementing each literal gives x’ + (y + z)(y’ + z’)… • …so f’(x,y,z) = x’ + (y + z)(y’ + z’) f(x,y,z) = x(y’z’ + yz) f’(x,y,z) = ( x(y’z’ + yz) )’ [ complement both sides ] = x’ + (y’z’ + yz)’ [ because (xy)’ = x’ + y’ ] = x’ + (y’z’)’ (yz)’ [ because (x + y)’ = x’ y’ ] = x’ + (y + z)(y’ + z’) [ because (xy)’ = x’ + y’, twice] CS231 Boolean Algebra

  14. Standard forms of expressions • We can write expressions in many ways, but some ways are more useful than others • A sum of products (SOP) expression contains: • Only OR (sum) operations at the “outermost” level • Each term that is summed must be a product of literals • The advantage is that any sum of products expression can be implemented using a two-level circuit • literals and their complements at the “0th” level • AND gates at the first level • a single OR gate at the second level • This diagram uses some shorthands… • NOT gates are implicit • literals are reused • this is not okay in LogicWorks! f(x,y,z) = y’ + x’yz’ + xz CS231 Boolean Algebra

  15. Minterms • A minterm is a special product of literals, in which each input variable appears exactly once. • A function with n variables has 2n minterms (since each variable can appear complemented or not) • A three-variable function, such as f(x,y,z), has 23 = 8 minterms: • Each minterm is true for exactly one combination of inputs: x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Minterm Is true when… Shorthand x’y’z’ x=0, y=0, z=0 m0 x’y’z x=0, y=0, z=1 m1 x’yz’ x=0, y=1, z=0 m2 x’yz x=0, y=1, z=1 m3 xy’z’ x=1, y=0, z=0 m4 xy’z x=1, y=0, z=1 m5 xyz’ x=1, y=1, z=0 m6 xyz x=1, y=1, z=1 m7 CS231 Boolean Algebra

  16. Sum of minterms form • Every function can be written as a sum of minterms, which is a special kind of sum of products form • The sum of minterms form for any function is unique • If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. f = x’y’z’ + x’y’z + x’yz’ + x’yz + xyz’ = m0 + m1 + m2 + m3 + m6 = m(0,1,2,3,6) f’ = xy’z’ + xy’z + xyz = m4 + m5 + m7 = m(4,5,7) f’ contains all the minterms not in f CS231 Boolean Algebra

  17. The dual idea: products of sums • Just to keep you on your toes... • A product of sums (POS) expression contains: • Only AND (product) operations at the “outermost” level • Each term must be a sum of literals • Product of sums expressions can be implemented with two-level circuits • literals and their complements at the “0th” level • OR gates at the first level • a single AND gate at the second level • Compare this with sums of products f(x,y,z) = y’ (x’ + y + z’) (x + z) CS231 Boolean Algebra

  18. Maxterms • A maxterm is a sum of literals, in which each input variable appears exactly once. • A function with n variables has 2n maxterms • The maxterms for a three-variable function f(x,y,z): • Each maxterm is false for exactly one combination of inputs: x’ + y’ + z’ x’ + y’ + z x’ + y + z’ x’+ y + z x + y’ + z’ x + y’ + z x + y + z’ x + y + z Maxterm Is false when… Shorthand x + y + z x=0, y=0, z=0 M0 x + y + z’ x=0, y=0, z=1 M1 x + y’ + z x=0, y=1, z=0 M2 x + y’ + z’ x=0, y=1, z=1 M3 x’ + y + z x=1, y=0, z=0 M4 x’ + y + z’ x=1, y=0, z=1 M5 x’ + y’ + z x=1, y=1, z=0 M6 x’ + y’ + z’ x=1, y=1, z=1 M7 CS231 Boolean Algebra

  19. Product of maxterms form • Every function can be written as a uniqueproduct of maxterms • If you have a truth table for a function, you can write a product of maxterms expression by picking out the rows of the table where the function output is 0. (Be careful if you’re writing the actual literals!) f = (x’ + y + z)(x’ + y + z’)(x’ + y’ + z’) = M4 M5 M7 = M(4,5,7) f’ = (x + y + z)(x + y + z’)(x + y’ + z) (x + y’ + z’)(x’ + y’ + z) = M0 M1 M2 M3 M6 = M(0,1,2,3,6) f’ contains all the maxterms not in f CS231 Boolean Algebra

  20. Minterms and maxterms are related • Any minterm mi is the complement of the corresponding maxterm Mi • For example, m4’ = M4 because (xy’z’)’ = x’ + y + z Minterm Shorthand x’y’z’ m0 x’y’z m1 x’yz’ m2 x’yz m3 xy’z’ m4 xy’z m5 xyz’ m6 xyz m7 Maxterm Shorthand x + y + z M0 x + y + z’ M1 x + y’ + z M2 x + y’ + z’ M3 x’ + y + z M4 x’ + y + z’ M5 x’ + y’ + z M6 x’ + y’ + z’ M7 CS231 Boolean Algebra

  21. Converting between standard forms • We can convert a sum of minterms to a product of maxterms • In general, just replace the minterms with maxterms, using maxterm numbers that don’t appear in the sum of minterms: • The same thing works for converting from a product of maxterms to a sum of minterms From before f = m(0,1,2,3,6) and f’ = m(4,5,7) = m4 + m5 + m7 complementing (f’)’ = (m4 + m5 + m7)’ so f = m4’ m5’ m7’ [ DeMorgan’s law ] = M4 M5 M7 [ By the previous page ] = M(4,5,7) f = m(0,1,2,3,6) = M(4,5,7) CS231 Boolean Algebra

  22. Summary so far • So far: • A bunch of Boolean algebra trickery for simplifying expressions and circuits • The algebra guarantees us that the simplified circuit is equivalent to the original one • Introducing some standard forms and terminology • Next: • An alternative simplification method • We’ll start using all this stuff to build and analyze bigger, more useful, circuits CS231 Boolean Algebra

  23. Karnaugh maps • So far, we saw applications of Boolean logic to circuit design. • The basic Boolean operations are AND, OR and NOT. • These operations can be combined to form complex expressions, which can also be directly translated into a hardware circuit. • Boolean algebra helps us simplify expressions and circuits. • Next, we’ll look at a graphical technique for simplifying an expression into a minimal sum of products(MSP) form: • There are a minimal number of product terms in the expression. • Each term has a minimal number of literals. • Circuit-wise, this leads to a minimal two-level implementation. CS231 Boolean Algebra

  24. Review: Standard forms of expressions • We can write expressions in many ways, but some ways are more useful than others • A sum of products (SOP) expression contains: • Only OR (sum) operations at the “outermost” level • Each term that is summed must be a product of literals • The advantage is that any sum of products expression can be implemented using a two-level circuit • literals and their complements at the “0th” level • AND gates at the first level • a single OR gate at the second level • This diagram uses some shorthands… • NOT gates are implicit • literals are reused • this is not okay in LogicWorks! f(x,y,z) = y’ + x’yz’ + xz CS231 Boolean Algebra

  25. Terminology: Minterms • A minterm is a special product of literals, in which each input variable appears exactly once. • A function with n variables has 2n minterms (since each variable can appear complemented or not) • A three-variable function, such as f(x,y,z), has 23 = 8 minterms: • Each minterm is true for exactly one combination of inputs: x’y’z’ x’y’z x’yz’ x’yz xy’z’ xy’z xyz’ xyz Minterm Is true when… Shorthand x’y’z’ x=0, y=0, z=0 m0 x’y’z x=0, y=0, z=1 m1 x’yz’ x=0, y=1, z=0 m2 x’yz x=0, y=1, z=1 m3 xy’z’ x=1, y=0, z=0 m4 xy’z x=1, y=0, z=1 m5 xyz’ x=1, y=1, z=0 m6 xyz x=1, y=1, z=1 m7 CS231 Boolean Algebra

  26. Terminology: Sum of minterms form • Every function can be written as a sum of minterms, which is a special kind of sum of products form • The sum of minterms form for any function is unique • If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. f = x’y’z’ + x’y’z + x’yz’ + x’yz + xyz’ = m0 + m1 + m2 + m3 + m6 = m(0,1,2,3,6) f’ = xy’z’ + xy’z + xyz = m4 + m5 + m7 = m(4,5,7) f’ contains all the minterms not in f CS231 Boolean Algebra

  27. Re-arranging the truth table • A two-variable function has four possible minterms. We can re-arrange these minterms into a Karnaugh map. • Now we can easily see which minterms contain common literals. • Minterms on the left and right sides contain y’ and y respectively. • Minterms in the top and bottom rows contain x’ and x respectively. CS231 Boolean Algebra

  28. Karnaugh map simplifications • Imagine a two-variable sum of minterms: x’y’ + x’y • Both of these minterms appear in the top row of a Karnaugh map, which means that they both contain the literal x’. • What happens if you simplify this expression using Boolean algebra? x’y’ + x’y = x’(y’ + y) [ Distributive ] = x’  1 [ y + y’ = 1 ] = x’ [ x  1 = x ] CS231 Boolean Algebra

  29. More two-variable examples • Another example expression is x’y + xy. • Both minterms appear in the right side, where y is uncomplemented. • Thus, we can reduce x’y + xy to just y. • How about x’y’ + x’y + xy? • We have x’y’ + x’y in the top row, corresponding to x’. • There’s also x’y + xy in the right side, corresponding to y. • This whole expression can be reduced to x’ + y. CS231 Boolean Algebra

  30. A three-variable Karnaugh map • For a three-variable expression with inputs x, y, z, the arrangement of minterms is more tricky: • Another way to label the K-map (use whichever you like): CS231 Boolean Algebra

  31. Why the funny ordering? • With this ordering, any group of 2, 4 or 8 adjacent squares on the map contains common literals that can be factored out. • “Adjacency” includes wrapping around the left and right sides: • We’ll use this property of adjacent squares to do our simplifications. x’y’z + x’yz = x’z(y’ + y) = x’z  1 = x’z • x’y’z’ + xy’z’ + x’yz’ + xyz’ • = z’(x’y’ + xy’ + x’y + xy) • = z’(y’(x’ + x) + y(x’ + x)) • = z’(y’+y) • = z’ CS231 Boolean Algebra

  32. f(x,y,z) = x’y’z + xy’z + xyz’ + xyz = m1 + m5 + m6 + m7 Example K-map simplification • Let’s consider simplifying f(x,y,z) = xy + y’z + xz. • First, you should convert the expression into a sum of minterms form, if it’s not already. • The easiest way to do this is to make a truth table for the function, and then read off the minterms. • You can either write out the literals or use the minterm shorthand. • Here is the truth table and sum of minterms for our example: CS231 Boolean Algebra

  33. Unsimplifying expressions • You can also convert the expression to a sum of minterms with Boolean algebra. • Apply the distributive law in reverse to add in missing variables. • Very few people actually do this, but it’s occasionally useful. • In both cases, we’re actually “unsimplifying” our example expression. • The resulting expression is larger than the original one! • But having all the individual minterms makes it easy to combine them together with the K-map. xy + y’z + xz = (xy  1) + (y’z  1) + (xz  1) = (xy  (z’ + z)) + (y’z  (x’ + x)) + (xz  (y’ + y)) = (xyz’ + xyz) + (x’y’z + xy’z) + (xy’z + xyz) = xyz’ + xyz + x’y’z + xy’z CS231 Boolean Algebra

  34. f(x,y,z) = x’y’z + xy’z + xyz’ + xyz f(x,y,z) = m1 + m5 + m6 + m7 Making the example K-map • Next up is drawing and filling in the K-map. • Put 1s in the map for each minterm, and 0s in the other squares. • You can use either the minterm products or the shorthand to show you where the 1s and 0s belong. • In our example, we can write f(x,y,z) in two equivalent ways. • In either case, the resulting K-map is shown below. CS231 Boolean Algebra

  35. K-maps from truth tables • You can also fill in the K-map directly from a truth table. • The output in row i of the table goes into square mi of the K-map. • Remember that the rightmost columns of the K-map are “switched.” CS231 Boolean Algebra

  36. Grouping the minterms together • The most difficult step is grouping together all the 1s in the K-map. • Make rectangles around groups of one, two, four or eight 1s. • All of the 1s in the map should be included in at least one rectangle. • Do not include any of the 0s. • Each group corresponds to one product term. For the simplest result: • Make as few rectangles as possible, to minimize the number of products in the final expression. • Make each rectangle as large as possible, to minimize the number of literals in each term. • It’s all right for rectangles to overlap, if that makes them larger. CS231 Boolean Algebra

  37. Reading the MSP from the K-map • Finally, you can find the MSP. • Each rectangle corresponds to one product term. • The product is determined by finding the common literals in that rectangle. • For our example, we find that xy + y’z + xz = y’z +xy. (This is one of the additional algebraic laws from last time.) CS231 Boolean Algebra

  38. Practice K-map 1 • Simplify the sum of minterms m1 + m3 + m5 + m6. CS231 Boolean Algebra

  39. Solutions for practice K-map 1 • Here is the filled in K-map, with all groups shown. • The magenta and green groups overlap, which makes each of them as large as possible. • Minterm m6 is in a group all by its lonesome. • The final MSP here is x’z + y’z +xyz’. CS231 Boolean Algebra

  40. Four-variable K-maps • We can do four-variable expressions too! • The minterms in the third and fourth columns, and in the third and fourth rows, are switched around. • Again, this ensures that adjacent squares have common literals. • Grouping minterms is similar to the three-variable case, but: • You can have rectangular groups of 1, 2, 4, 8 or 16 minterms. • You can wrap around all four sides. CS231 Boolean Algebra

  41. Example: Simplify m0+m2+m5+m8+m10+m13 • The expression is already a sum of minterms, so here’s the K-map: • We can make the following groups, resulting in the MSP x’z’ + xy’z. CS231 Boolean Algebra

  42. y’z + yz’ + xy y’z + yz’ + xz K-maps can be tricky! • There may not necessarily be a unique MSP. The K-map below yields two valid and equivalent MSPs, because there are two possible ways to include minterm m7. • Remember that overlapping groups is possible, as shown above. CS231 Boolean Algebra

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