Outcomes. Unit 9. Logic Function Synthesis KARNAUGH MAPS. Implementing Combinational Functions with Karnaugh Maps


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1 .. Outcomes Unit I can use Karnaugh maps to synthesize combinational functions with several outputs I can determine the appropriate size and contents of a memory to implement any logic function (i.e. truth table) Implementing Combinational Functions with.. Logic Function Synthesis A new way to synthesize your logic functions KARNAUGH MAPS Given a function description as a T.T. or canonical form, how can we arrive at a circuit implementation or equation (i.e. perform logic synthesis)? First method Minterms / maxterms Can simplify to find minimal level implementation Use a decoder + gate per output New, second method Minimal level implementation (though not necessarily minimal , , level implementation)
2 . Gray Code. If used correctly, will always yield a minimal, implementation There may be a more minimal level, level,  level implementation but Kmaps produce the minimal twolevel (SOP or POS) implementation Represent the truth table graphically as a series of adjacent that allows a human to see where variables will cancel Different than normal binary ordering Reflective code When you add the (n+) th bit, reflect all the previous nbit combinations Consecutive code words differ by only bit differ by only bit bit Gray code when you move to the next bit, reflect the previous combinations differ by only bit differ by only bit bit Gray code Karnaugh Map Construction.. Every square represents input combination Must label axes in Gray code order Fill in squares with given function values XY Z F=Σ XYZ (,,,) G=Σ YZ (,,,,,,,,,,) Variable Karnaugh Map YZ Variable Karnaugh Map YZ
3 .. Squares with a '' represent minterms Squares with a '' represent maxterms YZ Maxterm: w + x + y + z Maxterm: w + x + y + z Minterm: w x y z Minterm: w x y z Groups (of,,, etc.) of adjacent s will always simplify to smaller product term than just individual minterms XY Z F=Σ XYZ (,,,,) Variable Karnaugh Map.. Groups of adjacent s will always simplify to smaller product term than just individual minterms Adjacent squares differ by variable This will allow us to use T = AB + AB = A or T = (A+B )(A+B) = A XY Z F=Σ XYZ (,,,,) Variable Karnaugh Map = m + m + m + m = x y z + x yz + xyz + xy z = z (x y + x y + xy + xy ) = z (x (y +y) + x(y+y )) = z (x +x) = z = m + m = xy z + xy z = xy (z +z) = xy Variable Karnaugh Map Variable Karnaugh Map Difference in X: & XY x yz + xyz Z = yz Difference in Y: & x yz + x y z = x z Difference in Z: & x yz + x yz = = x y = Adjacent squares = differ by bit = YZ = = = = = Adjacent squares differ by bit
4 . KMap Grouping Rules. adjacent s (or s) differ by only one variable adjacent s (or s) differ by two variables,, adjacent s (or s) differ by,, variables By grouping adjacent squares with s (or s) in them, we can come up with a simplified expression using T (or T for s) w x y z + w x y z + w x y z + w x y z = w z w z are constant while all combos of x and y are present (x y, x y, xy, xy) YZ (w +x +y+z) (w +x +y+z ) = (w +x +y) w x y z + w x y z = w y z Cover the 's [=onset] or 's [=offset] with groups as possible, but make those groups as possible Make them as large as possible even if it means "covering" a (or ) that's already a member of another group Make groups of,... and they must be rectangular or square in shape. Wrapping is legal Group These KMaps.. XY Z XY Z YZ YZ Cover the remaining with the largest group possible even if it reuses already covered s
5 .. Group This Groups can wrap around from: Right to left Top to bottom Corners YZ YZ YZ F = X Z + Z F = X Z.. KMap Translation Rules (SOP) When translating a group of s, find the variable values that are constant for each square in the group and translate only those variables values to a product term Grouping s yields SOP When translating a group of s, again find the variable values that are constant for each square in the group and translate only those variable values to a sum term Grouping s yields POS YZ F =
6 .. (SOP) (SOP) Y YZ W YZ Z F = Y F = Y + W Z +.. (SOP) (POS) YZ Z X X YZ F = Y + W Z + X Z F =
7 .. (POS) (POS) Y,Z YZ Y YZ F = (Y+Z) F = (Y+Z)(W +X +Y).. Exercises Groups can wrap around from: Right to left Top to bottom Corners X X YZ Z X X YZ Z YZ YZ Z F SOP = F POS = Z Z P=Σ XYZ (,,,) F = X Z + Z F = X Z P=
8 .. No Redundant Groups Multiple Minimal Expressions YZ This group does not cover new squares that are not already part of another essential grouping For some functions, groupings exist which will lead to alternate minimal Pick one DD DD Best way to cover this??. Terminology. If time permits FORMAL TERMINOLOGY FOR KMAPS Implicant: A product term (grouping of s) that covers a subset of cases where F= the product term is said to imply F because if the product term evaluates to then F= Prime Implicant: The largest grouping of s (smallest product term) that can be made Essential Prime Implicant: A prime implicant (product term) that is needed to cover all the s of F
9 .. Implicant Examples Implicant Examples YZ Not PRIME because not as large as possible YZ Not PRIME because not as large as possible.. Implicant Examples Implicant Examples YZ Not PRIME because not as large as possible An essential prime implicant An essential prime implicant (largest grouping possible, that must be included to cover all s) An essential prime implicant YZ Not PRIME because not as large as possible An essential prime implicant An essential prime implicant (largest grouping possible, that must be included to cover all s)
10 .. Implicant Examples Implicant Examples An essential prime implicant A prime implicant, but not an ESSENTIAL implicant because it is not needed to cover all s in the function YZ Not PRIME because not as large as possible An essential prime implicant An essential prime implicant (largest grouping possible, that must be included to cover all s), but not a PRIME implicant because it is not as large as possible (should expand to combo s and ) YZ An essential prime implicant (largest grouping possible, that must be included to cover all s) An essential prime implicant KMap Grouping Rules.. Make groups (implicants) of,,,,... and they must be rectangular or square in shape. Include the minimum number of essential prime implicants Use only essential prime implicants (i.e. as few groups as possible to cover all s) Ensure that you are using prime implicants (i.e. Always make groups as large as possible reusing squares if necessary) & VARIABLE KMAPS
11 .. Variable KMap Variable If we have a variable function we need a square KMap. Will an x matrix work? Recall Kmaps work because adjacent squares differ by bit How many adjacencies should we have for a given square?!! But drawn in dimensions we can t have adjacencies. V YZ To represent the adjacencies of a variable function [e.g. f(v,w,x,y,z)], imagine two x KMaps stacked on top of each other Adjacency across the two maps YZ These are adjacent YZ Traditional adjacencies still apply (Note: v is constant for that group and should be included) => v xy Adjacencies across the two maps apply (Now v is not constant) => w xy V= V= F = v xy + w xy adjacencies for variables (Stack of four x maps) Variable YZ YZ Group of U,V=, Not adjacent. YZ YZ U,V=, DON'T CARE OUTPUTS. Group of U,V=, U,V=,
12 .. Don tcares Invalid Input Combinations Sometimes there are certain input combinations that are illegal (due to physical or other external constraints) The outputs for the illegal inputs are don tcares The output can either be or since the inputs can never occur Don tcares can be included in groups of or groups of when grouping in KMaps Use them to make as big of groups as possible Given intermediate functions F and F, how could you use AND, OR, NOT to make G Notice certain F,F combinations never occur in G(x,y,z) what should we make their output in the T.T. X Y Z F F G X Y Z F F G F F G Use 'Don't care' outputs as wildcards (e.g. the blank tile in Scrabble TM ). They can be either or whatever helps make bigger groups to cover the ACTUAL 's.. Invalid Input Combinations An example of where Don'tCares may come into play is Binary Coded Decimal (BCD) Rather than convert a decimal number to unsigned binary (i.e. summing increasing powers of ) we can represent each decimal digit as a separate group of bits (with weights,,, for each group of bits) Combinations  cannot occur! BCD Representation: () Important: BCD represent each decimal digit with a separate group of bits This is not the binary representation of, it is the Binary Coded Decimal (BCD) representation Don t Care Example DD D D D D GT DD d d d d d d d d d d d d DD DD d d d d d d GT SOP = GT POS =
13 .. Don t Cares Don t Cares Reuse d s to make as large a group as possible to cover,, & You can use d s when grouping s and converting to POS YZ d YZ d d d d d d d d d d d d d d d d d d d F = Z + Y Use these d s to make a group of d d d d F = Y+Z.. Designing Circuits w/ KMaps bit Number Decrementer Given a description Block Diagram Truth Table KMap for each output bit (each output bit is a separate function of the inputs) bit unsigned decrementer (Z = X) If X[:] = then Z[:] =, etc. X X X Z Z Z X[:] bit Unsigned Decrementer Z[:] X X X X X X X X X Z = X X + X X + X X X Z = X X + X X Z = X
14 .. Squaring Circuit Design a combinational circuit that accepts a bit number and generates an output binary number equal to the square of the input number. (B = A ) Using bits we can represent the numbers from to. The possible squared values range from to. Thus to represent the possible outputs we need how many bits? bit Squaring Circuit Inputs Outputs AA A A A A B B B B B B B=A A B = AA A B = AA A B = bit Squaring Circuit A A A. B B B B B B
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