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Signal Expressions
•Multiply out:
F = ((X + Y) Z) + (X Y Z)
= (X Z) + (Y Z) + (X Y Z)
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New circuit, same functionSum of Products
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Product of Sums Form
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Sum-of-products form
AND-OR
NAND-NAND
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Product-of-sums form
OR-AND
NOR-NOR
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Shortcut: Symbol substitution
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Different circuit, same function
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Combinational-Circuit Analysis
• Combinational circuits -- outputs depend only on current inputs (not on history).
• Kinds of combinational analysis:– exhaustive (truth table)
– algebraic (expressions)
– simulation / test bench• Write functional description in HDL
• Define test conditions / test vectors
• Compare circuit output with functional description (or known-good realization)
• Repeat for “random” test vectors
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Combinational-Circuit Design
• Sometimes you can write an equation or equations directly using “logic” (the kind in your brain).
• Example (alarm circuit):
• Corresponding circuit:
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Alarm-circuit transformation
• Sum-of-products form– Useful for programmable logic devices
• “Multiply out”:
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Brute-force design
• Truth table --> canonical sum (sum of minterms)
• Example:prime-number & 1 detector– 4-bit input, N3N2N1N0
row N3 N2 N1 N0 F 0 0 0 0 0 0 1 0 0 0 1 1 2 0 0 1 0 1 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 1 6 0 1 1 0 0 7 0 1 1 1 1 8 1 0 0 0 0 9 1 0 0 1 010 1 0 1 0 011 1 0 1 1 112 1 1 0 0 013 1 1 0 1 114 1 1 1 0 015 1 1 1 1 0
F = (1,2,3,5,7,11,13)
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Minterm list --> canonical sum
Recall T10
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Algebraic simplification
• Theorem T10
• Reduce number of gates and gate inputs
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Resulting circuit
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Circuit Optimization
• Goal: To obtain the simplest implementation for a given function
• Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm
• Optimization requires a cost criterion to measure the simplicity of a circuit
• Two distinct cost criteria we will use:– Literal cost (L)– Gate input cost (G) and cost with NOTs (GN)
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• Literal – a variable or it complement• Literal cost – the number of literal appearances in
a Boolean expression corresponding to the logic circuit diagram
• Examples:– F = BD + AB’C + AC’D’ L = 8
– F = BD + AB’C + AB’D’ + ABC’ L =
– F = (A + B)(A + D)(B + C +D’)(B’ + C’ + D) L =
– Which solution is best?
Literal Cost
1110
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Gate Input Cost• Gate input costs - the number of inputs to the gates in the implementation
corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted)
• For SOP and POS equations, it can be found from the equation(s) by finding the sum of:– all literal appearances
– the number of terms excluding terms consisting only of a single literal,(G) and
– optionally, the number of distinct complemented single literals (GN).
• Example:
– F = BD + AB’C + AC’D’ G = 11, GN = 14
– F = BD + AB’C + AB’D’ + ABC’ G = , GN =
– F = (A + B)(A + D)(B + C +D’)(B’ + C’ + D) G = , GN =
– Which solution is best?
15 19
14 17
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• Example 1:
• F = A + B C + B’C’
Cost Criteria (continued)
A
BC
F
L = 5
L (literal count) counts the AND inputs and the single literal OR input.
G = L + 2 = 7
G (gate input count) adds the remaining OR gate inputs
GN = G + 2 = 9
GN(gate input count with NOTs) adds the inverter inputs
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• Example 2:
• F = A B C + A’B’C’
• L = 6 G = 8 GN = 11
• F = (A + C’ )( B’ + C)( A’ + B)
• L = 6 G = 9 GN = 12• Same function and same
literal cost
• But first circuit has bettergate input count and bettergate input count with NOTs
• Select it!
Cost Criteria (continued)ABC
F
F
ABC
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Boolean Function Optimization
• Minimizing the gate input (or literal) cost of a (a set of) Boolean equation(s) reduces circuit cost.
• We choose gate input cost.
• Boolean Algebra and graphical techniques are tools to minimize cost criteria values.
• Some important questions:– When do we stop trying to reduce the cost?
– Do we know when we have a minimum cost?
• Treat optimum or near-optimum cost functionsfor two-level (SOP and POS) circuits first.
• Introduce a graphical technique using Karnaugh maps (K-maps, for short)
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Karnaugh Maps (K-map)
• A K-map is a collection of squares– Each square represents a minterm– The collection of squares is a graphical representation of a
Boolean function– Adjacent squares differ in the value of one variable– Alternative algebraic expressions for the same function are
derived by recognizing patterns of squares
• The K-map can be viewed as– A reorganized version of the truth table– A topologically-warped Venn diagram as used to visualize sets in
algebra of sets
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Some Uses of K-Maps
• Provide a means for:– Finding optimum or near optimum
• SOP and POS standard forms, and• two-level AND/OR and OR/AND circuit implementations
for functions with small numbers of variables– Visualizing concepts related to manipulating
Boolean expressions, and– Demonstrating concepts used by computer-aided
design programs to simplify large circuits
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Two Variable Maps• A 2-variable Karnaugh Map:
– Note that minterm m0 andminterm m1 are “adjacent”and differ in the value of thevariable y
– Similarly, minterm m0 and
minterm m2 differ in the x variable.
– Also, m1 and m3 differ in the x variable as well. – Finally, m2 and m3 differ in the value of the variable y
y = 0 y = 1
x = 0 m
0 = m1 =
x = 1 m2 =
m
3 =
yx yx
yx yx
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K-Map and Truth Tables
• The K-Map is just a different form of the truth table.
• Example – Two variable function:
– We choose a,b,c and d from the set {0,1} to implement a
particular function, F(x,y).
Function TableK-MapInput
Values (x,y)
Function Value F(x,y)
0 0 a 0 1 b 1 0 c 1 1 d
y = 0 y = 1 x = 0 a b x = 1 c d
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K-Map Function Representation
• Example: F(x,y) = x
• For function F(x,y), the two adjacent cells containing 1’s can be combined using the Minimization Theorem:
F = x y = 0 y = 1
x = 0 0 0
x = 1 1 1
xyxyx)y,x(F
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K-Map Function Representation
• Example: G(x,y) = x + y
• For G(x,y), two pairs of adjacent cells containing 1’s can be combined using the Minimization Theorem:
G = x+y y = 0 y = 1
x = 0 0 1
x = 1 1 1
yxyxxyyxyx)y,x(G
Duplicate x y