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EE 365

Combinational-Circuit Synthesis

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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, including corner cases• 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 (next lec.)

• “Multiply out”:

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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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Brute-force design

• Truth table --> canonical sum (sum of minterms)

• Example:prime-number 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 0 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

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Algebraic simplification

• Theorem T8,

• Reduce number of gates and gate inputs

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Resulting circuit

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Visualizing T10 -- Karnaugh maps

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3-variable Karnaugh map

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Example: F = (1,2,5,7)

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Karnaugh-map usage

• Plot 1s corresponding to minterms of function.• Circle largest possible rectangular sets of 1s.

– # of 1s in set must be power of 2– OK to cross edges

• Read off product terms, one per circled set.– Variable is 1 ==> include variable– Variable is 0 ==> include complement of variable– Variable is both 0 and 1 ==> variable not included

• Circled sets and corresponding product terms are called “prime implicants”

• Minimum number of gates and gate inputs

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Prime-number detector (again)

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• When we solved algebraically, we missed one simplification -- the circuit below has three less gate inputs.

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Another example

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Yet another example

• Distinguished 1 cells• Essential prime implicants

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Quine-McCluskey algorithm

• This process can be made into a program, using appropriate algorithms and data structures.– Guaranteed to find “minimal” solution

• Required computation has exponential complexity (run time and storage)-- works well for functions with up to 8-12 variables, but quickly blows up for larger problems.

• Heuristic programs (e.g., Espresso) used for larger problems, usually give minimal results.

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Lots of possibilities

• Can follow a “dual” procedure to find minimal products of sums (OR-AND realization)

• Can modify procedure to handle don’t-care input combinations.

• Can draw Karnaugh maps with up to six variables.

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Real-World Logic Design

• Lots more than 6 inputs -- can’t use Karnaugh maps

• Design correctness more important than gate minimization– Use “higher-level language” to specify logic

operations

• Use programs to manipulate logic expressions and minimize logic.

• PALASM, ABEL, CUPL -- developed for PLDs• VHDL, Verilog -- developed for ASICs