Digital Integrated Circuits - week nine -
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Transcript of Digital Integrated Circuits - week nine -
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Gheorghe M. Ştefan http://arh.pub.ro/gstefan/
- 2014 -
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Veitch-Karnaugh diagramsm0 = a’b’c’d’ Are based on the minimal Hamming distance
m1 = a’b’c’d between adjacent cells
m2 = a’b’cd’
m3 = a’b’cd
m4 = a’bc’d’
m5 = a’bc’d
m6 = a’bcd’
m7 = a’bcd
m8 = ab’c’d’
m9 = ab’c’d
. . .m15 = abcd
Veitch version2014 Digital Integrated Circuits - week nine 2
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Karnaugh version
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Minimizing logic functionssmallest number, of rectangular surfaces of 1’s, with maximal area,containing 2i 1’s, and including all 1’s
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“don’t care”s
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a’b +a’c b + c
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V-K’s with included functionsIs a two-step process:1.Only 1’s2.1’s become “don’t’ care”s
(bc’d + a’bc’ + b’c) + (acde + c’de’)
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Transition diagrams
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Moore version
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Mealy version
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Verilog modulesImmediate Moore version for ‘bb
detector’
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State transition function:
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Output transition function:
For the delayed version: always @(posedge clock)
case(state)
init_state : out <= no;
. . .
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For the immediate Mealy version of ‘bb detector’ only the output function is different:
For the delayed version the change is similar as for
Moore version 2014 Digital Integrated Circuits - week nine 15
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Automata are complex circuitsThe number of lines in the Verilog description:
Are equal with the number of states for the behavioral descriptions
Are equal with the number of gates used for the structural descriptions
The transition diagrams have the size in O(|Q| log |Q|)
Because they are complex we must keep them small
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State coding
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First coding
Second coding
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First version: Second version:
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Coding stylesMinimal variation encoding:
successive states are coded with minimal Hamming distance
Reduced dependency encoding:minimal Hamming distance for states
preceded by the same stateIncremental encoding:
whenever possible successive state are coded by incremented values
One-hot state encoding: one bit per state
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Minimal variation encoding
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Reduced dependency encoding
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Reduced dependency on both, X0 and X1 has no solution
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Parasitic effects in automata
Automata with asynchronous inputs
Automata with asynchronous outputsHazard generated by asynchronous inputs
Propagation hazardDynamic hazard2014 Digital Integrated Circuits - week nine 22
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Automata with asynchronous inputs
If the input variable switches asynchronously the input of the state register can vary in the prohibited time – tsu + t+ +th
No problems if only one bit switches:00 -> 01 => 00 -> 01 or 00 -> 00
Big problems if more than one bit switch:01 -> 10 => 01 -> 10 or 01 -> 00 or 01 -> 11 or 01 -> 01
The automaton evolves wrongly in the state space
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The solution: reduced dependency state coding
What can be done if there are state transitions depending on two or more asynchronous input variable ?
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Asynchronous outputs:hazard due to asynchronous inputs
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Asynchronous outputs:propagation hazard
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Avoiding propagation hazard Partial protection Full
protection
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Dynamic hazard
Due to many-level combinational circuits (ex: carry propagation)
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Fundamental limitsThe asynchronous input bits can be considered only
independently in distinct states Immediate Mealy automata with asynchronous
inputs can not have actual implementations (unpredictable outputs)
Delayed Mealy automata with asynchronous inputs can not have actual implementations (the state and outputs could evolve uncorrelated)
Hazard free Moore with asynchronous inputs have no actual solution (state must be encoded simultaneously with minimal variation and reduced dependency)
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Home work 9 Problem 1:
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