Digital Integrated Circuits - week nine -

Gheorghe M. Ştefan http://arh.pub.ro/gstefan/

- 2014 -

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

2014 Digital Integrated Circuits - week nine 3

Karnaugh version


Minimizing logic functionssmallest number, of rectangular surfaces of 1’s, with maximal area,containing 2i 1’s, and including all 1’s


“don’t care”s


a’b +a’c b + c

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’)


Transition diagrams


Moore version


Mealy version


Verilog modulesImmediate Moore version for ‘bb

detector’


State transition function:


Output transition function:

For the delayed version: always @(posedge clock)

case(state)

init_state : out <= no;

. . .


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

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


State coding


First coding

Second coding

First version: Second version:


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


Minimal variation encoding


Reduced dependency encoding


Reduced dependency on both, X0 and X1 has no solution

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

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


The solution: reduced dependency state coding

What can be done if there are state transitions depending on two or more asynchronous input variable ?


Asynchronous outputs:hazard due to asynchronous inputs


Asynchronous outputs:propagation hazard


Avoiding propagation hazard Partial protection Full

protection


Dynamic hazard

Due to many-level combinational circuits (ex: carry propagation)


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)


Home work 9 Problem 1:


Digital Integrated Circuits - week nine -

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