Transcript Combinational logic minimization : Covering, Multiple Outputs, Hazards

The covering procedure
The covering procedure
The covering procedure
• Remove rows with essential PI’s and any
columns with x’s in those rows
The covering procedure
• Remove rows that are covered by other rows
• Remove columns that cover other columns
• Why?
The covering procedure
• Remove rows that are covered by other rows
• Remove columns that cover other columns
The covering procedure
• Rows  PI’s
– Covering row takes care of more minterms
– Minterms included in a smaller (covered) row are
also included in the bigger (covering) one
– Can discard the small ones and use only the
covering row; minterm coverage is preserved
• Columns  min/max terms
– Whenever a min/max term corresponding to a
covered (smaller) column is included by some PI,
the min/max term corresponding to the covering
(bigger) column also gets included
– Covering column can be dropped
– Reduces # of PI’s that include this min/max term
Cyclic PI charts
• Cyclic PI charts have no essential PI’s
– Cannot be reduced by rules 1 and 2
• Example of cyclic PI chart of 3 variables
BC
A
00 01 11 10
0
1 1 1
1 1 1
1
Cyclic PI chart
chosen PI
• Cyclic PI charts have no essential PI’s
– Select the row with max number of x’s (randomly if
more than one); PI1 in this example
Cyclic PI chart
After removing PI1,
apply rules 1 and 2.
Remove covered
PI2 and PI6
PI3 and PI5 cover the
resulting chart.
Minimal cover: PI1, PI3, PI5
Cyclic PI charts
• Example of cyclic PI chart of 4 variables
1
1
1
1
1 1
• Q: if PI’s covering 4 minterms are allowed, can
one create a cyclic PI chart where no PIs are
essential?
Cyclic PI charts
• A: yes
1
1
1
1
1
1
1
1
1 1
1 1
• Q: what about a 4 variable K-map and groups
of 8 ones?
– In general n variable functions with a K-map and
PIs covering 2n-1 min/max terms – can there be a
cyclic PI chart?
Incompletely specified functions
• When some of the minterms can be either 0 or
1, we can denote them by ‘d’ (don’t care)
• When simplifying, we use ‘d’s to generate PIs,
but do not include them in the PI chart
Incompletely specified functions
Incompletely specified functions
Multiple simultaneous outputs
Multiple simultaneous outputs
Multiple simultaneous outputs
• In general, # of lists ≤ n+1 (n = # variables)
‘d’s are not in the charts,
but are used for PIs
• List 1, group 1
• Group 2, list 2
• Group 3, list 3
Multiple simultaneous outputs
Why select
PI3 over
PI11?
• PIs from higher-numbered list are likely to
cover more PIs (not always true: don’t cares)
Multiple simultaneous outputs
Hazards and K-maps
• Static hazard or glitch: unwanted output
transition when inputs change and the output
should have remained the same
• For simplicity consider only a single input
changes at a time
• Different gates have different propagation
delays
Hazards and K-maps
t1
t3
t2
t1 = t2 = t3
Hazards and K-maps
t1
t3
t2
t1 > t2 > t3
Hazards and K-maps
• A hazard exists when a changing input
requires corresponding minterms/maxterms to
be covered by different product/sum terms
• Remove hazards by bridging the gaps on the
K-map:
Hazards and K-maps
• Hazard-free circuit:
• Cover each pair
of adjacent
minterms by a
different product term
• Deliberate redundancy like this makes circuits
impossible to test completely
• Static 1 hazards: in SOP circuits: 1  0  1
• Static 0 hazards: in POS circuits: 0  1  0
Hazards and K-maps
• Static 0 hazards in POS circuits:
• Identify the hazard(s): how many? Where?
Hazards and K-maps
• Hazards identified and fixed? What is missing?
Hazards and K-maps
• Dynamic hazards:
– When input change requires output change
– Occur when output makes more than one transition
• Always result from static hazards elsewhere
– Eliminating the static hazards eliminates the
dynamic ones as well
Prime number detector: F =  (1, 2, 3, 5, 7, 11, 13)
00
N3 N2
01
11
10
00
01
x
x
x
N1 N0
11
x
10
x
x
x
0001
0011
00-1
0101
0111
01-1
0--1
Karnaugh maps: 2, 3, and 4 variable
Example:
F = X’YZ’ + XZ + Y’Z
Another example:
Prime implicants
(maximal clusters)
Prime number detector
Prime number detector
Another example:
distinguished cell: covered by only one prime implicant
essential prime implicant: contains distinguished cell
Another example:
primes, distinguished cells, essentials
Selecting essentials leaves an uncovered cell
cover with simpler implicant: W’Z
Eclipsing (in reduced map)
P eclipses Q if P covers all of Q’s ones
if P is no more expensive (same or fewer literals),
then choose P over Q
Alas, no essential prime implicants
branching: choose a cell and examine all implicants
that cover that cell
Don’t cares....
Multiple functions
can use separate Karnaugh maps
...or can manage to find common terms...
For more than 6 input variables,
Karnaugh maps are difficult to manipulate
Need computer program....
Quine-McCluskey algorithm
typedef unsigned short WORD; /* assume 16-bit registers */
struct cube {
WORD t; /* marks uncomplemented variables */
WORD f; /* marks complemented variables */
}
typedef struct cube CUBE;
CUBE P1, x, y, z;
X15 X12 X10 X 9 X 4 X1 X 0
Example in four variables
Equation:
w x’ y z’ + w’ x’ y z’ = x’ y z’
Karnaugh map:
wx
00
01
yz 00
01
11
10 1
11
10
1
Cubes (last four bits):
1010  0010 = 1000 ==> single one in common position ==> combinable
0101  1101 = 1000
1010 & 0010 = 0010 ==> w now missing, new cube corresponds to z’ y z’
0101 & 1101 = 0101
Start with minterms (0-cubes)
Combine when possible to form (1-cubes)....
Example: w’xy’z + wxy’z + w’xyz + wxyz = xz
wx
00
yz 00
01
11
10
01
11
1
1
1
1
Cubes: 0101
1010
0101
1010
1101
0010
0111
1000
1111
0000
10
1101
0010
0101  1101 = 1000
1010  0010 = 1000
0111 1111
1000 0000
0101 & 1101 = 0101
1010 & 0010 = 0010
Start with minterms (0-cubes)
Combine when possible to form (1-cubes)....
Example: w’xy’z + wxy’z + w’xyz + wxyz = xz
wx
00
yz 00
01
11
10
01
11
1
1
1
1

Cubes: 0101
1010
10
1101
0010
0101  1101 = 1000
1010  0010 = 1000
0101 & 1101 = 0101
1010 & 0010 = 0010
0111 1111
1000 0000
0101
1010
 1101 0101
0010 0010
0111
1000
1111
0000
wx
00
yz 00
01
11
10
01
11
1
1
1
1
10
Start with minterms (0-cubes)
Combine when possible to form (1-cubes)....
Example: w’xy’z + wxy’z + w’xyz + wxyz = xz
wx
00
yz 00
01
11
10
01
11
1
1
1
1

Cubes: 0101
1010
10

1101
0010

0111 1111
1000 0000
0101
1010
wx
00
 1101 0101
0010 0010
yz 00
01
11
10
 0111 0101
1000 1000
 1111
0000
1101
0000
0111
0000
01
11
1
1
1
1
10
Continue to form 2-cubes
Example: w’xy’z + wxy’z + w’xyz + wxyz = xz
wx
00
yz 00
01
11
10
01
11
1
1
1
1

Cubes: 0101
0010
10

0101
1000
0101
0010
0101
1000
 1101
0000
 0111 0101
0000 0000
0101
0000
1101
0000
0111
0000
Quine-McCluskey Algorithm:
Read in all minterms (0-cubes);
mark all 0-cubes “uncovered”;
for (m = 0; m < Nvar; m++)
for (j = 0; j < Ncubes[m]; j++)
for (k = j + 1; k < Ncubes[m]; k++)
if (combinable(cube[m][j], cube[m][k])) {
mark cube[m][j] and cube[m][k] “covered”
temp = combined cube;
if (temp not already at level m + 1) {
add temp to level m + 1;
mark temp “uncovered”
}
}
Manual algorithm: F = (2, 5, 7, 9, 13, 15) (variables WXYZ)
uncovered terms
0010
0101
1001
01-1
-101
1-01
0111
1101
-111
11-1
1111
-1-1
Manual algorithm: F = (2, 5, 7, 9, 13, 15) (variables WXYZ)
WY’Z
0010
0101 x
1001 x
0111 x
1101 x
1111
01-1 x
-101 x
1-01
-111
11-1
-1-1
XZ
x
x
WX
01
11
10
01
1
1
1
11
1
1
x
00
00
YZ
W’X’YZ’
10
1
Prime
Implicants
0010
1-01
-1-1
Minterms
2
5
7
9
13
15
x
x
x
x
x
x
x
distinguished minterms (cells): 2, 5, 7, 9, 15
essential prime implicants: 0010, 1-01, -1-1 (all)
F = 0010 + 1-01 + -1-1 = W’X’YZ’ + WY’Z + XZ
Not all prime implicants are necessarily essential
distinguished cells
essential implicants
remainder
Minimal form: A + E + C
C eclipses B and D
Not all prime implicants are necessarily essential
distinguished cells
essential implicants
remainder
Minimal form: A + E + C
C eclipses B and D
Not all prime implicants are necessarily essential
distinguished cells
essential implicants
remainder
Minimal form: A + E + C
C eclipses B and D
Static hazard: X = Y = 1, Z falls from 1 to 0
XZ’
Z’
Z’
XZ’
Reconstruct Karnaugh map:
F = XZ’ + YZ = XYZ’ + XY’Z’ + XYZ + X’YZ
Consensus term
Solution: add consensus term
XZ’
Z’
Z’
XZ’
XZ’
Z’
Consensus term