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Transcript of Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section...
![Page 1: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/1.jpg)
1
Gate-Level Minimization
section instructor:Ufuk Çelikcan
![Page 2: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/2.jpg)
2
Complexity of Digital Circuits• Directly related to the complexity of the algebraic
expression we use to build the circuit.
• Truth table
– may lead to different implementations
– Question: which one to use?
• Optimization techniques of algebraic expressions
– So far, ad hoc.
– Need more systematic (algorithmic) way
• Karnaugh (K-) map technique
• Quine-McCluskey
• Espresso
![Page 3: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/3.jpg)
3
Two-Variable K-Map• Two variables: x and y
4 minterms:• m0 = x’y’ 00
• m1 = x’y 01
• m2 = xy’ 10
• m3 = xy 11
y
x 0 1
0 m0 m1
1 m2 m3
y
x 0 1
0 x’y’ x’y
1 xy’ xy
![Page 4: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/4.jpg)
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Example: Two-Variable K-Map
F = m0 + m1 + m2 = x’y’ + x’y + xy’
F = …
F = …
F = …
y
x 0 1
0 1 1
1 1 0
![Page 5: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/5.jpg)
Remember the Shortcuts
x + x = x ↔ x · x = x
x + 1 = 1 ↔ x · 0 = 0
x + xy = x ↔ x · xy = x [Absorption]
(x + y)’ = x’ · y’ ↔ (x.y)’=x’+y’ [DeMorgan]
![Page 6: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/6.jpg)
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Example: Two-Variable K-Map
F = m0 + m1 + m2 = x’y’ + x’y + xy’
F = …
F = …
F = …
F = x’ + y’
• We can do the same optimization by combining adjacent cells.
y
x 0 1
0 1 1
1 1 0
![Page 7: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/7.jpg)
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Three-Variable K-Map
• Adjacent squares: they differ by only one variable, which is primed in one square and not primed in the other
m2 m6 , m3 m7
m2 m0 , m6 m4
yz
x 00 01 11 10
0 m0 m1 m3 m2
1 m4 m5 m7 m6
![Page 8: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/8.jpg)
8
Example: Three-Variable K-Map
• F1(x, y, z) = (2, 3, 4, 5)
1
0
10110100
yz
x
• F1(x, y, z) =
• F2(x, y, z) = (3, 4, 6, 7)
yz
x 00 01 11 10
0
1
• F2(x, y, z) =
![Page 9: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/9.jpg)
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Example: Three-Variable K-Map
• F1(x, y, z) = (2, 3, 4, 5)
11
11
00
00
1
0
10110100
yz
x
• F1(x, y, z) =
• F2(x, y, z) = (3, 4, 6, 7)
yz
x 00 01 11 10
0 0 0 1 0
1 1 0 1 1
• F2(x, y, z) =
![Page 10: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/10.jpg)
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• F1(x, y, z) = (2, 3, 4, 5)
11
11
00
00
1
0
10110100
yz
x
• F1(x, y, z) = xy’ + x’y
• F2(x, y, z) = (3, 4, 6, 7)
yz
x 00 01 11 10
0 0 0 1 0
1 1 0 1 1
• F2(x, y, z) = xz’ + yz
Example: Three-Variable K-Map
![Page 11: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/11.jpg)
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In 3-Variable Karnaugh Maps
• 1 alone square represents one minterm with 3 literals
• 2 adjacent squares represent a term with 2 literals
• 4 adjacent squares represent a term with 1 literal
• 8 adjacent squares produce a function that is always equal to 1.
![Page 12: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/12.jpg)
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Example
• F1(x, y, z) = (0, 2, 4, 5, 6)
z
1
0
10110100
yz
x
y
x
F1(x, y, z) =
![Page 13: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/13.jpg)
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Example
111
1
0
001
• F1(x, y, z) = (0, 2, 4, 5, 6)
z
1
0
10110100
yz
x
y
x
F1(x, y, z) =
![Page 14: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/14.jpg)
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Example
111
1
0
001
• F1(x, y, z) = (0, 2, 4, 5, 6)
z
1
0
10110100
yz
x
y
x
F1(x, y, z) =
![Page 15: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/15.jpg)
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Finding Sum of Minterms
• If a function is not expressed in sum of minterms
form, it is possible to get it using K-maps
– Example: F(x, y, z) = x’z + x’y + xy’z + yz
1
0
10110100
yz
x
![Page 16: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/16.jpg)
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Finding Sum of Minterms
• If a function is not expressed in sum of minterms
form, it is possible to get it using K-maps
– Example: F(x, y, z) = x’z + x’y + xy’z + yz
1
0
10110100
yz
x
F(x, y, z) = x’y’z + x’yz + x’yz’ + xy’z + xyz
![Page 17: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/17.jpg)
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Four-Variable K-Map• Four variables: x, y, z, t
– 4 literals– 16 minterms
zt
xy 00 01 11 10
00 m0 m1 m3 m2
01 m4 m5 m7 m6
11 m12 m13 m15 m14
10 m8 m9 m11 m10
z
t
x
y
![Page 18: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/18.jpg)
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Example: Four-Variable K-Map
F(x,y,z,t) = (0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14)
10
11
01
00
10110100
zt
xy
F(x,y,z,t) =
![Page 19: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/19.jpg)
19
Example: Four-Variable K-Map
F(x,y,z,t) = (0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14)
00
0
0
0
11
111
111
111
10
11
01
00
10110100
zt
xy
F(x,y,z,t) =
![Page 20: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/20.jpg)
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Example: Four-Variable K-Map
F(x,y,z,t) = (0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14)
00
0
0
0
11
111
111
111
10
11
01
00
10110100
zt
xy
F(x,y,z,t) =
![Page 21: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/21.jpg)
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Example: Four-Variable K-Map
• F(x,y,z,t) = x’y’z’ + y’zt’ + x’yzt’ + xy’z’
10
11
01
00
10110100
zt
xy
• F(x,y,z,t) =
![Page 22: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/22.jpg)
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Example: Four-Variable K-Map
• F(x,y,z,t) = x’y’z’ + y’zt’ + x’yzt’ + xy’z’
10
11
01
1011
0000
1000
101100
10110100
zt
xy
• F(x,y,z,t) =
![Page 23: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/23.jpg)
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Example: Four-Variable K-Map
• F(x,y,z,t) = x’y’z’ + y’zt’ + x’yzt’ + xy’z’
10
11
01
1011
0000
1000
101100
10110100
zt
xy
• F(x,y,z,t) =
![Page 24: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/24.jpg)
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Prime Implicants
• Prime Implicant: is a product term obtained by combining maximum possible number of adjacent squares in the map
• If a minterm can be covered by only one prime implicant, that prime implicant is said to be an essential prime implicant.– A single 1 on the map represents a prime implicant if it is
not adjacent to any other 1’s.
– Two adjacent 1’s form a prime implicant, provided that they are not within a group of four adjacent 1’s.
– So on
![Page 25: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/25.jpg)
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Example: Prime Implicants• F(x,y,z,t) = (0, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15)
10
11
01
00
10110100
zt
xy
![Page 26: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/26.jpg)
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Example: Prime Implicants• F(x,y,z,t) = (0, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15)
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
![Page 27: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/27.jpg)
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Example: Prime Implicants• F(x,y,z,t) = (0, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15)
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
F(x,y,z,t) = y’t’ + yt + xy’ + zt
• Which ones are the essential prime implicants?
![Page 28: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/28.jpg)
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Example: Prime Implicants• F(x,y,z,t) = (0, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15)
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
• Why are these the essential prime implicants?
![Page 29: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/29.jpg)
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Example: Prime Implicants• F(x,y,z,t) = (0, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15)
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
• y’t’ – essential since m0 is covered only in it
• yt - essential since m5 is covered only in it
• They together cover m0, m2, m8, m10, m5, m7, m13, m15
zt
xy 00 01 11 10
00 m0 m1 m3 m2
01 m4 m5 m7 m6
11 m12 m13 m15 m14
10 m8 m9 m11 m10
![Page 30: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/30.jpg)
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Example: Prime Implicants
• m3, m9, m11 are not yet covered.
• How do we cover them?
• There is actually more than one way.
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
zt
xy 00 01 11 10
00 m0 m1 m3 m2
01 m4 m5 m7 m6
11 m12 m13 m15 m14
10 m8 m9 m11 m10
![Page 31: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/31.jpg)
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Example: Prime Implicants
• Both y’z and zt covers m3 and m11.
• m9 can be covered in two different prime implicants:
xt or xy’• m3, m11 zt or y’z
• m9 xy’ or xt
zt
xy 00 01 11 10
00 1 0 1 101 0 1 1 011 0 1 1 010 1 1 1 1
1
2
3
4
![Page 32: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/32.jpg)
32
Example: Prime Implicants• F(x, y, z, t) = yt + y’t’ + zt + xt or
• F(x, y, z, t) = yt + y’t’ + zt + xy’ or
• F(x, y, z, t) = yt + y’t’ + y’z + xt or
• F(x, y, z, t) = yt + y’t’ + y’z + xy’
• Therefore, what to do1. Find out all the essential prime implicants
2. Then find the other prime implicants that cover the minterms that are not covered by the essential prime implicants. There are more than one way to choose those.
3. Simplified expression is the logical sum of the essential implicants plus the other implicants
![Page 33: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/33.jpg)
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Five-Variable Map
• Downside:– Karnaugh maps with more than four variables are not
simple to use anymore.
– 5 variables 32 squares, 6 variables 64 squares
– Somewhat more practical way for F(x, y, z, t, w) :
tw
yz 00 01 11 10
00 m0 m1 m3 m2
01 m4 m5 m7 m6
11 m12 m13 m15 m14
10 m8 m9 m11 m10
tw
yz 00 01 11 10
00 m16 m17 m19 m18
01 m20 m21 m23 m22
11 m28 m29 m31 m30
10 m24 m25 m27 m26
x = 0 x = 1
![Page 34: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/34.jpg)
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Many-Variable Maps• Adjacency:
– Each square in the x = 0 map is adjacent to the corresponding square in the x = 1 map.
– For example, m4m20 and m15 m31
• 6-variables: Use four 4-variable maps to obtain 64 squares required for six variable optimization
• Alternative way: Use computer programs
– Quine-McCluskey method
– Espresso method
![Page 35: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/35.jpg)
37
Example: Five-Variable Map
F(x, y, z, t, w) = (0, 2, 4, 6, 9, 13, 21, 23, 25, 29, 31)
tw
yz 00 01 11 10
00
01
11
10
tw
yz 00 01 11 10
00
01
11
10
x = 0 x = 1
• F(x,y,z,t,w) =
![Page 36: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/36.jpg)
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Example: Five-Variable Map
F(x, y, z, t, w) = (0, 2, 4, 6, 9, 13, 21, 23, 25, 29, 31)
tw
yz 00 01 11 10
00 1 101 1 111 110 1
tw
yz 00 01 11 10
00
01 1 111 1 110 1
x = 0 x = 1
• F(x,y,z,t,w) =
![Page 37: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/37.jpg)
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Product of Sums Simplification
• So far– simplified expressions from Karnaugh maps are in sum of
products form.
• Simplified product of sums can also be derived from Karnaugh maps.
• Method:– A square with 1 actually represents a “minterm”– Similarly an empty square (a square with 0) represents a
“maxterm”.– Treat the 0’s in the same manner as we treat 1’s – The result is a simplified expression in product of sums
form.
![Page 38: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/38.jpg)
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Example: Product of Sums
• F(x, y, z, t) = (0, 1, 2, 5, 8, 9, 10)– Simplify this function in
a. sum of products
b. product of sumszt
xy 00 01 11 10
00 1 1 101 111
10 1 1 1
F(x, y, z, t) =
![Page 39: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/39.jpg)
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Example: Product of Sums
• F(x, y, z, t) = (0, 1, 2, 5, 8, 9, 10)– Simplify this function in
a. sum of products
b. product of sumszt
xy 00 01 11 10
00 1 1 101 111
10 1 1 1
F(x, y, z, t) =
![Page 40: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/40.jpg)
42
Example: Product of Sums
• F(x, y, z, t) = (0, 1, 2, 5, 8, 9, 10)– Simplify this function in
a. sum of products
b. product of sumszt
xy 00 01 11 10
00 1 1 101 111
10 1 1 1
F(x,y,z,t) = y’t’ + y’z’ + x’z’t
![Page 41: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/41.jpg)
43
Example: Product of Sums1. Find F’(x,y,z,t) in Sum of Products form by
grouping 0’s2. Apply DeMorgan’s theorem to F’ (use dual
theorem) to find F in Product of Sums formzt
xy 00 01 11 10
00 1 1 0 101 0 1 0 011 0 0 0 010 1 1 0 1
![Page 42: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/42.jpg)
44
Example: Product of Sums
zt
xy 00 01 11 10
00 1 1 0 101 0 1 0 011 0 0 0 010 1 1 0 1
1. Find F’(x,y,z,t) in Sum of Products form by grouping 0’s
2. Apply DeMorgan’s theorem to F’ (use dual theorem) to find F in Product of Sums form
![Page 43: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/43.jpg)
45
Example: Product of Sums
zt
xy 00 01 11 10
00 1 1 0 101 0 1 0 011 0 0 0 010 1 1 0 1
F’ = yt’ + zt + xy
1. Find F’(x,y,z,t) in Sum of Products form by grouping 0’s
2. Apply DeMorgan’s theorem to F’ (use dual theorem) to find F in Product of Sums form
![Page 44: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/44.jpg)
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Example: Product of Sums
zt
xy 00 01 11 10
00 1 1 0 101 0 1 0 011 0 0 0 010 1 1 0 1
F(x,y,z,t) = (y’+t)(z’+t’)(x’+y’)
1. Find F’(x,y,z,t) in Sum of Products form by grouping 0’s
2. Apply DeMorgan’s theorem to F’ (use dual theorem) to find F in Product of Sums form
![Page 45: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/45.jpg)
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Example: Product of Sums
z’
y’
t’
y’
z’
x’
F
t
y’
t
x’
y’
z’
F
t’
F(x,y,z,t) = y’t’ + y’z’ + x’z’t: sum of products implementation
F = (y’ + t)(x’ + y’)(z’ + t’): product of sums implementation
![Page 46: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/46.jpg)
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Product of Maxterms
• If the function is originally expressed in the product of maxterms canonical form, the procedure is also valid
• Example: – F(x, y, z) = (0, 2, 5, 7)
1
0
10110100
yz
x
F(x, y, z) =
![Page 47: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/47.jpg)
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Product of Maxterms
• If the function is originally expressed in the product of maxterms canonical form, the procedure is also valid
• Example: – F(x, y, z) = (0, 2, 5, 7)
1
0
10110100
yz
x
F(x, y, z) =
0 0
0 0
![Page 48: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/48.jpg)
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Product of Maxterms
• If the function is originally expressed in the product of maxterms canonical form, the procedure is also valid
• Example: – F(x, y, z) = (0, 2, 5, 7)
1
0
10110100
yz
x
F(x, y, z) =
0 0
0 0
F(x, y, z) = x’z + xz’
![Page 49: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/49.jpg)
51
Product of Sums• To enter a function F, expressed in product of sums,
in the map1. take its complement, F’
2. Find the squares corresponding to the terms in F’,
3. Fill these square with 0’s and others with 1’s.
• Example: – F(x, y, z, t) = (x’ + y’ + z’)(y + t)
– F’(x, y, z, t) =
![Page 50: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/50.jpg)
52
Product of Sums• To enter a function F, expressed in product of sums,
in the map1. take its complement, F’
2. Find the squares corresponding to the terms in F’,
3. Fill these square with 0’s and others with 1’s.
• Example: – F(x, y, z, t) = (x’ + y’ + z’)(y + t)
– F’(x, y, z, t) = zt
xy 00 01 11 10
00 0 001
11 0 010 0 0
![Page 51: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/51.jpg)
53
Don’t Care Conditions 1/2
• Some functions are not defined for certain input combinations– Such function are referred as incompletely
specified functions
– therefore, the corresponding output values do not have to be defined
– This may significantly reduces the circuit complexity
– Example: A circuit that takes the 10’s complement of decimal digits
![Page 52: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/52.jpg)
54
Unspecified Minterms
• For unspecified minterms, we do not care what the value the function produces.
• Unspecified minterms of a function are called don’t care conditions.
• We use “X” symbol to represent them in Karnaugh map.
• Useful for further simplification• The symbol X’s in the map can be taken 0 or 1 to
make the Boolean expression even more simplified
![Page 53: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/53.jpg)
55
Example: Don’t Care Conditions
• F(x, y, z, t) = (1, 3, 7, 11, 15) – function
• d(x, y, z, t) = (0, 2, 5) – don’t care conditions
zt
xy 00 01 11 10
00 X 1 1 X01 0 X 1 011 0 0 1 010 0 0 1 0
F =
F1 = or
F2 =
![Page 54: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/54.jpg)
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Example: Don’t Care Conditions• F1 = zt + x’y’ = (0, 1, 2, 3, 7, 11, 15)
• F2 = zt + x’t = (1, 3, 5, 7, 11, 15)
• The two functions are algebraically unequal– But as far as the function F is concerned: both
functions are acceptable
• Look at the simplified product of sums expression for the same function F.zt
xy 00 01 11 10
00 X 1 1 X01 0 X 1 011 0 0 1 010 0 0 1 0
F’ =
F =
![Page 55: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/55.jpg)
Quine-McCluskey Method
- Better than kmap for computers because a computer cant break down a graphical thing like K-map, but it can easily solve by QM
- It is functionally identical to Karnaugh mapping, - but the tabular form makes it more efficient for use in computer algorithms,
- and it also gives a deterministic way to check that the minimal form of a Boolean function has been reached.
- It is sometimes referred to as the tabulation method.
![Page 56: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/56.jpg)
Quine-McCluskey Method
• F(x1,x2,x3,x4)= 2,4,6,8,9,10,12,13,15
mi x1 x2 x3 x4
2 0 0 1 0
4 0 1 0 0
8 1 0 0 0
6 0 1 1 0
9 1 0 0 1
10 1 0 1 0
12 1 1 0 0
13 1 1 0 1
15 1 1 1 1
![Page 57: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/57.jpg)
Quine-McCluskey Method
List 1 List 2 List 3
mi x1 x2 x3 x4 mi x1 x2 x3 x4 mi x1 x2 x3 x4
2 0 0 1 0 ok 2,6 0 - 1 0 8,9,12,13 1 - 0 -
4 0 1 0 0 ok 2,10 - 0 1 0 8,12,9,13 1 - 0 -
8 1 0 0 0 ok 4,6 0 1 - 0 Finished
6 0 1 1 0 ok 4,12 - 1 0 0
9 1 0 0 1 ok 8,9 1 0 0 - ok
10 1 0 1 0 ok 8,10 1 0 - 0
12 1 1 0 0 ok 8,12 1 - 0 0 ok
13 1 1 0 1 ok 9,13 1 - 0 1 ok
15 1 1 1 1 ok 12,13 1 1 0 - ok
13,15 1 1 - 1
![Page 58: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/58.jpg)
Quine-McCluskey Method
List 1 List 2 List 3
mi x1 x2 x3 x4 mi x1 x2 x3 x4 mi x1 x2 x3 x4
2,6 0 - 1 0 t2 8,9,12,13 1 - 0 -
2,10 - 0 1 0 t3
4,6 0 1 - 0 t4 Finished
4,12 - 1 0 0 t5
8,10 1 0 - 0 t6
13,15 1 1 - 1 t7
t1
![Page 59: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/59.jpg)
Quine-McCluskey Method2 4 6 8 9 10 12 13 15
t1 X X X X
t2 X X
t3 X X
t4 X X
t5 X X
t6 X X
t7 X X
2 4 6 10
t2 X X
t3 X X
t4 X X
t5 X t5 is a subset of t4
t6 X t6 is a subset of t3
F(x1,x2,x3,x4)=t1+t7+t3+t4=x1x3’ + x1x2x4 + x2’x3x4’ + x1’x2x4’
![Page 60: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/60.jpg)
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NAND and NOR Gates • NAND and NOR gates are easier to fabricate
VDD
A
B
C = (AB)’
CMOS 2-input AND gates requires 6 CMOS transistors
CMOS 3-input NAND gates requires 6 CMOS transistors
![Page 61: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/61.jpg)
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Design with NAND or NOR Gates• It is beneficial to derive conversion rules from
Boolean functions given in terms of AND, OR, an NOT gates into equivalent NAND or NOR implementations
x (x x)’ = x’ NOT
[ (x y)’ ]’ = x y ANDxy
(x’ y’ )’ = x + y OR
x
y
![Page 62: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/62.jpg)
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New Notation
• Implementing a Boolean function with NAND gates is easy if it is in sum of products form.
• Example: F(x, y, z, t) = xy + zt
xyz
(xyz)’
AND-invert
x’ + y’ + z’xy
z
Invert-OR
xy
zt
xy
zt
F(x, y, z, t) = xy + zt
F(x, y, z, t) = ((xy)’)’ + ((zt)’)’
=
ALWAYS USE THIS INTERMEDIATE
FORM WITH BUBBLES TO AVOID
CONFUSION
![Page 63: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/63.jpg)
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The Conversion Method
• Example: F(x, y, z) = (1, 3, 4, 5, 7)
xy
zt
yz
x 00 01 11 10
0 1 11 1 1 1
F = z + xy’
F = (z’)’ + ((xy’)’)’
y
zt
x
[ (xy)’ (zt)’ ] ’((xy)’)’ + ((zt)’)’ = xy + zt =
![Page 64: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/64.jpg)
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Example: Design with NAND Gates
• Summary1. Simplify the function
2. Draw a NAND gate for each product term
3. Draw a NAND gate for the OR gate in the 2nd level,
4. A product term with single literal needs an inverter in the first level. Assume single, complemented literals are available.
F = z + xy’
x
z’
y’
F
x
z’
y’
F
F = (z’)’ + ((xy’)’)’
![Page 65: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/65.jpg)
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Multi-Level NAND Gate Designs
• The standard form results in two-level implementations
• Non-standard forms may raise a difficulty
• Example: F = x(zt + y) + yz’– 4-level implementation
zt
x
yF
yz’
![Page 66: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/66.jpg)
Example: Multilevel NAND…
F
F = x(zt + y) + yz’
zt
F
y’
x
y
z’
z
t
y’
x
y
z’
MUST COMPENSATE FOR EVERY NON-COUPLED BUBBLE
![Page 67: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/67.jpg)
84
Design with Multi-Level NAND Gates
Rules
1. Convert all AND gates to NAND gates
2. Convert all OR gates to NAND gates
3. Insert an inverter (one-input NAND gate) at the output if the final operation is AND
4. Check the bubbles in the diagram. For every bubble along a path from input to output there must be another bubble. If not so, complement the input literal
![Page 68: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/68.jpg)
85
Another (Harder) Example
• Example: F = (xy’ + xy)(z + t’)
– (three-level implementation)
x
z
y’
yx
F
t’
![Page 69: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/69.jpg)
86
Example: Multi-Level NAND Gates
G = [ (xy’ + xy)(z’ + t) ]’
F = (xy’ + xy)(z + t’)
F = (xy’ + xy)(z + t’)
x
z’
y’
yx
t
F = (xy’ + xy)(z + t’)
x
z’
y’
yx
t
![Page 70: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/70.jpg)
87
Design with NOR Gates
• NOR is the dual operation of NAND.– All rules and procedure we used in the design with
NAND gates apply here in a similar way.
– Function is implemented easily if it is in product of sums form.
(x + x)’ = x’ NOT
[ (x+ y)’ ]’ = x + y OR
(x’ + y’ )’ = x · y AND
x
xy
x
y
![Page 71: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/71.jpg)
=
![Page 72: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/72.jpg)
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Example: Design with NOR Gates• F = (x+y) (z+t) w
F = (x + y) (z + t) w
zt
xy
F
w
zt
xy
w’
![Page 73: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/73.jpg)
Example: Design with NOR Gates
• F = (xy’ + zt) (z + t’)
zt
xy’
F
zt’
F = [((x’ + y)’ + (z’ + t’)’)’ + (z + t’)’]’= ((x’ + y)’ + (z’ + t’)’)(z + t’)= (xy’ + zt) (z + t’)
z’t’
x’y
zt’
![Page 74: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/74.jpg)
Harder Example• Example: F = x(zt + y) + yz’zt
x
yF
yz’
F
z’
t’
y
x’
z
y’
![Page 75: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/75.jpg)
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Exclusive-OR Function• The symbol:
• x y = xy’ + x’y• (x y)’ = xy + x’y’
• Properties1. x 0 = x2. x 1 = x’3. x x = 04. x x’ = 15. x y’ = x’ y = (x y)’ - XNOR
• Commutative & Associative• x y = y x • (x y) z = x (y z)
![Page 76: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/76.jpg)
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Exclusive-OR Function
• XOR gate is NOT universal– Only a limited number of Boolean functions can be
expressed in terms of XOR gates
• XOR operation has very important application in arithmetic and error-detection circuits.
• Odd Function– (x y) z = (xy’ + x’y) z
= (xy’ + x’y) z’ + (xy’ + x’y)’ z= xy’z’ + x’yz’ + (xy + x’y’) z= xy’z’ + x’yz’ + xyz + x’y’z= (4, 2, 7, 1)
![Page 77: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/77.jpg)
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Odd Function
• If an odd number of variables are equal to 1, then the function is equal to 1.
• Therefore, multivariable XOR operation is referred as “odd” function.
yz
x 00 01 11 10
0 0 1 0 11 1 0 1 0
Odd function
yz
x 00 01 11 10
0 1 0 1 01 0 1 0 1
Even function
![Page 78: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/78.jpg)
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Odd & Even Functions
• (x y z)’ = ((x y) z)’
xy
z
x y z Odd function
x
y
z
(x y z)’
![Page 79: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/79.jpg)
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Adder Circuit for Integers
• Addition of two-bit numbers Z = X + Y
X = (x1 x0) and Y = (y1 y0)
Z = (z2 z1 z0)
• Bitwise addition1. z0 = x0 y0 (sum)
c1 = x0 y0 (carry)
2. z1 = x1 y1 c1
c2 = x1 y1 + x1 c1 + y1 c1
3. z2 = c2
![Page 80: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/80.jpg)
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FA
Adder Circuit
c2= z2
x0y0
z0
c1
x1
z1
y1
z0 = x0 y0
c1 = x0 y0
z1 = x1 y1 c1
c2 = x1 y1 + x1 c1 + y1 c1z2 = c2
![Page 81: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/81.jpg)
Comparator Circuit with NAND gates• F(X>Y)
X = (x1 x0) and Y = (y1 y0)
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y1 y0
x1 x0 00 01 11 10
00 0 0 0 0
01 1 0 0 0
11 1 1 0 1
10 1 1 0 0
F(x1, x0, y1, y0) = x1y1‘ + x1x0y0‘ + x0y0‘y1‘
![Page 82: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/82.jpg)
Comparator Circuit - Schematic
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![Page 83: Gate-Level Minimization - Hacettepecelikcan/231/bolum04.pdf · Gate-Level Minimization section instructor: ... –A single 1 on the map represents a prime implicant if it is ... –Quine-McCluskey](https://reader030.fdocuments.in/reader030/viewer/2022021504/5aacddc17f8b9a693f8d9b10/html5/thumbnails/83.jpg)
Comparator Circuit - Simulation
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