1 hamming code, k-maps-multiplexer midterm 1 revision prof. sin-min lee department of computer...
TRANSCRIPT
![Page 1: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/1.jpg)
1
Hamming Code, K-maps-MultiplexerMidterm 1 Revision
Prof. Sin-Min Lee
Department of Computer Science
![Page 2: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/2.jpg)
04/19/23 2
Let us play a game !
A volunteer from the audience?
Pick a number, any number, between 1 and 50
![Page 3: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/3.jpg)
04/19/23 3
Is the Number in Here?
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
39 41 43 45 47 49
![Page 4: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/4.jpg)
04/19/23 4
Is the Number in Here?
2 3 6 7 10 11 14 15 18 19 22 23 26 27 30 31 34 35 38
39 42 43 46 47 50
![Page 5: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/5.jpg)
04/19/23 5
Is the Number in Here?
4 5 6 7 12 13 14 15 20 21 22 23 28 29 30 31 36 37
38 39 44 45 46 47
![Page 6: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/6.jpg)
04/19/23 6
Is the Number in Here?
8 9 10 11 12 13 14 15 24 25 26 27 28 29 30 31 40 41
42 43 44 45 46 47
![Page 7: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/7.jpg)
04/19/23 7
Is the Number in Here?
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 49
50
![Page 8: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/8.jpg)
04/19/23 8
Is the Number in Here?
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
49 50
![Page 9: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/9.jpg)
04/19/23 9
And the Number is ….
![Page 10: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/10.jpg)
04/19/23 10
![Page 11: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/11.jpg)
04/19/23 11
Richard Hamming
Richard Wesley Hamming, mathematician, pioneer computer scientist, and professor, died of a heart attack on January 7, 1998, in Monterey, California, at the age of 82. His research career began at Bell Laboratories in the 1940s, in the early days of electronic computers, and included the invention of the Hamming error-correcting codes. In the 1970s he shifted to teaching, and at his death he was Distinguished Professor Emeritus of computer science at the Naval Postgraduate School. He is survived by his wife Wanda, a niece, and a nephew.
![Page 12: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/12.jpg)
04/19/23 12
![Page 13: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/13.jpg)
04/19/23 INFS 515 Digital Logic Level 13
1948: Error Correction
Error-detecting coding, first developed for telephone switching, is now used throughout the computing and telecommunications industries. In 1948 , R.W. Hamming (left) of Bell Labs developed a general theory for error-correcting schemes in which "check-bits" are interspersed with information bits to form binary words in patterns. When a single error occurs in transmission, the word becomes invalid, but the error is automatically located and corrected.
![Page 14: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/14.jpg)
04/19/23 14
![Page 15: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/15.jpg)
04/19/23 INFS 515 Digital Logic Level 15
![Page 16: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/16.jpg)
04/19/23 INFS 515 Digital Logic Level 16
![Page 17: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/17.jpg)
04/19/23 INFS 515 Digital Logic Level 17
![Page 18: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/18.jpg)
04/19/23 INFS 515 Digital Logic Level 18
![Page 19: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/19.jpg)
04/19/23 INFS 515 Digital Logic Level 19
![Page 20: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/20.jpg)
04/19/23 INFS 515 Digital Logic Level 20
![Page 21: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/21.jpg)
04/19/23 21
![Page 22: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/22.jpg)
04/19/23 INFS 515 Digital Logic Level 22
![Page 23: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/23.jpg)
04/19/23 INFS 515 Digital Logic Level 23
![Page 24: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/24.jpg)
04/19/23 INFS 515 Digital Logic Level 24
![Page 25: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/25.jpg)
04/19/23 INFS 515 Digital Logic Level 25
![Page 26: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/26.jpg)
04/19/23 INFS 515 Digital Logic Level 26
![Page 27: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/27.jpg)
04/19/23 INFS 515 Digital Logic Level 27
![Page 28: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/28.jpg)
04/19/23 INFS 515 Digital Logic Level 28
![Page 29: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/29.jpg)
04/19/23 INFS 515 Digital Logic Level 29
![Page 30: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/30.jpg)
04/19/23 30
Truth table to K-Map
A B P
0 0 1
0 1 1
1 0 0
1 1 1
B
A 0 1
0 1 1
1 1
minterms are represented by a 1 in the corresponding location in the K map.
The expression is:
A.B + A.B + A.B
![Page 31: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/31.jpg)
04/19/23 31
K-Maps
• Adjacent 1’s can be “paired off”
• Any variable which is both a 1 and a zero in this pairing can be eliminated
• Pairs may be adjacent horizontally or vertically
B
A 0 1
0 1 1
1 1
a pair
another pair
B is eliminated, leaving A as the term
A is eliminated, leaving B as the term
The expression becomes A + B
![Page 32: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/32.jpg)
32
• Two Variable K-Map
AB
C P
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 0
A.B.C + A.B.C + A.B.C
BC
A 00 01 11 10
0 1
1 1 1One square filled in for each minterm.
Notice the code sequence: 00 01 11 10 – a Gray code.
![Page 33: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/33.jpg)
04/19/23 33
Grouping the Pairs
BC
A 00 01 11 10
0 1
1 1 1
equates to B.C as A is eliminated.
Here, we can “wrap around” and this pair equates to A.C as B is eliminated.
Our truth table simplifies to
A.C + B.C as before.
![Page 34: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/34.jpg)
04/19/23 34
Groups of 4
BC
A00 01 11 10
0 1 1
1 1 1
Groups of 4 in a block can be used to eliminate two variables:
The solution is B because it is a 1 over the whole block
(vertical pairs) = BC + BC = B(C + C) = B.
![Page 35: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/35.jpg)
Karnaugh Maps
• Three Variable K-Map
– Extreme ends of same row considered adjacent
A BC
00 01 11 10
0
1
A.B.C A.B.C A.B.C A.B.C
A.B.C A.B.C A.B.C A.B.C
0010A.B.C
A.B.C
A.B.C
A.B.C
![Page 36: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/36.jpg)
Karnaugh Maps
• Three Variable K-Map example X A.B.C A.B.C A.B.C A.B.C
A BC
00 01 11 10
0
1
X =
![Page 37: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/37.jpg)
04/19/23 37
The Block of 4, again
A BC
00 01 11 10
0 1 1
1 1 1
X = C
![Page 38: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/38.jpg)
04/19/23 38
Returning to our car example, once more• Two Variable K-Map
A B
C P
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 0
A.B.C + A.B.C + A.B.C
AB
C 00 01 11 10
0 1 1 1
1There is more than one way to label the axes of the K-Map, some views lead to groupings which are easier to see.
![Page 39: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/39.jpg)
Karnaugh Maps• Four Variable K-Map
– Four corners adjacent
AB CD
00 01 11 10
00
01
11
10
A.B.C.D A.B.C.D A.B.C.D A.B.C.D
A.B.C.D A.B.C.D A.B.C.D A.B.C.D
A.B.C.D A.B.C.D A.B.C.D A.B.C.D
A.B.C.D A.B.C.D A.B.C.D A.B.C.D
A.B.C.D
A.B.C.D
A.B.C.D
A.B.C.D
![Page 40: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/40.jpg)
Karnaugh Maps
• Four Variable K-Map example F A.B.C.DA.B.C.D+A.B.C.DA.B.C.DA.B.C.DA.B.C.DA.B.C.D
AB CD
00 01 11 10
00
01
11
10
F =
![Page 41: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/41.jpg)
41
Product-of-SumsWe have populated the maps with 1’s using sum-of-products extracted from the truth table.
We can equally well work with the 0’s
AB
C00 01 11 10
0 1 1 1
1 1
A B C P
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 0
AB
C00 01 11 10
0 0
1 0 0 0P = (A + B).(A + C)
P = A.B + A.Cequivalent
![Page 42: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/42.jpg)
04/19/23 42
Inverted K Maps
• In some cases a better simplification can be obtained if the inverse of the output is considered– i.e. group the zeros instead of the ones– particularly when the number and patterns of
zeros is simpler than the ones
![Page 43: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/43.jpg)
Karnaugh Maps• Example: Z5 of the Seven Segment Display
0 0 0 0 1
0 0 0 1 0
0 0 1 1 0
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 0
1 0 0 0 1
X1 X2 X3 X4 Z5
1 0 0 1 0
1 0 1 0 X
1 0 1 1 X
1 1 0 0 X1 1 0 1 X1 1 1 0 X1 1 1 1 X
0
1
2
3
4
5
6
7
8
9
0 0 1 0 1X1X2
X3 X4 00 01 11 10
00
01
11
10
Z5 =
• Better to group 1’s or 0’s?
![Page 44: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/44.jpg)
04/19/23 INFS 515 Digital Logic Level 44
![Page 45: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/45.jpg)
04/19/23 INFS 515 Digital Logic Level 45
![Page 46: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/46.jpg)
04/19/23 46
Karnaugh Map Method of Multiplexer Implementation
Consider the function:
A is taken to be the data variable and B,C to be the select variables.
![Page 47: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/47.jpg)
04/19/23 INFS 515 Digital Logic Level 47
Example of MUX combo circuit • F(X,Y,Z) = m(1,2,6,7)
![Page 48: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/48.jpg)
04/19/23 INFS 515 Digital Logic Level 48
![Page 49: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/49.jpg)
49
Implementing the Canonical Sum
• The binary decoder generates all minterms of n-variable logic function.
• The canonical sum ( sum of minterms ) of a logic functions is obtained by adding all minterms of that function:
-Match the order of input bits-Activate Enable inputs
• Example : G2A
Y0
Y1
Y2
Y3
A
B
Z
Y
74x138
Y4
Y5
Y6
Y7CX
G2B
G1
FX Y Z
( , , ), ,
2 4 7F
+5V
![Page 50: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/50.jpg)
50
Design Canonical Form w/ MUX
7) 6, 2, m(1,C)B,F(A,
ABCCABCBACBAC)B,F(A,
F
A0
A1
A2
A3
S1 S0
8-to-18-to-1MuxMux
S2
A4
A5
A6
A7
00
00
00
00
11
11
11
11
Each input in a MUX is a minterm
AA BB CC
![Page 51: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/51.jpg)
51
Design Canonical Form w/ MUX
7) 6, 2, m(1,C)B,F(A,
ABCCABCBACBAF
A B F
0 0
0 1
1 0
1 1
![Page 52: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/52.jpg)
52
Design Canonical Form w/ MUX
7) 6, 2, m(1,F
ABCCABCBACBAF
A B F
0 0 C
0 1 C
1 0 0
1 1 1
F
A0
A1
A2
A3S1 S0
En
4-to-14-to-1MuxMux
AA BB
CC
CC
00
11
Vdd
![Page 53: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/53.jpg)
53
Design Canonical Form w/ MUX
7) 6, 2, m(1,F
ABCCABCBACBAF
B C F
0 0
0 1
1 0
1 1
![Page 54: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/54.jpg)
54
Design Canonical Form w/ MUX
7) 6, 2, m(1,F
ABCCABCBACBAF
B C F
0 0 0
0 1 A
1 0 1
1 1 A
F
A0
A1
A2
A3S1 S0
En
4-to-14-to-1MuxMux
BB CC
AA
AA
Vdd
![Page 55: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/55.jpg)
55
Demultiplexers (DeMux)
F
A0
A1
A2
A3S1 S0
4-to-14-to-1MuxMux
A
D0
D1
D2
D3S1 S0
1-to-41-to-4DeMuxDeMux
![Page 56: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/56.jpg)
56
DeMux Operations
S1 S0 D3 D2 D1 D0
0 0 0 0 0 A
0 1 0 0 A 0
1 0 0 A 0 0
1 1 A 0 0 0
A
D0
D1
D2
D3S1 S0
1-to-41-to-4DeMuxDeMux
ASSD
ASSD
ASSD
ASSD
013
012
011
010
![Page 57: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/57.jpg)
57
DeMux Operations
S1 S0 D3 D2 D1 D0
0 0 0 0 0 A
0 1 0 0 A 0
1 0 0 A 0 0
1 1 A 0 0 0
ASSD
ASSD
ASSD
ASSD
013
012
011
010
D0
D1
D2
D3
A
S1
S0
![Page 58: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/58.jpg)
58
Implementing Functions Using Implementing Functions Using DecodersDecoders
• Example: Full adderS(x, y, z) = (1,2,4,7)
C(x, y, z) = (3,5,6,7)
x
y
z
3-to-8Decoder
S2
S1
S0
0
1
2
3
4
5
6
7
S
C
![Page 59: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/59.jpg)
59
EncoderEncoder
• Function: given 2n inputs, encode the index of input with 1 as the output.
::
noutputs
2n to n
encoder2n
inputs
![Page 60: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/60.jpg)
60
Encoders & Priority EncodersEncoders & Priority Encoders
• Encoders
• Priority encoders
X Y I 0 I 1 I 2 I 3 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1
X Y I 0 I 1 I 2 I 3 1 0 0 0 0 0 x 1 0 0 0 1 x x 1 0 1 0 x x x 1 1 1
Logic functionsX = I0’I1’I2I3’ + I0’I1’I2’I3
Y = I0’I1I2’I3’ + I0’I1’I2’I3
Logic functionsX = I2I3’ + I3
Y = I1I2’I3’ + I3
![Page 61: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/61.jpg)
61
Binary AddersBinary AddersFor example: add 1011012 and 110102,
ie., 4510 + 2610 = 7110
1 1 11 0 1 1 0 1
+ 1 1 0 1 0
_______________________________________1 0 0 0 1 1 1
![Page 62: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/62.jpg)
62
Half AdderHalf AdderTruth table Logic function
![Page 63: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/63.jpg)
63
Full AdderFull Adder
![Page 64: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/64.jpg)
Binary AddersBinary Adders
• Arithmetic Circuit– a combinational circuit for arithmetic operations
such as addition, subtraction, multiplication, and division with binary numbers or decimal numbers in a binary code
• Addition of 2 binary inputs, 'Half Adder‘– 0+0=0, 0+1=1, 1+0=1, 1+1 = 10
S = X'Y + XY' = X Y;
C = XY
![Page 65: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/65.jpg)
Binary AddersBinary Adders• Addition of 3 binary inputs, 'Full Adder'
Logic Diagram of Full Adder
![Page 66: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/66.jpg)
Binary AddersBinary Adders
• Binary Ripple Carry Adder
– sum of two n-bit binary numbers in parallel
– 4-bit parallel adder
A = 1011, B = 0011
![Page 67: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/67.jpg)
Binary AddersBinary Adders
• Carry Lookahead Adder– The ripple carry adder has a long circuit delay
• the longest delay: 2 n + 2 gate delay
Carry Lookahead Adder• reduced delay at the price of complex hardware
– a new logic hierarchy
Pi: propagate function
Gi: generate function
![Page 68: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/68.jpg)
3.8 3.8 Binary AddersBinary Adders
Development of Carry Lookahead Adder
![Page 69: 1 Hamming Code, K-maps-Multiplexer Midterm 1 Revision Prof. Sin-Min Lee Department of Computer Science](https://reader030.vdocuments.net/reader030/viewer/2022032523/56649d745503460f94a53947/html5/thumbnails/69.jpg)