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Introduction to Computer Engineering
ECE 203
Northwestern University
Department of Electrical Engineering and Computer Science
Teacher: Robert DickOffice: L477 TechEmail: [email protected]: 467–2298Webpage: http://www.ece.northwestern.edu/˜dickrp/ece203Teaching Assistants Zhenyu Gu (L470)
Debasish Das (M314)
1
Homework index
1 Reading assignment . . . . . . . . . . . . . . 42
2
Today’s topics
• Lab and lecture comments
• Review: Minimization overview
• Review: Karnaugh map SOP minimization
• POS using SOP K-map trick
• Quine-McCluskey optimal two-level minimization method
3
Lab and lecture comments
• Anybody falling behind?
• If something isn’t making sense, stop me and I’ll elaborate using
the chalkboard
– I’m glad to do it!
• Lab expectations (lab two and above)
– Complete schematics
– Easy to debug, color-coded wiring
– Terse but clear description
4
Review: Minimization techniques
Advantages and disadvantages?
• Algebraic manipulation
• Karnaugh maps
• Quine-McCluskey
• Advanced topic: Kernel extraction
• Advanced topic: Heuristic minimization, e.g., Espresso
5
Deriving POS
00 11 1001
01
00
11
10
1 1 1
11
11 1
0 0
0
0 0
10
0
f(a,b,c,d)ab
cd
Find SOP form for zeros:
f = ab z + c d + a b d
6
Deriving POS
Apply DeMorgan’s theorem
f = ab d + c d + a b d (1)
f = ab d + c d + a b d (2)
f =(
ab d)
· ( c d) ·(
a b d)
(3)
f =(
a + b +d)(
c+ d)(
a+b+ d)
(4)
• Advanced topic: Read the POS expression directly from the
Karnaugh map
– More difficult
7
Quine-McCluskey two-level logic minimization
• Compute prime implicants with a well-defined algorithm
– Start from minterms
– Merge adjacent implicants until further merging impossible
• Select minimal cover from prime implicants
– Unate covering problem
• What is happening?
– ab+a b = a
8
Computing prime implicants
0000
000100101000
10011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
9
Computing prime implicants
000X0000
000100101000
10011010
1111
11011110
0000
0001
0000 000X
0001
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
10
Computing prime implicants
000X0000
000100101000
10011010
1111
11011110
0000
0001
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
11
Computing prime implicants
00X0000X0000
000100101000
10011010
1111
11011110
0000
00010010
000000X0
0010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
12
Computing prime implicants
00X0000X0000
000100101000
10011010
1111
11011110
0000
00010010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
13
Computing prime implicants
00X0X000
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
0000
X000
1000
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
14
Computing prime implicants
00X0X000
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
15
Computing prime implicants
00X0X000X001
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
1001
X0010001
1001
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
16
Computing prime implicants
00X0X000X001
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
1001
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
17
Computing prime implicants
00X0X000X001X010
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
0010 X010
1010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
18
Computing prime implicants
00X0X000X001X010
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
19
Computing prime implicants
00X0X000X001X010100X
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
1000 100X
1001
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
20
Computing prime implicants
00X0X000X001X010100X
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
21
Computing prime implicants
00X0X000X001X010100X10X0
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
100010X0
1010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
22
Computing prime implicants
00X0X000X001X010100X10X0
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
23
Computing prime implicants
00X0X000X001X010100X10X01X01
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
100110101101
1001 1X01
1101
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
24
Computing prime implicants
00X0X000X001X010100X10X01X01
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
100110101101
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
25
Computing prime implicants
00X0X000X001X010100X10X01X011X10
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
1001101011011110
1010 1X10
1110∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
26
Computing prime implicants
00X0X000X001X010100X10X01X011X10
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
1001101011011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
27
Computing prime implicants
00X0X000X001X010100X10X01X011X10111X
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
1111
110111101111
111X1110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
28
Computing prime implicants
00X0X000X001X010100X10X01X011X10111X
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
29
Computing prime implicants
00X0X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
1111
110111101111
11X11101
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
30
Computing prime implicants
00X0X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000
000100101000
10011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
31
Computing prime implicants
X00X00X0X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X
000100101000 100X
10011010
1111
11011110
000X
100X
X00X
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
32
Computing prime implicants
X00X00X0X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X
000100101000 100X
10011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
33
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0
000100101000 100X
10X010011010
1111
11011110
00X0
10X0
X0X0
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
34
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0
000100101000 100X
10X010011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
35
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0X000X0010001
00101000 100X
10X010011010
1111
11011110
X000X001
X00X
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
36
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0X000X0010001
00101000 100X
10X010011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
37
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0X000X0010001
00101000
X010100X10X0
10011010
1111
11011110
X000
X010
X0X0
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
38
Computing prime implicants
X00XX0X000X0
X000X001X010100X10X01X011X10111X11X1
000X0000
000100101000
10011010
1111
11011110
0000 000X00X0X000X0010001
00101000
X010100X10X0
10011010
1111
11011110
∑ = 4
∑ = 0
∑ = 1
∑ = 2
∑ = 3
39
Summary
• Review: Minimization overview
• Review: Karnaugh map SOP minimization
• POS using SOP K-map trick
• Quine-McCluskey optimal two-level minimization method
40
Next lecture – More advanced building blocks
• Encoders and decoders
• MUXs
• Advanced TG techniques
41
Reading assignment
• M. M. Mano and C. R. Kime, Logic and Computer Design
Fundamentals. Prentice-Hall, Englewood Cliffs, NJ, third ed.,
2004
• Sections 2.7–2.10
• Sections 4.1–4.5
• Section 4.6 (decoders and multiplexers only)
• M. M. Mano and C. R. Kime, Web supplements to Logic and
Computer Design Fundamentals. Prentice-Hall, Englewood
Cliffs, NJ, 2004. http://www.writphotec.com/mano/Supplements
handout on tabular method (Quine-McCluskey)
42
Additional references
• If QM doesn’t click, please also see the following references
• R. H. Katz, Contemporary Logic Design. The
Benjamin/Cummings Publishing Company, Inc., 1994: pp. 85–88
• J. P. Hayes, Introduction to Digital Logic Design. Addison-Wesley,
Reading, MA, 1993 pp. 320, 321
• You can get these from me or the library
43
Computer geek culture reference
• http://www.deepchip.com
44