CprE 281: Digital Logic
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1 CprE 281: Digital Logic Instructor: Alexander Stoytchev
2 Signed Numbers CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
3 Administrative Stuff HW5 is out It is due on Monday Oct 4pm. Please write clearly on the first page (in block capital letters) the following three things: Your First and Last Name Your Student ID Number Your Lab Section Letter
4 Labs Next Week Administrative Stuff Mini-Project This one is worth 3% of your grade. Make sure to get all the points _Fall_281/labs/Project-Mini/
5 Quick Review
6 Adding two bits (there are four possible cases) [ Figure 3.1a from the textbook ]
7 Adding two bits (the truth table) [ Figure 3.1b from the textbook ]
8 Adding two bits (the logic circuit) [ Figure 3.1c from the textbook ]
9 The Half-Adder [ Figure 3.1c-d from the textbook ]
10 Addition of multibit numbers Bit position i [ Figure 3.2 from the textbook ]
11 Analogy with addition in base 10 + c 3 c 2 c 1 c 0! x 2 x 1 x 0! y 2 y 1 y 0! s 2 s 1 s 0!!
12 Analogy with addition in base 10! ! 1 5 7! 5 4 6!
13 Analogy with addition in base 10 carry ! ! 1 5 7! 5 4 6!
14 Analogy with addition in base 10 + c 3 c 2 c 1 c 0! x 2 x 1 x 0! y 2 y 1 y 0! s 2 s 1 s 0!!
15 Problem Statement and Truth Table [ Figure 3.2b from the textbook ] [ Figure 3.3a from the textbook ]
16 Let s fill-in the two K-maps [ Figure 3.3a-b from the textbook ]
17 Let s fill-in the two K-maps [ Figure 3.3a-b from the textbook ]
18 The circuit for the two expressions [ Figure 3.3c from the textbook ]
19 This is called the Full-Adder [ Figure 3.3c from the textbook ]
20 XOR Magic (s i can be implemented in two different ways)
21 A decomposed implementation of the full-adder circuit c s i s i s HA c x i HA y c c i + 1 i (a) Block diagram c i s i x i y i c i + 1 (b) Detailed diagram [ Figure 3.4 from the textbook ]
22 The Full-Adder Abstraction c s i s i s HA c x i HA c y c i + 1 i
23 The Full-Adder Abstraction c i x i FA c i + 1 y i s i
24 We can place the arrows anywhere x i y i c i+1 FA c i s i
25 n-bit ripple-carry adder x n 1 y n 1 x 1 y 1 x 0 y 0 c n FA c n 1 c 2 FA c 1 FA c 0 s n 1 s 1 s 0 MSB position LSB position [ Figure 3.5 from the textbook ]
26 n-bit ripple-carry adder abstraction x n 1 y n 1 x 1 y 1 x 0 y 0 c n FA c n 1 c 2 FA c 1 FA c 0 s n 1 s 1 s 0 MSB position LSB position
27 n-bit ripple-carry adder abstraction x n 1 y n 1 x 1 y 1 x 0 y 0 c n c 0 s n 1 s 1 s 0
28 The x and y lines are typically grouped together for better visualization, but the underlying logic remains the same x n 1 x 1 x 0 y 1 y n 1 y 0 c n c 0 s n 1 s 1 s 0
29
30 Math Review: Subtraction 39-15??
31 Math Review: Subtraction
32 Math Review: Subtraction ??????
33 Math Review: Subtraction
34
35 Math Review: Subtraction ??????
36 Math Review: Subtraction
37
38 The problems in which row are easier to calculate? ?????? ??????
39 The problems in which row are easier to calculate? Why?
40 Another Way to Do Subtraction =
41 Another Way to Do Subtraction = = 82 + (100 64) - 100
42 Another Way to Do Subtraction = = 82 + (100 64) = 82 + ( ) - 100
43 Another Way to Do Subtraction = = 82 + (100 64) = 82 + ( ) = 82 + (99 64)
44 Another Way to Do Subtraction = = 82 + (100 64) = 82 + ( ) Does not require borrows = 82 + (99 64)
45 9 s Complement (subtract each digit from 9)
46 10 s Complement (subtract each digit from 9 and add 1 to the result) = 36
47 Another Way to Do Subtraction = 82 + (99 64)
48 Another Way to Do Subtraction 9 s complement = 82 + (99 64)
49 Another Way to Do Subtraction 9 s complement = 82 + (99 64) =
50 Another Way to Do Subtraction 9 s complement = 82 + (99 64) = s complement
51 Another Way to Do Subtraction 9 s complement = 82 + (99 64) = s complement = // add the first two
52 Another Way to Do Subtraction 9 s complement = 82 + (99 64) = s complement = = // add the first two // delete the leading 1 = 18
53
54 Formats for representation of integers b n 1 b 1 b 0 MSB Magnitude (a) Unsigned number b n 1 b n 2 b 1 b 0 Sign 0 denotes + 1 denotes MSB Magnitude (b) Signed number [ Figure 3.7 from the textbook ]
55 Negative numbers can be represented in following ways Sign and magnitude 1 s complement 2 s complement
56 1 s complement Let K be the negative equivalent of an n-bit positive number P. Then, in 1 s complement representation K is obtained by subtracting P from 2 n 1, namely K = (2 n 1) P This means that K can be obtained by inverting all bits of P.
57 Find the 1 s complement of
58 Find the 1 s complement of Just flip 1's to 0's and vice versa.
59 A) Example of 1 s complement addition ( + 5 ) + ( + 2 ) ( + 7 ) [ Figure 3.8 from the textbook ]
60 A) Example of 1 s complement addition ( + 5 ) + ( + 2 ) ( + 7 )
61 B) Example of 1 s complement addition (- 5 ) + ( + 2 ) (- 3 ) [ Figure 3.8 from the textbook ]
62 B) Example of 1 s complement addition (- 5 ) + ( + 2 ) (- 3 )
63 C) Example of 1 s complement addition ( + 5 ) + ( 2 ) ( + 3 ) [ Figure 3.8 from the textbook ]
64 C) Example of 1 s complement addition ( + 5 ) + ( 2 ) ( + 3 )
65 C) Example of 1 s complement addition ( + 5 ) + ( 2 ) ( + 3 ) But this is 2!
66 C) Example of 1 s complement addition ( + 5 ) + ( 2 ) ( + 3 ) We need to perform one more addition to get the result.
67 C) Example of 1 s complement addition ( + 5 ) + ( 2 ) ( + 3 ) We need to perform one more addition to get the result.
68 D) Example of 1 s complement addition ( 5 ) + ( 2 ) ( 7 ) [ Figure 3.8 from the textbook ]
69 D) Example of 1 s complement addition ( 5 ) + ( 2 ) ( 7 )
70 D) Example of 1 s complement addition ( 5 ) + ( 2 ) ( 7 ) But this is +7!
71 D) Example of 1 s complement addition ( 5 ) + ( 2 ) ( 7 ) We need to perform one more addition to get the result.
72 D) Example of 1 s complement addition ( 5 ) + ( 2 ) ( 7 ) We need to perform one more addition to get the result.
73 2 s complement Let K be the negative equivalent of an n-bit positive number P. Then, in 2 s complement representation K is obtained by subtracting P from 2 n, namely K = 2 n P
74 Deriving 2 s complement For a positive n-bit number P, let K 1 and K 2 denote its 1 s and 2 s complements, respectively. K 1 = (2 n 1) P K 2 = 2 n P Since K 2 = K 1 + 1, it is evident that in a logic circuit the 2 s complement can computed by inverting all bits of P and then adding 1 to the resulting 1 s-complement number.
75 Find the 2 s complement of
76 Find the 2 s complement of
77 Quick Way to find 2 s complement Scan the binary number from right to left Copy all bits that are 0 from right to left Stop at the first 1 Copy that 1 as well Invert all remaining bits
78 Interpretation of four-bit signed integers [ Table 3.2 from the textbook ]
79 A) Example of 2 s complement addition ( + 5 ) + ( + 2 ) ( + 7 ) [ Figure 3.9 from the textbook ]
80 B) Example of 2 s complement addition + ( 5 ) ( + 2 ) ( 3 ) [ Figure 3.9 from the textbook ]
81 C) Example of 2 s complement addition ( + 5 ) + ( 2 ) ( + 3 ) ignore [ Figure 3.9 from the textbook ]
82 D) Example of 2 s complement addition ( 5 ) + ( 2 ) ( 7 ) ignore [ Figure 3.9 from the textbook ]
83
84 Example of 2 s complement subtraction ( + 5 ) ( + 2 ) ( + 3 ) ignore [ Figure 3.10 from the textbook ]
85 Example of 2 s complement subtraction ( 5 ) ( + 2 ) ( 7 ) ignore [ Figure 3.10 from the textbook ]
86 Example of 2 s complement subtraction ( + 5 ) ( 2 ) ( + 7 ) [ Figure 3.10 from the textbook ]
87 Example of 2 s complement subtraction ( 5 ) ( 2 ) ( 3 ) [ Figure 3.10 from the textbook ]
88 Graphical interpretation of four-bit 2 s complement numbers [ Figure 3.11 from the textbook ]
89 Take-Home Message Subtraction can be performed by simply adding the 2 s complement of the second number, regardless of the signs of the two numbers. Thus, the same adder circuit can be used to perform both addition and subtraction!!!
90 Adder/subtractor unit y n 1 y 1 y 0 Add Sub control x n 1 x 1 x 0 c n n -bit adder c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
91 XOR Tricks control out y
92 XOR as a repeater 0 y y
93 XOR as an inverter 1 y y
94 Addition: when control = 0 y n 1 y 1 y 0 Add Sub control x n 1 x 1 x 0 c n n -bit adder c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
95 Addition: when control = 0 y n 1 0 y 1 0 y Add Sub control x n 1 x 1 x 0 c n n -bit adder c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
96 Addition: when control = 0 y n 1 0 y 1 0 y Add Sub control x n 1 x 1 x 0 c n n -bit adder y n-1 y 1 y 0 c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
97 Subtraction: when control = 1 y n 1 y 1 y 0 Add Sub control x n 1 x 1 x 0 c n n -bit adder c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
98 Subtraction: when control = 1 y n 1 1 y 1 1 y Add Sub control x n 1 x 1 x 0 c n n -bit adder c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
99 Subtraction: when control = 1 y n 1 1 y 1 1 y Add Sub control x n 1 x 1 x 0 c n n -bit adder y n-1 y 1 y 0 c 0 s n 1 s 1 s 0 [ Figure 3.12 from the textbook ]
100 Subtraction: when control = 1 y n 1 1 y 1 1 y Add Sub control x n 1 x 1 x 0 c n n -bit adder y n-1 y 1 y 0 1 c 0 s n 1 s 1 s 0 carry for the first column! [ Figure 3.12 from the textbook ]
101 Examples of determination of overflow + ( + 7 ) ( + 2 ) ( 7 ) ( + 2 ) ( + 9 ) c 4 = 0 c 3 = 1 ( 5 ) c 4 = 0 c 3 = 0 ( + 7 ) + ( 2 ) ( 7 ) + ( 2 ) ( + 5 ) c 4 = 1 c 3 = 1 ( 9 ) c 4 = 1 c 3 = 0 [ Figure 3.13 from the textbook ]
102 Examples of determination of overflow + ( + 7 ) ( + 2 ) ( 7 ) ( + 2 ) ( + 9 ) ( + 7 ) + ( 2 ) c 4 = 0 c 3 = ( 5 ) Overflow occurs only in these two cases. ( 7 ) + ( 2 ) c 4 = 0 c 3 = ( + 5 ) c 4 = 1 c 3 = 1 ( 9 ) c 4 = 1 c 3 = 0 [ Figure 3.13 from the textbook ]
103 Examples of determination of overflow + ( + 7 ) ( + 2 ) ( 7 ) ( + 2 ) ( + 9 ) ( + 7 ) + ( 2 ) c 4 = 0 c 3 = ( 5 ) Overflow occurs only in these two cases. ( 7 ) + ( 2 ) c 4 = 0 c 3 = ( + 5 ) c 4 = 1 c 3 = 1 ( 9 ) c 4 = 1 c 3 = 0 Overflow = c 3 c 4 + c 3 c 4 [ Figure 3.13 from the textbook ]
104 Examples of determination of overflow + ( + 7 ) ( + 2 ) ( 7 ) ( + 2 ) ( + 9 ) ( + 7 ) + ( 2 ) c 4 = 0 c 3 = ( 5 ) Overflow occurs only in these two cases. ( 7 ) + ( 2 ) c 4 = 0 c 3 = ( + 5 ) c 4 = 1 c 3 = 1 ( 9 ) c 4 = 1 c 3 = 0 Overflow = c 3 c 4 + c 3 c 4 XOR [ Figure 3.13 from the textbook ]
105 Calculating overflow for 4-bit numbers with only three significant bits
106 Calculating overflow for n-bit numbers with only n-1 significant bits
107 Another way to look at the overflow issue
108 Another way to look at the overflow issue If both numbers that we are adding have the same sign but the sum does not, then we have an overflow.
109 Questions?
110 THE END
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