CprE 281: Digital Logic
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1 CprE 281: Digital Logic Instructor: Alexander Stoytchev
2 Multiplication CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
3 HW 6 is out Administrative Stuff It is due on Monday Oct 4pm
4 Quick Review
5 The Full-Adder Circuit [ Figure 3.3c from the textbook ]
6 The Full-Adder Circuit Let's take a closer look at this. [ Figure 3.3c from the textbook ]
7 Another Way to Draw the Full-Adder Circuit c i y i x i c i+1 = x i y i + (x i + y i )c i
8 Decomposing the Carry Expression c i+1 = x i y i + x i c i + y i c i
9 Decomposing the Carry Expression c i+1 = x i y i + x i c i + y i c i c i+1 = x i y i + (x i + y i )c i
10 Decomposing the Carry Expression c i+1 = x i y i + x i c i + y i c i c i+1 = x i y i + (x i + y i )c i c i y i x i c i+1
11 Another Way to Draw the Full-Adder Circuit c i+1 = x i y i + x i c i + y i c i c i+1 = x i y i + (x i + y i )c i c i y i s i x i c i+1
12 Another Way to Draw the Full-Adder Circuit c i+1 = x i y i + (x i + y i )c i c i y i s i x i c i+1
13 Another Way to Draw the Full-Adder Circuit c i+1 = x i y i + (x i + y i )c i g i p i c i y i s i x i c i+1
14 Another Way to Draw the Full-Adder Circuit c i+1 = x i y i + (x i + y i )c i g i p i c i y i s i x i p i g i c i+1
15 Yet Another Way to Draw It (Just Rotate It) x i y i g i p i c i c i+1 s i
16 Now we can Build a Ripple-Carry Adder x 1 y 1 x 0 y 0 g 1 p 1 g 0 p 0 c 2 c 1 c 0 Stage 1 Stage 0 s 1 s 0 [ Figure 3.14 from the textbook ]
17 Now we can Build a Ripple-Carry Adder x 1 y 1 x 0 y 0 g 1 p 1 g 0 p 0 c 2 c 1 c 0 Stage 1 Stage 0 s 1 s 0 [ Figure 3.14 from the textbook ]
18 The delay is 5 gates (1+2+2) x 1 y 1 x 0 y 0 g 1 p 1 g 0 p 0 c 2 c 1 c 0 Stage 1 Stage 0 s 1 s 0
19 n-bit ripple-carry adder: 2n+1 gate delays x 1 y 1 x 0 y 0 g 1 p 1 g 0 p 0 c 2 c 1 c 0... Stage 1 Stage 0 s 1 s 0
20 Decomposing the Carry Expression c i+1 = x i y i + x i c i + y i c i c i+1 = x i y i + (x i + y i )c i g i p i c i+1 = g i + p i c i c i+1 = g i + p i (g i-1 + p i-1 c i-1 ) = g i + p i g i-1 + p i p i-1 c i-1
21 Carry for the first two stages c 1 = g 0 + p 0 c 0 c 2 = g 1 + p 1 g 0 + p 1 p 0 c 0
22 The first two stages of a carry-lookahead adder x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 1 s 0 [ Figure 3.15 from the textbook ]
23 It takes 3 gate delays to generate c 1 x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 1 s 0
24 It takes 3 gate delays to generate c 2 x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 1 s 0
25 The first two stages of a carry-lookahead adder x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 1 s 0
26 It takes 4 gate delays to generate s 1 x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 1 s 0
27 It takes 4 gate delays to generate s 2 x 1 y 1 x 0 y 0 x 0 y 0 g 1 p 1 g 0 p 0 c 0 c 2 c 1 s 2 s 1 s 0
28 N-bit Carry-Lookahead Adder It takes 3 gate delays to generate all carry signals It takes 1 more gate delay to generate all sum bits Thus, the total delay through an n-bit carry-lookahead adder is only 4 gate delays!
29 Expanding the Carry Expression c i+1 = g i + p i c i c 1 = g 0 + p 0 c 0 c 2 = g 1 + p 1 g 0 + p 1 p 0 c 0 c 3 = g 2 + p 2 g 1 + p 2 p 1 g 0 + p 2 p 1 p 0 c 0... c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0
30 Expanding the Carry Expression c i+1 = g i + p i c i c 1 = g 0 + p 0 c 0 c 2 = g 1 + p 1 g 0 + p 1 p 0 c 0 c 3 = g 2 + p 2 g 1 + p 2 p 1 g 0 + p 2 p 1 p 0 c 0... c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 Even this takes only 3 gate delays + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0
31 A hierarchical carry-lookahead adder with ripple-carry between blocks x y x 15 8 y 15 8 x 7 0 y 7 0 c 32 Block c 3 24 c 16 Block 1 c 8 Block 0 c 0 s s 15 8 s 7 0 [ Figure 3.16 from the textbook ]
32 A hierarchical carry-lookahead adder x y x 15 8 y 15 8 x 7 0 y 7 0 Block 3 c 24 Block 1 Block 0 c 0 G 3 P 3 G 1 P 1 G 0 P 0 s s 15 8 s 7 0 c 32 c 16 c 8 Second-level lookahead [ Figure 3.17 from the textbook ]
33 The Hierarchical Carry Expression c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0
34 The Hierarchical Carry Expression c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0
35 The Hierarchical Carry Expression c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 G 0 + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0 P 0
36 The Hierarchical Carry Expression c 8 = g 7 + p 7 g 6 + p 7 p 6 g 5 + p 7 p 6 p 5 g 4 G 0 + p 7 p 6 p 5 p 4 g 3 + p 7 p 6 p 5 p 4 p 3 g 2 + p 7 p 6 p 5 p 4 p 3 p 2 g 1 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 g 0 + p 7 p 6 p 5 p 4 p 3 p 2 p 1 p 0 c 0 P 0 c 8 = G 0 + P 0 c 0
37 The Hierarchical Carry Expression c 8 = G 0 + P 0 c 0 c 16 = G 1 + P 1 c 8 = G 1 + P 1 G 0 + P 1 P 0 c 0 c 24 = G 2 + P 2 G 1 + P 2 P 1 G 0 + P 2 P 1 P 0 c 0 c 32 = G 3 + P 3 G 2 + P 3 P 2 G 1 + P 3 P 2 P 1 G 0 + P 3 P 2 P 1 P 0 c 0
38 A hierarchical carry-lookahead adder x y x 15 8 y 15 8 x 7 0 y 7 0 Block 3 c 24 Block 1 Block 0 c 0 G 3 P 3 G 1 P 1 G 0 P 0 s s 15 8 s 7 0 c 32 c 16 c 8 Second-level lookahead [ Figure 3.17 from the textbook ]
39 Hierarchical CLA Adder Carry Logic Block 2 P1 P0 c0 p7 x7 y7 g7 x6 y6 p7g6 g6 p7p6g5 p7p6p5g4 Block 1 p7p6p5p4g3 p7p6p5p4p3g2 p7p6p5p4p3p2g1 p7p6p5p4p3p2p1g0 P0 c0 C8 5 gate delays C16 5 gate delays C24 5 Gate delays C32 5 Gate delays SECOND LEVEL HIERARCHY c16 G1 P1 G0 P0 c0 c8 P1 G0 G0 P1 P0 c0 Block 3 Block 2 Block 1 c0 p17 c17 p9 c9 p0 c1 g17 g9 g0 FIRST LEVEL HIERARCHY
40 Hierarchical CLA Critical Path Block 2 p7 x7 y7 g7 x6 y6 p7g6 g6 p7p6g5 p7p6p5g4 Block 1 p7p6p5p4g3 p7p6p5p4p3g2 p7p6p5p4p3p2g1 p7p6p5p4p3p2p1g0 G1 G0 P0 c0 c8 P1 P1 G0 G0 P1 P0 c0 c16 P1 P0 c0 P0 c0 C9 7 gate delays C17 7 gate delays C25 7 Gate delays SECOND LEVEL HIERARCHY Block 3 Block 2 Block 1 c0 p17 c17 p9 c9 p0 c1 g17 g9 g0 FIRST LEVEL HIERARCHY
41 Total Gate Delay Through a Hierarchical Carry-Lookahead Adder Is 8 gates 3 to generate all Gj and Pj +2 to generate c8, c16, c24, and c32 +2 to generate internal carries in the blocks +1 to generate the sum bits (one extra XOR)
42
43 Decimal Multiplication by 10 What happens when we multiply a number by 10? 4 x 10 =? 542 x 10 =? 1245 x 10 =?
44 Decimal Multiplication by 10 What happens when we multiply a number by 10? 4 x 10 = x 10 = x 10 = 12450
45 Decimal Multiplication by 10 What happens when we multiply a number by 10? 4 x 10 = x 10 = x 10 = You simply add a zero as the rightmost number
46 Decimal Division by 10 What happens when we divide a number by 10? 14 / 10 =? 540 / 10 =? 1240 x 10 =?
47 Decimal Division by 10 What happens when we divide a number by 10? 14 / 10 = 1 //integer division 540 / 10 = x 10 = 124 You simply delete the rightmost number
48 Binary Multiplication by 2 What happens when we multiply a number by 2? 011 times 2 =? 101 times 2 =? times 2 =?
49 Binary Multiplication by 2 What happens when we multiply a number by 2? 011 times 2 = times 2 = times 2 = You simply add a zero as the rightmost number
50 Binary Multiplication by 4 What happens when we multiply a number by 4? 011 times 4 =? 101 times 4 =? times 4 =?
51 Binary Multiplication by 4 What happens when we multiply a number by 4? 011 times 4 = times 4 = times 4 = add two zeros in the last two bits and shift everything else to the left
52 Binary Multiplication by 2 N What happens when we multiply a number by 2 N? 011 times 2 N = // add N zeros 101 times 4 = // add N zeros times 4 = // add N zeros
53 Binary Division by 2 What happens when we divide a number by 2? 0110 divided by 2 =? 1010 divides by 2 =? divides by 2 =?
54 Binary Division by 2 What happens when we divide a number by 2? 0110 divided by 2 = divides by 2 = divides by 2 = You simply delete the rightmost number
55 Decimal Multiplication By Hand [
56 Binary Multiplication By Hand [Figure 3.34a from the textbook]
57 Binary Multiplication By Hand [Figure 3.34b from the textbook]
58 Binary Multiplication By Hand [Figure 3.34c from the textbook]
59 Figure A 4x4 multiplier circuit.
60 Figure A 4x4 multiplier circuit.
61 Positive Multiplicand Example [Figure 3.36a in the textbook]
62 Positive Multiplicand Example add an extra bit to avoid overflow [Figure 3.36a in the textbook]
63 Negative Multiplicand Example [Figure 3.36b in the textbook]
64 Negative Multiplicand Example add an extra bit to avoid overflow but now it is 1 [Figure 3.36b in the textbook]
65 What if the Multiplier is Negative? Convert both to their 2's complement version This will make the multiplier positive Then Proceed as normal This will not affect the result Example: 5*(-4) = (-5)*(4)= -20
66 Binary Coded Decimal
67 Table of Binary-Coded Decimal Digits
68 Addition of BCD digits [Figure 3.38a in the textbook]
69 Addition of BCD digits The result is greater than 9, which is not a valid BCD number [Figure 3.38a in the textbook]
70 Addition of BCD digits add 6 [Figure 3.38a in the textbook]
71 Addition of BCD digits [Figure 3.38b in the textbook]
72 Addition of BCD digits The result is 1, but it should be 7 [Figure 3.38b in the textbook]
73 Addition of BCD digits add 6 [Figure 3.38b in the textbook]
74 Why add 6? Think of BCD addition as a mod 16 operation Decimal addition is mod 10 operation
75 BCD Arithmetic Rules Z = X + Y If Z <= 9, then S=Z and carry-out = 0 If Z > 9, then S=Z+6 and carry-out =1
76 Block diagram for a one-digit BCD adder [Figure 3.39 in the textbook]
77 How to check if the number is > 9?
78 A four-variable Karnaugh map x1 x2 x3 x m m m m m m m m m m m m m m m m15 x 1 x 2 x 3 x x m 0 m 1 m 5 m 3 m 2 m 6 m 4 m 12 x 2 m 13 m 7 m 15 m 14 x 1 m 8 m 9 m 11 m 10 x 4
79 How to check if the number is > 9? z3 z2 z1 z m m m m m m m m m m m m m m m m15 z 3 z z 2 1 z
80 How to check if the number is > 9? z3 z2 z1 z m m m m m m m m m m m m m m m m15 z 3 z z 2 1 z f = z 3 z 2 + z 3 z 1
81 How to check if the number is > 9? z3 z2 z1 z m m m m m m m m m m m m m m m m15 z 3 z z 2 1 z f = z 3 z 2 + z 3 z 1 In addition, also check if there was a carry f = carry-out + z 3 z 2 + z 3 z 1
82 Verilog code for a one-digit BCD adder module bcdadd(cin, X, Y, S, Cout); input Cin; input [3:0] X, Y; output reg [3:0] S; output reg Cout; reg [4:0] Z; (X, Y, Cin) begin Z = X + Y + Cin; if (Z < 10) {Cout, S} = Z; else {Cout, S} = Z + 6; end endmodule [Figure 3.40 in the textbook]
83 Circuit for a one-digit BCD adder [Figure 3.41 in the textbook]
84 Circuit for a one-digit BCD adder carry-out + z 3 z 2 + z 3 z 1 [Figure 3.41 in the textbook]
85 Circuit for a one-digit BCD adder 1 [Figure 3.41 in the textbook]
86 Circuit for a one-digit BCD adder [Figure 3.41 in the textbook]
87 Circuit for a one-digit BCD adder [Figure 3.41 in the textbook]
88 Circuit for a one-digit BCD adder 1 implicit add 6 1 [Figure 3.41 in the textbook]
89 Questions?
90 THE END
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