Binary Arithmetic of Signed Binary Numbers Summer 2012

Binary Arithmetic of Signed Binary Numbers Summer 2012 ETE 204 - Digital Electronics 1

2's Complement Addition of n-bit signed binary numbers is straightforward using the 2's Complement number system. Addition is carried out in the same way as for n -bit positive numbers. Carry from the sign bit (leftmost bit) is ignored. Overflow occurs if the correct result (including the sign bit) cannot be represented in n bits. Summer 2012 ETE 204 - Digital Electronics 2

2's Complement Addition: Example Using 2's Complement addition and 8 -bit representation, add the following numbers: -47 + 83 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 3

2's Complement Addition: Example Using 2's Complement addition and 8 -bit representation, add the following numbers: -32 + -105 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 4

2's Complement Addition: Example Using 2's Complement addition and 8 -bit representation, add the following numbers: 19 + 52 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 5

2's Complement Addition: Example Using 2's Complement addition and 8 -bit representation, add the following numbers: 64 + 78 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 6

2's Complement Subtraction can be implemented using addition. A – B = A + (-B) Summer 2012 Determine the 2's Complement representation for the negative number -B. Use 2's Complement addition to add A and -B. ETE 204 - Digital Electronics 7

2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8 -bit representation: 64 – 78 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 8

2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8 -bit representation: -35 – 62 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 9

2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8 -bit representation: 14 – (-59) Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 10

2's Complement Subtraction: Example Subtract the following numbers, using binary subtraction and 8 -bit representation: 27 – 45 Can this subtraction be carried out? Summer 2012 ETE 204 - Digital Electronics 11

1's Complement Addition Similar to 2's Complement Addition of n-bit signed binary numbers. However, rather than ignore the carry-out from the sign (leftmost) bit, add it to the least significant bit (LSB) of the n-bit sum. Summer 2012 Known as the end-around carry. ETE 204 - Digital Electronics 12

1's Complement Addition: Example Using 1's Complement addition and 8 -bit representation, add the following numbers: -31 + -84 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 13

1's Complement Addition: Example Using 1's Complement addition and 8 -bit representation, add the following numbers: 52 + 73 Did overflow occur? Summer 2012 ETE 204 - Digital Electronics 14

Overflow The general rule for detecting overflow when performing 2's Complement or 1's Complement Addition: Summer 2012 An overflow occurs when the addition of two positive numbers results in a negative number. An overflow occurs when the addition of two negative numbers results in a positive number. Overflow cannot occur when adding a positive number to a negative number. ETE 204 - Digital Electronics 15

Binary Codes Summer 2012 ETE 204 - Digital Electronics 16

Binary Codes Weighted Codes Each position in the code has a specific weight Decimal value of code can be determined Unweighted Codes Summer 2012 Positions of code do not have a specific weight Decimal value assigned to each code ETE 204 - Digital Electronics 17

Binary Codes n-bit Weighted Codes Code: an-1 an-2 an-3. . . a 1 a 0 Weights: wn-1, wn-2, wn-3, . . . , w 1, w 0 Decimal Value: an-1 x wn-1 + an-2 x wn-2 + … + a 1 x w 1 + a 0 x w 0 4 -bit Weighted Code Summer 2012 Code: a 3 a 2 a 1 a 0 ETE 204 - Digital Electronics 18

Binary Codes Examples of 4 -bit weighted codes 8 -4 -2 -1 6 -3 -1 -1 4 bits → 16 code words Excess-3 (obtained from 8 -4 -2 -1) Summer 2012 4 bits → 16 code words Only 10 code words required to represent decimal digits 4 bits → 16 code words ETE 204 - Digital Electronics 19

Binary Codes Examples of unweighted codes 2 -out-of-5 Code Gray Code Summer 2012 Exactly 2 of the 5 bits are “ 1” for a valid code word. 10 valid code words. Code values for successive decimal digits differ in exactly one bit. 4 bits → 16 code words. ETE 204 - Digital Electronics 20

Binary Codes Summer 2012 ETE 204 - Digital Electronics 21

Binary Coded Decimal (BCD) 4 -bit binary number used to represent each decimal digit. Weighted code: 8 -4 -2 -1 Binary values 0000 … 1001 used to represent decimal values 0 … 9. Binary values 1010 … 1111 not used. Very different from binary representation. Summer 2012 ETE 204 - Digital Electronics 22

Binary Coded Decimal In BCD, each decimal digit is replaced by its binary equivalent value. Example: Binary: Summer 2012 937. 2510 = 1110101001. 012 ETE 204 - Digital Electronics 23

ASCII American Standard Code for Information Interchange Common code for the storage and transfer of alphanumeric characters. 7 -bit Weighted Code Can represent 128 characters Used to represent letters, numbers, and other characters Any word or number can be represented using its ASCII code. Summer 2012 ETE 204 - Digital Electronics 24

ASCII Code (incomplete) Summer 2012 ETE 204 - Digital Electronics 25

Questions? Summer 2012 ETE 204 - Digital Electronics 26