Summer 2012ETE Digital Electronics1 Binary Arithmetic of Signed Binary Numbers

2 2's Complement Addition 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 2012ETE Digital Electronics

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

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

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

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

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

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

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

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

11 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 2012ETE Digital Electronics

12 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.  Known as the end-around carry. Summer 2012ETE Digital Electronics

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

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

15 Overflow The general rule for detecting overflow when performing 2's Complement or 1's Complement Addition:  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. Summer 2012ETE Digital Electronics

16 Binary Codes Summer 2012ETE Digital Electronics

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

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

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

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

21 Binary Codes Summer 2012ETE Digital Electronics

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

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

24 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 2012ETE Digital Electronics

25 ASCII Code (incomplete) Summer 2012ETE Digital Electronics

26 Questions? Summer 2012ETE Digital Electronics