1. ECE 331 – Digital System Design Representation and Binary Arithmetic of Negative Numbers and Binary Codes (Lecture #10) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. Fall 2010ECE 331 - Digital System Design2 Representation of Negative Numbers (continued)

3. Fall 2010ECE 331 - Digital System Design3 Signed Binary Numbers Three representations for signed binary numbers: 1. Sign and Magnitude 2. 1's Complement 3. 2's Complement

4. Fall 2010ECE 331 - Digital System Design4 1's Complement An n-bit positive number (P) is represented in the same way as in the Sign and Magnitude representation.  The sign bit (MSB) = 0.  The remaining n-1 bits represent the magnitude.

5. Fall 2010ECE 331 - Digital System Design5 1's Complement An n-bit negative number (N) is represented using the “1's Complement” of the equivalent positive number (P).  N' = 1's Complement representation for the negative number N.  N' = (2 n – 1) – P where P = |N|  The sign bit (MSB) = 1 for all negative numbers using the 1's Complement representation.

6. Fall 2010ECE 331 - Digital System Design6 1's Complement Example: Determine the 1's Complement representation for the following negative numbers, using 8 bits: - 11 - 107 - 74 Hint: (2 n – 1) = (2 8 – 1) = 255

7. Fall 2010ECE 331 - Digital System Design7 1's Complement The 1's Complement representation of N can also be determined using the bit-wise complement of P.  N = n-bit negative number  P = |N|  N' = 1's Complement representation of N.  N' = bit-wise complement of P i.e. complement P, bit-by-bit.

8. Fall 2010ECE 331 - Digital System Design8 1's Complement Example: Determine the 1's Complement representation (using the bit-wise complement) for the following negative numbers, using 8 bits: - 74 - 11 - 107

9. Fall 2010ECE 331 - Digital System Design9 1's Complement For an n-bit signed binary number, - (2 n-1 – 1) <= D <= + (2 n-1 – 1) Includes a representation for -0 and +0. Represents an equal number of positive and negative values.

10. Fall 2010ECE 331 - Digital System Design10 1's Complement Given a negative number (N), represented using the 1's Complement representation (N'), the magnitude of the number (P) can be determined as follows: P = (2 n – 1) – N' or P = bit-wise complement of N'

11. Fall 2010ECE 331 - Digital System Design11 2's Complement An n-bit positive number (P) is represented in the same way as in the Sign and Magnitude representation.  The sign bit (MSB) = 0.  The remaining n-1 bits represent the magnitude.

12. Fall 2010ECE 331 - Digital System Design12 2's Complement An n-bit negative number (N) is represented using the “2's Complement” of the equivalent positive number (P).  N* = 2's Complement representation for the negative number N.  N* = (2 n ) – P where P = |N|  The sign bit (MSB) = 1 for all negative numbers using the 2's Complement representation.

13. Fall 2010ECE 331 - Digital System Design13 2's Complement Example: Determine the 2's Complement representation for the following negative numbers, using 8 bits: - 11 - 107 - 74 Hint: (2 n ) = (2 8 ) = 256

14. Fall 2010ECE 331 - Digital System Design14 2's Complement The 2's Complement representation is related to the 1's Complement representation as follows: N' = (2 n – 1) – P N* = (2 n ) – P N* = N' + 1

15. Fall 2010ECE 331 - Digital System Design15 2's Complement The 2's Complement representation of N can also be determined by adding 1 to the 1's Complement representation of N.  N = n-bit negative number  P = |N|  N' = One's Complement representation of N. N' = bit-wise complement of P.  N* = N' + 1

16. Fall 2010ECE 331 - Digital System Design16 2's Complement Example: Determine the 2's Complement representation (using the 1's Complement) for the following negative numbers, using 8 bits: - 74 - 11 - 107

17. Fall 2010ECE 331 - Digital System Design17 2's Complement For an n-bit signed binary number, - (2 n-1 ) <= D <= + (2 n-1 – 1) Includes only one representation for 0. Represents an additional negative value.

18. Fall 2010ECE 331 - Digital System Design18 2's Complement Given a negative number (N), represented using the 2's Complement representation (N*), the magnitude of the number (P) can be determined as follows: P = (2 n ) – N* or P = bit-wise complement of N* + 1

19. Fall 2010ECE 331 - Digital System Design19 Signed Binary Numbers

20. Fall 2010ECE 331 - Digital System Design20 Binary Arithmetic of Signed Binary Numbers

21. Fall 2010ECE 331 - Digital System Design21 2's Complement Addition Addition of n-bit signed binary numbers is straightforward using the 2's Complement system. Addition is carried out in the same way as the addition of n-bit positive numbers. Carry from the sign position (MSB) is ignored. Overflow occurs if the correct result (including the sign) cannot be represented in n bits.

22. Fall 2010ECE 331 - Digital System Design22 2's Complement Addition Example:

23. Fall 2010ECE 331 - Digital System Design23 2's Complement Addition Example:

24. Fall 2010ECE 331 - Digital System Design24 2's Complement Addition Example:

25. Fall 2010ECE 331 - Digital System Design25 2's Complement Addition Exercise: Add the following numbers using 2's Complement Addition: 32 + 45 -17 + 63 Does overflow occur for either addition?

26. Fall 2010ECE 331 - Digital System Design26 Two'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)

27. Fall 2010ECE 331 - Digital System Design27 2's Complement Subtraction Exercise: Subtract the following numbers using 2's Complement Addition: 32 – 45 -17 – 63 Does overflow occur for either subtraction?

28. Fall 2010ECE 331 - Digital System Design28 1's Complement Addition Similar to the addition of n-bit numbers using 2's Complement Addition. Instead of discarding the carry from the sign position (MSB), it must be added to the least significant bit (LSB) of the n-bit sum.  Referred to as an end-around carry.

29. Fall 2010ECE 331 - Digital System Design29 1's Complement Addition Example:

30. Fall 2010ECE 331 - Digital System Design30 1's Complement Addition Example:

31. Fall 2010ECE 331 - Digital System Design31 1's Complement Addition Exercise: Add the following numbers using 1's Complement Addition: 32 + 45 -17 + 63 Does overflow occur for either addition?

32. Fall 2010ECE 331 - Digital System Design32 Overflow General rule for detecting overflow when adding two n-bit numbers using either 1's Complement or 2's Complement Addition  An overflow occurs when the addition of two positive numbers results in a negative value or the addition of two negative numbers results in a positive value.  Cannot occur when adding a positive number and a negative number.

33. Fall 2010ECE 331 - Digital System Design33 Binary Codes

34. Fall 2010ECE 331 - Digital System Design34 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

35. Fall 2010ECE 331 - Digital System Design35 Binary Codes 4-bit Weighted Codes  Code:a 3 a 2 a 1 a 0  Weights:w 3, w 2, w 1, w 0  Decimal Value:a 3 x w 3 + a 2 x w 2 + a 1 x w 1 + a 0 x w 0 Examples  8-4-2-1  6-3-1-1  Excess-3 (obtained from 8-4-2-1)

36. Fall 2010ECE 331 - Digital System Design36 Binary Codes Examples of unweighted codes  2-out-of-5 Code Exactly 2 of the 5 bits are “1” for a valid code word.  Gray Code Code values for successive decimal digits differ in exactly one bit.

37. Fall 2010ECE 331 - Digital System Design37 Binary Codes

38. Fall 2010ECE 331 - Digital System Design38 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.

39. Fall 2010ECE 331 - Digital System Design39 In the simplest form of binary code, each decimal digit is replaced by its binary equivalent. For example, 937.25 is represented by: Binary Coded Decimal

40. Fall 2010ECE 331 - Digital System Design40 Binary Codes ASCII Code  American Standard Code for Information Interchange  Common code used for the storage and transfer of alphanumeric characters.  7-bit Weighted Code Can represent a total of 128 characters  Used to represent letters, numbers and other characters (e.g. special control characters)  Any word or number can be represented (and stored or transferred) using its ASCII Code.

41. Fall 2010ECE 331 - Digital System Design41 ASCII Code (incomplete)

42. Fall 2010ECE 331 - Digital System Design42 Questions?