ece 331 – digital system design representation and binary arithmetic of negative numbers and...

ECE 331 – Digital System Design

Representation and Binary Arithmeticof Negative Numbers

andBinary Codes

(Lecture #10)

The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,

and were used with permission from Cengage Learning.

Fall 2010 ECE 331 - Digital System Design 2

Representation of Negative Numbers

(continued)


Signed Binary Numbers

Three representations for signed binary numbers:

1. Sign and Magnitude2. 1's Complement3. 2's Complement


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.


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' = (2n – 1) – P where P = |N|

The sign bit (MSB) = 1 for all negative numbers using the 1's Complement representation.


1's Complement

Example:

Determine the 1's Complement representation for the following negative numbers, using 8 bits:

- 11- 107- 74

Hint: (2n – 1) = (28 – 1) = 255


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.


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


1's Complement

For an n-bit signed binary number,

- (2n-1 – 1) <= D <= + (2n-1 – 1)

Includes a representation for -0 and +0. Represents an equal number of positive and

negative values.


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 = (2n – 1) – N'or

P = bit-wise complement of N'


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.


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* = (2n) – P where P = |N|

The sign bit (MSB) = 1 for all negative numbers using the 2's Complement representation.


2's Complement

Example:

Determine the 2's Complement representation for the following negative numbers, using 8 bits:

- 11- 107- 74

Hint: (2n) = (28) = 256


2's Complement

The 2's Complement representation is related to the 1's Complement representation as follows:

N' = (2n – 1) – PN* = (2n) – P

N* = N' + 1


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


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


2's Complement

For an n-bit signed binary number,

- (2n-1) <= D <= + (2n-1 – 1)

Includes only one representation for 0. Represents an additional negative value.


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 = (2n) – N*or

P = bit-wise complement of N* + 1


Binary Arithmeticof



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.



Example:



Exercise:

Add the following numbers using 2's Complement Addition:

32 + 45-17 + 63

Does overflow occur for either addition?


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)


2's Complement Subtraction

Exercise:

Subtract the following numbers using 2's Complement Addition:

32 – 45 -17 – 63

Does overflow occur for either subtraction?



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.



Example:



Exercise:

Add the following numbers using 1's Complement Addition:

32 + 45-17 + 63

Does overflow occur for either addition?


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.


Binary Codes


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


Binary Codes

4-bit Weighted Codes Code: a3a2a1a0 Weights: w3, w2, w1, w0 Decimal Value: a3 x w3 + a2 x w2 + a1 x w1 + a0 x w0

Examples 8-4-2-1 6-3-1-1 Excess-3 (obtained from 8-4-2-1)


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.


Decimal Digit

8-4-2-1 Code (BCD)

6-3-1-1 Code

Excess-3 Code

2-out-of-5 Code

Gray Code

0 0000 0000 0011 00011 0000

1 0001 0001 0100 00101 0001

2 0010 0011 0101 00110 0011

3 0011 0100 0110 01001 0010

4 0100 0101 0111 01010 0110

5 0101 0111 1000 01100 1110

6 0110 1000 1001 10001 1010

7 0111 1001 1010 10010 1011

8 1000 1011 1011 10100 1001

9 1001 1100 1100 11000 1000

Binary Codes


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.


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


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.


ASCII Code (incomplete)


Questions?

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