Download - ECE 331 – Digital System Design Representation of Negative Numbers, Binary Arithmetic of Negative Numbers, and Binary Codes (Lecture #11) The slides included

ECE 331 – Digital System Design

Representation of Negative Numbers,Binary Arithmetic of Negative Numbers,

andBinary Codes

(Lecture #11)

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.

Spring 2011 ECE 331 - Digital System Design 2

Representation of Negative Numbers

(continued)


Signed Binary Numbers

Representations for signed binary numbers:

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



Arithmetic circuits are difficult to design for Sign and Magnitude binary numbers.

Consequently, this number system is not typically used in digital (computer) systems.

Instead other number systems, namely the 1's and 2's Complements, are more commonly used.

As we will see, it is rather easy to design arithmetic circuits for binary numbers represented in these number systems.


1's Complement

A positive number, N, is represented in the same way as in the Sign and Magnitude representation.

For an n-bit number, The leftmost bit (sign bit) = 0.

Indicating a positive number. The remaining n-1 bits represent the

magnitude.


1's Complement

A negative number, -N, is represented by the “1's complement” of the positive number, N.

N' = 1's complement representation for -N. For an n-bit signed binary number,

The leftmost bit (sign bit) = 1 for all negative numbers in the 1's Complement system.

N' = (2n – 1) – N


1's Complement: Examples

Using 8 bits, determine the 1's Complement representation for the following negative

numbers:

-15-102


1's Complement

The 1's Complement representation for -N can also be determined by taking the bit-wise complement of N.

N' = 1's Complement representation for -N. For an n-bit signed binary number,

i.e. complement N, bit-by-bit

N' = bit-wise complement of N




numbers:

-15-102

(Use the bit-wise complement)


1's Complement

For an n-bit 1's Complement binary number,

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

negative values.

- (2n-1 – 1) <= N <= + (2n-1 – 1)


1's Complement

To determine the magnitude of a negative number, -N, that is represented by its 1's Complement, N', simply take the “1's complement” of the 1's Complement.

N = (2n – 1) – N'

N = bit-wise complement of N'orpositive #

1's complement rep.for negative #


2's Complement

A positive number, N, is represented in the same way as in the Sign and Magnitude representation.

For an n-bit number, The leftmost bit (sign bit) = 0.

Indicating a positive number. The remaining n-1 bits represent the

magnitude.


2's Complement

A negative number, -N, is represented by the “2's complement” of the positive number, N.

N* = 2's complement representation for -N. For an n-bit signed binary number,

The leftmost bit (sign bit) = 1 for all negative numbers in the 2's Complement system.

N* = (2n) – N




numbers:

-12-95


2's Complement

The 1's and 2's Complement representations for a negative number, -N, are related as follows:

N* = (2n) – N

N' = (2n - 1) – N

N* = N' +1


2's Complement

Thus, the 2's Complement representation for -N can also be determined by adding 1 to the 1's Complement representation for -N.

N' = 1's Complement representation for -N. N* = 2's Complement representation for -N.

For an n-bit signed binary number,

N* = N' + 1




numbers:

-12-95

(Use the 1's complement)


2's Complement

For an n-bit 2's Complement binary number,

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

- (2n-1) <= N <= + (2n-1 – 1)


2's Complement

To determine the magnitude of a negative number, -N, that is represented by its 2's Complement, N*, simply take the “2's complement” of the 2's Complement.

N = (2n) – N*

N = (N*)' + 1orpositive #

2's complement rep. for negative #

bit-wise complement of 2's complement


Binary Arithmeticof



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.


2's Complement Addition: Example

Using 2's Complement addition and 8-bit representation, add the following numbers:

-47 + 83

Did overflow occur?




-32 + -105

Did overflow occur?




19 + 52

Did overflow occur?




64 + 78

Did overflow occur?


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)


2's Complement Subtraction: Example

Subtract the following numbers, using 2's Complement addition and 8-bit representation:

64 – 78

Did overflow occur?




-35 – 62

Did overflow occur?




14 – (-59)

Did overflow occur?



Subtract the following numbers, using binary subtraction and 8-bit representation:

27 – 45

Can this subtraction be carried out?


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.




-31 + -84

Did overflow occur?




52 + 73

Did overflow occur?


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.


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

n-bit Weighted Codes Code: an-1an-2an-3...a1a0 Weights: wn-1, wn-2, wn-3, ..., w1, w0 Decimal Value: an-1 x wn-1 + an-2 x wn-2 + …

+ a1 x w1 + a0 x w0 4-bit Weighted Code

Code: a3a2a1a0


Binary Codes

Examples of 4-bit weighted codes 8-4-2-1

4 bits → 16 code words Only 10 code words required to represent

decimal digits 6-3-1-1

4 bits → 16 code words Excess-3 (obtained from 8-4-2-1)

4 bits → 16 code words


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.


Binary Codes

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 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.


Binary Coded Decimal

In BCD, each decimal digit is replaced by its binary equivalent value.

Example:

Binary: 937.2510 = 1110101001.012


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.


ASCII Code (incomplete)


Questions?

Download - ECE 331 – Digital System Design Representation of Negative Numbers, Binary Arithmetic of Negative Numbers, and Binary Codes (Lecture #11) The slides included

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