digital logic design lecture 3 complements , number codes and registers

Digital Logic DesignLecture 3

Complements, Number Codes and Registers

Overview

° Complement of numbers• Addition and subtraction

° Binary coded decimal

° Gray codes for binary numbers

° ASCII characters

° Moving towards hardware• Storing data

• Processing data

Complements

• In general, we (human beings) express negative numbers by placing a minus (-) sign at the left end of the number. Similarly while representing the integers in binary format, we can leave the left-most bit be the sign bit. If the left-most bit is a zero, the integer is positive; if it is a one, it is negative.

• Zero is positive and -0 = 0

• The top-most bit should tell us the sign of the integer.

• The negative of a negative integer is the original integer ie., -(-55) is 55.

• x - y should give the same result as x + (-y). That is, 8 - 3 should give us the same result as 8 + (-3).

• Negative and positive numbers shouldn't be treated in different ways when we do multiplication and division with them.

Complement

° If we consider only positive numbers, this would allow for all numbers from 0 (naturally represented by 00000000) to and inclusively 255 (represented by 11111111)

° The most obvious solution is to spare the first bit as a sign indicator, thus leaving the 7 last bits to represent the numbers.

° This simple way to represent negative numbers has for it the equally simple way to compute the negative of a given number: just invert the first bit!. Hence, the number 3, for example, represented as (00000011) will give (10000011) for -3 (we have just toggled the first bit).

° This simple way has a drawback: the negative of 0 (00000000) is now (10000000), known as -0

Two’s Complement Representation

• The two’s complement of a binary number involves inverting all bits and adding 1.

• 2’s comp of 00110011 is 11001101

• 2’s comp of 10101010 is 01010110

• For an n bit number N the 2’s complement is (2n-1) – N + 1.

• Called radix complement by Mano since 2’s complement for base (radix 2).

• To find negative of 2’s complement number take the 2’s complement.

000011002 = 1210

Sign bit Magnitude

111101002 = -1210

Sign bit Magnitude

Two’s Complement Shortcuts° Algorithm 1 – Simply complement each bit and

then add 1 to the result.• Finding the 2’s complement of (01100101)2 and of its 2’s

complement…

N = 01100101 [N] = 10011011

10011010 01100100

+ 1 + 1

--------------- ---------------

10011011 01100101

° Algorithm 2 – Starting with the least significant bit, copy all of the bits up to and including the first 1 bit and then complementing the remaining bits.• N = 0 1 1 0 0 1 0 1

[N] = 1 0 0 1 1 0 1 1

Finite Number Representation

° Machines that use 2’s complement arithmetic can represent integers in the range

-2n-1 <= N <= 2n-1-1

where n is the number of bits available for representing N. Note that 2n-1-1 = (011..11)2 and –2n-1 = (100..00)2

oFor 2’s complement more negative numbers than positive.

oFor 1’s complement two representations for zero.

oFor an n bit number in base (radix) z there are zn

different unsigned values.

(0, 1, …zn-1)

1’s Complement Addition

° Using 1’s complement numbers, adding numbers is easy.

° For example, suppose we wish to add +(1100)2 and +(0001)2.

° Let’s compute (12)10 + (1)10.• (12)10 = +(1100)2 = 011002 in 1’s comp.

• (1)10 = +(0001)2 = 000012 in 1’s comp. 0 1 1 0 0 + 0 0 0 0 1-------------- 0 0 1 1 0 1 0-------------- 0 1 1 0 1

Add carry

Final Result

Step 1: Add binary numbersStep 2: Add carry to low-order bit

Add

1’s Complement Subtraction

° Using 1’s complement numbers, subtracting numbers is also easy.

° For example, suppose we wish to subtract +(0001)2 from +(1100)2.

° Let’s compute (12)10 - (1)10.• (12)10 = +(1100)2 = 011002 in 1’s comp.

• (-1)10 = -(0001)2 = 111102 in 1’s comp.

0 1 1 0 0 - 0 0 0 0 1--------------

0 1 1 0 0 + 1 1 1 1 0-------------- 1 0 1 0 1 0 1-------------- 0 1 0 1 1

Add carry

Final Result

Step 1: Take 1’s complement of 2nd operandStep 2: Add binary numbersStep 3: Add carry to low order bit

1’s comp

Add

2’s Complement Addition

° Using 2’s complement numbers, adding numbers is easy.

° For example, suppose we wish to add +(1100)2 and +(0001)2.

° Let’s compute (12)10 + (1)10.• (12)10 = +(1100)2 = 011002 in 2’s comp.

• (1)10 = +(0001)2 = 000012 in 2’s comp. 0 1 1 0 0 + 0 0 0 0 1-------------- 0 0 1 1 0 1

FinalResult

Step 1: Add binary numbersStep 2: Ignore carry bit

Add

Ignore


° Using 2’s complement numbers, follow steps for subtraction

° For example, suppose we wish to subtract +(0001)2 from +(1100)2.

° Let’s compute (12)10 - (1)10.• (12)10 = +(1100)2 = 011002 in 2’s comp.

• (-1)10 = -(0001)2 = 111112 in 2’s comp.

0 1 1 0 0 - 0 0 0 0 1--------------

0 1 1 0 0 + 1 1 1 1 1-------------- 1 0 1 0 1 1

Final Result

Step 1: Take 2’s complement of 2nd operandStep 2: Add binary numbersStep 3: Ignore carry bit

2’s comp

Add

IgnoreCarry

2’s Complement Subtraction: Example #2

° Let’s compute (13)10 – (5)10.• (13)10 = +(1101)2 = (01101)2

• (-5)10 = -(0101)2 = (11011)2

° Adding these two 5-bit codes…

° Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result. Indeed,

(01000)2 = +(1000)2 = +(8)10.

0 1 1 0 1 + 1 1 0 1 1-------------- 1 0 1 0 0 0

carry

2’s Complement Subtraction: Example #3° Let’s compute (5)10 – (12)10.

• (-12)10 = -(1100)2 = (10100)2

• (5)10 = +(0101)2 = (00101)2


° Here, there is no carry bit and the sign bit is 1. This indicates a negative result, which is what we expect. (11001)2 = -(7)10.

0 0 1 0 1 + 1 0 1 0 0-------------- 1 1 0 0 1


° Let’s compute (13)10 - (5)10.• (13)10 = +(1101)2 = (01101)2

• (-5)10 = -(0101)2 = (11011)2


° Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result.

0 1 1 0 1 + 1 1 0 1 1-------------- 1 0 1 0 0 0

carry

Data Representation and Communication° Human communication

Includes language, images and sounds

° Computers Process and store all forms of data in binary format

° Conversion to computer-usable representation using data formats

Define the different ways human data may be represented, stored and processed by a computer

° Numbers are important to computers as they • Represent information precisely

• Can be processed

° Decimal numbers are favored by humans. Binary numbers are natural to computers. Hence, conversion is required.

° If little calculation is required, we can use some coding schemes to store decimal numbers, for data transmission purposes.

° Examples: BCD (or 8421), Excess-3, 8-4-2-1, 2421, etc.

° Each decimal digit is represented as a 4-bit code.

° The number of digits in a code is also called the length of the code.

Data Representation and Communication

BCD

Digit Bit pattern

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

Note: the following 6 bit patterns are not used:

101010111100110111101111

° Binary Coded Decimal (BCD) represents each decimal digit with four bits• Ex. 0011 0010 1001 = 32910

° This is NOT the same as 0011001010012

° Why do this? Because people think in decimal.

Putting It All Together

° BCD not very efficient

° Used in early computers (40s, 50s)

° Used to encode numbers for seven-segment displays.

° Easier to read?

Gray Code

° Gray code is not a number system.• It is an alternate way to represent

four bit data

° Only one bit changes from one decimal digit to the next

° Useful for reducing errors in communication.

° Can be scaled to larger numbers.

Digit Binary Gray Code

0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101 10 1010 1111 11 1011 1110 12 1100 1010 13 1101 1011 14 1110 1001 15 1111 1000

ASCII Code

° American Standard Code for Information Interchange

° ASCII is a 7-bit code, frequently used with an 8th bit for error detection (more about that in a bit).

Character ASCII (bin) ASCII (hex) Decimal OctalA 1000001 41 65 101B 1000010 42 66 102C 1000011 43 67 103…

Z

a

…

1

‘

21

000 001 010 011 100 101 110 1110000 NULL DLE 0 @ P ` p0001 SOH DC1 ! 1 A Q a q0010 STX DC2 " 2 B R b r0011 ETX DC3 # 3 C S c s0100 EDT DC4 $ 4 D T d t0101 ENQ NAK % 5 E U e u0110 ACK SYN & 6 F V f v0111 BEL ETB ' 7 G W g w1000 BS CAN ( 8 H X h x1001 HT EM ) 9 I Y i y1010 LF SUB * : J Z j z1011 VT ESC + ; K [ k {1100 FF FS , < L \ l |1101 CR GS - = M ] m }1110 SO RS . > N ^ n ~1111 SI US / ? O _ o DEL

Most significant bit

Least significant bit

ASCII Code

22


e.g., ‘a’ = 1100001

ASCII Code

23

Alphabetic codes


ASCII Code

24

Numeric codes


ASCII Code

25


Punctuation, etc.

ASCII Code

ASCII Codes and Data Transmission

° ASCII Codes

° A – Z (26 codes), a – z (26 codes)

° 0-9 (10 codes), others (@#$%^&*….)

° Complete listing in Mano text

° Transmission susceptible to noise

° Typical transmission rates (1500 Kbps, 56.6 Kbps)

° How to keep data transmission accurate?

Parity Codes

° Parity codes are formed by concatenating a parity bit, P to each code word of C.

° In an odd-parity code, the parity bit is specified so that the total number of ones is odd.

° In an even-parity code, the parity bit is specified so that the total number of ones is even.

Information BitsP

1 1 0 0 0 0 1 1

Added even parity bit

0 1 0 0 0 0 1 1

Added odd parity bit

Parity Code Example

° Concatenate a parity bit to the ASCII code for the characters 0, X, and = to produce both odd-parity and even-parity codes.

Character ASCII Odd-Parity ASCII

Even-Parity ASCII

0 0110000 10110000 00110000

X 1011000 01011000 11011000

= 0111100 10111100 00111100

Binary Data Storage

• Binary cells store individual bits of data

• Multiple cells form a register.

• Data in registers can indicate different values

• Hex (decimal)

• BCD

• ASCII

Binary Cell

0 0 1 0 1 0 1 1

Register Transfer

° Data can move from register to register.

° Digital logic used to process data

° We will learn to design this logic

Register A Register B

Register C

Digital Logic Circuits

Transfer of Information

° Data input at keyboard

° Shifted into place

° Stored in memory

NOTE: Data input in ASCII

Building a Computer

° We need processing

° We need storage

° We need communication

° You will learn to use and design these components.

Summary° 2’s complement most important (only 1

representation for zero).

° Important to understand treatment of sign bit for 1’s and 2’s complement.

° Although 2’s complement most important, other number codes exist

° ASCII code used to represent characters (including those on the keyboard)

° Registers store binary data

digital logic design lecture 3 complements , number codes and registers

Documents

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