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Outline

Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements Signed Binary Numbers Binary Codes Binary Storage and Registers Binary Logic

Digital Systems

Digital computer is the best-known example of a digital system Others are telephone switching exchanges, digital voltmeters, digital calculators, etc. A digital system manipulates discrete elements of information Discrete elements: electric impulses, decimal digits, letters of an alphabet, any other set of meaningful symbols

Digital Systems

In a digital system, discrete elements of information are represented by signals Electrical signals (voltages & currents) are the most common Present day systems have only two discrete values (binary) Alternative, many-valued circuits are less reliable A lot of information is already discrete and continuous values can be quantized (sampled)

Digital Systems

Digital Systems

A digital computer is an interconnection of digital modules

To understand each module, it is necessary to have a basic knowledge of digital systems

Binary Numbers

7392 represents a quantity that is equal to

Decimal number system is of base (or radix) 10 In binary system, possible values are 0 and 1 and each digit is multiplied by E.g. 11010.11 is

Binary Numbers

Hexadecimal (base 16) numbers use digits 09 and letters A, B, C, D, E, F to represent values 10-15

Operations work similarly in all bases

Number-Base Conversions

Converting a number from base x to decimal is simple (as shown before) Decimal to base x is easier if number is separated into integer and fraction parts Convert 41 to binary

Divide 41 by 2, quotient is 20 and remainder is 1. Continue dividing the quotient until it becomes 0. Remainders give us the binary number as follows:

Number-Base Conversions

Number-Base Conversions

Conversion of a fraction is similar but the number is multiplied by to instead of dividing

Octal and Hexadecimal Numbers

Conversions between binary, octal and hexadecimal numbers are easier Each octal digit corresponds to 3 binary digits and each hexadecimal digit corresponds to 4 binary digits

Complements

Simplifies

the subtraction operation Logical operations

Two types exist

The radix complement (r’s complement)

10’s complement, 2’s complement

The diminished radix complement ((r-1)’s complement)

9’s complement, 1’s complement

Diminished Radix (r-1) complement

Given a number N in base r having n digits:

When r=10, (r-1)’s complement is called 9’s complement. 10n-1 is a number represented by n 9’s. 9’s complement of 546700 is (n=6)

(r-1)’s complement of N is (rn-1)-N

999999-546700=453299

9’s complement of 012398 is (n=6)

999999-012398=987601

1’s complement

For binary numbers, r=2 and r-1=1. 1’s complement of N is (2n-1)-N If n=4, 2n=10000. So, 2n-1=1111. To determine the 1’s complement of a number, subtract each digit from 1. Or, bit flip!!! Replace 0’s with 1’s, 1’s with 0’s!!!

Example:

If N= 1011000, 1’s comp.= 0100111 If N= 010110, 1’s comp.= 101001

Radix (r’s) complement

Given a number N in base r having n digits:

When r=10, r’s complement is called 10’s complement. 10’s complement of 546700 is (n=6)

r’s complement of N is rn-N

1000000-546700=453300

10’s complement of 012398 is (n=6)

1000000-012398=987602

2’s complement

For binary numbers, r=2, 2’s complement of N is 2n-N To determine the 2’s complement of a number, determine 1’s complement and add 1 to it. Example:

If N= 1011001, 1’s comp.= 0100110, 2’s comp.=0100111 If N= 1101100, 2’s comp.= 0010100

Another way of finding 2’s comp.: Leave all least significant 0’s and the first 1 unchanged, bit flip the remaning digits.

Subtraction with Complements Minuend: 101101 Subtrahend: 100111 Difference: 000110 1.

2.

3.

Add the minuend M to r’s complement of the subtrahend N. M + (rn-N) = M - N+rn If M>=N, the sum will produce an end carry. Discard it and what is left is the result M-N. If M

Example

Example

Example

Signed Binary Numbers

Negative numbers is shown with a minus sign in math. In digital systems, the first bit decides the sign of the number.

If the first bit 0, the number is positive. If the first bit 1, the number is negative.

This is called signed magnitude convention.

Signed complement systems

To represent negative number, 1’s complement and 2’s complements are also used.

Example

Represent +9 and -9 in eight bit system

+9 is same for all systems: 00001001 -9

To determine negative number

Signed magnitute: Take the positive number, change the most significant bit to 1 One’s complement: Take the one’s complement of the positive number. Two’s complement: Take the two’s complement of the positive number. (Or add 1 to one’s complement)

Arithmetic Addition

Aritmetic subtraction

Take the 2’s complement of subtrahend. Add it to the minuend. Discard cary if there is any.

Examples: 10-5 (8 bits), -3-5, 18-(-9)

Binary Codes – BCD Codes

n bit can code upto 2n combinations.

BCD Addition

Example

Other Decimal Codes

Gray Codes

Only one bit changes when going from one number to the next. How to determine the gray code equivalent of a number:

Add 0 to the left of number. XOR every two neigboring pair in order. The result is the gray code.

Example: 1 1 0 0 0 0 -> 0 1 1 0 0 0 0 101000

Error Detecting Codes

Add an extra bit (parity bit) to make the total number of one’s either even or odd.

Binary logic

Truth tables

Gate sysbols

Timing diagrams

More than two inputs

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