Digitaltechnik 1

Digital Codes

• Digitalization encompasses all integer codes.
• Unsigned integers are used.
• Integer conversion is a process.
• Digits greater than 'q' are considered.
• Negative numbers are represented.
• One's complement is a method handling negative numbers, can overflow.
• Two's complement is another arithmetic representation for negative numbers.
• Decimal representation is a way to express values.
• BCD (Binary-Coded Decimal) addition is a specific way of adding numbers represented in this way.
• Binary code rotary disc is a type of encoder.
• Gray code is a coding system.
• Detection and correction of errors are different tasks in coding.
• Even/odd parity is used for error detection in bit streams

Binary Logic

• Binary logic is easily implemented electronically.
• Values are clearly defined; it's either 0 or 1.
• Binary switches are used.
• Using positive binary logic in the digital world.
• Binary values are easily converted into electronic signals.
• Electronic switches can be made of transistors.

Binary Numbers

• The number of bits determines the maximum representable values; 2^n
• 8-bit values can represent 256 distinct values (0 to 255).

Binary Counting/Addition

• Counting and adding in binary is similar to the decimal system, with only two digits (0 and 1).
• Addition of two n-bit numbers produces an n+1 bit number.
• Overflow can occur in hardware operations.

Hamming Weight

• Hamming weight represents the number of '1's in a binary number.
• Example: 1011 has a Hamming weight of 3.

Hamming Distance

• The Hamming distance is the number of differing bits between two binary number strings of the same length.
• Example: For A = 1011 and B = 1101, the Hamming distance is 2.

Coding/Decoding

• A semantic symbol can be translated into a digital form and represented by a sequence of 0s and 1s.
• A sequence of bits with a specific meaning is known as a codeword.
• For proper interpretation of a bitstream, the utilized encoding technique must be known.
• Out of band (OOB) information is data exchanged outside the main communication channel.

Ambiguity

• Ambiguity in codes should be avoided.
• Example: Morse code, where the same combination of dots and dashes can represent different letters.

Prefix Codes

• Prefix codes ensure that no codeword is a prefix of another.

Redundancy

• Redundancy is a deliberate or accidental duplication of information.
• Redundancy can improve error tolerance, reliability, and security.
• QR codes contain redundant information to ensure complete information exchange.

Unsigned Integers

• The simplest representation of a binary sequence is an unsigned integer.
• The formula calculates the value of a binary string given its bits.

Integer Conversion

• Any integer system can be converted to any other base (as long as its an integer base).
• Repeating division by m is used.
• Conversion is simpler when the conversion bases are powers of each other.

Digits greater than 9 (Hexadecimal)

• Numbers like 16 (hexadecimal) use letters for digits greater than 9 (e.g., A=10, B=11).

Negative Numbers: One's Complement

• The most significant bit (MSB) designates the number's sign. A zero indicates a positive number, and a one indicates a negative number.
• To negate a number, invert its bits.
• Example calculations for converting a decimal value to one's complement using a specific number of bits are presented.

Negative Numbers: Two's Complement

• The MSB indicates the number's sign. It's the most consistent way to represent negative numbers.
• No special case for overflow.
• Converting to and from two's complement is shown.
• Example calculations show converting to and from two's complement using a specific bit length are presented.

Overflow

• Overflow in addition can happen, when the result of a sum is too large to fit in the space of the word or result length.
• In addition, any carry over the upper bit in the result of the sum is discarded.
• A correction or extra step is usually required when adding numbers in one's complement method.

Decimal Representation

• Converting between decimal (base-10) and binary (base-2) involves several steps and considerations.

Pseudotetrades

• Values 0-9 are represented as tetrades.
• Values beyond 9 are pseudotetrades.
• Conversions and examples are given.

BCD Addition

• BCD addition is similar to binary addition with manual corrections to maintain the BCD properties.

Gray Code

• Gray code is a coding system where consecutive numbers only differ by a single bit.
• It's useful for reducing errors during transitions.

Error Detection and Correction

• Error detecting codes identify potential errors.
• Error correcting codes not only detect but also correct specific types of errors.

Parity

• Parity (even or odd) is a simple method of error detection in data using bits to check.