Data Representation

uint8_t

Unsigned 8-bit type, same as unsigned char.

Carry flag

Set if the result of an addition exceeds the maximum value that can be represented in the given number of bits.

Borrow flag

Set if the result of a subtraction is negative.

Unsigned Byte

Range: 0 to 255, Storage Size: 1 byte, Powers of 2: 0 to 28-1.

Unsigned Halfword

Range: 0 to 65,535, Storage Size: 2 bytes, Powers of 2: 0 to 216-1.

Unsigned Word

Range: 0 to 4,294,967,295, Storage Size: 4 bytes, Powers of 2: 0 to 232-1.

Unsigned Double-word

Range: 0 to 18,446,744,073,709,551,615, Storage Size: 8 bytes.

Binary to Decimal Conversion

For example, 0101102 = 0×24 + 1×23 + 0×22 + 1×21 + 1×20 = 8 + 2 + 1 = 11.

Decimal to Binary Conversion

Example: 52 = 32 + 20 = 32 + 16 + 4 = 1000002 + 0100002 + 0001002 = 1101002.

Five-bit binary code

A binary representation that uses 5 bits.

Carry in addition

Occurs if the result exceeds the maximum value for the given bits (i.e., c > 2n-1).

Borrow in subtraction

Occurs if the result is less than zero.

ARM Cortex-M processors

On these processors, the carry flag and the borrow flag are physically the same flag bit in the status register.

Addition crossing boundary

If the traverse crosses the boundary between 0 and 2n-1, the carry flag is set on addition.

Subtraction crossing boundary

If the traverse crosses the boundary between 0 and 2n-1, the carry flag is cleared on subtraction.

Example of carry flag

In the case of uint8_t a = 255 and b = 2, c = a + b results in c being 1 0000 0001.

Example of borrow flag

In the case of uint8_t a = 1 and b = 2, c = a - b results in c being 1 1111 1111.

Bit weight

For bit i, the weight is 2i.

Data representation

The method of representing data in binary form.

Number systems

Includes binary, octal, decimal, and hexadecimal.

Hexadecimal representation

Uses base 16, represented with digits 0-9 and letters A-F.

Octal representation

Uses base 8, represented with digits 0-7.

Decimal representation

Uses base 10, represented with digits 0-9.

Unsigned Integer Representation

Natural representation of unsigned integer

ARM Carry

On ARM, Carry = NOT Borrow, for subtraction

Signed Integer Representation

Three ways to represent signed binary integers

Signed Magnitude

value= (-1)sign × Magnitude

One's Complement

α + ᾱ = 2ⁿ−1

Two's Complement

α + ᾱ = 2ⁿ

Range of Signed Integers

Range [−2ⁿ−1 + 1, 2ⁿ−1 −1] for Signed Magnitude and One's Complement; [−2ⁿ−1, 2ⁿ−1 −1] for Two's Complement

Zero Representation

Two zeroes (±0) for Signed Magnitude and One's Complement; One zero for Two's Complement

Unique Numbers

2ⁿ−1 for Signed Magnitude and One's Complement; 2ⁿ for Two's Complement

Signed Byte Range

Range -128 to +127; Powers of 2: -2⁷ to 2⁷-1

Signed Halfword Range

Range -32,768 to +32,767; Powers of 2: -2¹⁵ to 2¹⁵-1

Signed Word Range

Range -2,147,483,648 to +2,147,483,647; Powers of 2: -2³¹ to 2³¹-1

Signed Double-word Range

Range -9,223,372,036,854,775,808 to +9,223,372,036,854,775,807

Two's Complement Weight

The weight of the sign bit is -2(n-1)

Convert to Negative

A number can be converted to its negative counterpart by: 1) bitwise NOT, and 2) plus one

Example of Two's Complement

Example 1: TC(3) Binary 0b00011 Decimal 3 Step 1: Invert every bit 0b11100 Step 2: Add 1 + 0b00001 Two's complement 0b11101 -3

Overflow in Addition

Overflow happens in addition if and only if positive + positive => negative or negative + negative => positive

Overflow in Subtraction

Overflow happens in subtraction if and only if positive - negative => negative or negative - positive => positive

Overflow Condition

Overflow cannot occur when adding operands with different signs or when subtracting operands with the same signs

8-bit Signed Addition Example

int8_t a, b, c; a = 127; b = 3; c = a + b; What is the value of c as signed decimal?

8-bit Signed Subtraction Example

int8_t a, b, c; a = -128; b = 2; c = a - b; What is the value of c as signed decimal?

Overflow flag for signed numbers

Indicates that the result of an operation has exceeded the range that can be represented with the given number of bits.

Two's complement

A method for representing signed integers in binary, where the most significant bit indicates the sign of the number.

Sign bit weight

The weight of the sign bit is negative, specifically -2^(n-1), where n is the total number of bits.

Overflow flag

A flag set by the ALU to indicate that an arithmetic operation has resulted in a value that cannot be represented within the given number of bits.

ALU

Arithmetic and Logic Unit, a digital circuit used to perform arithmetic and logic operations.

High-level languages

Programming languages that provide abstraction from the hardware, allowing programmers to write code without needing to manage hardware details.

Expression: 0100 + 0010

Results in 0110 with no carry or overflow.

Expression: 0100 + 0110

Results in 1010 with no carry but an overflow.

Expression: 1100 + 1110

Results in 1010 with a carry but no overflow.

Expression: 1100 + 1010

Results in 0110 with both a carry and an overflow.

Hardware adder for signed numbers

The same hardware adder designed for unsigned numbers can also correctly add signed numbers.

Simple Addition example

For unsigned addition: 23 + 6 = 29.

Signed Addition example

For signed addition: -9 + 6 = -3.

Unsigned number representation

If 11101 represents an unsigned number, then 11101 in decimal is 29.

Signed number representation

If 11101 represents a signed number, then 11101 in decimal is -3.

Subtraction hardware

The same hardware designed for subtraction works correctly for both signed and unsigned subtraction.

Minuend and subtrahend

In subtraction, the minuend is the number from which another number (the subtrahend) is to be subtracted.

Signed Integer Representation summary

The CPU and ALU do not differentiate between signed and unsigned types for addition and subtraction.

Condition Flags in ALU

The ALU sets up both the carry flag and the overflow flag, but it is the programmer's responsibility to interpret these flags.

Unsigned Subtraction

minuend - subtrahend

Difference

The result of subtraction

Cortex-M4 Condition Flags

Flags that indicate the result of arithmetic/logic operations: N (negative), Z (zero), V (overflow), C (carry).

N Flag

Indicates the result is negative.

Z Flag

Indicates the result is zero.

V Flag

Indicates signed overflow.

C Flag

Indicates unsigned overflow.

ASCII

American Standard Code for Information Interchange.

String Comparison

Strings are compared based on their ASCII values.

String Length

Determined by counting characters until the null character (NUL, ASCII value 0x00) is reached.

Convert to Upper Case

Changing lowercase letters to uppercase by adjusting their ASCII values.

Memory Address

The location in memory where data is stored.

Character Array

An array of characters, often used to store strings.

Null Terminator

A special character (ASCII value 0x00) that indicates the end of a string.

ADDS Instruction

An instruction that adds two registers and sets condition flags.

Subtraction Conversion

Subtraction can be converted to addition by inversion and plus-one.

Unsigned Multiplication Hardware

Works correctly for signed numbers if the product keeps the same number of bits as operands.

ASCII Character Set

A set of 128 characters represented by specific decimal and hexadecimal values.

Example of ASCII Values

For instance, 'A' is 65 in decimal and 41 in hexadecimal.

Pointer Dereference Operator

Used to access the value at the address pointed to by a pointer.

Array Subscript Operator

Used to access elements in an array using an index.

Loop Until NULL

A method to iterate through a string until the null character is encountered.