# Bitwise Manipulation & Masks ## Logical Shifts - Logical shifts move all bits in a binary number left or right by a specified number of positions - Used extensively in low-level programming and digital electronics - Two main types: **left shift** and **right shift**

A logical shift is a bitwise operation that moves each bit in a binary number by a specified number of positions, filling vacated positions with zeros

### Left Shift - Moves all bits to the left by a specified number of places - Zeros fill the vacated positions on the right - Each left shift effectively **multiplies the number by 2** #### Example: Left Shift by 2 Places Original binary number: `00010110` (decimal 22) | Step | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |------|-----|----|----|----|----|----|----|---| | Original | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | | After shift | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | **Result:** `01011000` = 88 Mathematical effect: - Original: 22 - After 1 left shift: 44 (22 × 2) - After 2 left shifts: 88 (22 × 2 × 2)

Each left shift doubles the value: shifting left by $n$ positions multiplies by $2^n$

### Right Shift - Moves all bits to the right by a specified number of places - Zeros fill the vacated positions on the left - Each right shift effectively **divides the number by 2** (integer division) #### Example: Right Shift by 3 Places Original binary number: `11010000` (decimal 208) | Step | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |------|-----|----|----|----|----|----|----|---| | Original | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | | After shift | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | **Result:** `00011010` = 26 Mathematical effect: - Original: 208 - After 1 right shift: 104 (208 ÷ 2) - After 2 right shifts: 52 (208 ÷ 4) - After 3 right shifts: 26 (208 ÷ 8)

Each right shift halves the value: shifting right by $n$ positions divides by $2^n$ (rounded down)

## Bitwise Manipulation - Bitwise manipulation uses logical operators to directly manipulate individual bits - Operators include **AND**, **OR**, and **XOR** - Essential for efficient low-level operations and hardware control ### Masks

A mask is a binary pattern used with bitwise operators to isolate, modify, or test specific bits within another binary value

#### Purpose of Masks Masks allow programmers to: - **Extract** specific bits from a value (e.g., isolate the lower 4 bits) - **Set** particular bits to 1 - **Clear** particular bits to 0 - **Toggle** bits (flip their state) - **Test** whether specific bits are set ### Bitwise AND Operation - Returns 1 only when **both** corresponding bits are 1 - All other combinations return 0 - Commonly used to **isolate or extract** specific bits

Use AND with a mask to extract specific bit patterns while zeroing out unwanted bits

#### Truth Table for AND | Bit A | Bit B | A AND B | |-------|-------|---------| | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 | #### Example: AND Operation Extracting the middle 4 bits from a byte: | Description | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |-------------|-----|----|----|----|----|----|----|---| | Binary value | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | | Mask | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | | **Result** | **0** | **0** | **1** | **1** | **0** | **1** | **0** | **0** | The mask with AND successfully isolated bits in positions 32, 16, 8, and 4. ### Bitwise OR Operation - Returns 1 when **either** (or both) corresponding bits are 1 - Returns 0 only when both bits are 0 - Commonly used to **set specific bits** to 1

Use OR with a mask to set specific bits to 1 while leaving other bits unchanged

#### Truth Table for OR | Bit A | Bit B | A OR B | |-------|-------|--------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 | #### Example: OR Operation Setting specific bits to 1: | Description | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |-------------|-----|----|----|----|----|----|----|---| | Binary value | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | | Mask | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | | **Result** | **1** | **1** | **1** | **1** | **1** | **1** | **1** | **0** | The mask with OR forced bits in positions 64, 32, and 8 to become 1. ### Bitwise XOR Operation - Returns 1 when bits are **different** (one is 1, the other is 0) - Returns 0 when bits are **the same** (both 0 or both 1) - Commonly used to **toggle bits** or detect differences

Use XOR with a mask to flip specific bits: 0 becomes 1 and 1 becomes 0

#### Truth Table for XOR | Bit A | Bit B | A XOR B | |-------|-------|---------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 | #### Example: XOR Operation Toggling specific bits: | Description | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |-------------|-----|----|----|----|----|----|----|---| | Binary value | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | | Mask | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | | **Result** | **1** | **0** | **0** | **0** | **1** | **1** | **0** | **1** | The mask with XOR flipped bits in positions 32, 16, and 8. ## Summary of Operations | Operation | When Result = 1 | Primary Use | |-----------|-----------------|-------------| | **AND** | Both bits are 1 | Extracting/isolating specific bits | | **OR** | At least one bit is 1 | Setting specific bits to 1 | | **XOR** | Bits are different | Toggling bits or comparing values |

The result of bitwise operations depends on applying the operation to each corresponding bit position independently

## Practical Applications ### Extracting Flag Bits Imagine an 8-bit status register where each bit represents a different flag: - Use AND with mask `00001000` to check if bit 3 is set - If result is `00001000`, the flag is active; if `00000000`, it's inactive ### Enabling Hardware Features When configuring a device: - Use OR with mask `10100000` to enable features controlled by bits 7 and 5 - Other feature bits remain in their current state ### Toggling LED States In embedded systems programming: - Use XOR with mask `00010001` to toggle LEDs connected to bits 4 and 0 - LEDs that were on turn off, and vice versa ### Efficient Multiplication and Division - Left shift by 3 positions = multiply by 8 (much faster than arithmetic multiplication) - Right shift by 2 positions = divide by 4 (faster than arithmetic division)

Bitwise operations execute extremely quickly in hardware, making them ideal for performance-critical code