multiply_by_n

The multiply_by_n function multiplies an 8-bit unsigned integer x by another 8-bit unsigned integer n using bitwise operations and addition, without using the multiplication operator (*). This method is efficient and often useful in embedded systems where multiplication instructions might be slow or unavailable.

Function Code

uint8_t multiply_by_n(uint8_t x, uint8_t n) 
{
    uint8_t result = 0;
    uint8_t multiplier = x;

    /* Iterate through the bits of n */
    for (int i = 0; i < 8; ++i) {
        if (n & (1 << i)) {
            /* If the i-th bit of n is set, add the current multiplier to the result */
            result += multiplier;
        }
        /* Update the multiplier by left-shifting it by 1 */
        multiplier <<= 1;
    }

    return result;
}

How It Works

1. Initialize Variables:

   • result: To store the final product.
   • multiplier: Starts with the value of x and will be updated during the iteration.

2. Iterate Through the Bits of n:

   • The loop runs from 0 to 7, checking each bit of n.

3. Check Each Bit:

   • If the i-th bit of n is 1, the current value of multiplier is added to result.

4. Update Multiplier:

   • The multiplier is left-shifted by 1 to prepare for the next bit check.

Example

Let's consider an example where x = 5 and n = 3.

1. Binary Representation:

   • x = 5 (binary 00000101)
   • n = 3 (binary 00000011)

2. Step-by-Step Execution:

Initial Values:
x = 5 (binary 00000101)
n = 3 (binary 00000011)

Iterate through the bits of n:
Bit 0: n & (1 << 0) = 1 (bit is set)
    result = result + multiplier (0 + 5) = 5
    multiplier <<= 1 (5 << 1) = 10
Bit 1: n & (1 << 1) = 1 (bit is set)
    result = result + multiplier (5 + 10) = 15
    multiplier <<= 1 (10 << 1) = 20
Bit 2: n & (1 << 2) = 0 (bit is not set)
    multiplier <<= 1 (20 << 1) = 40
Bit 3: n & (1 << 3) = 0 (bit is not set)
    multiplier <<= 1 (40 << 1) = 80
Bit 4: n & (1 << 4) = 0 (bit is not set)
    multiplier <<= 1 (80 << 1) = 160
Bit 5: n & (1 << 5) = 0 (bit is not set)
    multiplier <<= 1 (160 << 1) = 320
Bit 6: n & (1 << 6) = 0 (bit is not set)
    multiplier <<= 1 (320 << 1) = 640
Bit 7: n & (1 << 7) = 0 (bit is not set)
    multiplier <<= 1 (640 << 1) = 1280

Final Result:
result = 15 (binary 00001111)

Use Cases

When to Use

• Lack of Hardware Multiplication: When the embedded processor does not have a hardware multiplier or the multiplication instruction is slow.
• Optimization: To optimize code for speed and size in embedded systems where resources are limited.
• Educational Purposes: To understand and implement multiplication using basic bitwise operations.

Use Cases in Embedded Systems

• Signal Processing: Efficiently multiplying values in digital filters or signal processing algorithms.
• Control Systems: Multiplying control parameters or sensor data in real-time systems.
• Graphics Processing: Efficiently handling operations in low-power graphics processors or microcontrollers used in display systems.