Radix Conversions, Complement Arithmetic, Binary Codes & Error Detection

Explanation

Digital computation relies on positional number systems where the weight of each digit is a power of base r. In n-bit 2's complement representation, values span from -2^{n-1} to +(2^{n-1} - 1). Gray codes minimize electrical switching noise and eliminate transient decoding spikes in rotary optical shaft encoders. Error-detecting and correcting codes introduce controlled redundancy: a Hamming code with k parity bits protects m data bits provided 2^k ≥ m + k + 1.

Key points

• Signed numbers: Signed-magnitude has dual zero (+0 and -0); 1's complement has end-around carry; 2's complement (1's comp + 1) is standard in microprocessors because subtraction is executed via adder hardware.
• Hardware overflow rule: In 2's complement addition, overflow occurs if and only if two numbers of the same sign yield a result with the opposite sign (V = C_{in} ⊕ C_{out} on the MSB).
• Binary Codes: BCD (8421, invalid codes 1010-1111); Excess-3 (self-complementing BCD + 3); Gray code (unit-distance code where adjacent values change by exactly 1 bit).
• Hamming Code (Error correction): Parity bits placed at powers of 2 (positions 1, 2, 4, 8...). Detects and pinpoints single-bit errors via parity syndrome checks.

Example

Code examples

// C++ Bitwise Verification of 2's Complement Addition & Overflow Detection
#include <iostream>
using namespace std;

bool addWithOverflow(int8_t a, int8_t b, int8_t &result) {
    int16_t sum = (int16_t)a + (int16_t)b;
    result = (int8_t)sum;
    // Overflow occurs when both operands have same sign, but result has opposite sign
    bool overflow = ((a > 0 && b > 0 && result < 0) || (a < 0 && b < 0 && result > 0));
    return overflow;
}

int main() {
    int8_t x = 100, y = 50, res;
    if (addWithOverflow(x, y, res)) {
        cout << "Arithmetic Overflow Detected! Result wrapped to: " << (int)res << endl;
    }
    return 0;
}

Exam questions

• [8 Marks] Perform subtraction using 2's complement arithmetic: (i) (42)₁₀ - (68)₁₀, (ii) (-35)₁₀ - (-18)₁₀ using 8-bit registers. State how overflow is detected.
• [7 Marks] Encode data bits 1011 using 7-bit even parity Hamming Code. Show how a single-bit error in the 5th bit position is detected and corrected.
• [7 Marks] Convert (523.6875)₁₀ to Binary, Octal, and Hexadecimal representations.