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.ser-calc-header h2 { color: #1a73e8; margin-bottom: 10px; font-size: 28px; }
.ser-calc-header p { color: #666; font-size: 16px; }
.ser-calc-form-group { margin-bottom: 20px; }
.ser-calc-label { display: block; font-weight: 600; margin-bottom: 8px; color: #444; }
.ser-calc-input { width: 100%; padding: 12px; border: 2px solid #e0e0e0; border-radius: 8px; font-size: 16px; box-sizing: border-box; transition: border-color 0.3s; }
.ser-calc-input:focus { border-color: #1a73e8; outline: none; }
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function erfc(x) {
var z = Math.abs(x);
var t = 1.0 / (1.0 + 0.5 * z);
var ans = t * Math.exp(-z * z – 1.26551223 + t * (1.00002368 + t * (0.37409196 + t * (0.09678418 + t * (-0.18628806 + t * (0.27886807 + t * (-1.13520398 + t * (1.48851587 + t * (-0.82215223 + t * 0.17087277)))))))));
return x >= 0 ? ans : 2.0 – ans;
}
function qFunction(x) {
return 0.5 * erfc(x / Math.sqrt(2));
}
function calculateSER() {
var ebNoDb = document.getElementById("ebno_input").value;
var errorDiv = document.getElementById("ser-calc-error-msg");
var outputDiv = document.getElementById("ser-calc-output");
if (ebNoDb === "" || isNaN(ebNoDb)) {
errorDiv.style.display = "block";
outputDiv.style.display = "none";
return;
}
errorDiv.style.display = "none";
var ebNoLinear = Math.pow(10, (parseFloat(ebNoDb) / 10));
// Formula for 16-QAM: Ps = 3 * Q(sqrt(0.8 * Eb/N0)) – 2.25 * Q^2(sqrt(0.8 * Eb/N0))
var arg = Math.sqrt(0.8 * ebNoLinear);
var qVal = qFunction(arg);
var ser = (3 * qVal) – (2.25 * Math.pow(qVal, 2));
var ber = ser / 4;
document.getElementById("res-ser").innerText = ser.toExponential(4);
document.getElementById("res-ber").innerText = ber.toExponential(4);
document.getElementById("res-linear").innerText = ebNoLinear.toFixed(2);
outputDiv.style.display = "block";
}
16-QAM Symbol Error Rate Calculator
Calculate theoretical Symbol Error Rate (SER) and Bit Error Rate (BER) for 16-state Quadrature Amplitude Modulation.
Please enter a valid numeric value for Eb/N0.
Typical values range from 0 dB (noisy) to 20 dB (clean).
Symbol Error Rate (SER):
–
Bit Error Rate (BER) Approx:
–
Eb/N0 (Linear Ratio):
–
Understanding 16-Symbol Error Rate Calculation
In digital communications, 16-QAM (Quadrature Amplitude Modulation) is a modulation scheme that transmits 4 bits per symbol. Calculating the Symbol Error Rate (SER) is critical for assessing the performance of a wireless or wired communication link under the influence of Additive White Gaussian Noise (AWGN).
The Mathematical Model
For a rectangular 16-QAM constellation, the probability of symbol error ($P_s$) is calculated based on the $Q$-function, which represents the tail probability of the Gaussian distribution. The standard formula used in this calculator is:
$P_s = 3 Q\left(\sqrt{\frac{4}{5} \frac{E_b}{N_0}}\right) – 2.25 Q^2\left(\sqrt{\frac{4}{5} \frac{E_b}{N_0}}\right)$
Key Variables:
- Eb/N0: The ratio of energy per bit to the noise power spectral density. This is the standard figure of merit for comparing different modulation schemes.
- Q-Function: Defined as $Q(x) = \frac{1}{2} \text{erfc}(\frac{x}{\sqrt{2}})$. It calculates the probability that a normal random variable exceeds $x$ standard deviations.
- BER Approximation: Since 16-QAM uses Gray coding, a single symbol error most likely results in only one bit error out of the 4 bits. Thus, $BER \approx \frac{SER}{\log_2(16)} = \frac{SER}{4}$.
Practical Example
If your system operates at an Eb/N0 of 12 dB:
- Convert dB to linear: $10^{(12/10)} \approx 15.85$.
- Calculate the argument for the Q-function: $\sqrt{0.8 \times 15.85} \approx 3.56$.
- The resulting SER will be approximately $5.56 \times 10^{-4}$.
- The BER will be roughly $1.39 \times 10^{-4}$.