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BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)


Bit Error Rate (BER) & SNR Guide

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What is Bit Error Rate (BER)?

The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figure of merit for any digital communication link.

BER = (number of bits received in error) / (total number of transmitted bits)

What is Signal-to-Noise Ratio (SNR)?

SNR is the ratio of signal power to noise power, typically expressed in decibels (dB) to handle large variations in signal strength.

SNR (dB) = 10 * log10(Signal Power / Noise Power)

Example: An SNR of 3 dB means signal power is 2x stronger than noise.


Theoretical Error Probability of Digital Modulation

Key Concept: The performance of a modulation scheme is measured by how many errors occur for a given signal strength. We distinguish between Bit Error Rate (BER)—the probability of a single bit being wrong—and Symbol Error Rate (SER)—the probability of a group of bits (a symbol) being wrong.

1. Bit Error Rate (BER) Formulas

BER indicates the end-to-end performance of a data link. For higher-order modulation like QAM, we use approximations based on Gray Coding (where adjacent symbols differ by only one bit).

Binary Phase Shift Keying (BPSK)

The baseline for performance; very robust against noise.

\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] Quadrature Phase Shift Keying (QPSK)

Carries 2 bits per symbol but maintains the same BER as BPSK.

\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] M-ary PSK (e.g., 8-PSK) \[ P_b \approx \frac{2}{\log_2(M)} Q\left( \sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{M}\right) \right) \] Rectangular M-ary QAM (16, 64, 256-QAM)

As M increases, the distance between points decreases, making them more sensitive to noise.

16-QAM: \[ P_b \approx \frac{3}{4} Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]

64-QAM: \[ P_b \approx \frac{7}{12} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]

256-QAM: \[ P_b \approx \frac{15}{32} Q\left( \sqrt{\frac{8}{85}\frac{E_b}{N_0}} \right) \]

2. Symbol Error Rate (SER) Formulas

SER is useful for designing the physical layer and hardware. One symbol error usually results in only one bit error if Gray Coding is used.

BPSK & QPSK

BPSK: \[ P_s = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

QPSK: \[ P_s \approx 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

General M-QAM Symbol Error Rate

A symbol is "lost" if the noise pushes it outside its decision boundary.

16-QAM: \[ P_s \approx 3Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]

64-QAM: \[ P_s \approx \frac{7}{2} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]

Summary of Parameters

  • \(E_b\): Energy required to send one bit.
  • \(E_s\): Energy required to send one symbol (\(E_s = E_b \cdot \log_2 M\)).
  • \(N_0\): Noise power density.
  • AWGN Channel: These formulas assume a static environment with only Gaussian noise (no fading).
  • Gray Coding: A technique where adjacent constellation points differ by only one bit to minimize BER.

Interactive BER Calculator

Calculate theoretical BER for PSK and QAM systems instantly.

M-ary PSK

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M-ary QAM

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Performance Comparison

BER vs. SNR Graph
Download MATLAB BER Code →
Technique Bits/Symbol
BPSK1
QPSK2
8-PSK3
16-QAM4
64-QAM6

Modulation Scheme and Bandwidth Requirement

Modulation Bits/Symbol Bandwidth
BPSK 1 Rb
QPSK 2 Rb / 2
8-PSK 3 Rb / 3
16-QAM 4 Rb / 4

Read more about Modulation Scheme & Bandwidth Requirement



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BER Requirements by Application

Technology Typical Modulation Target BER
5G Mobile Data 256-QAM 10-4 to 10-6
Satellite Comms (Deep Space) BPSK / QPSK 10-3 (Pre-FEC)
Fiber Optic Networks Coherent QAM 10-12
Submarine Cables 8-QAM / 16-QAM 10-15

Quick Knowledge Check

If you double the signal power (increase by 3dB), what happens to the SNR?


QPSK vs. 4-QAM: Why They Should Have Same BER?

In digital communications, QPSK and 4-QAM exhibit identical Bit Error Rate (BER) performance. Although they are categorized differently, they share the same constellation geometry and power efficiency.

Read why they are identical →

📚 Also Read About

Frequently Asked Questions

What is a good BER for wireless communication?

For voice transmission, a BER of 10^-3 is often acceptable. For high-speed data services, a BER of 10^-6 or lower is typically required.

How does SNR affect BER?

As SNR increases, the signal becomes clearer relative to the noise, resulting in a significantly lower Bit Error Rate.

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