Skip to main content

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


Bit Error Rate (BER) & SNR Guide

Analyze communication system performance with our interactive simulators and MATLAB tools.

📘 Theory 🧮 Simulators 💻 MATLAB Code 📚 Resources


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

---

M-ary QAM

---

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



Try Interactive Online Simulators


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.

BER vs SNR Main Page →
Modulation Techniques Main Page >
Online Signal Processing Simulations Main Page >
📊 MATLAB & ONLINE SIMULATIONS
🎓 GATE SOLUTIONS
📡 WIRELESS & THEORY

Contact Us

Name

Email *

Message *

Popular Posts

Q-function in BER vs SNR Calculation (with Simulation)

Q-function in BER vs. SNR Calculation In digital communications and signal processing, the Q-function plays a significant role in predicting system reliability. It allows engineers to quantify the probability that Gaussian noise will exceed a specific threshold, causing a bit error. What is the Q-function? The Q-function is a mathematical function representing the tail probability of the standard normal (Gaussian) distribution. It is the complementary cumulative distribution function (CCDF) of a standard Gaussian distribution. Q(x) = (1 / √(2Ï€)) ∫â‚“∞ e^(-t² / 2) dt Q-Function Interactive Simulator Move the slider to see how the "Tail Probability" (the area in red) changes. This area represents the Probability of Error (BER) . Threshold Distance ( x ) — (Simulates Increasing SNR) x = 1.0 Q(x) = 0.1587 ...
Read more

RMS Delay Spread, Excess Delay Spread and Multi-path ...(with MATLAB + Simulator)

📘 Overview of Delay Spread and Multi-path 🧮 Excess Delay spread 🧮 Power delay Profile 🧮 RMS Delay Spread 📚 Further Reading 📂 Other Topics on RMS Delay Spread, Excess Delay ... 🧮 Multipath Components or MPCs 🧮 Online Simulator for Calculating RMS Delay Spread 🧮 Why is there significant multipath in the case of very high frequencies? 🧮 Why RMS Delay Spread is essential for wireless communication? 🧮 Why the Power Delay Profile is essential? 🧮 MATLAB Codes for Calculating Different Types of delay Spreads Delay Spread, Excess Delay Spread, and Multipath (MPCs) The fundamental distinction between wireless and wired connections is that in wireless connections signal reaches at receiver thru multipath signal propagation rather than directed transmission like co-axial cable. Wireless Communication has no set communication path between the transmitter and the receiver. The line...
Read more

FM Bandwidth and FM Band Explained

FM radio uses the frequency band from 88 MHz to 108 MHz , which is a 20 MHz-wide spectrum . This is the range of carrier frequencies available to stations. 108 MHz − 88 MHz = 20 MHz However, a single FM station occupies only about 200 kHz . This is the bandwidth of the modulated FM signal. 1. Why One FM Station Needs ~200 kHz FM uses frequency modulation . The bandwidth depends on how far the carrier swings. Carson's Rule gives the approximate FM bandwidth: B = 2 ( Δf + f m ) ...
Read more

OFDM Symbols and Subcarriers Explained

This article explains how OFDM (Orthogonal Frequency Division Multiplexing) symbols and subcarriers work. It covers modulation, mapping symbols to subcarriers, subcarrier frequency spacing, IFFT synthesis, cyclic prefix, and transmission. Step 1: Modulation First, modulate the input bitstream. For example, with 16-QAM , each group of 4 bits maps to one QAM symbol. Suppose we generate a sequence of QAM symbols: s0, s1, s2, s3, s4, s5, …, s63 Step 2: Mapping Symbols to Subcarriers Assume N sub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier): Mapping (example) OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7 OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15 … OFDM sym...
Read more

Frequency Shift Keying (FSK) Modulation & Demodulation (with Simulation)

Frequency Shift Keying (FSK) Theoretical Foundations: Frequency Shift Keying (FSK) is a discrete frequency modulation scheme wherein the digital information is encoded via instantaneous shifts in the carrier signal's frequency. The fundamental implementation is Binary FSK (BFSK), which maps binary data onto two distinct, discrete spectral states. A binary '1' (the "mark" state) is represented by a carrier frequency \( f_1 \), while a binary '0' (the "space" state) corresponds to frequency \( f_2 \). Each symbol is sustained for a bit interval denoted by \( T_b \). FSK Transmitter Characterization: The mathematical model for the modulated BFSK output \( s(t) \) is defined as: \[ s(t) = \begin{cases} A_c \cos(2\pi f_1 t), & \text{for } m = 1 \\ A_c \cos(2\pi f_2 t), & \text{for } m = 0 \end{cases} \] ...
Read more

Orthogonal Time Frequency Space (OTFS) (with MATLAB)

In OTFS (Orthogonal Time Frequency Space) modulation — a scheme designed for high-Doppler and time-varying wireless channels — the terms ISFFT and SFFT are key mathematical transformations used to move between different representation domains. Figure: OTFS block diagram 1. ISFFT — Inverse Symplectic Finite Fourier Transform Purpose: Transforms data symbols from the delay-Doppler domain to the time-frequency domain . \[ X[n, m] = \frac{1}{\sqrt{NM}} \sum_{k=0}^{N-1} \sum_{l=0}^{M-1} x[k, l] \, e^{j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)} \] Here, \( N \) is the number of Doppler bins (time slots), and \( M \) is the number of delay bins (subcarriers). The ISFFT maps each data symbol from the delay-Doppler grid (where the channel is sparse and easier to equalize) to the time-frequency grid (where standard multicarrier modulation like OFDM can be applied). 2. SFFT — Symplectic Finite Fourier Transform Purpose: Performs the reverse operation ...
Read more