Skip to main content

Theoretical BER vs SNR for m-ary PSK and QAM


📊 BER vs SNR 📡 Modulation 🧮 MATLAB 📘 Simulations

Relationship Between Bit Error Rate (BER) and Signal-to-Noise Ratio (SNR)

The relationship between Bit Error Rate (BER) and Signal-to-Noise Ratio (SNR) is a fundamental concept in digital communication systems. Here’s a detailed explanation:

  • BER (Bit Error Rate): The ratio of the number of bits incorrectly received to the total number of bits transmitted. It measures the quality of the communication link.
  • SNR (Signal-to-Noise Ratio): The ratio of the signal power to the noise power, indicating how much the signal is corrupted by noise.

Relationship

The BER typically decreases as the SNR increases. This relationship helps evaluate the performance of various modulation schemes.

BPSK (Binary Phase Shift Keying)

  • Simple and robust.
  • BER in AWGN channel: BER = 0.5 × erfc(√SNR)
  • Performs well at low SNR.

QPSK (Quadrature Phase Shift Keying)

  • Transmits 2 bits per symbol.
  • BER: BER = 0.5 × erfc(√(SNR))
  • More spectrally efficient than BPSK, slightly higher BER at same SNR.

M-ary PSK (Phase Shift Keying)

  • Encodes multiple bits using M phases.
  • General BER expression is complex and depends on cos and sin terms.
  • Higher-order schemes (e.g., 8-PSK, 16-PSK) have higher BER for same SNR.

M-ary PSK is a modulation technique where each symbol represents log₂(M) bits by shifting the phase of a carrier. The phases are spaced evenly in a circle, and the distance between points decreases with higher M, making it more error-prone at low SNRs.

BER (approximate for large M and high SNR in AWGN):


BER ≈ (2 / log₂(M)) × erfc(√SNR × sin(Ï€ / M))
  • As M increases, spectral efficiency improves but BER performance degrades.
  • Gray coding is typically used to minimize BER.
  • Exact BER uses summation expressions, but approximations are common for analysis.

M-ary QAM (Quadrature Amplitude Modulation)

  • Uses both amplitude and phase variations.
  • BER: BER = (log₂(M)/2) × (1 - 1/√M) × erfc(√((3SNR)/(M - 1)))
  • Higher-order QAM (e.g., 16-QAM, 64-QAM) is more spectrally efficient, but BER increases with order.

Practical Considerations

  • AWGN Channel: Additive White Gaussian Noise affects the signal; SNR is key to system performance.
  • Simulation: SNR is varied to analyze BER behavior using tools like MATLAB.
  • Error Correction: Techniques like Forward Error Correction (FEC) help reduce BER.

Example

For a BPSK system in an AWGN channel:

Formula: BER = 0.5 × erfc(√SNR)

At SNR = 0 dB: BER ≈ 0.078649603525

Conclusion

Understanding the BER vs. SNR relationship is essential for designing efficient digital communication systems. Different modulation schemes provide trade-offs between spectral efficiency and BER performance.

BER vs Eb/N0 Simulation



*Changes update the plot in real-time.

Access full simulation →
other simulations →

Further Reading

  1. MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, ...
  2. Theoretical BER vs SNR for ASK, FSK, and PSK
  3. Theoretical vs. simulated (using MATLAB) BER vs. SNR for ASK, FSK, and PSK

🚀 Explore BER vs SNR Tools

Run simulations, view graphs, and deepen your understanding.



📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...
Read more

MUSIC Algorithm Explained (with MATLAB + Simulator)

Practical Implementation of the MUSIC Algorithm The focus is on how the algorithm works computationally , not just theory, and it explains the denominator (a H E n E n H a) mathematically and intuitively. 1. Introduction The MUSIC (Multiple Signal Classification) algorithm is a high-resolution method used in signal processing and array processing to estimate the Direction of Arrival (DOA) of signals received by a sensor array. Unlike classical beamforming methods, MUSIC uses eigenvector decomposition of the covariance matrix to separate the signal subspace and noise subspace , allowing it to achieve much higher angular resolution. In practical implementations, MUSIC works by: Simulating or collecting array signals Computing the covariance matrix Performing eigenvalue decomposition Separating signal and noise subspaces Scanning possible angles using a steering vector Constructing a pseudo-spectrum where peaks indicate signal directions 2. Signal Mo...
Read more

PSD Calculation with FFT: MATLAB Tutorial for Signal Analysis

  Implementation Steps 1. FFT Computes the Frequency Content of a Signal FFT converts a time-domain signal to the frequency domain. If: The signal is sampled at rate $f_s$ You compute an $N_{\text{FFT}}$-point FFT Then each FFT bin corresponds to a frequency resolution of: $$\Delta f = \frac{f_s}{N_{\text{FFT}}}$$ So the FFT gives you accurate frequency content, assuming the signal is stationary and adequately sampled (Nyquist criterion met).  2. Magnitude Squared Gives Power (Not Amplitude) $$P[k] = |X[k]|^2$$ This gives power at each frequency bin, not just amplitude. It represents how much energy is present at each frequency. It's a key step for PSD.  3. Normalization Makes the PSD Physically Meaningful The equation: $$\text{PSD}[k] = \frac{|X[k]|^2}{N_{\text{FFT}} \cdot f_s \cdot U}$$ is derived from first principles and ensures that the u...
Read more

MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSK, BPSK (with Simulation)

🧮 MATLAB Code for BPSK, M-ary PSK, and M-ary QAM Together 🧮 MATLAB Code for M-ary QAM 🧮 MATLAB Code for M-ary PSK 📚 Further Reading MATLAB Script for BER vs. SNR for M-QAM, M-PSK, QPSK, BPSK % Written by Salim Wireless clc; clear; close all; snr_db = -5:2:25; psk_orders = [2, 4, 8, 16, 32]; qam_orders = [4, 16, 64, 256]; ber_psk_results = zeros(length(psk_orders), length(snr_db)); ber_qam_results = zeros(length(qam_orders), length(snr_db)); for i = 1:length(psk_orders) ber_psk_results(i, :) = berawgn(snr_db, 'psk', psk_orders(i), 'nondiff'); end for i = 1:length(qam_orders) ber_qam_results(i, :) = berawgn(snr_db, 'qam', qam_orders(i)); end figure; semilogy(snr_db, ber_psk_results(1, :), 'o-', 'LineWidth', 1.5, 'DisplayName', 'BPSK'); hold on; for i = 2:length(psk_orders) semilogy(snr_db, ber_psk_results(i, :), 'o-', 'DisplayName', sprintf('%d-PSK', psk_or...
Read more

Theoretical BER vs SNR for BPSK

Theoretical Bit Error Rate (BER) vs Signal-to-Noise Ratio (SNR) for BPSK in AWGN Channel Let’s simplify the explanation for the theoretical Bit Error Rate (BER) versus Signal-to-Noise Ratio (SNR) for Binary Phase Shift Keying (BPSK) in an Additive White Gaussian Noise (AWGN) channel. Key Points Fig. 1: Constellation Diagrams of BASK, BFSK, and BPSK [↗] BPSK Modulation Transmits one of two signals: +√Eb or −√Eb , where Eb is the energy per bit. These signals represent binary 0 and 1 . AWGN Channel The channel adds Gaussian noise with zero mean and variance N₀/2 (where N₀ is the noise power spectral density). Receiver Decision The receiver decides if the received signal is closer to +√Eb (for bit 0) or −√Eb (for bit 1) . Bit Error Rat...
Read more

MATLAB Code for ASK, FSK, and PSK (with Online Simulator)

MATLAB Code for ASK, FSK, and PSK Comprehensive implementation of digital modulation and demodulation techniques with simulation results. 📘 Theory 📡 ASK Code 📶 FSK Code 🎚️ PSK Code 🕹️ Simulator 📚 Further Reading Amplitude Shift Frequency Shift Phase Shift Live Simulator ASK, FSK & PSK HomePage MATLAB Code MATLAB Code for ASK Modulation and Demodulation COPY % The code is written by SalimWireless.Com clc; clear all; close all; % Parameters Tb = 1; fc = 10; N_bits = 10; Fs = 100 * fc; Ts = 1/Fs; samples_per_bit = Fs * Tb; rng(10); binar...
Read more

Constellation Diagrams of ASK, PSK, and FSK (with MATLAB Code + Simulator)

Constellation Diagrams: ASK, FSK, and PSK Comprehensive guide to signal space representation, including interactive simulators and MATLAB implementations. 📘 Overview 🧮 Simulator ⚖️ Theory 📈 Q-function 📚 Resources BASK Modulation Transmits one of two signals: 0 or $\sqrt{E_b}$, representing binary 0 and 1. Simple but sensitive to noise. BFSK Modulation Transmits one of two signals: $\sqrt{E_b}$ on the Y-axis or $\sqrt{E_b}$ on the X-axis. These are orthogonal signals. BPSK Modulation Transmits $+\sqrt{E_b}$ or $-\sqrt{E_b}$ (antipodal signaling). Most efficient binary scheme. ...
Read more