BER vs SNR: Constellation Analysis BER Derivation from Constellation Points The probability of a bit error in an AWGN channel is determined by the minimum Euclidean distance (\(d_{min}\)) between symbols in a constellation. The generic formula for Bit Error Rate (BER) is: \[ BER = Q \left( \frac{d_{min}}{\sqrt{2N_0}} \right) \] 1. Binary PSK (BPSK) BPSK is antipodal . The points are located at \(+\sqrt{E_b}\) and \(-\sqrt{E_b}\). Because the carrier is always "on" at full strength, the Peak Power equals the Average Power . Distance (\(d_{min}\)): \(2\sqrt{E_b}\) BER Formula: \(Q(\sqrt{2E_b/N_0})\) At 0 dB SNR: \(Q(\sqrt{2}) \approx 0.078\) 2. Orthogonal FSK FSK symbols are perpendicular in signal space. Like PSK, FSK has a constant envelope , meaning the power never fluctuates regardless of which frequency is sent. Distance (\(d_{min}\)): \(\sqrt{2E_b}\) BER Formula: \(Q(\sqrt{E_b/N_0...

Frequency Shift Keying (FSK) FSK is a digital modulation technique where each bit is represented by a different carrier frequency: 0 → f 0 ,   1 → f 1 The transmitted FSK signal can be represented as: s(t) = A cos(2π f i t),   i ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) spreads the received signal. This simulation shows the effect of noise on the FSK constellation by plotting each symbol based on its frequency and amplitude. Number of Random Bits: Signal-to-Noise Ratio (SNR) in dB:

Binary Phase Shift Keying (BPSK) Simulation BPSK is a type of digital modulation where each bit is represented by one of two possible phases of a carrier signal: 0 → -1,   1 → +1 The transmitted BPSK signal can be written as: s(t) = √(2E_b) * cos(2πf_c t + πb),   b ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) affects the signal. At the receiver, the demodulated signal is compared with a threshold (0) to detect the bit. This simulation demonstrates the effect of noise on BPSK constellation points and allows you to calculate the Bit Error Rate (BER). Number of Bits: SNR (dB):

QPSK Constellation with AWGN Visualizing modulation symbols under different SNR conditions Input Parameters Number of Random Bits: Signal-to-Noise Ratio (SNR) in dB: Theory In 4-Phase Shift Keying (4-PSK), each symbol encodes 2 bits using one of four possible phases: 0, π/2, π, 3π/2 . The transmitted complex baseband signal is: x[n] = A · exp(jθ_n) where A is the symbol amplitude, and θ_n ∈ {0, π/2, π, 3π/2} is the phase corresponding to the transmitted bits. When the signal passes through an additive white Gaussian noise (AWGN) channel, the received signal is: r[n] = x[n] + w[n] where w[n] ~ N(0, σ²) is the complex AWGN. The signal-to-noise ratio (SNR) in decibels is defined as: SNR (...

Constellation Point Distance and Power Requirement in M-ary PSK In digital communication systems, especially M-ary PSK (Phase Shift Keying) , the arrangement of constellation points plays a crucial role in determining error performance and power efficiency. 1. Constellation Point Distance In an M-ary PSK system, the constellation points are equally spaced on a circle of radius √E s , where: E s = Energy per symbol M = Number of constellation points The minimum distance between adjacent constellation points is: d min = 2 √E s sin(π / M) Key Observations: As M increases, sin(π/M) decreases. The minimum distance between points decreases. Smaller distance → constellation points are closer → higher probability of error. 2. Probability of Error and Symbol Energy The proba...

Ultimate Guide to QPSK: Modulation, Formulas, and Interactive Simulators 📊 Modulations 🧪 Simulations 💻 MATLAB 🚀 Simulator 🧮 Formulas Quadrature Phase Shift Keying (QPSK) What is QPSK? QPSK is a digital modulation scheme that conveys data by changing the phase of a carrier wave. Unlike BPSK, which uses two phases, QPSK uses four distinct phases separated by \(\pi/2\) (90°). This allows each symbol to represent two bits , effectively doubling the data rate within the same bandwidth. The Mathematical Representation: \(s(t) = A \cos(2\pi f_c t + \theta_n)\) where \(\theta_n \in \{0, \pi/2, \pi, 3\pi/2\}\) '00' : \(A \cos(2\pi f_c t)\) '01' : \(A \cos(2\pi f_c t + 90^\circ)\) '11' : \(A \cos(2\pi f_c t + 180^\circ)\) '10' : \(A \cos(2\pi f_c t + 270^\circ)\) ...

Simulator for BPSK Constellation Diagram SNR (dB): 15 Add AWGN Noise Run Simulation Constellation Diagram of BPSK Fig 1: Constellation Diagram of BPSK BPSK uses a single orthonormal basis function : \[ \phi_1(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_c t), \quad 0 \le t \le T_b \] Binary '1': \( s_1(t) = \sqrt{E_b} \, \phi_1(t) \) Binary '0': \( s_2(t) = -\sqrt{E_b} \, \phi_1(t) \) Energy of both symbols: \( E_b \) Distance between points: \( d_{12} = 2\sqrt{E_b} \) The constellation lies on a single axis (in-phase), since BPSK only varies phase. Read More about Constellation Diagram of PSK Constellation Diagrams Main Page → Other Simulations →

📘 Overview & Theory 🧮 MATLAB Code for Constellation Diagrams of ASK, FSK, and PSK 🧮 Online Simulator for Constellation diagrams of ASK, FSK, and PSK 📚 Further Reading   MATLAB Script % The code is developed by SalimWireless.Com clc; clear; close all; % Parameters numSymbols = 1000; % Number of symbols to simulate symbolIndices = randi([0 1], numSymbols, 1); % Random binary symbols (0 or 1) % ASK Modulation (BASK) askAmplitude = [0, 1]; % Amplitudes for binary ASK askSymbols = askAmplitude(symbolIndices + 1); % Modulated BASK symbols % FSK Modulation (Modified BFSK with 90-degree offset) fs = 100; % Sampling frequency symbolDuration = 1; % Symbol duration in seconds t = linspace(0, symbolDuration, fs*symbolDuration); fBase = 1; % Base frequency frequencies = [fBase, fBase]; % Same frequency for both % Generate FSK symbols with 90° phase offset fskSymbols = arrayfun(@(idx) ...     cos(2*pi*frequencies(1)*t) * (1-idx) + ...     ...

📘 Overview & Theory 🧮 MATLAB Codes 🧮 Q-function 📚 Further Reading Bit Error Rate (BER) Equations In ASK, noise directly affects the signal amplitude, making it the most vulnerable since the data is carried in amplitude changes. In FSK, data is represented by frequency variations, and because noise typically impacts amplitude more than frequency, FSK is more robust than ASK. In PSK, data is encoded in the signal phase, and BPSK specifically uses 180-degree phase shifts, creating the greatest separation between signal points and therefore achieving the lowest bit error rate (BER) for the same power level. BER formulas for ASK, FSK, and PSK modulation schemes. ASK BER = 0.5 × erfc(0.5 × √SNR) FSK BER = 0.5 × erfc(√(SNR / 2)) PSK BER = 0.5 × erfc(√SNR) ...

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...

MATLAB Code % The code is written by SalimWireless.Com % Clear previous data and plots clc; % Clear the command window clear all; % Remove all variables from the workspace close all; % Close all figure windows % Parameters Tb = 1; % Bit duration in seconds fc = 10; % Carrier frequency in Hz N = 10; % Number of bits to transmit % Generate carrier signal t = 0:Tb/100:1; % Time vector for one bit duration with 100 samples carrier_signal = sqrt(2/Tb) * sin(2*pi*fc*t); % Normalized carrier signal % Generate message signal rng(10); % Set random seed for reproducibility binary_data = rand(1, N); % Generate random binary data between 0 and 1 % Initialize time intervals for message bits t_start = 0; t_end = Tb; % Loop to generate and modulate each bit for i = 1:N t = t_start:0.01:t_end; % Time vector for current bit duration % Generate binary message signal (1 or 0) if binary_data(i) > 0.5 binary_data(i) = 1; message_signal = ones(1, length(t)); % Bit value 1 else binary_data(i) = 0;...

📘 Overview 🧮 Simulator for constellation diagram of FSK 🧮 Theory 🧮 MATLAB Code 📚 Further Reading 📚 BER vs SNR from Constellation   Binary bits '0' and '1' can be mapped to 'j' and '1' to '1', respectively, for Baseband Binary Frequency Shift Keying (BFSK) . Signals are in phase here. These bits can be mapped into baseband representation for a number of uses, including power spectral density (PSD) calculations. For passband BFSK transmission, we can modulate signal 'j' with a lower carrier frequency and signal '1' with a higher carrier frequency while transmitting over a wireless channel. Let's assume we are transmitting carrier signal fc1 for the transmission of binary bit '1' and carrier signal fc2 for the transmission of binary bit '0'. Simulator for 2-FSK Constellation Diagram Simulator for 2-FSK Constellation Diagram ...