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 effici...

                          Received Signal Time Domain : y(t)=[ h(t)] ∗ s(t)+w(t) Where: ·           h(t) = Channel Impulse Response (due to Multi-path Rayleigh Fading) s(t) : Transmitted signal ·          ∗ : Convolution operator ·          w(t) : Additive noise   Frequency Domain : Y(ω) =  H(ω) ⋅ S(ω)+W(ω) Where: ·          Y(ω) : Received signal spectrum ·          H(ω),S(ω),W(ω) : Fourier Transforms of the respective time-domain signals   Digital Communication with Channel Equalization and Demodulation: Overcoming Rayleigh Fading and AWGN Digital communication system with channel equalization and demodulation involves transmitting a modulated signal through a channel affecte...

📘 Overview 🧮 MATLAB Codes 🧮 Online Simulator for Calculating BER of M-ary PSK and QAM 🧮 QPSK vs BPSK and QAM: A Comparison of Modulation Schemes in Wireless Communication 🧮 Are QPSK and 4-PSK same? 📚 Further Reading   QPSK offers double the data rate of BPSK while maintaining a similar bit error rate at low SNR when Gray coding is used. It shares spectral efficiency with 4-QAM and can outperform 4-QAM or 16-QAM in very noisy channels. QPSK is widely used in practical wireless systems, often alongside QAM in adaptive modulation schemes [Read more...] What is the Gray Code? Gray Code: Gray code is a binary numeral system where two successive values differ in only one bit. This property is called the single-bit difference or unit distance code. It is also known as reflected binary code. Let's convert binary 111 to Gray code: Binary bits: B = 1 1 1 Apply the rule: G[0] = B[0] = 1...

📘 Overview 🧮 MATLAB Code for Estimation of SNR from FFT bins of a Noisy Signal 🧮 MATLAB Code for Estimation of Signal-to-Noise Ratio from Power Spectral Density Using FFT and Kaiser Window Periodogram from real signal data 📚 Further Reading   Here, you can find the SNR of a received signal from periodogram / FFT bins using the Kaiser operator. The beta (β) parameter characterizes the Kaiser window, which controls the trade-off between the main lobe width and the side lobe level in the frequency domain. For that you should know the sampling rate of the signal.  The Kaiser window is a type of window function commonly used in signal processing, particularly for designing finite impulse response (FIR) filters and performing spectral analysis. It is a general-purpose window that allows for control over the trade-off between the main lobe width (frequency resolution) and side lobe levels (suppression of spectral leakage). The Kaiser window is defined...

In this MATLAB code it is described how to calculate 'BER vs SNR' and 'SER vs SNR' from the Channel Impulse Response (CIR) .  SER =  Symbol error rate For a typical communication system, if we transmit the signal x[n], then received signal will be, y[n] = h[n]*x[n]  + noise Where, h[n] is channel impulse response and '*' indicates convolution operation MATLAB Code clc; clear all; close all; % Channel Impulse Response (CIR) channel_response = [0.85]; % Example channel impulse response coefficients %channel_response = [0.8, 0.1]; % Modulation parameters modulation_order = 4; % QPSK modulation num_symbols = 1000; % Number of symbols to transmit num_bits = 1000*log2(modulation_order); % Signal generation tx_symbols = randi([0, modulation_order-1], 1, num_symbols); tx_signal = qammod(tx_symbols, modulation_order); % Channel simulation rx_signal = conv(tx_signal, channel_response); % Add AWGN (Additive White Gaussian Noise) snr_dB = 0:2:20; ber = zeros(size(snr_dB))...

  How do we calculate BER vs. SNR for real systems? In real systems such as BPSK over an AWGN channel , bits are transmitted as signals (e.g., +√P or −√P ). Due to noise, the received signal becomes: y = √P · x + n where n ∼ N(0, σ²) , and x ∈ {−1, +1} . At the receiver, we perform detection (e.g., using sign detection). The probability of a bit error is given by the Gaussian Q-function: BER = Q(√(2P / σ²)) = Q(√(2·SNR)) This gives the theoretical BER vs. SNR curve. BER vs. SNR from Gaussian Q factor Linear Equalization (e.g., Zero Forcing) When you're using an equalizer like Zero Forcing (ZF) , you get: Effective noise enhancement depending on the channel matrix H The diagonal elements of W = (H H H) −1 (in ZF) tell you how noise variance is scaled after equalization. So the BER becomes: BER i = Q( √(2·Ω s / (σ²·W ii )) ) Where: Ω s = symbol power W ii = noise amplifica...

  For practical communication systems, we calculate the signal-to-noise ratio (SNR) per channel like this, SNR =  Ωd / Ωn where,  Ωd =  (P' P) where P is the channel impulse response. For example, P = [p(0) p(1) p(2) ... p(M-1)] If the signal is being received at the receiver through multiple channels, then the average SNR per channel is calculated as, Average SNR = [SNR(1) + SNR(2) + SNR(3) + ... + SNR(n)] / n

🧮 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; num_symbols = 1e5; snr_db = -20:2:20; 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) psk_order = psk_orders(i); for j = 1:length(snr_db) data_symbols = randi([0, psk_order-1], 1, num_symbols); modulated_signal = pskmod(data_symbols, psk_order, pi/psk_order); received_signal = awgn(modulated_signal, snr_db(j), 'measured'); demodulated_symbols = pskdemod(received_signal, psk_order, pi/psk_order); ber_psk_results(i, j) = sum(data_symbols ~= demodulated_symbols) / num_symbols; end end for ...

📘 Overview of BER and SNR 🧮 Online Simulator for BER calculation of m-ary QAM and m-ary PSK 🧮 MATLAB Code for BER calculation of M-ary QAM, M-ary PSK, QPSK, BPSK, ... 📚 Further Reading 📂 View Other Topics on M-ary QAM, M-ary PSK, QPSK ... 🧮 Online Simulator for Constellation Diagram of m-ary QAM 🧮 Online Simulator for Constellation Diagram of m-ary PSK 🧮 MATLAB Code for BER calculation of ASK, FSK, and PSK 🧮 MATLAB Code for BER calculation of Alamouti Scheme 🧮 Different approaches to calculate BER vs SNR What is Bit Error Rate (BER)? The abbreviation BER stands for bit error rate, which indicates how many corrupted bits are received (after the demodulation process) compared to the total number of bits sent in a communication process. It is defined as,  In mathematics, BER = (number of bits received in error / total number of transmitted bits)  On the other hand, SNR ...