MIMO Beamforming: Physics to Math MIMO Beamforming & Channel Analysis for (N X 2) How spatial separation determines Matrix Rank and Condition Number (assuming number of users are 2) 1. Physical Config Antennas at Base Station (\(N\)): 4 User 1 Angle (\(\theta_1\)): 60 ° User 2 Angle (\(\theta_2\)): 120 ° Channel Matrix \(H\) (Derived) ...

  MATLAB Code %% 2D Conceptual + Electronically Steered Beamforming clc; clear; close all ; %% PARAMETERS Fs = 1000; % Sampling frequency [Hz] T = 1/Fs; % Sampling period t = 0:T:1-T; % 1-second time vector f_sig = 50; % Signal frequency [Hz] %% SIMULATE ANTENNA SIGNALS p = sin(2*pi*f_sig*t); % Reference signal (antenna p) vx = 0.8*sin(2*pi*f_sig*t + pi/6); % vx antenna vy = 0.6*sin(2*pi*f_sig*t + pi/4); % vy antenna %% NORMALIZE SIGNALS pnor = p / max(abs(p)); vxnor = vx / max(abs(vx)); vynor = vy / max(abs(vy)); %% PASSIVE ROTATION (Conceptual Beamforming) I1 = real(fft(pnor) .* conj(fft(vxnor))); I2 = real(fft(pnor) .* conj(fft(vynor))); theta = 360 * atan(sum(I2)/sum(I1)) / (2*pi); % degrees vc = vxnor * cosd(theta) + vynor * sind(theta); resultant_passive = pnor + 2*vc; %% ELECTRONICALLY STEERED BEAM theta_steer = 60; % Desired steering angle in degrees lambda = 1; % Normalized wavelength d = 0.5*lambda; ...

Why Half-Power (−3 dB) Is Often Used The short answer is: half-power is used because it is mathematically natural, physically meaningful, and robust in real systems. Below is the intuition—built step by step, without hand-waving. 1. Power vs Amplitude: Why “Half” Appears Naturally Most physical signals behave as: Amplitude → field, voltage, pressure Power / intensity ∝ (amplitude) 2 If power drops to one-half, amplitude becomes: \[ \sqrt{\tfrac{1}{2}} \approx 0.707 \] In decibels: \[ 10\log_{10}(1/2) \approx -3.01\ \text{dB} \] This is why the −3 dB point universally corresponds to half power. 2. Why Half-Power Defines a Natural Beamwidth (HPBW) Near its maximum, most radiation or filter responses can be approximated by a second-order Taylor expansion: \[ P(\theta) \approx P_{\max}(1 - a\theta^2) \] Half-power occurs when: \[ 1 - a\theta_{1/2}^2 ...

This document explains the structure and equations of hybrid analog–digital MIMO systems, emphasizing how digital beamforming works alongside analog beamforming for mmWave systems. 1. Complete Hybrid MIMO Signal Model In a hybrid MIMO transmitter, processing is split into two stages: Digital (baseband) precoder: \( \mathbf{F}_{\text{BB}} \) Analog (RF) precoder: \( \mathbf{F}_{\text{RF}} \) The transmitted signal is: \( \mathbf{x} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} \) Subject to the power constraint: \( \| \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \|_F^2 = N_s \) At the receiver, the signal is: \( \mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{n} = \mathbf{H} \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathbf{n} \) After analog and digital combining: \( \mathbf{r} = \mathbf{W}_{\text{BB}}^H \mathbf{W}_{\text{RF}}^H \mathbf{H} \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathb...

MATLAB Code for Zero-Forcing (ZF) Beamforming in 4×4 MIMO Systems clc; clear; close all; %% Parameters Nt = 4; % Transmit antennas Nr = 4; % Receive antennas (must be >= Nt for ZFBF) numBits = 1e4; % Number of bits per stream SNRdB = 0; % SNR in dB numRuns = 100; % Number of independent runs for averaging %% Precompute noise standard deviation noiseSigma = 10^(-SNRdB / 20); %% Accumulator for total errors totalErrors = 0; for run = 1:numRuns % Generate random bits: [4 x 10000] bits = randi([0 1], Nt, numBits); % BPSK modulation: 0 → +1, 1 → -1 txSymbols = 1 - 2 * bits; % Rayleigh channel matrix: [4 x 4] H = (randn(Nr, Nt) + 1j * randn(Nr, Nt)) / sqrt(2); %% === Zero Forcing Beamforming at Transmitter === W_zf = pinv(H); % Precoding matrix: [Nt x Nr] txPrecoded = W_zf * txSymbols; % Apply ZF precoding % Normalize transmit power (optional but useful) txPrecoded = txPrecoded / sqrt(mean(abs(txPrecoded(:)).^2)); %% Channel transmission with AWGN noise = noiseSigma * (randn(...

📘 Overview of Beamforming in MIMO 🧮 Hybrid Beaforming 🧮 Spectral Efficiency in Hybrid Beamforming 📚 Further Reading         Fig 1: Hybrid Beamforming    In digital beamforming , each antenna element has its own radio frequency (RF) chain, consisting of analog components (such as amplifiers, filters, and mixers) and digital components (such as analog-to-digital converters (ADCs) and digital signal processors (DSPs)). On the other hand, Only a subset of antennas (or antenna elements) has their own RF chains, while the remaining antennas share a common RF chain. In hybrid beamforming, the beamforming process is divided into analog (say, A) and digital (say, D) beamforming domains.  The goal is to optimise the overall throughput or sum rate R (A, D) obtained over Gaussian signalling on MMwave channels by designing (A, D).     The related sum rate optimization problem looks like this:     Here, set F...

      Fig 1: Digital Beamforming    The main working principle between the beamforming (or analog beamforming) is to maximize the signal strength in a particular direction towards the receiver. For example, you can steer the antenna manually towards the transmitter to maximize the signal strength, like dish antennas. However, the approach could be more practical for mobile communications. With the help of a phase shifter, we do it electronically. On the other hand, for example, a dish antenna has an aperture that adds some gain to the received signal.Similarly, placing many antennas at a particular space instant creates a beam in a specific direction, minimizing signal strength in the rest of the directions. Here, a combination of antennas creates virtual apertures. On the transmitter side, it transmits a more robust signal toward the receiver. Oppositely, it makes a virtual aperture at the receiver and captures the signal more efficiently. The main advantage o...