Skip to main content
Home Wireless Communication Modulation MATLAB Beamforming Project Ideas MIMO Computer Networks Lab 🚀

Mathematical Aspects of Beamforming in MIMO



Beam steering, which permits strong directed beams towards the receiver to combat excessive pathloss, especially for higher frequency bands, immediately comes to mind when discussing mathematical aspects of Beamforming in MIMO antennas. On the other side, it also lessens signal interference and improves the effectiveness of spatial multiplexing in Massive MIMO communication. Let's go right to the mathematical parts of Beamforming, which will make it easier for you to code in Python and MATLAB.



1. Beam Steering (Analog Beamforming)




In the first stage, the BS applies beam steering at the side of the mobile station (MS) while the MS enables omnidirectional transmission. In the following step, the MS uses beam steering while the BS is an omnidirectional transmitter. The best beamformer and combiner pair are then identified at BS & MS, and they make communication available. Following is an outline of the codebook:


 

 
Let's say a small town or village has a cell tower in the midst of it. Now everybody can understand the cell tower's 360-degree coverage area (if not, you restrict the coverage to a particular direction or sector). The codebook above specifies what the signal intensity will be different at a specific coverage zone defined by the azimuth angle or elevation angular ranges from the transmitter (here, cell tower).

Assume that the first element in the given set, f, indicates the coverage zone from 0 to 10 degrees.
The second element depicts the coverage area between 10 and 20 degrees from the base station.
Additionally, every component in the codebook has directions, or azimuth angle ranges from 0 to 360 degree.
A similar procedure is applicable for mobile stations (MS) to identify the strongest beam between them by determining the optimum path (here, beam) from MS to BS.


2. Digital Beamforming


Fig: Digital Beamforming

Each antenna element, in this case, is connected to a separate RF chain during digital Beamforming. Filters, mixers, amplifiers, etc., make up RF chains. Each RF chain controls a particular data stream between TX and RX.
Any signal or data stream transmitted by transmitter side antenna T1 is typically received by all receiver side antennas. There are four different user equipment (UEs) or mobile stations shown in the above diagram. All four UEs receive any signal that is sent by antenna T1. Assume that receiver R1 was the only one for which the signal was intended. It will then be regarded as interference for receiver side antennas R1, R2, R3,..., and R8. In this situation, a digital beamforming matrix is crucial to eliminate interference at all undesirable receivers while transmitting the signal from T1. and permit R1 to only receive the signal. The individual data streams between T2 and R2, T3 and R3, and so forth can be assumed similarly.

At the receiver side, the signal received by users vector y, 
                                                                y = √ρHDs + n
                                                               where H=Channel Matrix
                                                               s = transmitted symbol/signal
                                                               n = additive white Gaussian noise (AWGN)
                                                               ρ = average received power
                                                               D = digital precoding/beamforming matrix

For a multiuser scenario, the hybrid beamforming equation looks like
                                                               y = √ρ.H.[D1 D2 ... Dn].s + n
                                                               Where 'Dn' denotes the digital precoder 
                                                                for u-th user

Now cancel interference at u-th user due to other users; we must design the baseband precoder so that HuDn for nǂ u should be zero at the u-th mobile station (MS). Therefore, HuDn =0 cancels interferences at u-th MS.
 
 
 ------------------------------------------------------------------------------------------------------------
. - - -  - - - - - - beamforming
                            - -  - Analog Beamforming
.                           - -  - Digital Beamforming
.                                      - - Equations related to Spectral Efficiency in Digital Beamforming
.                           - -  - Hybrid Beamforming
.                                      - - Equations related to Spectral Efficiency in Hybrid Beamforming
--------------------------------------------------------------------------------------------------------------

3. Hybrid Beamforming


First, we connect multiple antenna elements in hybrid Beamforming to increase gain, which is crucial for today's higher-frequency wireless communication systems. Then, precisely as illustrated in the above figure, we apply digital Beamforming to those RF chains. The key advantages of hybrid Beamforming are that
Less interference than digital Beamforming without sacrificing a significant difference in a MIMO system's throughput.
The transmitted signal has a large amount of gain added by analog Beamforming or beam steering to extend its range.
For lower-dimensional MIMO systems, digital Beamforming works well, but massive MIMO systems are where the future of communication is headed. Compared to digital Beamforming, hybrid Beamforming is less complicated and more cost-effective.

People are good at skipping over material they already know!

View Related Topics to







Admin & Author: Salim

profile

  Website: www.salimwireless.com
  Interests: Signal Processing, Telecommunication, 5G Technology, Present & Future Wireless Technologies, Digital Signal Processing, Computer Networks, Millimeter Wave Band Channel, Web Development
  Seeking an opportunity in the Teaching or Electronics & Telecommunication domains.
  Possess M.Tech in Electronic Communication Systems.


Contact Us

Name

Email *

Message *

Popular Posts

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...

Modulation Constellation Diagrams BER vs. SNR BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ... 1. 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 refers to the signal-to-noise power ratio. For ease of calculation, we commonly convert it to dB or decibels.   2. What is Signal the signal-to-noise ratio (SNR)? SNR = signal power/noise power (SNR is a ratio of signal power to noise power) SNR (in dB) = 10*log(signal power / noise power) [base 10] For instance, the SNR for a given communication system is 3dB. So, SNR (in ratio) = 10^{SNR (in dB) / 10} = 2 Therefore, in this instance, the signal power i

Comparisons among ASK, PSK, and FSK | And the definitions of each

Modulation ASK, FSK & PSK Constellation MATLAB Simulink MATLAB Code Comparisons among ASK, PSK, and FSK    Comparisons among ASK, PSK, and FSK Comparison among ASK,  FSK, and PSK Performance Comparison: 1. Noise Sensitivity:    - ASK is the most sensitive to noise due to its reliance on amplitude variations.    - PSK is less sensitive to noise compared to ASK.    - FSK is relatively more robust against noise, making it suitable for noisy environments. 2. Bandwidth Efficiency:    - PSK is the most bandwidth-efficient, requiring less bandwidth than FSK for the same data rate.    - FSK requires wider bandwidth compared to PSK.    - ASK's bandwidth efficiency lies between FSK and PSK. Bandwidth Calculator for ASK, FSK, and PSK The baud rate represents the number of symbols transmitted per second Select Modulation Type: ASK FSK PSK Baud Rate (Hz):

MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...

Modulation Constellation Diagrams BER vs. SNR MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...   MATLAB Script for  BER vs. SNR for M-QAM, M-PSK, QPSk, BPSK %Written by Salim Wireless %Visit www.salimwireless.com for study materials on wireless communication %or, if you want to learn how to code in MATLAB clc; clear; close all; % Parameters num_symbols = 1e5; % Number of symbols snr_db = -20:2:20; % Range of SNR values in dB % PSK orders to be tested psk_orders = [2, 4, 8, 16, 32]; % QAM orders to be tested qam_orders = [4, 16, 64, 256]; % Initialize BER arrays ber_psk_results = zeros(length(psk_orders), length(snr_db)); ber_qam_results = zeros(length(qam_orders), length(snr_db)); % BER calculation for each PSK order and SNR value for i = 1:length(psk_orders) psk_order = psk_orders(i); for j = 1:length(snr_db) % Generate random symbols data_symbols = randi([0, psk_order-1]

Difference between AWGN and Rayleigh Fading

Wireless Signal Processing Gaussian and Rayleigh Distribution Difference between AWGN and Rayleigh Fading 1. Introduction Rayleigh fading coefficients and AWGN, or additive white gaussian noise [↗] , are two distinct factors that affect a wireless communication channel. In mathematics, we can express it in that way.  Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading. y = hx + n ... (i) The transmitted signal  x  is multiplied by the channel coefficient or channel impulse response (h)  in the equation above, and the symbol  "n"  stands for the white Gaussian noise that is added to the signal through any type of channel (here, it is a wireless channel or wireless medium). Due to multi-paths the channel impulse response (h) changes. And multi-paths cause Rayleigh fading. 2. Additive White Gaussian Noise (AWGN) The mathematical effect involves adding Gauss

FFT Magnitude and Phase Spectrum using MATLAB

MATLAB Code clc; clear; close all; % Parameters fs = 100;           % Sampling frequency t = 0:1/fs:1-1/fs;  % Time vector % Signal definition x = cos(2*pi*15*t - pi/4) - sin(2*pi*40*t); % Compute Fourier Transform y = fft(x); z = fftshift(y); % Frequency vector ly = length(y); f = (-ly/2:ly/2-1)/ly*fs; % Compute phase phase = angle(z); % Plot magnitude of the Fourier Transform figure; subplot(2, 1, 1); stem(f, abs(z), 'b'); xlabel('Frequency (Hz)'); ylabel('|y|'); title('Magnitude of Fourier Transform'); grid on; % Plot phase of the Fourier Transform subplot(2, 1, 2); stem(f, phase, 'b'); xlabel('Frequency (Hz)'); ylabel('Phase (radians)'); title('Phase of Fourier Transform'); grid on;   Output  Copy the MATLAB Code from here % The code is written by SalimWireless.Com clc; clear; close all; % Parameters fs = 100; % Sampling frequency t = 0:1/fs:1-1/fs; % Time vector % Signal definition x = cos(2*pi*15*t -

Channel Impulse Response (CIR)

Channel Impulse Response (CIR) Wireless Signal Processing CIR, Doppler Shift & Gaussian Random Variable  The Channel Impulse Response (CIR) is a concept primarily used in the field of telecommunications and signal processing. It provides information about how a communication channel responds to an impulse signal.   What is the Channel Impulse Response (CIR) ? It describes the behavior of a communication channel in response to an impulse signal. In signal processing,  an impulse signal has zero amplitude at all other times and amplitude  ∞ at time 0 for the signal. Using a Dirac Delta function, we can approximate this.  ...(i) δ( t) now has a very intriguing characteristic. The answer is 1 when the Fourier Transform of  δ( t) is calculated. As a result, all frequencies are responded to equally by  δ (t). This is crucial since we never know which frequencies a system will affect when examining an unidentified one. Since it can test the system for all freq

MATLAB Code for Pulse Amplitude Modulation (PAM) and Demodulation

  Pulse Amplitude Modulation (PAM) & Demodulation MATLAB Script clc; clear all; close all; fm= 10; % frequency of the message signal fc= 100; % frequency of the carrier signal fs=1000*fm; % (=100KHz) sampling frequency (where 1000 is the upsampling factor) t=0:1/fs:1; % sampling rate of (1/fs = 100 kHz) m=1*cos(2*pi*fm*t); % Message signal with period 2*pi*fm (sinusoidal wave signal) c=0.5*square(2*pi*fc*t)+0.5; % square wave with period 2*pi*fc s=m.*c; % modulated signal (multiplication of element by element) subplot(4,1,1); plot(t,m); title('Message signal'); xlabel ('Time'); ylabel('Amplitude'); subplot(4,1,2); plot(t,c); title('Carrier signal'); xlabel('Time'); ylabel('Amplitude'); subplot(4,1,3); plot(t,s); title('Modulated signal'); xlabel('Time'); ylabel('Amplitude'); %demdulated d=s.*c; % At receiver, received signal is multiplied by carrier signal filter=fir1(200,fm/fs,'low'); % low-pass FIR fi