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

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, ... 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.   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 s...
Read more

Antenna Gain-Combining Methods - EGC, MRC, SC, and RMSGC

 There are different antenna gain-combining methods. They are as follows. 1. Equal gain combining (EGC) 2. Maximum ratio combining (MRC) 3. Selective combining (SC) 4. Root mean square gain combining (RMSGC) 1. Equal gain combining method We add the correlated data streams from different antennas in the equal gain combining method. Then we multiply the resultant data with (1/(number of antennas)) For example, for two antenna gain-combining  If the received symbols are y1 and y2, then  Equal combing gain, y_egc = 0.5 * (y1 + y2) 2. Maximum ratio combining method We multiply the individual data streams with weights in the maximum ratio combining method. More weightage is multiplied by those data streams with maximum {|h|^2}, where h denotes the channel impulse response. And less weightage is multiplied by those data streams with corresponding small value of  {|h|^2}.  Then we sum the data streams to improve SNR. In the case of Maximum Ratio Combining, if y1 an...
Read more

MATLAB Code for Pulse Width Modulation (PWM) and Demodulation

   Pulse Width Modulation (PWM) MATLAB Script clc; clear all; close all; fs=30; %frequency of the sawtooth signal fm=3; %frequency of the message signal sampling_frequency = 10e3; a=0.5; % amplitide t=0:(1/sampling_frequency):1; %sampling rate of 10kHz sawtooth=2*a.*sawtooth(2*pi*fs*t); %generating a sawtooth wave subplot(4,1,1); plot(t,sawtooth); % plotting the sawtooth wave title('Comparator Wave'); msg=a.*sin(2*pi*fm*t); %generating message wave subplot(4,1,2); plot(t,msg); %plotting the sine message wave title('Message Signal'); for i=1:length(sawtooth) if (msg(i)>=sawtooth(i)) pwm(i)=1; %is message signal amplitude at i th sample is greater than %sawtooth wave amplitude at i th sample else pwm(i)=0; end end subplot(4,1,3); plot(t,pwm,'r'); title('PWM'); axis([0 1 0 1.1]); %to keep the pwm visible during plotting. %% Demodulation % Demodulation: Measure the pulse width to reconstruct the signal demodulated_signal = zeros(size(msg)); for i = 1:leng...
Read more

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...
Read more

Constellation Diagrams of ASK, PSK, and FSK

BASK (Binary ASK) Modulation: Transmits one of two signals: 0 or -√Eb, where Eb​ is the energy per bit. These signals represent binary 0 and 1.    BFSK (Binary FSK) Modulation: Transmits one of two signals: +√Eb​ ( On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset ) or √Eb (on x-axis), where Eb​ is the energy per bit. These signals represent binary 0 and 1.  BPSK (Binary PSK) Modulation: Transmits one of two signals: +√Eb​ or -√Eb (they differ by 180 degree phase shift), where Eb​ is the energy per bit. These signals represent binary 0 and 1.  Key Points For Binary Amplitude Shift Keying (BASK), binary bit '0' can be represented as lower level voltage or no signal and bit '1' as higher level voltage.  For Binary Frequency Shift Keying (BFSK), you can map binary bit '0' to 'j' and bit '1' to '1'. So, signals are in phase.  A phase shift of 0 degrees could represent a binary '1...
Read more

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...
Read more

Gaussian minimum shift keying (GMSK)

Dive into the fascinating world of GMSK modulation, where continuous phase modulation and spectral efficiency come together for robust communication systems! Core Process of GMSK Modulation Phase Accumulation (Integration of Filtered Signal) After applying Gaussian filtering to the Non-Return-to-Zero (NRZ) signal, we integrate the smoothed NRZ signal over time to produce a continuous phase signal: θ(t) = ∫ 0 t m filtered (τ) dτ This integration is crucial for avoiding abrupt phase transitions, ensuring smooth and continuous phase changes. Phase Modulation The next step involves using the phase signal to modulate a high-frequency carrier wave: s(t) = cos(2πf c t + θ(t)) Here, f c is the carrier frequency, and s(t) represents the continuous-phase modulated carrier wave. Quadrature Modulation (Optional) ...
Read more

MATLAB Code for QAM (Quadrature Amplitude Modulation)

  One of the best-performing modulation techniques is QAM [↗] . Here, we modulate the symbols by varying the carrier signal's amplitude and phase in response to the variation in the message signal (or voltage variation). So, we may say that QAM is a combination of phase and amplitude modulation. Additionally, it performs better than ASK or PSK [↗] . In fact, any constellation for any type of modulation, signal set (or, symbols) is structured in a way that prevents them from interacting further by being distinct by phase, amplitude, or frequency. MATLAB Script (for 4-QAM) % This code is written by SalimWirelss.Com % This is an example of 4-QAM. Here constellation size is 4 % or total number of symbols/signals is 4 % We need 2 bits once to represent four constellation points % QAM modulation is the combination of Amplitude modulation plus % Phase Modulation. We map the decimal value of the input symbols, i.e., % 00, 01, 10, 11 to 1 + 1i, -1 + 1i, 1 - 1i, and -1 - 1i, respectively. cl...
Read more