Skip to main content

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.

 

Further Reading

People are good at skipping over material they already know!

View Related Topics to







Contact Us

Name

Email *

Message *

Popular Posts

Online Simulator for ASK, FSK, and PSK

Try our new Digital Signal Processing Simulator!   Start Simulator for binary ASK Modulation Message Bits (e.g. 1,0,1,0) Carrier Frequency (Hz) Sampling Frequency (Hz) Run Simulation Simulator for binary FSK Modulation Input Bits (e.g. 1,0,1,0) Freq for '1' (Hz) Freq for '0' (Hz) Sampling Rate (Hz) Visualize FSK Signal Simulator for BPSK Modulation ...
Read more

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

📘 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. BER = (number of bits received in error) / (total number of tran...
Read more

Constellation Diagrams of ASK, PSK, and FSK

📘 Overview of Energy per Bit (Eb / N0) 🧮 Online Simulator for constellation diagrams of ASK, FSK, and PSK 🧮 Theory behind Constellation Diagrams of ASK, FSK, and PSK 🧮 MATLAB Codes for Constellation Diagrams of ASK, FSK, and PSK 📚 Further Reading 📂 Other Topics on Constellation Diagrams of ASK, PSK, and FSK ... 🧮 Simulator for constellation diagrams of m-ary PSK 🧮 Simulator for constellation diagrams of m-ary QAM 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...
Read more

What is - 3dB Frequency Response? Applications ...

📘 Overview & Theory 📘 Application of -3dB Frequency Response 🧮 MATLAB Codes 🧮 Online Digital Filter Simulator 📚 Further Reading Filters What is -3dB Frequency Response?   Remember, for most passband filters, the magnitude response typically remains close to the peak value within the passband, varying by no more than 3 dB. This is a standard characteristic in filter design. The term '-3dB frequency response' indicates that power has decreased to 50% of its maximum or that signal voltage has reduced to 0.707 of its peak value. Specifically, The -3dB comes from either 10 Log (0.5) {in the case of power} or 20 Log (0.707) {in the case of amplitude} . Viewing the signal in the frequency domain is helpful. In electronic amplifiers, the -3 dB limit is commonly used to define the passband. It shows whether the signal remains approximately flat across the passband. For example, in pulse shapi...
Read more

BER performance of QPSK with BPSK, 4-QAM, 16-QAM, 64-QAM, 256-QAM, etc

📘 Overview 📚 QPSK vs BPSK and QAM: A Comparison of Modulation Schemes in Wireless Communication 📚 Real-World Example 🧮 MATLAB Code 📚 Further Reading   QPSK provides twice the data rate compared to BPSK. However, the bit error rate (BER) is approximately the same as BPSK at low SNR values when gray coding is used. On the other hand, QPSK exhibits similar spectral efficiency to 4-QAM and 16-QAM under low SNR conditions. In very noisy channels, QPSK can sometimes achieve better spectral efficiency than 4-QAM or 16-QAM. In practical wireless communication scenarios, QPSK is commonly used along with QAM techniques, especially where adaptive modulation is applied. Modulation Bits/Symbol Points in Constellation Usage Notes BPSK 1 2 Very robust, used in weak signals QPSK 2 4 Balanced speed & reliability 4-QAM ...
Read more

Theoretical BER vs SNR for binary ASK, FSK, and PSK

📘 Overview & Theory 🧮 MATLAB Codes 📚 Further Reading Theoretical BER vs SNR for Amplitude Shift Keying (ASK) The theoretical Bit Error Rate (BER) for binary ASK depends on how binary bits are mapped to signal amplitudes. For typical cases: If bits are mapped to 1 and -1, the BER is: BER = Q(√(2 × SNR)) If bits are mapped to 0 and 1, the BER becomes: BER = Q(√(SNR / 2)) Where: Q(x) is the Q-function: Q(x) = 0.5 × erfc(x / √2) SNR : Signal-to-Noise Ratio N₀ : Noise Power Spectral Density Understanding the Q-Function and BER for ASK Bit '0' transmits noise only Bit '1' transmits signal (1 + noise) Receiver decision threshold is 0.5 BER is given by: P b = Q(0.5 / σ) , where σ = √(N₀ / 2) Using SNR = (0.5)² / N₀, we get: BER = Q(√(SNR / 2)) Theoretical BER vs ...
Read more

Channel Impulse Response (CIR)

📘 Overview & Theory 📘 How CIR Affects the Signal 🧮 Online Channel Impulse Response Simulator 🧮 MATLAB Codes 📚 Further Reading What is the Channel Impulse Response (CIR)? 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. 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. Fig: Dirac Delta Function The result of this calculation is that all frequencies are responded to equally by δ(t) . This is crucial since we never know which frequenci...
Read more

Coherence Bandwidth and Coherence Time

🧮 Coherence Bandwidth 🧮 Coherence Time 🧮 MATLAB Code s 📚 Further Reading For Doppler Delay or Multi-path Delay Coherence time T coh ∝ 1 / v max (For slow fading, coherence time T coh is greater than the signaling interval.) Coherence bandwidth W coh ∝ 1 / τ max (For frequency-flat fading, coherence bandwidth W coh is greater than the signaling bandwidth.) Where: T coh = coherence time W coh = coherence bandwidth v max = maximum Doppler frequency (or maximum Doppler shift) τ max = maximum excess delay (maximum time delay spread) Notes: The notation v max −1 and τ max −1 indicate inverse proportionality. Doppler spread refers to the range of frequency shifts caused by relative motion, determining T coh . Delay spread (or multipath delay spread) determines W coh . Frequency-flat fading occurs when W coh is greater than the signaling bandwidth. Coherence Bandwidth Coherence bandwidth is...
Read more