Skip to main content

Massive MIMO | SVD, Multiplexing, Rank and Condition Number


Today, we'll talk about the importance of massive MIMO in modern 5G communication systems. We are aware that MIMO technology has been used in the past for 4G LTE. Massive MIMO has a number of advantages over traditional MIMO systems.

Basic benefits:
1. MIMO is a technology that allows for spatial multiplexing;
2. Higher beamforming gain via numerous antennas;
3. Allows for space, frequency, and time diversity.

Singular Value Decomposition (SVD):

Go through the process of singular value decomposition (SVD)

H = U∑VH

Mathematically, SVD denotes: Here in massive MIMO, we basically factorize the channel matrix,

U and V are unitary matrices
= diagonal singular value matrix

The values of the unitary matrices U and V are arranged in such a way that the singular values of the matrix ∑ are in decreasing order. SVD aids in the optimal allocation of power to each singular value. It also has something to do with spatial multiplexing.

Master Massive MIMO Technology

Explore our comprehensive simulator and deep-dive technical guides for 5G Beamforming.

Launch Simulator >

Spatial Multiplexing (SM):

Spatial multiplexing allows us to deliver multiple data streams to the transmitter and receiver at the same time. The number of simultaneous and independent data streams between TX and RX is determined by the singular values in matrix ∑ above. The number of simultaneous data streams is determined by the rank of a wireless communication channel matrix. In MIMO communication, capacity of the system increases with the number of antenna elements and the log of the signal to noise ratio, or SNR.

Signal Coherency at receiver side:

Now I'll talk about how we can go from simple MIMO to massive MIMO for 5G connectivity. We already know that increasing the antenna array size in MIMO improves spectral efficiency. When the number of antenna elements in a huge MIMO system is increased, however, the signal phase alignment at the receiver side improves. It basically focuses the resulting strong signal in a single direction.

Massive MIMO communication – Uplink and Downlink

Users directly transmit their symbols via the large MIMO UL link. To reduce interference in one's transmitted symbol from symbols of other users, the BS must recover each individual's symbol using linear decoding. We employ a pre-coding (beamforming) technique for downlink or DL communication to cancel interference between users using correct baseband and RF pre-coding and a combining (or weighting) matrix.

Rank and Condition number of a massive MIMO channel matrix

The number of independent rows or columns in a matrix determines its rank. When we determine the rank of a channel matrix, we may determine how many independent data streams are possible between the TX and RX MIMO antennas. In most circumstances, the rank of a channel matrix in massive MIMO is very small, especially when operating at the millimetre wave band. As a result, it generates a sparse channel matrix.

% Example Matlab Rank Calculation
H = [1 0.5; 0.5 1];
rank_H = rank(H);
cond_H = cond(H);
disp(['Rank: ', num2str(rank_H)]);

The condition number is a statistic used to characterise the quality of MIMO channels. It is defined as the ratio of the greatest to lowest singular value. In MIMO, a low condition number (below 20 dB) usually indicates good orthogonality. However, the condition number is substantially higher here during extremely high frequency operation. As a result, we employ beamforming to overcome constraints.

Read more about Rank and Condition number

#beamforming


Contact Us

Name

Email *

Message *

Popular Posts

MIMO Channel Matrix | Rank and Condition Number

MIMO / Massive MIMO MIMO Channel Matrix | Rank and Condition...   The channel matrix in wireless communication is a matrix that describes the impact of the channel on the transmitted signal. The channel matrix can be used to model the effects of the atmospheric or underwater environment on the signal, such as the absorption, reflection or scattering of the signal by surrounding objects. When addressing multi-antenna communication, the term "channel matrix" is used. Let's assume that only one TX and one RX are in communication and there's no surrounding object. Here, in our case, we can apply the proper threshold condition to a received signal and get the original transmitted signal at the RX side. However, in real-world situations, we see signal path blockage, reflections, etc.,  (NLOS paths [↗]) more frequently. The obstruction is typically caused by building walls, etc. Multi-antenna communication was introduced to address this issue. It makes diversity app...

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

Bit Error Rate (BER) & SNR Guide Analyze communication system performance with our interactive simulators and MATLAB tools. 📘 Theory 🧮 Simulators 💻 MATLAB Code 📚 Resources BER Definition SNR Formula BER Calculator MATLAB Comparison 📂 Explore M-ary QAM, PSK, and QPSK Topics ▼ 🧮 Constellation Simulator: M-ary QAM 🧮 Constellation Simulator: M-ary PSK 🧮 BER calculation for ASK, FSK, and PSK 🧮 Approaches to BER vs SNR Calculation What is Bit Error Rate (BER)? The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figur...

Online Simulator for ASK, FSK, and PSK Signal Generation

Interactive Digital Signal Processing (DSP) Tutorial and Simulator for ASK, FSK, and BPSK modulation techniques. Try our new Digital Signal Processing Simulator!   •   Interactive ASK, FSK, and BPSK tools updated for 2025. Start Now Digital Modulation Visualizer: ASK, FSK, & BPSK Simulator Learn and visualize binary modulation techniques (ASK, FSK, BPSK) in real-time with adjustable carrier and sampling parameters. Perfect for DSP students and engineers. 📡 ASK Simulator 📶 FSK Simulator 🎚️ BPSK Simulator 📚 More Topics ASK Modulator FSK Modulator BPSK Modulator More Topics 1. ASK (Amplitude Shift Keying) Simulat...

MATLAB Code for OTFS (Orthogonal Time Frequency Space)

MATLAB Code for OTFS (Orthogonal Time Frequency Space) %% Clear workspace clc; clear; close all ; %% Step 1: OTFS Parameters N_delay = 4; % Number of delay bins (rows) N_doppler = 4; % Number of Doppler bins (columns) N_sym = N_delay * N_doppler; modOrder = 4; % QPSK SNR_dB = 20; % Noise level %% Step 2: Generate random data symbols data = randi([0 modOrder-1], N_sym, 1); txSymbols = pskmod(data, modOrder, pi/4); disp( 'Transmitted Delay-Doppler symbols:' ); disp(reshape(txSymbols, N_delay, N_doppler)); %% Step 3: Map Delay-Doppler → Time-Frequency (ISFFT) % ISFFT: Inverse Symplectic Finite Fourier Transform % 1. Take IDFT along Doppler (columns) % 2. Take DFT along Delay (rows) ddSymbols = reshape(txSymbols, N_delay, N_doppler); % Step 3a: IDFT along columns (Doppler) tfGrid = ifft(ddSymbols, N_doppler, 2); %IFFT (accross columns) along Doppler → spreads in time (Delay → Time) %FFT (accross rows)along Delay → spreads in frequency (Delay → Frequency) % Step 3b: DFT along ...

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

📘 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...

FFT Butterfly Method Explained (with Simulations)

4-Point FFT Using Butterfly Method Given: x[n] = {0, 1, 2, 3} Step 1: Split into Even & Odd Even indices: x e = {x[0], x[2]} = {0, 2} Odd indices: x o = {x[1], x[3]} = {1, 3} Step 2: 2-point DFT For any {a, b}: DFT = {a + b, a - b} Even Part (E): {0+2, 0-2} = {2, -2} Odd Part (O): {1+3, 1-3} = {4, -2} Step 3: Combine Using Butterfly X[k] = E[k] + W 4 k O[k] X[k + 2] = E[k] - W 4 k O[k] Twiddle Factors (N=4): W 4 0 = 1, W 4 1 = -j Final Calculations: X[0] = E[0] + W 4 0 O[0] = 2 + (1)(4) = 6 X[2] = E[0] - W 4 0 O[0] = 2 - (1)(4) = -2 X[1] = E[1] + W 4 1 O[1] = -2 + (-j)(-2) = -2 + 2j X[3] = E[1] - W 4 1 O[1] = -2 - (-j)(-2) = -2 - 2j Final Answer: X[k] = {6, -2 + 2j, -2, -2 - 2j} 8-Point FFT Using Butterfly Method Given: x[n] = {0,1,2,3,4,5,6,7} Step 1: Split into Bit-Reversed Order To perform DIT-FFT, split the 8 points into pairs of two: Group A: {x[0], x[4]} = {0, 4}...

OFDM Symbols and Subcarriers Explained

This article explains how OFDM (Orthogonal Frequency Division Multiplexing) symbols and subcarriers work. It covers modulation, mapping symbols to subcarriers, subcarrier frequency spacing, IFFT synthesis, cyclic prefix, and transmission. Step 1: Modulation First, modulate the input bitstream. For example, with 16-QAM , each group of 4 bits maps to one QAM symbol. Suppose we generate a sequence of QAM symbols: s0, s1, s2, s3, s4, s5, …, s63 Step 2: Mapping Symbols to Subcarriers Assume N sub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier): Mapping (example) OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7 OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15 … OFDM sym...