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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.

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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.

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