Spatial Multiplexing

#1The MIMO Promise — Why Spatial Multiplexing

MIMO capacity scales linearly with min(N_Tx, N_Rx) — same bandwidth, same power, just more antennas:

#2MIMO Channel Model Foundation

2.1 The Channel Matrix H

  H (N_Rx × N_Tx):    Received signal:  y = H × x + n

  ┌ h₁₁  h₁₂  h₁₃  h₁₄ ┐    x = Tx signal vector
  │ h₂₁  h₂₂  h₂₃  h₂₄ │    y = Rx signal vector
  │ h₃₁  h₃₂  h₃₃  h₃₄ │    n = noise
  └ h₄₁  h₄₂  h₄₃  h₄₄ ┘    h_ij = complex gain from Tx_j to Rx_i

2.2 Rich Scattering vs Limited Scattering

#3SVD — Parallel Spatial Channels

The Singular Value Decomposition decomposes any MIMO channel into independent parallel sub-channels:

Water-Filling Power Allocation

#4Channel Rank & Degrees of Freedom

The rank of H determines how many independent spatial streams can be transmitted:

#5SU-MIMO — Single-User Spatial Multiplexing

Multiple independent data streams sent to a single UE over the same time-frequency resource:

#6MU-MIMO — Multi-User Spatial Multiplexing

Multiple UEs share the same time-frequency resource, separated spatially by precoding:

#7Precoding for Spatial Multiplexing

Precoding adjusts transmitted signals to cancel inter-stream interference at the receiver:

#85G NR Layer Mapping & DMRS

#9CSI Feedback — RI, PMI, CQI

Rank Adaptation Simulation

Why Codebook-Based Precoding?

In FDD and FR2 mmWave, the gNB does not have direct access to the downlink channel matrix H. Sending the full H matrix would require enormous overhead — codebooks solve this by pre-defining a finite set of precoding matrices known to both ends (TS 38.214).

DFT Beam Grid & Two-Stage Precoding: W = W1 × W2

5G NR codebooks are built from DFT beams — a grid of fixed beams covering all spatial directions. The precoding matrix uses a two-stage structure:

Type I Codebook — Single Beam Selection

Type I is the basic codebook for SU-MIMO. It selects a single dominant DFT beam direction per layer:

  Type I — Single DFT Beam Per Layer
  ──────────────────────────────────
  Rank 1: One beam per polarization, co-phased
  Rank 2–4: Different beam per layer (from same W1 subset)

  PMI Payload: ~10–30 bits (W1: 4–8 bits wideband + W2: 3–6 bits/subband)
  Used when:   SU-MIMO, ≤ 4 layers, FR1 sub-6 GHz
  3GPP spec:   TS 38.214 Section 5.2.2.2

Type II Codebook — Linear Combination of Beams

Type II is the advanced codebook for MU-MIMO and high-resolution precoding. It combines L = 2, 3, or 4 beams with individual amplitude and phase coefficients:

  Type II — L Beams Combined Per Layer
  ─────────────────────────────────────
  Precoder: W_l = W1 × Σ (p_l,i × c_l,i) × b_i    (i = 1..L)

  p_l,i = amplitude {1, √0.5, √0.25, √0.125}   (3 bits/beam)
  c_l,i = phase: QPSK (4 phases) or 8-PSK (8 phases)

  PMI Payload: ~50–300 bits (10–20× more than Type I!)
  Used when:   MU-MIMO, high-order SU-MIMO, massive MIMO
  3GPP spec:   TS 38.214 Section 5.2.2.2.3

Type I vs Type II — Head-to-Head

Wideband vs Subband Feedback

CSI feedback can be reported with different frequency granularity:

CSI Reporting Modes — Periodic, Aperiodic, Semi-Persistent

5G NR supports three CSI reporting modes, each triggered differently and carried on different physical channels:

#10Diversity vs Multiplexing Trade-off

#11Massive MIMO Multiplexing

Favorable Propagation & Channel Hardening

#12Beam-Domain Multiplexing (6G)

Transform to the beam domain via lens array or DFT — channel becomes sparse, reducing processing complexity:

#13AI/ML-Driven Multiplexing (6G)

#14Complete Evolution — LTE → 5G → 6G

#15Practical Field Considerations

#16Further Reading

3GPP Specifications

• TS 38.211 — NR Physical Channels (layer mapping, DMRS)
• TS 38.212 — NR Multiplexing and Channel Coding
• TS 38.213 — NR Physical Layer Procedures (CSI reporting triggers, periodic/aperiodic/semi-persistent)
• TS 38.214 — NR Physical Layer Procedures for Data (Type I/Type II codebooks, precoding, spatial multiplexing)

Research Papers

• "MIMO Wireless Communications" — Biglieri et al., Cambridge
• "Noncooperative Cellular Wireless with Unlimited Numbers of BS Antennas" — Marzetta, 2010
• "Diversity and Multiplexing: A Fundamental Tradeoff" — Zheng & Tse, IEEE 2003
• "Ultra-Massive MIMO in Spatial and Beam Domains" — Southeast University / IEEE 2025

Companion Resources

Telecom Leaders Podcast Beamforming Page → ☆ Cell-Free MAC →