Questions: MIMO Capacity

Question 1

A MIMO system has 4 transmit and 4 receive antennas in a rich scattering environment with high SNR. Approximately how does its capacity compare to a single-antenna (SISO) system at the same total power?

• A. About the same — more antennas don't help because total power is fixed
• B. About 4x the SISO capacity — MIMO capacity scales linearly with min(n_T, n_R) at high SNR
• C. About 16x — capacity scales with n_T * n_R
• D. About 2x — capacity scales logarithmically with antenna count

Answer: (B)

At high SNR with rich scattering, H has min(n_T, n_R) = 4 significant singular values, creating 4 parallel spatial channels. Each carries roughly (1/4) of the total power and achieves roughly (1/4) of what a SISO system would with full power. But there are 4 of them, and at high SNR the log(1 + SNR/4) ≈ log(SNR/4) ≈ log(SNR) - 2, so total capacity ≈ 4*log(SNR) - 8 versus SISO's log(SNR). The ratio approaches 4 = min(n_T, n_R). This linear scaling in antenna count, without requiring more bandwidth or power, is why MIMO is transformative.

Question 2

Water-filling power allocation across MIMO eigenmodes allocates more power to weaker channels.

• T. True
• F. False

Answer: False

Water-filling allocates MORE power to STRONGER eigenmodes and less (or zero) to weaker ones. The name comes from the analogy of pouring water into a vessel: the channel gains are the bottom (inverted — weak channels have higher 'floors'), and water (power) fills the gaps, naturally putting more water where the floor is lower (stronger channels). Weak eigenmodes may receive zero power if their channel gain is below a threshold. This is optimal because a bit of power yields more capacity on a strong channel than on a weak one.

Question 3

Explain why MIMO capacity depends critically on the channel matrix H, and describe the conditions under which MIMO fails to provide a capacity gain over SISO.

Model answer: MIMO capacity depends on the singular values of H. If H has rank r, only r independent spatial streams can be supported. In rich scattering (urban, indoor), H is typically full-rank with comparable singular values, yielding near-maximum capacity gain. In line-of-sight (LOS) with no scattering, H may be rank-1 regardless of antenna count — all antennas see essentially the same path, and MIMO degenerates to beamforming (array gain only, no multiplexing gain). Capacity is then similar to SISO plus a power gain from beamforming. Correlated fading (antennas too close together, or poor scattering) reduces the effective rank and diminishes MIMO's multiplexing advantage.

This is why 5G massive MIMO systems use large numbers of antennas (64-256) — even in partially correlated channels, the effective rank is large enough to serve many users simultaneously via spatial multiplexing (MU-MIMO), approaching the sum capacity of the multi-user channel.