System Model
In massive MIMO, the base station has \( M \) antennas (large, e.g. 100+) and serves \( K \) single-antenna users. Received vector: \[ y = \sum_{i=1}^K \sqrt{p_u}\, g_i x_i + n \] where \( g_i \sim \mathcal{CN}(0, \beta_i I_M) \) is user-\( i \) channel and \( n \sim \mathcal{CN}(0, I) \).
Combining (MRC / Matched Filter)
For user 1, the matched filter (normalized) is: \[ w_1 = \frac{g_1}{\|g_1\|} \] Projection:
Matched Filter with Components \( g_0, g_1, g_2, \dots \)
Suppose the channel vector is \[ g = [ g_0, g_1, g_2, \dots, g_{M-1} ]^T \]
- Combining weight: \( w = g / \|g\| \)
- Matched filter output: \[ r = w^H y = \frac{g_0^* y_0 + g_1^* y_1 + g_2^* y_2 + \dots + g_{M-1}^* y_{M-1}}{\|g\|} \]
- Signal term: \( \sqrt{p_u}\, \|g\| x \)
- Noise term: still \( \mathcal{CN}(0,1) \) after projection.
⇒ The matched filter aligns and adds each antenna branch using the conjugate channel coefficients, boosting the desired signal by \( \|g\| \) while keeping noise variance constant.
Statistics in Massive MIMO
- Desired signal: power grows with \( \|g_1\|^2 \approx M \beta_1 \) (linear in antennas).
- Noise: after projection → still \( \mathcal{CN}(0,1) \).
- Interference: each term \( (g_1^H/\|g_1\|) g_i \sim \mathcal{CN}(0, \beta_i) \). With large M, cross terms become nearly orthogonal (favorable propagation).
Asymptotic SINR (Large M)
Average SINR for user 1:
As \( M \to \infty \), the desired signal dominates → interference/noise vanish relative to signal.
Massive MIMO Effects
- Array gain: desired signal ∝ M.
- Channel hardening: random variations of \( \|g_1\|^2 \) average out → behaves almost deterministic.
- Favorable propagation: inner products \( g_1^H g_i \approx 0 \) when M is large → interference suppressed.
- Matched filter = project received vector onto channel vector.
- Signal adds coherently across antennas, noise remains white.
- In massive MIMO, signal ∝ M but interference/noise grow slower → huge SINR gains.