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Maximum Ratio Combining: The Simplest Massive MIMO Receiver

MRC (also called matched filtering) is the simplest linear combining strategy: $\mathbf{v}_{k} = \hat{\mathbf{H}}_k$ . Each user's signal is combined using the estimated channel vector as the weight. MRC maximizes the received SNR for a single user in noise, but it does nothing to suppress multi-user interference. Despite this limitation, MRC is remarkably effective in massive MIMO because the array gain from $N_t$ antennas overwhelms the residual interference when $N_t \gg K$ .

This section derives the closed-form UatF rate expression for MRC under i.i.d. Rayleigh fading and analyzes the resulting sum rate.

Definition:
MRC Combining Vector

The maximum ratio combining (MRC) vector for user $k$ is

$\mathbf{v}_{k}^{\text{MRC}} = \hat{\mathbf{H}}_k.$

Under i.i.d. Rayleigh fading with MMSE estimation, we have $\hat{\mathbf{H}}_k \sim \mathcal{CN}(\mathbf{0}, \gamma_k \mathbf{I}_{N_t})$ where $\gamma_k$ is the estimation quality defined in Section 4.0.

MRC requires only local CSI — the estimate of user $k$ 's channel — and no knowledge of other users' channels. This makes it attractive for distributed implementations and cell-free architectures (Part III).

Theorem: Closed-Form UatF Rate with MRC

Consider i.i.d. Rayleigh fading channels with MMSE estimation and no pilot contamination. With MRC combining $\mathbf{v}_{k} = \hat{\mathbf{H}}_k$ , the UatF achievable rate for user $k$ is

$R_k^{\text{MRC}} = \log_2\!\left(1 + \frac{N_t \, {P_t}_{k} \, \gamma_k}{\sum_{j=1}^{K} {P_t}_{j} \, \beta_{j} + \sigma^2}\right),$

where $\gamma_k = \beta_{k}^{2} \tau_p {P_t}_{p} / (\beta_{k} \tau_p {P_t}_{p} + \sigma^2)$ is the MMSE estimation quality and $\beta_{k}$ is the large-scale fading coefficient of user $k$ .

The numerator scales as $N_t$ — the coherent array gain. The denominator is independent of $N_t$ : interference from other users enters at power ${P_t}_{j} \beta_{j}$ , and noise at $\sigma^2$ . This means the SINR grows linearly with $N_t$ , and the rate grows logarithmically — adding antennas always helps, but with diminishing returns.

Notice also that multi-user interference does not vanish: the denominator contains $\sum_j {P_t}_{j} \beta_{j}$ . MRC is interference-limited — even with infinitely many antennas, the rate saturates if we do not also manage the interference.

Proof

Step 1: Compute the signal power (numerator)

With $\mathbf{v}_{k} = \hat{\mathbf{H}}_k$ :

$\mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}] = \mathbb{E}[\hat{\mathbf{H}}_k^H (\hat{\mathbf{H}}_k + \tilde{\mathbf{H}}_k)] = \mathbb{E}[\|\hat{\mathbf{H}}_k\|^2] = N_t \gamma_k.$

The numerator of the UatF SINR is ${P_t}_{k} (N_t \gamma_k)^2$ .

Step 2: Compute the interference terms

For $j \neq k$ (no pilot contamination): $\hat{\mathbf{H}}_k$ and $\mathbf{H}_{j}$ are independent, so

$\mathbb{E}[|\hat{\mathbf{H}}_k^H \mathbf{H}_{j}|^2] = \mathbb{E}[\hat{\mathbf{H}}_k^H \mathbf{H}_{j} \mathbf{H}_{j}^{H} \hat{\mathbf{H}}_k] = \text{tr}(\mathbb{E}[\hat{\mathbf{H}}_k \hat{\mathbf{H}}_k^H] \, \mathbb{E}[\mathbf{H}_{j} \mathbf{H}_{j}^{H}]) = N_t \gamma_k \beta_{j}.$

Step 3: Compute the desired user's variance

$\mathbb{E}[|\hat{\mathbf{H}}_k^H \mathbf{H}_{k}|^2] = (N_t \gamma_k)^2 + N_t \gamma_k \beta_{k}.$ $The first term is$ |\mathbb{E}[\hat{\mathbf{H}}k^H \mathbf{H}{k}]|^2 $(already in the numerator). The remainder is$ N_t \gamma_k \beta_{k}$.

Step 4: Compute the noise term

$\sigma^2 \, \mathbb{E}[\|\hat{\mathbf{H}}_k\|^2] = \sigma^2 \, N_t \gamma_k.$ $

Step 5: Assemble the SINR

The denominator is:

$\sum_{j=1}^{K} {P_t}_{j} \, N_t \gamma_k \beta_{j} + \sigma^2 \, N_t \gamma_k - {P_t}_{k} (N_t \gamma_k)^2.$

Wait — we must subtract the signal power. The denominator of the UatF SINR is (from Definition DUatF Effective SINR):

$\sum_j {P_t}_{j} \, N_t \gamma_k \beta_{j} + \sigma^2 \, N_t \gamma_k.$

Dividing numerator and denominator by $N_t \gamma_k$ yields

$\text{SINR}_k^{\text{MRC}} = \frac{N_t \, {P_t}_{k} \, \gamma_k}{\sum_{j=1}^{K} {P_t}_{j} \, \beta_{j} + \sigma^2}. \quad \blacksquare$

,

MRC Is Interference-Limited

A crucial observation: as $N_t \to \infty$ , the MRC rate does not grow without bound. Instead,

$R_k^{\text{MRC}} \to \log_2\!\left(1 + \frac{N_t \, {P_t}_{k} \, \gamma_k}{\sum_j {P_t}_{j} \beta_{j} + \sigma^2}\right) \to \infty,$

so the per-user rate does grow with $N_t$ — but the denominator is constant. The sum rate $\sum_k R_k \sim K \log_2(N_t)$ grows only logarithmically. Moreover, if we try to increase $K$ proportionally to $N_t$ , the interference in the denominator grows and the per-user rate saturates. This is the fundamental limitation of MRC: it does not manage interference.

Definition:
Sum Spectral Efficiency

The sum spectral efficiency (sum rate) is

$S = \sum_{k=1}^{K} R_k \quad \text{[bits/s/Hz]}.$

For MRC with equal power ${P_t}_{k} = P_t$ and equal path loss $\beta_{k} = \beta$ for all users:

$S^{\text{MRC}} = K \log_2\!\left(1 + \frac{N_t \, P_t \, \gamma}{K \, P_t \, \beta + \sigma^2}\right).$

Example: Sum Rate Scaling with MRC

With equal power and equal path loss, show that the MRC sum rate scales as $O(K \log_2 N_t)$ when $K$ is fixed, and find the optimal number of users $K^*$ that maximizes the sum rate for a given $N_t$ .

Solution

Fixed K scaling

For fixed $K$ , as $N_t \to \infty$ :

$S^{\text{MRC}} \approx K \log_2\!\left(\frac{N_t \, P_t \, \gamma}{K \, P_t \, \beta + \sigma^2}\right) = K \log_2(N_t) + \text{const},$

confirming $O(K \log_2 N_t)$ growth.

Optimal K

Taking the derivative of $S$ with respect to $K$ (treating it as continuous) and setting it to zero:

$\frac{dS}{dK} = \log_2\!\left(1 + \frac{N_t \gamma}{K \beta + \sigma^2/P_t}\right) - \frac{K \beta \, N_t \gamma}{\ln 2 \left(K \beta + \sigma^2/P_t\right)\left(K \beta + \sigma^2/P_t + N_t \gamma\right)} = 0.$

This does not admit a simple closed form, but numerical evaluation shows that the optimal $K^*$ grows roughly as $N_t / c$ for some constant $c$ depending on the SNR. The key insight: the sum rate grows linearly in $N_t$ (i.e., $O(N_t)$ ) when $K$ is optimized.

MRC SINR and Rate vs. Number of Antennas

Explore how the per-user SINR and achievable rate scale with the number of base station antennas $N_t$ . Adjust the number of users, transmit SNR, and path loss to see the interference-limited behavior.

Parameters

K

(users)10

Number of single-antenna users

\text{SNR}

[dB]10

Transmit SNR per user (dB)

\beta

[dB]-10

Path loss coefficient (dB)

\tau_p / \tau_c

0.1

Fraction of coherence interval used for pilots

Common Mistake: MRC (Uplink) vs. MRT (Downlink)

Mistake:

Confusing MRC with MRT (Maximum Ratio Transmission). Some texts use "MRC" for both uplink and downlink, while others distinguish the two.

Correction:

MRC is the uplink (receive) combining scheme: $\hat{x}_k = \hat{\mathbf{H}}_k^H \mathbf{y}$ . It maximizes the receive SNR.

MRT is the downlink (transmit) precoding scheme: $\mathbf{x} = \sum_k \hat{\mathbf{H}}_k s_k / \|\hat{\mathbf{H}}_k\|$ . It maximizes the received power at each user.

The rate expressions are analogous due to uplink-downlink duality in TDD systems, but the SINR formulas differ because interference statistics differ. This chapter focuses on the uplink; Chapter 6 covers downlink precoding.

Common Mistake: Estimation Quality vs. Path Loss

Mistake:

Using $\beta_{k}$ (path loss) where $\gamma_k$ (estimation quality) is needed, or vice versa. In the MRC SINR expression, the numerator contains $\gamma_k$ (estimation quality), not $\beta_{k}$ .

Correction:

The estimation quality $\gamma_k$ incorporates both the path loss $\beta_{k}$ and the pilot training parameters:

$\gamma_k = \frac{\beta_{k}^{2} \tau_p {P_t}_{p}}{\beta_{k} \tau_p {P_t}_{p} + \sigma^2}.$

In the high-SNR pilot regime ( $\beta_{k} \tau_p {P_t}_{p} \gg \sigma^2$ ), $\gamma_k \approx \beta_{k}$ and the distinction becomes minor. But at low pilot SNR, $\gamma_k \ll \beta_{k}$ and the difference matters.

Quick Check

In the MRC SINR expression, does the denominator depend on $N_t$ ?

Yes, it grows linearly with $N_t$

No, it is independent of $N_t$

It depends on $N_t$ through the estimation quality $\gamma_k$

Correction:

No, it is independent of

N_t

Correct. This is why the SINR grows linearly with $N_t$ — only the numerator scales with the array size.

🔧Engineering Note

Computational Complexity of MRC

MRC requires only $N_t$ complex multiply-accumulate (MAC) operations per user per symbol: the inner product $\hat{\mathbf{H}}_k^H \mathbf{y}$ . For $K$ users, the total is $N_t K$ MACs. This scales linearly in both dimensions, making MRC attractive for real-time implementation in massive MIMO base stations.

By contrast, ZF and MMSE require a matrix inversion of size $K \times K$ , adding $O(K^{3})$ operations per coherence interval. For systems with $K \leq 20$ , this overhead is manageable; for $K \geq 100$ , it becomes a bottleneck.

Practical Constraints

•
Real-time processing at subframe level (~1 ms in 5G NR)
•
Per-antenna processing must be parallelizable
•
Memory bandwidth for storing all channel estimates

Key Takeaway

MRC achieves a per-user rate that grows as $\log_2(N_t)$ and a sum rate that scales as $O(N_t)$ when the number of users is optimized. Its simplicity (one inner product per user) makes it the default choice for cell-free massive MIMO and distributed architectures. The price: MRC is interference-limited and performs poorly when $K$ is comparable to $N_t$ .

Achievable Rate with MRC

Maximum Ratio Combining: The Simplest Massive MIMO Receiver

Definition: MRC Combining Vector

Theorem: Closed-Form UatF Rate with MRC

Step 1: Compute the signal power (numerator)

Step 2: Compute the interference terms

Step 3: Compute the desired user's variance

Step 4: Compute the noise term

Step 5: Assemble the SINR

MRC Is Interference-Limited

Definition: Sum Spectral Efficiency

Example: Sum Rate Scaling with MRC

Fixed K scaling

Optimal K

MRC SINR and Rate vs. Number of Antennas

Parameters

Common Mistake: MRC (Uplink) vs. MRT (Downlink)

Common Mistake: Estimation Quality vs. Path Loss

Quick Check

Computational Complexity of MRC

Key Takeaway

Definition:
MRC Combining Vector

Definition:
Sum Spectral Efficiency