Ferkans — Interactive Telecom Tutor

The Simplest Reasonable Receiver

Maximum ratio combining (MRC) — also called the matched filter — is the simplest linear receiver. It maximizes the received SNR for each user individually, ignoring the interference from all other users. This sounds hopelessly naive, and in conventional MIMO it would be. But in the massive MIMO regime where $N_t \gg K$ , the inter-user interference averages out, and MRC becomes a surprisingly effective strategy.

The point is that with many antennas, the channel vectors of different users become approximately orthogonal — favorable propagation — so ignoring interference costs little. This section quantifies exactly how little.

Definition:
MRC (Matched Filter) Receiver

The MRC receiver for user $k$ applies the combining vector

$\mathbf{g}_k^{\text{MRC}} = \mathbf{h}_k,$

i.e., the combining matrix is $\mathbf{G}^{\text{MRC}} = \mathbf{H}$ . The soft estimate of user $k$ 's symbol is

$\hat{x}_k^{\text{MRC}} = \mathbf{h}_k^H \mathbf{y} = \underbrace{\mathbf{h}_k^H \mathbf{h}_k}_{\text{desired}} x_k + \underbrace{\sum_{j \neq k} \mathbf{h}_k^H \mathbf{h}_j \, x_j}_{\text{interference}} + \underbrace{\mathbf{h}_k^H \mathbf{w}}_{\text{noise}}.$

MRC is also known as maximum ratio combining because it maximizes the output SNR when only user $k$ is transmitting (single-user bound). It is the receive-side dual of MRT (maximum ratio transmission) in the downlink (MIMO Ch. 6).

Theorem: MRC SINR Expression

Under the uplink model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ with equal power $P_k = P$ for all users, the post-detection SINR of user $k$ with MRC is

$\text{SINR}_k^{\text{MRC}} = \frac{P \|\mathbf{h}_k\|^4} {\sum_{j \neq k} P |\mathbf{h}_k^H \mathbf{h}_j|^2 + \sigma^2 \|\mathbf{h}_k\|^2}.$

The numerator is proportional to $\|\mathbf{h}_k\|^4$ (squared channel gain), the denominator has two terms: interference from other users (scaled by the squared inner products $|\mathbf{h}_k^H \mathbf{h}_j|^2$ ) and noise (scaled by $\|\mathbf{h}_k\|^2$ ).

Proof

Decompose the MRC output

The MRC output for user $k$ is $\hat{x}_k = \mathbf{h}_k^H \mathbf{y} = \mathbf{h}_k^H \mathbf{h}_k x_k + \sum_{j \neq k} \mathbf{h}_k^H \mathbf{h}_j x_j + \mathbf{h}_k^H \mathbf{w}$ .

Compute signal and interference powers

Since $\mathbb{E}[|x_k|^2] = P$ and the symbols are independent across users:

Desired signal power: $P |\mathbf{h}_k^H \mathbf{h}_k|^2 = P \|\mathbf{h}_k\|^4$
Interference from user $j$ : $P |\mathbf{h}_k^H \mathbf{h}_j|^2$
Noise power: $\mathbb{E}[|\mathbf{h}_k^H \mathbf{w}|^2] = \sigma^2 \|\mathbf{h}_k\|^2$

Form the SINR

Taking the ratio of signal power to the sum of interference and noise powers yields the stated expression. $\blacksquare$

Theorem: MRC in the Massive MIMO Limit

Consider i.i.d. Rayleigh fading: $\mathbf{h}_k \sim \mathcal{CN}(\mathbf{0}, \beta_k \mathbf{I})$ independently across users. As $N_t \to \infty$ with $K$ fixed, the per-user SINR with MRC satisfies

$\text{SINR}_k^{\text{MRC}} \xrightarrow{\text{a.s.}} \frac{P \beta_k}{\sigma^2 / N_t},$

i.e., the interference vanishes and the effective SNR scales linearly with $N_t$ .

This is the hallmark of massive MIMO. With infinitely many antennas, the channel vectors become orthogonal ( $\mathbf{h}_k^H \mathbf{h}_j / N_t \to 0$ for $k \neq j$ ), so interference disappears. Each user sees a single-user channel with SNR proportional to $N_t$ . Equivalently, we can reduce the transmit power as $P = E/N_t$ and maintain a finite rate — the $1/N_t$ power scaling law.

Proof

Apply the law of large numbers

For i.i.d. Rayleigh fading, $\frac{1}{N_t} \|\mathbf{h}_k\|^2 \xrightarrow{\text{a.s.}} \beta_k$ and $\frac{1}{N_t} \mathbf{h}_k^H \mathbf{h}_j \xrightarrow{\text{a.s.}} 0$ for $k \neq j$ .

Normalize the SINR

Dividing numerator and denominator by $N_t^{2}$ :

$\text{SINR}_k^{\text{MRC}} = \frac{P \left(\frac{1}{N_t}\|\mathbf{h}_k\|^2\right)^2} {\sum_{j \neq k} P \left|\frac{1}{N_t} \mathbf{h}_k^H \mathbf{h}_j\right|^2 + \frac{\sigma^2}{N_t} \cdot \frac{1}{N_t}\|\mathbf{h}_k\|^2}.$

Take the limit

As $N_t \to \infty$ , the interference terms in the denominator vanish (each is $\mathcal{O}(1/N_t)$ ), yielding

$\text{SINR}_k^{\text{MRC}} \to \frac{P \beta_k^2}{\frac{\sigma^2}{N_t} \beta_k} = \frac{P N_t \beta_k}{\sigma^2}.$

This is the single-user SNR with an $N_t$ -fold array gain. $\blacksquare$

Key Takeaway

MRC achieves linear SINR scaling. In the massive MIMO limit, the per-user SINR with MRC grows as $N_t P \beta_k / \sigma^2$ . This means the transmit power can be reduced as $P \propto 1/N_t$ with no loss in rate — the fundamental energy efficiency promise of massive MIMO.

Example: MRC SINR for a 2-User System

Consider a BS with $N_t = 64$ antennas serving $K = 2$ users with i.i.d. Rayleigh fading, $\beta_1 = 1$ , $\beta_2 = 0.5$ , $P = 10$ dBm, and $\sigma^2 = -94$ dBm (over 10 MHz bandwidth). Compute the expected MRC SINR for user 1 in the massive MIMO approximation.

Solution

Convert to linear scale

$P = 10^{10/10} \times 10^{-3} = 0.01$ W, $\sigma^2 = 10^{-94/10} \times 10^{-3} \approx 3.98 \times 10^{-13}$ W.

Apply the massive MIMO SINR formula

$\text{SINR}_1^{\text{MRC}} \approx \frac{P N_t \beta_1}{\sigma^2} = \frac{0.01 \times 64 \times 1}{3.98 \times 10^{-13}} \approx 1.61 \times 10^{12}.$ $

Convert to dB and compute rate

$\text{SINR}_1 \approx 122$ dB. Of course this is the interference-free limit with perfect CSI. In practice, pilot contamination, imperfect channel estimates, and finite $N_t$ reduce this significantly. The rate per user is $R_1 = \log_2(1 + \text{SINR}_1) \approx 40.5$ bits/s/Hz — well above practical modulation orders, confirming that the massive MIMO operating point is not SNR-limited but interference- and overhead-limited.

MRC SINR vs. Number of BS Antennas $N_t$

Explore how the per-user SINR with MRC scales with the number of BS antennas. Observe that as $N_t$ grows, the SINR increases linearly (in dB scale, a constant slope) and the gap between the finite- $N_t$ SINR and the asymptotic limit shrinks.

Parameters

K

(users)8

Number of single-antenna users

P/\sigma^2

(dB)10

Per-user transmit SNR

Common Mistake: MRC Does Not Suppress Interference

Mistake:

A common misconception is that MRC "cancels" inter-user interference. It does not — MRC is purely a single-user matched filter that maximizes the desired signal power while completely ignoring interference.

Correction:

Interference suppression in MRC happens passively through favorable propagation (channel orthogonality), not actively through the receiver design. When channels are correlated (e.g., users at similar angles), MRC suffers severe interference and ZF or MMSE should be used instead.

Historical Note: The Matched Filter: From Radar to MIMO

1943–2010

The matched filter dates back to radar signal processing in the 1940s, where it was shown to maximize the output SNR for detecting a known waveform in Gaussian noise (North, 1943; Turin, 1960). In the MIMO context, the per-user matched filter $\mathbf{g}_k = \mathbf{h}_k$ is the spatial analog: it coherently combines the signals from all $N_t$ antennas to maximize the SNR for user $k$ . The massive MIMO revolution showed that this simplest of all receivers becomes near-optimal when the number of antennas is large enough — a result that would have surprised the radar engineers of the 1940s, who always operated with a single receiver.

MRC (Maximum Ratio Combining)

A linear detection technique that uses $\mathbf{g}_k = \mathbf{h}_k$ as the combining vector for user $k$ . Maximizes the single-user SNR but ignores multi-user interference.

Quick Check

What is the per-symbol-vector computational complexity of MRC detection (applying $\hat{\mathbf{x}} = \mathbf{H}^{H} \mathbf{y}$ )?

$\mathcal{O}(N_t)$

$\mathcal{O}(N_t K)$

$\mathcal{O}(K^{3})$

$\mathcal{O}(N_t^{2} K)$