Ferkans — Interactive Telecom Tutor

Why Linear Precoding?

In the MU-MIMO downlink, the base station with $N_t$ antennas serves $K$ single-antenna users simultaneously. The transmitted signal is

$\mathbf{x} = \sum_{k=1}^{K} \mathbf{v}_{k} s_k$

where $\mathbf{v}_{k} \in \mathbb{C}^{N_t}$ is the precoding vector for user $k$ and $s_k$ is the data symbol. The design question is: how should we choose $\mathbf{v}_{1}, \ldots, \mathbf{v}_{\ntn{nusers}}$ ? Linear precoding answers this with a closed-form matrix operation on the channel state information, avoiding the combinatorial complexity of nonlinear schemes like DPC. The price is a gap to the broadcast channel capacity, but the simplicity gain is enormous.

Definition:
MU-MIMO Downlink System Model

Consider a base station with $N_t$ antennas serving $K \leq N_t$ single-antenna users. The channel from the BS to user $k$ is $\mathbf{h}_k \in \mathbb{C}^{N_t}$ , and the aggregate channel matrix is

$\mathbf{H} = [\mathbf{h}_1, \mathbf{h}_2, \ldots, \mathbf{h}_{K}]^H \in \mathbb{C}^{K \times N_t}.$

The received signal at user $k$ is

$y_k = \mathbf{h}_k^H \sum_{j=1}^{K} \sqrt{p_j}\,\mathbf{v}_{j} s_j + w_k$

where $p_j \geq 0$ is the power allocated to user $j$ , $\mathbf{v}_{j}$ is the unit-norm precoding vector ( $\|\mathbf{v}_{j}\| = 1$ ), $s_j \sim \mathcal{CN}(0, 1)$ is the data symbol, and $w_k \sim \mathcal{CN}(0, \sigma^2)$ is AWGN. The total power constraint is $\sum_{k=1}^{K} p_k \leq P_t$ .

Definition:
SINR for Linear Precoding

Under linear precoding with unit-norm vectors $\mathbf{v}_{k}$ , the SINR at user $k$ is

$\text{SINR}_k = \frac{p_k |\mathbf{h}_k^H \mathbf{v}_{k}|^2}{\sum_{j \neq k} p_j |\mathbf{h}_k^H \mathbf{v}_{j}|^2 + \sigma^2}$

and the achievable rate is $R_k = \log_2(1 + \text{SINR}_k)$ bits/s/Hz.

The numerator captures useful signal power; the denominator contains multi-user interference (MUI) plus noise. The three precoding strategies in this chapter differ in how they balance these two terms.

Definition:
Maximum Ratio Transmission (MRT)

MRT precoding (also called conjugate beamforming or matched-filter precoding) sets the precoding vector to be the normalised channel conjugate:

$\mathbf{v}_{k}^{\text{MRT}} = \frac{\mathbf{h}_k}{\|\mathbf{h}_k\|}$

The MRT precoding matrix is

$\mathbf{W}^{\text{MRT}} = \mathbf{H}^{H} \mathbf{D}^{-1}$

where $\mathbf{D} = \text{diag}(\|\mathbf{h}_1\|, \ldots, \|\mathbf{h}_{K}\|)$ ensures unit-norm columns.

MRT is the spatial analogue of the matched filter in detection theory: it maximises the inner product $|\mathbf{h}_k^H \mathbf{v}_{k}|$ by aligning the precoder with the channel direction. It ignores interference entirely.

,

Theorem: MRT Maximises Received SNR

For a single user ( $K = 1$ ) with channel $\mathbf{h} \in \mathbb{C}^{N_t}$ , the precoding vector that maximises the received SNR

$\text{SNR} = \frac{P_t |\mathbf{h}^H \mathbf{v}|^2}{\sigma^2}$

subject to $\|\mathbf{v}\| = 1$ , is $\mathbf{v}^{\star} = \mathbf{h}/\|\mathbf{h}\|$ . The maximum SNR is

$\text{SNR}^{\star} = \frac{P_t \|\mathbf{h}\|^2}{\sigma^2}.$

The Cauchy--Schwarz inequality tells us that the inner product $|\mathbf{h}^H \mathbf{v}|$ is maximised when $\mathbf{v}$ points in the same direction as $\mathbf{h}$ . This is exactly the matched filter principle: coherently combine the signals from all $N_t$ antennas.

Proof

Apply Cauchy--Schwarz

By the Cauchy--Schwarz inequality:

$|\mathbf{h}^H \mathbf{v}|^2 \leq \|\mathbf{h}\|^2 \|\mathbf{v}\|^2 = \|\mathbf{h}\|^2$

with equality if and only if $\mathbf{v} = e^{j\theta} \mathbf{h}/\|\mathbf{h}\|$ for some phase $\theta$ .

Conclude

Substituting $\mathbf{v}^{\star} = \mathbf{h}/\|\mathbf{h}\|$ into the SNR expression:

$\text{SNR}^{\star} = \frac{P_t \|\mathbf{h}\|^2}{\sigma^2}$

which equals $P_t/\sigma^2$ times the channel power gain $\|\mathbf{h}\|^2$ . With i.i.d. Rayleigh fading, $\|\mathbf{h}\|^2 \sim \chi^2_{2N_t}$ , so the array gain scales linearly in $N_t$ . $\blacksquare$

Theorem: MRT SINR in the Multi-User Regime

With MRT precoding and equal power allocation $p_k = P_t/K$ , the SINR at user $k$ is

$\text{SINR}_k^{\text{MRT}} = \frac{\frac{P_t}{K} \|\mathbf{h}_k\|^2}{\frac{P_t}{K} \sum_{j \neq k} \frac{|\mathbf{h}_k^H \mathbf{h}_j|^2}{\|\mathbf{h}_j\|^2} + \sigma^2}.$

In the massive MIMO regime ( $N_t \to \infty$ with $K$ fixed and i.i.d. Rayleigh fading), channel hardening gives $\|\mathbf{h}_k\|^2/N_t \to 1$ and favorable propagation gives $|\mathbf{h}_k^H \mathbf{h}_j|^2/(N_t \|\mathbf{h}_j\|^2) \to 0$ , so

$\text{SINR}_k^{\text{MRT}} \to \frac{P_t N_t}{K \sigma^2} \to \infty.$

MRT benefits from the array gain ( $N_t$ factor in the numerator) but pays the price of inter-user interference in the denominator. In the massive regime, favorable propagation kills the interference terms, and MRT becomes asymptotically optimal. For finite $N_t$ , the interference is the bottleneck: MRT is interference-limited.

Proof

Substitute MRT vectors

With $\mathbf{v}_{j}^{\text{MRT}} = \mathbf{h}_j/\|\mathbf{h}_j\|$ :

$|\mathbf{h}_k^H \mathbf{v}_{k}^{\text{MRT}}|^2 = \|\mathbf{h}_k\|^2, \quad |\mathbf{h}_k^H \mathbf{v}_{j}^{\text{MRT}}|^2 = \frac{|\mathbf{h}_k^H \mathbf{h}_j|^2}{\|\mathbf{h}_j\|^2}.$

Apply asymptotic results

By the law of large numbers with i.i.d. $\mathcal{CN}(0, \mathbf{I})$ entries:

$\frac{\|\mathbf{h}_k\|^2}{N_t} \xrightarrow{a.s.} 1, \qquad \frac{|\mathbf{h}_k^H \mathbf{h}_j|^2}{N_t} \xrightarrow{a.s.} 1 \quad (k \neq j).$

So the interference power scales as $O(1)$ while the signal power scales as $O(N_t)$ , giving SINR $\to \infty$ as $N_t \to \infty$ . $\blacksquare$

Example: MRT Precoding for Two Users

A BS with $N_t = 4$ antennas serves $K = 2$ users with channels

$\mathbf{h}_1 = [1, j, -1, -j]^T, \qquad \mathbf{h}_2 = [1, 1, 1, 1]^T.$

Compute the MRT precoding vectors, the resulting SINRs with equal power allocation and $P_t/\sigma^2 = 10$ dB, and the sum rate.

Solution

Normalise channels

$\|\mathbf{h}_1\| = 2$ , $\|\mathbf{h}_2\| = 2$ , so

$\mathbf{v}_{1}^{\text{MRT}} = \frac{1}{2}[1, j, -1, -j]^T, \qquad \mathbf{v}_{2}^{\text{MRT}} = \frac{1}{2}[1, 1, 1, 1]^T.$

Compute inner products

Signal: $|\mathbf{h}_1^H \mathbf{v}_{1}| = \|\mathbf{h}_1\| = 2$ , $|\mathbf{h}_2^H \mathbf{v}_{2}| = 2$ .

Interference: $\mathbf{h}_1^H \mathbf{v}_{2} = \frac{1}{2}(1 - j - 1 + j) = 0$ , $\mathbf{h}_2^H \mathbf{v}_{1} = \frac{1}{2}(1 + j - 1 - j) = 0$ .

Compute SINR and rates

With $p_k = P_t/2$ and $P_t/\sigma^2 = 10$ :

$\text{SINR}_1 = \frac{5 \times 4}{0 + 1} = 20, \qquad \text{SINR}_2 = 20.$

Sum rate $= 2 \log_2(1 + 20) = 8.72$ bits/s/Hz.

The zero interference occurs because $\mathbf{h}_1$ and $\mathbf{h}_2$ happen to be orthogonal. This is the favorable propagation condition; in general, with non-orthogonal channels, the interference would be nonzero.

MRT Beam Pattern

Visualise the spatial beam pattern $|(\mathbf{a}(\theta))^H \mathbf{v}_{k}^{\text{MRT}}|^2$ as a function of angle $\theta$ for a ULA with MRT precoding directed toward user $k$ . The beam always peaks at the user's direction but creates side lobes toward other users.

Parameters

N_t

16

Number of transmit antennas

\theta_{\text{user}}

(deg)30

User angle of arrival

K

1

Number of users (additional users at random angles)

Common Mistake: MRT Does Not Manage Interference

Mistake:

Assuming MRT is always a good choice because it maximises array gain.

Correction:

MRT maximises the signal power to the intended user but completely ignores the interference it causes to other users. When $K$ is comparable to $N_t$ and channels are not orthogonal, MRT becomes interference-limited: increasing $P_t$ does not improve the SINR because both signal and interference grow proportionally. Use ZF or RZF when $K/N_t$ is not small.

Quick Check

In the massive MIMO regime with i.i.d. Rayleigh fading and MRT precoding, how does the SINR scale with the number of antennas $N_t$ ?

$O(1)$ — it saturates

$O(N_t)$ — linear growth

$O(N_t^{2})$ — quadratic growth

$O(\log N_t)$ — logarithmic growth

Correction:

O(N_t)

— linear growth

The signal power grows as $N_t$ (array gain) while the interference from each user remains $O(1)$ due to favorable propagation. The net SINR scales as $O(N_t/K)$ .

Historical Note: From Radar to Cellular

1950s--1999

Maximum ratio transmission is the transmit-side dual of maximum ratio combining (MRC), which dates back to the 1950s work of Brennan on optimal diversity combining for radar and radio reception. The extension to the multi-user downlink was formalised by Lo (1999) and became the default precoding scheme in the early massive MIMO literature due to its simplicity: it requires only the channel conjugate and no matrix inversion.

Maximum Ratio Transmission (MRT)

A linear precoding strategy that sets each user's precoding vector to the normalised channel conjugate: $\mathbf{v}_{k} = \mathbf{h}_k/\|\mathbf{h}_k\|$ . Maximises SNR to the intended user but ignores multi-user interference.

Array Gain

The factor by which coherent combining across $N_t$ antennas increases the received SNR. With MRT, the array gain equals $\|\mathbf{h}_k\|^2$ , which concentrates around $N_t$ for i.i.d. Rayleigh channels.

Key Takeaway

MRT is the simplest linear precoder: align the beam with each user's channel. It delivers full array gain but is interference-limited when $K$ is not negligible relative to $N_t$ . In the massive regime ( $N_t \gg K$ ), favorable propagation makes MRT near-optimal.

Maximum Ratio Transmission (MRT)

Why Linear Precoding?

Definition: MU-MIMO Downlink System Model

Definition: SINR for Linear Precoding

Definition: Maximum Ratio Transmission (MRT)

Theorem: MRT Maximises Received SNR

Apply Cauchy--Schwarz

Conclude

Theorem: MRT SINR in the Multi-User Regime

Substitute MRT vectors

Apply asymptotic results

Example: MRT Precoding for Two Users

Normalise channels

Compute inner products

Compute SINR and rates

MRT Beam Pattern

Parameters

Common Mistake: MRT Does Not Manage Interference

Quick Check

Historical Note: From Radar to Cellular

Maximum Ratio Transmission (MRT)

Array Gain

Key Takeaway

Definition:
MU-MIMO Downlink System Model

Definition:
SINR for Linear Precoding

Definition:
Maximum Ratio Transmission (MRT)