Ferkans — Interactive Telecom Tutor

Practical Alternatives to Dirty Paper Coding

DPC achieves the BC capacity but is too complex for real systems. Linear precoding offers a practical middle ground: the base station multiplies each user's data symbol by a beamforming vector, and users decode their signals treating residual interference as noise. The transmitted signal is simply $\mathbf{x} = \sum_{k=1}^{K}\mathbf{w}_k s_k$ , where $\mathbf{w}_k$ is the precoding vector and $s_k$ is the data symbol for user $k$ . This section develops the two most important linear precoders — zero-forcing (ZF) and regularised zero-forcing (MMSE) — and quantifies their performance relative to DPC.

Definition:
Zero-Forcing Beamforming (Multi-User)

Zero-forcing (ZF) beamforming designs the precoding matrix to completely eliminate inter-user interference. For the $K$ -user MISO BC with channel matrix $\mathbf{H} = [\mathbf{h}_1, \ldots, \mathbf{h}_K]^H \in \mathbb{C}^{K \times N_t}$ (requires $N_t \geq K$ ):

$\mathbf{W}_{\text{ZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1} \boldsymbol{\Gamma}$

where $\boldsymbol{\Gamma} = \text{diag}(\sqrt{p_1}, \ldots, \sqrt{p_K})$ allocates power across users. The un-normalised ZF precoder $\tilde{\mathbf{W}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ satisfies $\mathbf{H}\tilde{\mathbf{W}} = \mathbf{I}_K$ , ensuring zero inter-user interference:

$y_k = \mathbf{h}_k^H\mathbf{x} + n_k = \sqrt{p_k}\,s_k + n_k$

The per-user rate is $R_k = \log_2(1 + p_k/\sigma^2)$ . The power constraint is $\sum_k p_k \|\tilde{\mathbf{w}}_k\|^2 \leq P$ .

The ZF power penalty is captured by $\|\tilde{\mathbf{w}}_k\|^2 = [(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}$ , which grows as users' channels become more aligned. When $K = N_t$ , the penalty can be severe; when $N_t \gg K$ , it vanishes and ZF approaches the performance of DPC.

Definition:
Regularised Zero-Forcing (MMSE) Precoding

Regularised ZF (RZF) precoding, also called MMSE precoding, adds a regularisation term to balance interference suppression against noise enhancement:

$\mathbf{W}_{\text{RZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H} + \alpha\mathbf{I}_K)^{-1}\boldsymbol{\Gamma}$

where $\alpha > 0$ is the regularisation parameter. The MMSE-optimal choice is:

$\alpha^* = \frac{K\sigma^2}{P}$

At high SNR ( $\alpha \to 0$ ), RZF converges to ZF. At low SNR ( $\alpha \to \infty$ ), it converges to matched-filter (MRT) beamforming $\mathbf{w}_k \propto \mathbf{h}_k$ .

Unlike ZF, RZF does not completely eliminate interference, but the residual interference is optimally traded against reduced noise amplification.

RZF consistently outperforms ZF at low-to-moderate SNR and is never worse at any SNR. In the massive MIMO regime ( $N_t \to \infty$ with $K/N_t$ fixed), both ZF and RZF achieve the same asymptotic performance due to channel hardening.

,

Zero-Forcing Beamforming Patterns

Visualise the beamforming patterns and per-user rates for ZF precoding as a function of user angles. Observe how the beamforming vectors adapt to steer nulls toward interfering users and how the sum rate degrades as users become closely spaced (nearly aligned channels).

Parameters

N_t

(BS antennas)4

User 1 angle

\theta_1

(degrees)30

User 2 angle

\theta_2

(degrees)-30

Theorem: ZF Sum Rate and Gap to DPC

For the $K$ -user MISO BC with i.i.d. Rayleigh fading channels and $N_t$ base-station antennas:

ZF sum rate: With equal power allocation $p_k = P/K$ :

$R_{\text{sum}}^{\text{ZF}} = \sum_{k=1}^{K} \mathbb{E}\!\left[\log_2\!\left(1 + \frac{P}{K\sigma^2} \cdot \frac{1}{[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}}\right)\right]$

The effective per-user SNR $1/[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}$ is chi-squared with $2(N_t - K + 1)$ degrees of freedom.

High-SNR gap to DPC: At high SNR:

$R_{\text{sum}}^{\text{DPC}} - R_{\text{sum}}^{\text{ZF}} = K \cdot \mathbb{E}\!\left[\log_2\!\left( [(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk} \cdot \|\mathbf{h}_k\|^2\right)\right]$

This gap is a constant (independent of SNR) and vanishes as $N_t/K \to \infty$ .

Multiplexing gain: Both ZF and DPC achieve the same multiplexing gain $\min(K, N_t)$ at high SNR.

At high SNR, the dominant effect is multiplexing gain (how many streams are sent), not power efficiency. Since ZF transmits the same $K$ streams as DPC, both achieve the same DoF. The constant gap arises from ZF's power penalty for nulling interference, which becomes negligible when $N_t \gg K$ (the base station has many more antennas than users).

Proof

Per-user ZF SNR

With ZF precoding, user $k$ sees no interference:

$\text{SNR}_{k}^{\text{ZF}} = \frac{p_k}{\sigma^2 \|\tilde{\mathbf{w}}_k\|^2} = \frac{p_k}{\sigma^2 [(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}}$

For Rayleigh fading, $1/[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk} \sim \chi^2_{2(N_t - K + 1)}$ , the same distribution as the ZF receiver SNR in the uplink (Section 16.3).

DPC per-user rate

With optimal DPC and power $p_k$ :

$R_k^{\text{DPC}} = \log_2\!\left(1 + \frac{p_k\|\mathbf{h}_k\|^2}{\sigma^2}\right)$

where the effective channel gain is $\|\mathbf{h}_k\|^2$ (no power penalty from interference pre-cancellation).

Gap computation at high SNR

At high SNR, $\log_2(1 + x) \approx \log_2(x)$ , so:

$R_k^{\text{DPC}} - R_k^{\text{ZF}} \approx \log_2\!\left(\frac{p_k\|\mathbf{h}_k\|^2} {p_k / [(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}}\right) = \log_2\!\left(\|\mathbf{h}_k\|^2 [(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}\right)$

Summing over $k$ and taking expectations yields the stated constant gap, which depends only on $N_t$ and $K$ (not SNR). As $N_t/K \to \infty$ , $[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk} \to 1/\|\mathbf{h}_k\|^2$ and the gap vanishes. $\blacksquare$

Example: ZF Precoder Design for a 3-Antenna 2-User System

A BS with $N_t = 3$ antennas serves $K = 2$ users. The channels are:

$\mathbf{h}_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{h}_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$

With total power $P = 20$ and $\sigma^2 = 1$ : (a) Design the ZF precoder. (b) Compute the sum rate with equal power allocation. (c) Compare with the MRT sum rate (treating interference as noise).

Solution

ZF precoder computation

$\mathbf{H} = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}KATEXPLACEHOLDER0END\mathbf{H}\mathbf{H}^{H} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}, \quad (\mathbf{H}\mathbf{H}^{H})^{-1} = \frac{1}{3} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}KATEXPLACEHOLDER1END\tilde{\mathbf{W}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1} = \frac{1}{3}\begin{bmatrix} 1 & 1 \\ 2 & -1 \\ -1 & 2 \end{bmatrix}$ $

ZF sum rate

Power penalty per user: $\|\tilde{\mathbf{w}}_1\|^2 = (1+4+1)/9 = 6/9 = 2/3$ , $\|\tilde{\mathbf{w}}_2\|^2 = (1+1+4)/9 = 6/9 = 2/3$ .

With equal power $p_1 = p_2 = P/2 = 10$ :

Effective power: $\sum_k p_k\|\tilde{\mathbf{w}}_k\|^2 = 10 \cdot 2/3 + 10 \cdot 2/3 = 40/3 \leq 20$ . Good.

Per-user SNR: $\text{SNR}_{k} = p_k/(\sigma^2\|\tilde{\mathbf{w}}_k\|^2) = 10/(2/3) = 15$

$R_{\text{sum}}^{\text{ZF}} = 2\log_2(1+15) = 2\log_2(16) = 8.0\;\text{bits/s/Hz}$

MRT comparison

MRT beamforming: $\mathbf{w}_k^{\text{MRT}} = \mathbf{h}_k/\|\mathbf{h}_k\|$ .

$\mathbf{w}_1^{\text{MRT}} = [1,1,0]^T/\sqrt{2}$ , $\mathbf{w}_2^{\text{MRT}} = [1,0,1]^T/\sqrt{2}$ .

User 1 signal: $|\mathbf{h}_1^H\mathbf{w}_1^{\text{MRT}}|^2 = (2/\sqrt{2})^2 = 2$

User 1 interference: $|\mathbf{h}_1^H\mathbf{w}_2^{\text{MRT}}|^2 = (1/\sqrt{2})^2 = 1/2$

User 1 SINR: $\text{SINR}_1 = \frac{10 \times 2}{1 + 10 \times 1/2} = 20/6 = 3.33$

By symmetry, $\text{SINR}_2 = 3.33$ .

$R_{\text{sum}}^{\text{MRT}} = 2\log_2(1+3.33) = 2\log_2(4.33) \approx 4.23\;\text{bits/s/Hz}$

ZF provides a $8.0 - 4.23 = 3.77$ bits/s/Hz improvement by eliminating inter-user interference. $\blacksquare$

Quick Check

For a heavily loaded system ( $K = N_t$ ) at moderate SNR, which linear precoding scheme generally achieves the highest sum rate?

ZF precoding

MRT (matched filter) precoding

Regularised ZF (MMSE) precoding

Random beamforming

Correction:

Regularised ZF (MMSE) precoding

RZF optimally balances interference suppression and noise amplification through the regularisation parameter $\alpha = K\sigma^2/P$ . It outperforms both ZF (too much noise amplification) and MRT (too much interference) at moderate SNR.

Why This Matters: Multi-User MIMO Precoding in 5G NR and Wi-Fi

Multi-user MIMO (MU-MIMO) precoding is a cornerstone of modern wireless standards. 5G NR supports up to 12 co-scheduled users with Type I/II CSI codebooks enabling ZF or RZF precoding at the gNB. The 5G NR specification allows both codebook-based (limited feedback) and non-codebook-based (reciprocity-based) precoding, with the latter preferred in TDD deployments where uplink channel estimates directly provide CSIT. Wi-Fi 6/7 (802.11ax/be) employs MU-MIMO in the downlink with explicit compressed beamforming feedback, supporting up to 8 spatial streams across multiple users. In both systems, RZF-type precoding is the dominant implementation choice due to its robustness and low complexity.

⚠️Engineering Note

CSI Acquisition Overhead in Multi-User MIMO

All multi-user MIMO precoding schemes require channel state information at the transmitter (CSIT). The overhead of acquiring CSIT is a critical practical constraint:

TDD (reciprocity-based): Users send uplink pilots; the BS estimates channels directly. Overhead scales as $O(K)$ pilot symbols per coherence interval. 5G NR TDD uses SRS (sounding reference signal) with up to 4 antenna ports per user.
FDD (feedback-based): Each user estimates its channel from downlink pilots, quantises it, and feeds back. Overhead scales as $O(KN_t)$ — each user must quantise an $N_t$ -dimensional vector. The feedback rate must grow as $B \geq (N_t-1)\text{SNR}_{\text{dB}}/3$ bits per user to maintain the ZF multiplexing gain.
Imperfect CSIT impact: With CSIT estimation error variance $\sigma_e^2$ , the ZF SINR degrades to $\text{SINR}_k \approx \frac{P/K}{(K-1)P\sigma_e^2 + \sigma^2}$ , creating an interference floor that limits the high-SNR gain.

In 5G NR, Type II CSI codebooks provide $\sim 10$ - $15$ dB of feedback gain over Type I by exploiting wideband/subband decomposition, but still fall $2$ - $4$ dB short of perfect CSIT performance for $N_t \geq 16$ .

Practical Constraints

•
FDD feedback rate must scale linearly with SNR (dB) to maintain DoF
•
TDD pilot overhead grows with K, limiting the number of co-scheduled users
•
CSI aging (Doppler) at 30 km/h and 3.5 GHz causes ~1 dB loss per ms delay

📋 Ref: 3GPP TS 38.214 v17, §5.2.2 — CSI reporting

⚠️Engineering Note

Numerical Conditioning of ZF Precoding

The ZF precoder $\mathbf{W} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ requires inverting the $K \times K$ Gram matrix $\mathbf{H}\mathbf{H}^{H}$ . Practical implementation considerations:

Condition number: When users' channels are nearly aligned, $\kappa(\mathbf{H}\mathbf{H}^{H}) \gg 1$ and the inversion amplifies numerical errors. For $K = N_t$ with Rayleigh fading, the median condition number is $\sim (N_t/2)^2$ , making inversion ill-conditioned for $N_t \geq 8$ .
Fixed-point arithmetic: In ASIC implementations (typical for base station baseband), 16-bit fixed-point precision suffices for $N_t \leq 8$ but requires 24-bit for $N_t = 16$ - $32$ .
Regularisation as stabiliser: RZF with $\alpha = K\sigma^2/P$ improves the condition number to $\kappa((\mathbf{H}\mathbf{H}^{H} + \alpha\mathbf{I})) \leq \kappa(\mathbf{H}\mathbf{H}^{H}) \cdot (\lambda_{\max} + \alpha)/(\lambda_{\min} + \alpha)$ , which is significantly smaller. This is another reason to prefer RZF over ZF in practice.
Computation: Direct inversion costs $O(K^3)$ . For massive MIMO ( $N_t = 64$ - $256$ , $K = 8$ - $16$ ), the matrix is small ( $K \times K$ ) and inversion is fast. The bottleneck is the $O(K N_t^2)$ multiplication $\mathbf{H}^{H}(\cdots)^{-1}$ .

Practical Constraints

•
Condition number of HH^H grows as (N_t/2)^2 for square Rayleigh channels
•
Fixed-point wordlength: 16 bits for N_t ≤ 8, 24 bits for N_t ≤ 32
•
Regularisation reduces condition number by factor of ~(λ_max+α)/(λ_min+α)

Key Takeaway

Linear precoding (ZF/RZF) achieves the same multiplexing gain as DPC at a fraction of the complexity. The performance gap is a constant number of bits/s/Hz that depends on $K/N_t$ but not on SNR. When $N_t \gg K$ (massive MIMO regime), this gap vanishes entirely — simple linear precoding becomes near-optimal, and the impractical DPC is no longer needed.

Why This Matters: Advanced MU-MIMO in the MIMO Book

The linear precoding and WMMSE techniques of this chapter extend in the MIMO book (Book MIMO) to far more complex scenarios: massive MIMO with hundreds of antennas (Ch. 3-5) where ZF/RZF become near-optimal, Joint Spatial Division and Multiplexing (JSDM, Ch. 10) exploiting channel covariance structure for FDD massive MIMO, multi-cell coordinated precoding with limited backhaul (Ch. 12), and cell-free distributed MIMO (Ch. 14) where precoding is distributed across geographically separated access points. The ITA book (Book ITA) provides the information-theoretic foundations of the MAC and BC capacity regions in greater depth.

Zero-Forcing Beamforming

A linear precoding strategy that completely eliminates inter-user interference by projecting each user's signal into the null space of all other users' channels: $\mathbf{W}_{\text{ZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ .

Regularised Zero-Forcing (RZF) Precoding

A linear precoding strategy that adds a regularisation term $\alpha\mathbf{I}$ to the channel Gram matrix to balance interference suppression against noise amplification: $\mathbf{W}_{\text{RZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H} + \alpha\mathbf{I})^{-1}$ . Also known as MMSE precoding.

Linear Precoding for Multi-User MIMO