Ferkans — Interactive Telecom Tutor

The Downlink Fronthaul Challenge

In the cell-free downlink, the CPU must convey each user's data and precoding instructions to the APs via the fronthaul. The CPU computes the precoded signals and compresses them for transmission to the APs, which reconstruct and transmit them over the air. This section develops the theory of compression-based precoding, where the precoded signals are jointly quantized to meet the fronthaul constraints.

Definition:
Downlink Signal Model with Fronthaul

The CPU computes the precoded signal for AP $l$ : $\mathbf{x}_l = \sum_{k=1}^{K} \mathbf{v}_{lk} s_k$ where $\mathbf{v}_{lk} \in \mathbb{C}^{N_t}$ is the precoding vector for user $k$ at AP $l$ and $s_k$ is user $k$ 's data symbol.

The CPU quantizes $\mathbf{x}_l$ to produce $\hat{\mathbf{x}}_l$ and forwards it to AP $l$ . AP $l$ transmits $\hat{\mathbf{x}}_l$ , so user $k$ receives: $y_k = \sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \hat{\mathbf{x}}_l + w_k = \underbrace{\sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \mathbf{x}_l}_{\text{intended signal}} + \underbrace{\sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \mathbf{e}_l}_{\text{compression noise}} + w_k$ where $\mathbf{e}_l = \hat{\mathbf{x}}_l - \mathbf{x}_l$ is the compression distortion at AP $l$ .

Theorem: Achievable Rate with Compression-Based Precoding

Under compression-based precoding with independent Gaussian compression at each AP, the achievable downlink rate for user $k$ is: $R_k^{\text{DL}} = \log_2\!\left(1 + \frac{\left|\sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \mathbf{v}_{lk}\right|^2}{\sum_{j \neq k} \left|\sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \mathbf{v}_{lj}\right|^2 + \sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \mathbf{R}_{e,l} \mathbf{H}_{lk} + \sigma^2}\right)$ where $\mathbf{R}_{e,l}$ is the compression noise covariance at AP $l$ , subject to the fronthaul constraint: $\log_2 \det\!\left(\mathbf{I} + \mathbf{R}_{e,l}^{-1} \mathbf{R}_{x,l}\right) \leq C_{\text{fh},l}$ with $\mathbf{R}_{x,l} = \sum_k \mathbf{v}_{lk} \mathbf{v}_{lk}^{H}$ being the precoded signal covariance.

The compression noise $\mathbf{R}_{e,l}$ appears as additional interference at each user. The fronthaul constraint creates a tradeoff: stronger precoding (larger $\mathbf{R}_{x,l}$ ) improves the desired signal but requires more fronthaul bits for accurate compression.

Proof

Model the effective received signal

User $k$ receives $y_k = \sum_l \mathbf{H}_{lk}^{H} (\mathbf{x}_l + \mathbf{e}_l) + w_k$ . Since $\mathbf{e}_l$ is independent of $\mathbf{x}_l$ and $w_k$ , the interference-plus-noise power is the sum of multi-user interference, compression noise, and thermal noise.

Compute the SINR

The desired signal power for user $k$ is $|\sum_l \mathbf{H}_{lk}^{H} \mathbf{v}_{lk}|^2$ . The total interference is the sum of: (i) multi-user interference $\sum_{j \neq k} |\sum_l \mathbf{H}_{lk}^{H} \mathbf{v}_{lj}|^2$ , (ii) compression noise $\sum_l \mathbf{H}_{lk}^{H} \mathbf{R}_{e,l} \mathbf{H}_{lk}$ , and (iii) thermal noise $\sigma^2$ .

Fronthaul constraint

The compression rate from the CPU to AP $l$ must satisfy $I(\mathbf{x}_l; \hat{\mathbf{x}}_l) \leq C_{\text{fh},l}$ . For Gaussian signals and distortion, this gives the log-det formula relating $\mathbf{R}_{e,l}$ to $C_{\text{fh},l}$ . $\blacksquare$

Definition:
Multivariate Compression for Fronthaul

Multivariate compression allows the CPU to jointly design the compression codebooks across all APs, exploiting the correlation in the precoded signals $\{\mathbf{x}_l\}_{l=1}^L$ .

The optimal multivariate compression minimizes the total distortion subject to the per-AP fronthaul constraints: $\min_{\{\mathbf{R}_{e,l}\}} \sum_{l=1}^{L} \text{tr}(\mathbf{R}_{e,l}) \quad \text{s.t.} \quad \log_2 \det\!\left(\mathbf{I} + \mathbf{R}_{e,l}^{-1} \mathbf{R}_{x_l | x_{\setminus l}}\right) \leq C_{\text{fh},l}, \; \forall l$ where $\mathbf{R}_{x_l | x_{\setminus l}}$ is the conditional covariance of AP $l$ 's precoded signal given all other APs' signals.

Multivariate compression achieves higher rates than independent compression because it avoids redundantly encoding the correlated components across APs. The gain is largest when APs serve overlapping sets of users.

Example: Compression-Based MRT with Two APs

Consider two single-antenna APs ( $N_t = 1$ ) serving one user. The channels are $h_1$ and $h_2$ with $|h_1|^2 = |h_2|^2 = 1$ . The CPU uses MRT precoding: $v_1 = h_1^*/|h_1|$ , $v_2 = h_2^*/|h_2|$ . Each AP has fronthaul capacity $C_{\text{fh}}$ bits per channel use. Compute the achievable rate as a function of $C_{\text{fh}}$ .

Solution

Compute precoded signal power

With MRT, the precoded signal at each AP has power $P_x = |v_l|^2 \mathbb{E}[|s|^2] = P_t/2$ (equal power split). The compression noise variance at each AP is: $\sigma_e^2 = \frac{P_x}{2^{C_{\text{fh}}} - 1}$

Compute the effective SINR

The desired signal power is $|h_1 v_1 + h_2 v_2|^2 = (|h_1| + |h_2|)^2 \cdot P_t/2 = 2P_t$ . The compression noise power is $\sum_l |h_l|^2 \sigma_e^2 = 2\sigma_e^2$ . The SINR is: $\text{SINR} = \frac{2P_t}{2\sigma_e^2 + \sigma^2} = \frac{2P_t}{\frac{P_t}{2^{C_{\text{fh}}} - 1} + \sigma^2}$

Analyze limiting cases

As $C_{\text{fh}} \to \infty$ : $\sigma_e^2 \to 0$ and $\text{SINR} \to 2P_t/\sigma^2$ (full MRT gain). As $C_{\text{fh}} \to 0$ : the compression noise dominates and $\text{SINR} \to 0$ . The rate is a monotonically increasing concave function of $C_{\text{fh}}$ .

Compression-Based Precoding Rate vs. Fronthaul Capacity

Visualize how the achievable downlink rate improves with fronthaul capacity for different precoding strategies (MRT, ZF) under compression-based forwarding.

Parameters

Number of APs

L

16

K

4

\text{SNR}

(dB)10

Precoding strategy

Theorem: Power-Fronthaul Tradeoff in Downlink

For compression-based precoding with per-AP power constraint $P_l$ and fronthaul capacity $C_{\text{fh},l}$ , the effective transmit power at AP $l$ satisfies: $P_l^{\text{eff}} = P_l \cdot \left(1 - 2^{-C_{\text{fh},l}}\right)$ The fraction $2^{-C_{\text{fh},l}}$ of the transmit power is "wasted" as compression noise that radiates as additional interference.

Finite fronthaul means that a portion of each AP's transmit power radiates as uncontrolled compression noise. At $C_{\text{fh}} = 1$ bit/dimension, half the power is wasted. At $C_{\text{fh}} = 5$ bits/dimension, only 3% is wasted.

Proof

Total transmit power decomposition

The AP transmits $\hat{\mathbf{x}}_l = \mathbf{x}_l + \mathbf{e}_l$ . The total radiated power is: $\mathbb{E}[\|\hat{\mathbf{x}}_l\|^2] = \mathbb{E}[\|\mathbf{x}_l\|^2] + \mathbb{E}[\|\mathbf{e}_l\|^2] = P_l + \text{tr}(\mathbf{R}_{e,l})$

Relate compression noise to fronthaul

For isotropic compression ( $\mathbf{R}_{e,l} = \sigma_e^2 \mathbf{I}$ ) and isotropic signal ( $\mathbf{R}_{x,l} = (P_l/N_t)\mathbf{I}$ ): $\sigma_e^2 = \frac{P_l/N_t}{2^{C_{\text{fh},l}/N_t} - 1}$

Compute effective power

The useful signal power (carrying intended data) is $P_l$ , while the compression noise power is $N_t \sigma_e^2$ . The fraction of power carrying useful information is $P_l / (P_l + N_t\sigma_e^2) = 1 - 2^{-C_{\text{fh},l}/N_t}$ , which for $N_t = 1$ simplifies to $1 - 2^{-C_{\text{fh},l}}$ . $\blacksquare$

Common Mistake: Violating Per-AP Power Constraints After Compression

Mistake:

Designing the precoder to satisfy the per-AP power constraint $\mathbb{E}[\|\mathbf{x}_l\|^2] \leq P_l$ without accounting for the compression noise. The actual transmit power $\mathbb{E}[\|\hat{\mathbf{x}}_l\|^2] = P_l + \text{tr}(\mathbf{R}_{e,l}) > P_l$ .

Correction:

The per-AP power constraint must be applied to the compressed signal: $\mathbb{E}[\|\hat{\mathbf{x}}_l\|^2] \leq P_l$ . This means the precoder must be scaled down to leave room for the compression noise: the useful signal power is at most $P_l - \text{tr}(\mathbf{R}_{e,l})$ .

Joint Precoding and Compression Optimization

Complexity: Each iteration requires solving a convex subproblem. Convergence is guaranteed but may be slow. Typical: 5--15 iterations for

<1\%

rate gap.

Input: Channels

\{\mathbf{H}_{lk}\}

, fronthaul capacities

\{C_{\text{fh},l}\}

, power budgets

\{P_l\}

Output: Precoding vectors

\{\mathbf{v}_{lk}\}

, compression covariances

\{\mathbf{R}_{e,l}\}

1. Initialize: Set

\mathbf{R}_{e,l} = \epsilon \mathbf{I}

for all

l

(low compression noise)

2. repeat

3.

\quad

Precoder update: Fix

\{\mathbf{R}_{e,l}\}

, optimize

\{\mathbf{v}_{lk}\}

to maximize weighted sum rate

4.

\quad

Compression update: Fix

\{\mathbf{v}_{lk}\}

, optimize

\{\mathbf{R}_{e,l}\}

subject to fronthaul and power constraints

5.

\quad

Compute sum rate

R^{(t)}

6. until

|R^{(t)} - R^{(t-1)}| < \epsilon

7. return

\{\mathbf{v}_{lk}\}

,

\{\mathbf{R}_{e,l}\}

The alternating optimization decouples the precoder and compression design, making each subproblem tractable. This is a practical algorithm suitable for quasi-static channels.

Quick Check

In downlink compression-based precoding, where does the compression noise appear from the user's perspective?

As additional thermal noise at the user

As additional interference that scales with the channel gain

It does not affect the user at all

As a reduction in the user's channel gain

Correction:

As additional interference that scales with the channel gain

The user receives $\sum_l \mathbf{H}_{lk}^{H} \mathbf{e}_l$ which passes through the channel. Unlike thermal noise, it scales with channel gain.

⚠️Engineering Note

Practical Fronthaul for Cell-Free Downlink

In practical deployments, the downlink fronthaul carries a mix of compressed I/Q samples and control signaling (scheduling, power control, timing). The useful data fraction is typically 85--90% of the total fronthaul capacity, with the rest consumed by headers, framing, and forward error correction.

Modern eCPRI implementations use 25 Gbps Ethernet links, providing approximately 22 Gbps of useful fronthaul capacity per AP. For a 100 MHz bandwidth with 4 antenna ports and 8-bit compression, this supports a single AP with about 3 bits/dimension of compression --- sufficient for 15--20 dB of compression SNR.

Practical Constraints

•
25G Ethernet: ~22 Gbps useful capacity per link
•
Compression overhead: 10--15% for headers and FEC
•
3 bits/dimension provides ~18 dB compression SNR

Key Takeaway

Compression-based precoding treats the downlink fronthaul as a rate-distortion problem: the CPU jointly designs the precoder and compression to maximize user rates under fronthaul constraints. Finite fronthaul wastes a fraction $2^{-C_{\text{fh}}}$ of each AP's transmit power as uncontrolled compression noise.

Compression-Based Precoding

A downlink fronthaul strategy where the central processor computes precoded signals, compresses them, and forwards the compressed versions to the APs for transmission. The compression noise acts as additional interference at the users.

Downlink Fronthaul Strategies