Ferkans — Interactive Telecom Tutor

Uplink Fronthaul: From Observations to Decisions

In the cell-free uplink, each AP $l$ observes a noisy superposition of all users' signals. The fundamental challenge is: how should AP $l$ compress its $N_t$ -dimensional observation $\mathbf{y}_l$ into at most $C_{\text{fh},l}$ bits/s to forward to the CPU?

We study three strategies of increasing sophistication: compress-and-forward (scalar quantization), quantize-and-forward (vector quantization matched to the source statistics), and estimate-and-forward (local MMSE estimation followed by forwarding the estimate).

Definition:
Uplink Signal Model with Fronthaul

The received signal at AP $l$ ( $l = 1, \ldots, L$ ) is: $\mathbf{y}_l = \sum_{k=1}^{K} \mathbf{H}_{lk} x_k + \mathbf{w}_{l}$ where $\mathbf{H}_{lk} \in \mathbb{C}^{N_t}$ is the channel from user $k$ to AP $l$ , $x_k$ is the transmit signal with power $\mathbb{E}[|x_k|^2] = {P_t}_{k}$ , and $\mathbf{w}_{l} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$ .

AP $l$ maps $\mathbf{y}_l$ to a fronthaul message $m_l \in \{1, \ldots, 2^{n C_{\text{fh},l}}\}$ , where $n$ is the block length. The CPU decodes all users' data from $(m_1, \ldots, m_L)$ .

Definition:
Quantize-and-Forward (QF)

In quantize-and-forward, AP $l$ applies vector quantization to its received signal $\mathbf{y}_l$ to produce a quantized version $\hat{\mathbf{y}}_l$ . The quantization is modeled as: $\hat{\mathbf{y}}_l = \mathbf{y}_l + \mathbf{q}_l$ where $\mathbf{q}_l \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_{q,l})$ is the quantization noise, independent of $\mathbf{y}_l$ .

The fronthaul constraint requires: $I(\mathbf{y}_l; \hat{\mathbf{y}}_l) = \log_2 \det\!\left(\mathbf{I} + \mathbf{R}_{q,l}^{-1} \mathbf{R}_{y,l}\right) \leq C_{\text{fh},l}$ where $\mathbf{R}_{y,l} = \sum_k {P_t}_{k} \mathbf{H}_{lk} \mathbf{H}_{lk}^{H} + \sigma^2 \mathbf{I}$ is the covariance of $\mathbf{y}_l$ .

The optimal quantization noise covariance $\mathbf{R}_{q,l}$ depends on the channel statistics and can be optimized jointly across APs.

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Theorem: Achievable Rate with Quantize-and-Forward

Under quantize-and-forward with optimal Gaussian quantization, the achievable rate for user $k$ is: $R_k^{\text{QF}} = \log_2\!\left(1 + {P_t}_{k} \sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \left(\sum_{j \neq k} {P_t}_{j} \mathbf{H}_{lj} \mathbf{H}_{lj}^{H} + \sigma^2 \mathbf{I} + \mathbf{R}_{q,l}\right)^{-1} \mathbf{H}_{lk}\right)$ The key difference from the unlimited-fronthaul expression is the additional term $\mathbf{R}_{q,l}$ in the interference-plus-noise covariance, representing the penalty of quantization.

Quantization noise $\mathbf{R}_{q,l}$ acts as additional interference. As $C_{\text{fh},l} \to \infty$ , we have $\mathbf{R}_{q,l} \to \mathbf{0}$ and recover the full-fronthaul rate. As $C_{\text{fh},l} \to 0$ , the quantization noise dominates and the rate vanishes.

Proof

Model the CPU's observation

The CPU observes $\hat{\mathbf{y}}_l = \mathbf{y}_l + \mathbf{q}_l$ for each AP. Stacking all APs: $\hat{\mathbf{y}} = \mathbf{H} \mathbf{x} + \mathbf{w} + \mathbf{q}$ where $\mathbf{q} = [\mathbf{q}_1^T, \ldots, \mathbf{q}_L^T]^T$ has block-diagonal covariance $\mathbf{R}_q = \text{blkdiag}(\mathbf{R}_{q,1}, \ldots, \mathbf{R}_{q,L})$ .

Apply the SINR formula

Treating $\mathbf{w} + \mathbf{q}$ as effective noise with covariance $\sigma^2 \mathbf{I} + \mathbf{R}_q$ , the SINR for user $k$ under linear MMSE decoding follows from the standard MMSE-SINR expression with modified noise covariance.

Relate quantization noise to fronthaul capacity

The fronthaul constraint $I(\mathbf{y}_l; \hat{\mathbf{y}}_l) \leq C_{\text{fh},l}$ determines the minimum achievable $\mathbf{R}_{q,l}$ . For scalar quantization, $\mathbf{R}_{q,l} = \alpha_l \mathbf{I}$ with $\alpha_l = \text{tr}(\mathbf{R}_{y,l}) / (N_t(2^{C_{\text{fh},l}/N_t} - 1))$ . $\blacksquare$

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Definition:
Estimate-and-Forward (EF)

In estimate-and-forward, AP $l$ first computes a local MMSE estimate of the users' signals, then quantizes and forwards the estimate. Specifically:

Local estimation: AP $l$ computes $\hat{\mathbf{x}}_l = \mathbf{A}_l^H \mathbf{y}_l$ where $\mathbf{A}_l$ is the local MMSE combining matrix.
Quantization: The estimate $\hat{\mathbf{x}}_l \in \mathbb{C}^{K}$ is quantized to produce $\tilde{\mathbf{x}}_l$ with quantization noise covariance $\mathbf{R}_{q,l}^{\text{EF}}$ .
Forwarding: $\tilde{\mathbf{x}}_l$ is forwarded to the CPU.

The advantage is that $\hat{\mathbf{x}}_l$ is $K$ -dimensional (one component per user) rather than $N_t$ -dimensional, so the fronthaul load scales with $K$ instead of $N_t$ .

When $N_t \gg K$ (the massive MIMO regime), EF provides substantial fronthaul savings over QF. This is the strategy adopted in most practical cell-free implementations.

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Quantize-and-Forward vs. Estimate-and-Forward

Property	Quantize-and-Forward (QF)	Estimate-and-Forward (EF)
What is forwarded	Quantized observation $\hat{\mathbf{y}}_l$	Quantized local estimate $\tilde{\mathbf{x}}_l$
Fronthaul dimension	$N_t$ per AP	$K$ per AP
Fronthaul scaling	Proportional to $N_t$	Proportional to $K$
Local processing	None (raw quantization)	MMSE combining at each AP
Information loss	Only quantization noise	Local estimation error + quantization noise
Optimality	Optimal as $C_{\text{fh}} \to \infty$	Near-optimal when $N_t \gg K$
Practical appeal	Simple but high fronthaul cost	Standard approach in cell-free deployments

Definition:
Wyner-Ziv Compression for Fronthaul

Wyner-Ziv compression exploits the fact that the CPU has side information from other APs when decoding the fronthaul message from AP $l$ . Instead of compressing $\mathbf{y}_l$ independently, AP $l$ can use distributed source coding to reduce the fronthaul rate.

The Wyner-Ziv rate-distortion function for Gaussian sources states that the minimum rate to describe $\mathbf{y}_l$ with distortion $D_l$ , given side information $\mathbf{y}_{\setminus l}$ at the decoder, is: $R_{\text{WZ},l}(D_l) = \frac{1}{2} \log_2 \frac{\det(\mathbf{R}_{y_l | y_{\setminus l}})}{\det(\mathbf{D}_l)}$ where $\mathbf{R}_{y_l | y_{\setminus l}}$ is the conditional covariance of $\mathbf{y}_l$ given all other APs' observations.

The Wyner-Ziv gain is largest when the APs' observations are highly correlated, which occurs when APs are closely spaced or when users are in the overlapping coverage region of multiple APs.

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Theorem: Achievable Rate with Estimate-and-Forward

Under estimate-and-forward with local MMSE combining at each AP, the achievable rate for user $k$ with the UatF bound is: $R_k^{\text{EF}} = \log_2\!\left(1 + \frac{\left|\sum_{l=1}^{L} \mathbb{E}[\mathbf{H}_{lk}^{H} \mathbf{a}_{lk}]\right|^2}{\sum_{j=1}^{K} {P_t}_{j} \sum_{l=1}^{L} \text{Var}\!\left(\mathbf{H}_{lj}^{H} \mathbf{a}_{lk}\right) + \sum_{l=1}^{L} \sigma^2 \|\mathbf{a}_{lk}\|^2 + \sum_{l=1}^{L} \mathbf{a}_{lk}^H \mathbf{R}_{q,l}^{\text{EF}} \mathbf{a}_{lk}}\right)$ where $\mathbf{a}_{lk}$ is the $k$ -th column of the local combining matrix $\mathbf{A}_l$ , and the last term in the denominator captures the fronthaul quantization penalty.

The rate expression has the same structure as the standard UatF bound from Chapter 4, with an additional quantization noise term. As fronthaul capacity increases, the quantization term vanishes and we recover the unlimited-fronthaul rate. The EF approach is attractive because the quantization operates on $K$ -dimensional estimates rather than $N_t$ -dimensional raw observations.

Proof

Start from the UatF bound

The UatF technique treats the channel estimate as deterministic and the estimation error as noise. With EF, the CPU receives $\tilde{x}_{lk} = \mathbf{a}_{lk}^H \mathbf{y}_l + q_{lk}$ where $q_{lk}$ is the quantization noise component.

Decompose the received signal

Writing $\tilde{x}_{lk} = \underbrace{\mathbb{E}[\mathbf{H}_{lk}^{H} \mathbf{a}_{lk}]}_{\text{desired signal gain}} x_k + \underbrace{(\mathbf{H}_{lk}^{H} \mathbf{a}_{lk} - \mathbb{E}[\mathbf{H}_{lk}^{H} \mathbf{a}_{lk}])x_k}_{\text{channel uncertainty}} + \underbrace{\sum_{j \neq k} \mathbf{H}_{lj}^{H} \mathbf{a}_{lk} x_j}_{\text{interference}} + \underbrace{\mathbf{a}_{lk}^H \mathbf{w}_{l}}_{\text{noise}} + \underbrace{q_{lk}}_{\text{quantization}}$

Compute the SINR

Summing across all APs and applying the UatF bound (treating all but the coherent signal as uncorrelated noise) yields the stated SINR expression. The quantization noise covariance $\mathbf{R}_{q,l}^{\text{EF}}$ enters the denominator as the fronthaul penalty. $\blacksquare$

Quantize-and-Forward vs. Estimate-and-Forward

Compare the achievable sum rate of QF and EF strategies as a function of per-AP fronthaul capacity. Observe how EF outperforms QF when the number of antennas per AP is much larger than the number of users.

Parameters

Number of APs

L

16

K

4

Antennas per AP8

\text{SNR}

(dB)10

Example: Scalar Quantization Fronthaul Rate

Consider a single-antenna AP ( $N_t = 1$ ) with received signal $y_l = \sum_{k=1}^{K} h_{lk} x_k + w_l$ where $|h_{lk}|^2 = \beta_{lk}$ and $\sigma^2 = 1$ . The total received power is $P_l = \sum_k {P_t}_{k} \beta_{lk} + 1$ . If the AP uses uniform scalar quantization with $b$ bits, what is the quantization noise variance and the required fronthaul rate?

Solution

Quantization noise variance

For Gaussian sources with optimal Lloyd-Max quantization at high resolution, the quantization noise variance is approximately: $\sigma_{q,l}^2 \approx P_l \cdot 2^{-2b}$

Fronthaul rate

The rate for transmitting $b$ bits per complex sample (I and Q components) over a bandwidth of $W$ Hz is: $R_{\text{fh},l} = 2b \cdot W \text{ bits/s}$

Rate-distortion interpretation

From rate-distortion theory for a Gaussian source with variance $P_l$ : $C_{\text{fh},l} = \log_2\!\left(\frac{P_l}{\sigma_{q,l}^2}\right) = \log_2\!\left(1 + \frac{P_l}{\sigma_{q,l}^2}\right) \approx 2b$ confirming that each bit of resolution reduces the quantization noise by 6 dB.

Common Mistake: Ignoring the Dimension Mismatch Between QF and EF

Mistake:

Comparing QF and EF at the same total fronthaul rate (in bits/s) without accounting for the dimensionality difference. QF forwards $N_t$ -dimensional observations while EF forwards $K$ -dimensional estimates. At the same total fronthaul rate, EF allocates more bits per dimension.

Correction:

When $N_t = 8$ and $K = 2$ , EF can allocate $4\times$ more bits per component than QF at the same total fronthaul rate. The correct comparison normalizes by the dimension of the forwarded vector: QF needs $C_{\text{fh}} \propto N_t$ while EF needs $C_{\text{fh}} \propto K$ .

Quick Check

In which scenario does estimate-and-forward (EF) provide the largest advantage over quantize-and-forward (QF)?

Single-antenna APs ( $N_t = 1$ ) serving many users

Many-antenna APs ( $N_t = 64$ ) serving few users ( $K = 4$ )

Equal number of antennas and users ( $N_t = K$ )

Very high fronthaul capacity (unlimited)

Correction:

Many-antenna APs (

N_t = 64

) serving few users (

K = 4

)

EF forwards $K$ -dimensional estimates instead of $N_t$ -dimensional observations, giving a $64/4 = 16\times$ dimension reduction.

Quick Check

What is the key advantage of Wyner-Ziv compression over independent compression for fronthaul?

It eliminates quantization noise entirely

It exploits side information at the CPU to reduce the required fronthaul rate

It requires no local processing at the AP

It works only with single-antenna APs

Correction:

It exploits side information at the CPU to reduce the required fronthaul rate

The CPU has observations from other APs, which serve as correlated side information for decoding the fronthaul message from AP $l$ .

Historical Note: The Wyner-Ziv Theorem and Its Wireless Renaissance

1976--2009

Aaron Wyner and Jacob Ziv published their foundational result on source coding with side information at the decoder in 1976. The theorem showed that for Gaussian sources, there is no rate penalty from not having side information at the encoder. This result remained largely theoretical for decades until the emergence of Cloud-RAN and distributed MIMO in the 2010s, where the fronthaul compression problem maps precisely onto Wyner-Ziv coding. Simeone, Somekh, Poor, and Shamai (2009) were among the first to make this connection explicit, launching a fruitful cross-fertilization between network information theory and wireless system design.

Key Takeaway

Estimate-and-forward is the practical workhorse for cell-free uplink fronthaul. By performing local MMSE combining before quantization, each AP reduces the fronthaul dimension from $N_t$ (antennas) to $K$ (users), enabling massive MIMO with affordable fronthaul infrastructure.

Quantize-and-Forward

An uplink fronthaul strategy where each AP directly quantizes its received signal vector and forwards the quantized version to the central processor. Optimal when fronthaul capacity is abundant.

Estimate-and-Forward

An uplink fronthaul strategy where each AP first computes local MMSE estimates of the users' signals, then quantizes and forwards the lower-dimensional estimates. Preferred when $N_t \gg K$ .

Uplink Fronthaul Strategies

Uplink Fronthaul: From Observations to Decisions

Definition: Uplink Signal Model with Fronthaul

Definition: Quantize-and-Forward (QF)

Theorem: Achievable Rate with Quantize-and-Forward

Model the CPU's observation

Apply the SINR formula

Relate quantization noise to fronthaul capacity

Definition: Estimate-and-Forward (EF)

Quantize-and-Forward vs. Estimate-and-Forward

Definition: Wyner-Ziv Compression for Fronthaul

Theorem: Achievable Rate with Estimate-and-Forward

Start from the UatF bound

Decompose the received signal

Compute the SINR

Quantize-and-Forward vs. Estimate-and-Forward

Parameters

Example: Scalar Quantization Fronthaul Rate

Quantization noise variance

Fronthaul rate

Rate-distortion interpretation

Common Mistake: Ignoring the Dimension Mismatch Between QF and EF

Quick Check

Quick Check

Historical Note: The Wyner-Ziv Theorem and Its Wireless Renaissance

Key Takeaway

Quantize-and-Forward

Estimate-and-Forward

Definition:
Uplink Signal Model with Fronthaul

Definition:
Quantize-and-Forward (QF)

Definition:
Estimate-and-Forward (EF)

Definition:
Wyner-Ziv Compression for Fronthaul