Ferkans — Interactive Telecom Tutor

Why Precise Rate Expressions Matter

Chapters 11-14 built the cell-free massive MIMO architecture from the ground up: distributed APs, user-centric clustering, cooperation levels, and fronthaul constraints. This chapter brings the story full circle by deriving precise, computable spectral efficiency expressions that account for all these practical realities. Without closed-form SE bounds, system design would require Monte Carlo simulation for every parameter choice — a luxury that 6G standardization timelines do not permit. The achievable rate expressions developed here are the engine that drives every comparison, optimization, and deployment decision in the rest of the chapter.

Definition:
Cell-Free Massive MIMO System Model

Consider a cell-free massive MIMO system with $L$ access points (APs), each equipped with $N$ antennas, jointly serving $K$ single-antenna users over the same time-frequency resource. The system operates in TDD mode with a coherence interval of $\tau_c$ samples, of which $\tau_p$ are used for uplink pilot transmission.

The channel from AP $l$ to user $k$ is modeled as

$\mathbf{H}_{lk} = \sqrt{\beta_{lk}} \, \tilde{\mathbf{H}}_{lk}, \quad \tilde{\mathbf{H}}_{lk} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_{lk})$

where $\beta_{lk}$ is the large-scale fading coefficient and $\mathbf{R}_{lk} \in \mathbb{C}^{N \times N}$ is the spatial correlation matrix with $\text{tr}(\mathbf{R}_{lk}) = N$ . The uplink received signal at AP $l$ during the data phase is

$\mathbf{y}_l = \sum_{k=1}^{K} \mathbf{H}_{lk} x_k + \mathbf{w}_{l}$

where $x_k$ is user $k$ 's transmitted symbol with power ${P_t}_{k}$ and $\mathbf{w}_{l} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_N)$ .

The model subsumes uncorrelated Rayleigh fading ( $\mathbf{R}_{lk} = \mathbf{I}_N$ ) as a special case. The spatial correlation structure becomes important when APs have multiple antennas and when exploiting covariance information for MMSE estimation.

Definition:
MMSE Channel Estimate

After receiving $\tau_p$ pilot symbols, AP $l$ forms the MMSE estimate of $\mathbf{H}_{lk}$ :

$\hat{\mathbf{H}}_{lk} = \sqrt{{P_t}_{k} \tau_p} \, \mathbf{R}_{lk} \boldsymbol{\Psi}_{lk}^{-1} \mathbf{y}_l^{\text{pilot}}$

where $\boldsymbol{\Psi}_{lk} = \sum_{j: \phi_j = \phi_k} {P_t}_{j} \tau_p \mathbf{R}_{lj} + \sigma^2 \mathbf{I}_N$ is the covariance of the received pilot signal and $\phi_k$ denotes the pilot index assigned to user $k$ . The estimation quality is captured by

$\gamma_{lk} \triangleq \frac{1}{N} \mathbb{E}\!\left[\|\hat{\mathbf{H}}_{lk}\|^2\right] = \frac{1}{N} \text{tr}\!\left({P_t}_{k} \tau_p \mathbf{R}_{lk} \boldsymbol{\Psi}_{lk}^{-1} \mathbf{R}_{lk}\right)$

The estimation error $\tilde{\mathbf{H}}_{lk} = \mathbf{H}_{lk} - \hat{\mathbf{H}}_{lk}$ is independent of $\hat{\mathbf{H}}_{lk}$ (by the orthogonality principle) with covariance $\mathbf{C}_{lk} = \mathbf{R}_{lk} - {P_t}_{k} \tau_p \mathbf{R}_{lk} \boldsymbol{\Psi}_{lk}^{-1} \mathbf{R}_{lk}$ .

Use-and-Then-Forget (UatF) Bound

A capacity lower bound obtained by treating the channel estimate as the true channel in the beamforming/combining operation and accounting for the estimation error as additional uncorrelated noise. The bound is tight when channel hardening holds and avoids requiring instantaneous effective channel knowledge at the receiver.

Theorem: UatF Bound for Uplink Spectral Efficiency

Consider a cell-free massive MIMO system where AP $l$ uses combining vector $\mathbf{v}_{lk} = \hat{\mathbf{H}}_{lk}$ (local MRC). The uplink spectral efficiency of user $k$ under the use-and-then-forget bound is

$\text{SE}_k^{\text{ul}} = \frac{\tau_c - \tau_p}{\tau_c} \log_2\!\left(1 + \text{SINR}_k^{\text{ul}}\right)$

where

$\text{SINR}_k^{\text{ul}} = \frac{{P_t}_{k} \left(\sum_{l=1}^{L} \sqrt{\eta_{lk}} \, N \gamma_{lk}\right)^2}{\sum_{j=1}^{K} {P_t}_{j} \sum_{l=1}^{L} \eta_{lk} \, \mathbb{E}\!\left[|\mathbf{v}_{lk}^{H} \mathbf{H}_{lj}|^2\right] - {P_t}_{k} \left(\sum_{l=1}^{L} \sqrt{\eta_{lk}} \, N \gamma_{lk}\right)^2 + \sigma^2 \sum_{l=1}^{L} \eta_{lk} \, N \gamma_{lk}}$

Here $\eta_{lk} \geq 0$ are the LSFD (large-scale fading decoding) weights applied at the CPU.

The numerator contains the coherent combining gain $(\sum_l \sqrt{\eta_{lk}} N \gamma_{lk})^2$ , which grows quadratically in the number of APs with good channels to user $k$ . This macro-diversity gain is the fundamental advantage of cell-free over co-located architectures: the sum extends over geographically distributed APs, each providing an independent fading path.

Proof

Signal decomposition

After local combining at AP $l$ , the CPU receives $\hat{s}_{lk} = \mathbf{v}_{lk}^{H} \mathbf{y}_l = \mathbf{v}_{lk}^{H} \mathbf{H}_{lk} x_k + \sum_{j \neq k} \mathbf{v}_{lk}^{H} \mathbf{H}_{lj} x_j + \mathbf{v}_{lk}^{H} \mathbf{w}_{l}$ . The CPU forms the global estimate $\hat{s}_k = \sum_{l=1}^{L} \sqrt{\eta_{lk}} \, \hat{s}_{lk}$ .

Desired signal extraction

Decompose the desired signal term: $\sum_l \sqrt{\eta_{lk}} \mathbf{v}_{lk}^{H} \mathbf{H}_{lk} = \sum_l \sqrt{\eta_{lk}} \mathbb{E}[\mathbf{v}_{lk}^{H} \mathbf{H}_{lk}] + \underbrace{\sum_l \sqrt{\eta_{lk}} (\mathbf{v}_{lk}^{H} \mathbf{H}_{lk} - \mathbb{E}[\mathbf{v}_{lk}^{H} \mathbf{H}_{lk}])}_{\text{beamforming gain uncertainty}}$ . With MRC, $\mathbb{E}[\hat{\mathbf{H}}_{lk}^H \mathbf{H}_{lk}] = \mathbb{E}[\|\hat{\mathbf{H}}_{lk}\|^2] = N \gamma_{lk}$ .

UatF bounding technique

Treat $\sum_l \sqrt{\eta_{lk}} \mathbb{E}[\mathbf{v}_{lk}^{H} \mathbf{H}_{lk}] \cdot x_k$ as the desired signal and everything else (beamforming uncertainty, inter-user interference, noise) as worst-case uncorrelated Gaussian noise. By the capacity of a Gaussian channel with known deterministic gain, the ergodic achievable rate follows. The pre-log factor $(\tau_c - \tau_p)/\tau_c$ accounts for pilot overhead.

SINR expression

Computing the signal and interference-plus-noise powers and applying the standard Gaussian lower bound yields the stated SINR expression. $\blacksquare$

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Definition:
Hardening-Based Bound

When channel hardening holds (i.e., $\mathbf{v}_{lk}^{H} \mathbf{H}_{lk} \approx \mathbb{E}[\mathbf{v}_{lk}^{H} \mathbf{H}_{lk}]$ with small variance), the effective channel gain is nearly deterministic. The hardening-based bound exploits this by using the deterministic effective channel gain in the rate expression:

$\text{SE}_k^{\text{hard}} = \frac{\tau_c - \tau_p}{\tau_c} \mathbb{E}\!\left[\log_2\!\left(1 + \text{SINR}_k^{\text{hard}}\right)\right]$

where the expectation is over the small-scale fading and the SINR is computed using the instantaneous effective channel gains rather than their expectations. This bound is tighter than UatF when channel hardening is strong but requires knowledge of the effective channel statistics at the receiver.

For co-located massive MIMO with many antennas, hardening is strong and both bounds are tight. For cell-free systems with single-antenna APs, hardening is weaker (averaging over independent large-scale fading coefficients instead of many co-located antenna elements), making the gap between the two bounds more significant.

Hardening-Based Bound

An achievable rate bound that exploits channel hardening to treat the effective channel gain as approximately deterministic. Tighter than UatF when hardening holds, but requires statistical CSI at the receiver.

Example: SE Under Uncorrelated Rayleigh Fading

Compute the closed-form UatF SE for a cell-free system with $L$ single-antenna APs ( $N = 1$ ), uncorrelated Rayleigh fading ( $\mathbf{R}_{lk} = 1$ ), and MRC combining, assuming no pilot contamination (orthogonal pilots for all users).

Solution

Channel estimation

With $N = 1$ and orthogonal pilots, $\boldsymbol{\Psi}_{lk} = {P_t}_{k} \tau_p \beta_{lk} + \sigma^2$ . The MMSE estimate is scalar: $\hat{h}_{lk} \sim \mathcal{CN}(0, \gamma_{lk})$ where $\gamma_{lk} = \frac{{P_t}_{k} \tau_p \beta_{lk}^{2}}{{P_t}_{k} \tau_p \beta_{lk} + \sigma^2}$ .

Desired signal power

With MRC ( $v_{lk} = \hat{h}_{lk}^*$ ), $\mathbb{E}[v_{lk}^* h_{lk}] = \mathbb{E}[|\hat{h}_{lk}|^2] = \gamma_{lk}$ . The coherent combining gain is $\left(\sum_{l=1}^{L} \sqrt{\eta_{lk}} \gamma_{lk}\right)^2$ .

Interference terms

For $j \neq k$ with orthogonal pilots: $\mathbb{E}[|v_{lk}^* h_{lj}|^2] = \gamma_{lk} \beta_{lj}$ (the estimate $\hat{h}_{lk}$ and the channel $h_{lj}$ are independent). For user $k$ itself: $\mathbb{E}[|v_{lk}^* h_{lk}|^2] = \gamma_{lk}^2 + \gamma_{lk}(\beta_{lk} - \gamma_{lk}) = \gamma_{lk} \beta_{lk}$ .

Closed-form SE

Substituting and setting $\eta_{lk} = 1$ (equal LSFD weights):

$\text{SE}_k = \frac{\tau_c - \tau_p}{\tau_c} \log_2\!\left(1 + \frac{{P_t}_{k} \left(\sum_{l} \gamma_{lk}\right)^2}{\sum_{j=1}^{K} {P_t}_{j} \sum_{l} \gamma_{lk} \beta_{lj} + \sigma^2 \sum_{l} \gamma_{lk}}\right)$

The numerator scales as $L^2$ (coherent gain from $L$ APs), while each interference term scales as $L$ — yielding an SINR that grows linearly in $L$ .

Theorem: Downlink SE with Conjugate Beamforming

Under conjugate beamforming (CB) at each AP, where AP $l$ uses precoding vector $\mathbf{v}_{lk} = \sqrt{\eta_{lk}} \hat{\mathbf{H}}_{lk}^*$ for user $k$ , the downlink SE under the UatF bound is

$\text{SE}_k^{\text{dl}} = \frac{\tau_c - \tau_p}{\tau_c} \log_2\!\left(1 + \text{SINR}_k^{\text{dl}}\right)$

with

$\text{SINR}_k^{\text{dl}} = \frac{\left(\sum_{l=1}^{L} \sqrt{\eta_{lk}} \, N \gamma_{lk}\right)^2}{\sum_{j=1}^{K} \sum_{l=1}^{L} \eta_{lj} \, \mathbb{E}\!\left[|\mathbf{H}_{lk}^{H} \hat{\mathbf{H}}_{lj}^*|^2\right] - \left(\sum_{l} \sqrt{\eta_{lk}} N \gamma_{lk}\right)^2 + \sigma^2}$

subject to the per-AP power constraint $\sum_{k=1}^{K} \eta_{lk} N \gamma_{lk} \leq {P_t}_{l}$ for all $l$ .

The structure mirrors the uplink: the numerator is the squared coherent combining gain, and the denominator contains beamforming gain uncertainty, inter-user interference, and noise. The per-AP power constraint couples the power control coefficients $\eta_{lk}$ across users, making the optimization problem more constrained than in co-located systems.

Proof

Received signal

User $k$ receives $y_k = \sum_{l=1}^{L} \mathbf{H}_{lk}^{H} \sum_{j=1}^{K} \sqrt{\eta_{lj}} \hat{\mathbf{H}}_{lj}^* s_j + w_k$ . Decompose the desired signal: $\sum_l \sqrt{\eta_{lk}} \mathbf{H}_{lk}^{H} \hat{\mathbf{H}}_{lk}^* = \underbrace{\sum_l \sqrt{\eta_{lk}} \mathbb{E}[\mathbf{H}_{lk}^{H} \hat{\mathbf{H}}_{lk}^*]}_{\text{deterministic}} + \text{random}$ .

Apply UatF

By the UatF technique, treat the deterministic part as the signal gain. By TDD reciprocity and the MMSE estimation properties, $\mathbb{E}[\mathbf{H}_{lk}^{H} \hat{\mathbf{H}}_{lk}^*] = N \gamma_{lk}$ . The SINR follows from computing the signal and interference-plus-noise powers. $\blacksquare$

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Impact of Pilot Contamination on SE

When users $k$ and $j$ share the same pilot ( $\phi_k = \phi_j$ ), the MMSE estimate $\hat{\mathbf{H}}_{lk}$ becomes correlated with $\mathbf{H}_{lj}$ . This introduces a coherent interference term that does not vanish as $L \to \infty$ :

$\mathbb{E}\!\left[\hat{\mathbf{H}}_{lk}^H \mathbf{H}_{lj}\right] = {P_t}_{j} \tau_p \, \text{tr}\!\left(\mathbf{R}_{lj} \boldsymbol{\Psi}_{lk}^{-1} \mathbf{R}_{lk}\right) \neq 0$

This is the cell-free version of pilot contamination. Unlike co-located massive MIMO where pilot contamination from users in other cells creates an asymptotic rate ceiling, in cell-free systems the impact is mitigated by (i) careful pilot assignment exploiting the geographic separation of users, and (ii) MMSE combining that suppresses contaminating users based on their spatial signatures.

Common Mistake: Confusing Coherent and Non-Coherent Combining Gains

Mistake:

Assuming that the combining gain from $L$ APs is always $L$ (linear). This confuses non-coherent combining (power addition) with coherent combining (amplitude addition).

Correction:

With coherent combining under perfect CSI, the signal power grows as $(\sum_l \sqrt{\beta_{lk}})^2$ , which is proportional to $L^2$ when all APs have equal path loss. Non-coherent combining gives only $\sum_l \beta_{lk} \propto L$ . The UatF bound achieves coherent combining for the desired signal (the numerator squares the sum), while interference combines non-coherently. This $L^2$ vs $L$ scaling is the fundamental reason cell-free outperforms alternatives.

Theorem: SE with Centralized MMSE (Level 4)

Under centralized MMSE combining (Level 4 cooperation), where the CPU has access to all received signals and channel estimates, the uplink SE of user $k$ is

$\text{SE}_k^{\text{L4}} = \frac{\tau_c - \tau_p}{\tau_c} \log_2\!\left(1 + {P_t}_{k} \hat{\mathbf{H}}_k^H \left(\sum_{j \neq k} {P_t}_{j} \hat{\mathbf{H}}_j \hat{\mathbf{H}}_j^H + \sum_{j=1}^{K} {P_t}_{j} \mathbf{C}_j + \sigma^2 \mathbf{I}_{LN}\right)^{-1} \hat{\mathbf{H}}_k\right)$

where $\hat{\mathbf{H}}_k = [\hat{\mathbf{H}}_{1k}^T, \ldots, \hat{\mathbf{H}}_{Lk}^T]^T \in \mathbb{C}^{LN}$ is the stacked channel estimate and $\mathbf{C}_j = \text{blkdiag}(\mathbf{C}_{1j}, \ldots, \mathbf{C}_{Lj})$ . This is an ergodic achievable rate (not just a bound) when the effective SINR is treated as a random variable and the expectation is taken over the channel realizations.

Level 4 gives the CPU access to the full $LN$ -dimensional received signal. The MMSE combining vector jointly suppresses inter-user interference using spatial signatures from all APs. This is the best achievable performance for linear processing but requires sending all raw baseband samples over the fronthaul — a cost of $LN$ complex scalars per sample.

Proof

Stacked system model

The stacked received signal is $\mathbf{y} = [\mathbf{y}_1^T, \ldots, \mathbf{y}_L^T]^T = \sum_k \hat{\mathbf{H}}_k x_k + \sum_k \tilde{\mathbf{H}}_k x_k + \mathbf{w}$ where $\tilde{\mathbf{H}}_k$ is the stacked estimation error.

MMSE combining

The MMSE combining vector for user $k$ is $\mathbf{v}_{k} = \left(\sum_{j} {P_t}_{j} (\hat{\mathbf{H}}_j \hat{\mathbf{H}}_j^H + \mathbf{C}_j) + \sigma^2 \mathbf{I}\right)^{-1} \hat{\mathbf{H}}_k$ . The resulting SINR follows from the matrix inversion lemma.

Rate expression

The standard capacity of a Gaussian channel with MMSE combining yields the stated expression. The expectation over fading gives the ergodic rate. $\blacksquare$

Spectral Efficiency vs Number of APs

Explore how the per-user spectral efficiency scales with the number of distributed APs under different cooperation levels (MRC, local MMSE, centralized MMSE) and with/without pilot contamination.

Parameters

K

(users)10

L_{\max}

(max APs)100

SNR (dB)10

Pilot scheme

Cooperation level

Quick Check

In the UatF bound, the beamforming gain uncertainty (the random fluctuation of $\mathbf{v}_{lk}^{H} \mathbf{H}_{lk}$ around its mean) is treated as:

Part of the desired signal, increasing the SINR

Worst-case uncorrelated Gaussian noise, decreasing the SINR

Ignored entirely, as it averages to zero

Quantified exactly using instantaneous CSI

Correction:

Worst-case uncorrelated Gaussian noise, decreasing the SINR

The UatF bound treats all random components — including beamforming gain uncertainty — as worst-case Gaussian noise uncorrelated with the desired signal. This gives a clean, computable lower bound.

Example: SE Loss from Pilot Contamination

In a cell-free system with $L = 64$ single-antenna APs and $K = 20$ users, only $\tau_p = 10$ orthogonal pilots are available. Two users $k$ and $j$ share the same pilot and are located such that $\beta_{lk} = \beta_{lj}$ for all $l$ (worst case: identical large-scale fading). Compare the SE of user $k$ with and without pilot contamination from user $j$ .

Solution

Without contamination

With orthogonal pilots, $\gamma_{lk} = \frac{{P_t}_{k} \tau_p \beta_{lk}^{2}}{{P_t}_{k} \tau_p \beta_{lk} + \sigma^2}$ . The SINR numerator is ${P_t}_{k} (\sum_l \gamma_{lk})^2$ , which grows as $L^2$ .

With contamination from user $j$

The contaminated estimate has $\boldsymbol{\Psi}_{lk} = ({P_t}_{k} + {P_t}_{j}) \tau_p \beta_{lk} + \sigma^2$ , so $\gamma_{lk}^{\text{cont}} = \frac{{P_t}_{k} \tau_p \beta_{lk}^{2}}{({P_t}_{k} + {P_t}_{j}) \tau_p \beta_{lk} + \sigma^2}$ . With equal powers, $\gamma_{lk}^{\text{cont}} \approx \gamma_{lk} / 2$ .

Coherent interference

The interference from user $j$ contains a coherent component $\mathbb{E}[\hat{h}_{lk}^* h_{lj}] = \gamma_{lk}^{\text{cont}}$ that adds constructively across APs. The interference term $(\sum_l \gamma_{lk}^{\text{cont}})^2$ scales as $L^2$ , same as the signal. As $L \to \infty$ , $\text{SINR}_k \to {P_t}_{k} \gamma_{lk}^2 / ({P_t}_{j} (\gamma_{lk}^{\text{cont}})^2)$ . With equal powers and $\gamma_{lk}^{\text{cont}} \approx \gamma_{lk}/2$ , the asymptotic SINR ceiling is $\gamma_{lk}^2 / (\gamma_{lk}/2)^2 = 4$ , or about 6 dB — a significant loss.

Key Takeaway

The spectral efficiency of cell-free massive MIMO is governed by the coherent combining gain $(\sum_l \gamma_{lk})^2$ in the SINR numerator, which grows quadratically in the number of contributing APs. This macro-diversity gain — absent in co-located architectures — is the fundamental performance advantage of cell-free systems.

Theorem: SE Under Fronthaul Capacity Constraints

When each AP-to-CPU fronthaul link has capacity $C_{\text{fh}}$ bits per sample and AP $l$ uses scalar quantization with $b_l$ bits per real dimension, the effective uplink SE becomes

$\text{SE}_k^{\text{fh}} = \frac{\tau_c - \tau_p}{\tau_c} \log_2\!\left(1 + \frac{\text{SINR}_k^{\text{ul}}}{1 + \alpha_q \cdot \text{SINR}_k^{\text{ul}}}\right)$

where $\alpha_q = \frac{\pi \sqrt{3}}{2} \cdot 2^{-2b_l}$ is the quantization distortion factor (from the Bussgang decomposition) and $b_l \leq C_{\text{fh}} / (2N)$ is the per-dimension bit budget. As $b_l \to \infty$ , $\alpha_q \to 0$ and the full SE is recovered.

Fronthaul quantization introduces an effective noise floor that cannot be overcome by adding more APs. With coarse quantization ( $b_l$ small), $\alpha_q$ is large and the SE saturates regardless of SNR — the fronthaul becomes the bottleneck. This motivates the fronthaul-aware designs of Chapter 14.

Proof

Bussgang decomposition

The quantized signal $\hat{y}_l = \mathcal{Q}(y_l)$ is decomposed as $\hat{y}_l = (1 - \alpha_q) y_l + d_l$ where $d_l$ is the quantization distortion uncorrelated with $y_l$ , with variance $\alpha_q (1 - \alpha_q) \sigma_{y_l}^2$ .

Effective SINR

The quantization distortion adds to the noise floor. The effective SINR after combining at the CPU becomes $\text{SINR}^{\text{fh}} = \text{SINR}^{\text{ul}} / (1 + \alpha_q \text{SINR}^{\text{ul}})$ , a monotonically increasing but saturating function of the original SINR. $\blacksquare$

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⚠️Engineering Note

Practical Fronthaul Bit-Width Selection

In eCPRI-based fronthaul (O-RAN 7.2x split), the typical quantization resolution is $b = 8$ -- $12$ bits per I/Q sample. At $b = 8$ , $\alpha_q \approx 2.4 \times 10^{-5}$ , meaning the SE loss from quantization is negligible for SINR below 46 dB. At $b = 4$ (aggressive compression), $\alpha_q \approx 0.01$ and the SE ceiling appears at SINR $\approx 20$ dB. For cell-free systems with many APs and moderate per-AP SINR, $b = 6$ -- $8$ bits suffices for less than 0.5 dB SE loss.

Practical Constraints

•
eCPRI fronthaul rate = 2 * N * b * f_s bits/s per AP, where f_s is the sampling rate
•
For N = 4 antennas, b = 8, f_s = 30.72 MHz: fronthaul = 1.97 Gbps per AP
•
25 Gbps Ethernet can support ~12 multi-antenna APs with b = 8

Spectral Efficiency (SE)

The achievable rate per unit bandwidth, measured in bits/s/Hz. In cell-free massive MIMO, the per-user SE accounts for pilot overhead via the pre-log factor $(\tau_c - \tau_p)/\tau_c$ and for imperfect CSI via the UatF or hardening-based bounds.

Large-Scale Fading Decoding (LSFD)

A technique where the CPU applies AP-specific weights $\eta_{lk}$ to the locally combined signals before aggregation. The weights depend only on large-scale fading coefficients (which change slowly), avoiding the need for instantaneous CSI at the CPU. Optimizing $\eta_{lk}$ maximizes the effective SINR.

Historical Note: Origins of the Use-and-Then-Forget Bound

2000s-2010s

The UatF bounding technique traces back to Medard (2000), who showed that treating the channel estimate as the true channel yields a valid achievable rate in fading channels. The technique was popularized in the massive MIMO context by Marzetta (2010) and Ngo et al. (2013), who used it to derive closed-form rate expressions that revealed the power scaling laws. The name "use-and-then-forget" was coined by Marzetta, Larsson, Yang, and Ngo in their 2016 textbook, emphasizing that the bound does not require the receiver to track the instantaneous effective channel — it "uses" the channel estimate for beamforming and then "forgets" it for detection.

Why This Matters: SE Bounds in 5G NR System Design

The UatF bound is not just an academic tool — it directly informs 5G NR system design. The 3GPP system-level evaluation methodology uses similar capacity bounds (treating CSI errors as noise) to compare MIMO configurations. The cell-free SE expressions derived here predict the performance gains that O-RAN distributed MIMO deployments achieve over conventional macro-cell architectures, guiding operators in their network densification strategies.

See full treatment in Chapter 22

Quick Check

In the fronthaul-limited SE expression, what happens as the number of APs $L$ increases while the per-AP fronthaul capacity $C_{\text{fh}}$ remains fixed?

SE grows without bound because more APs always help

SE saturates at a ceiling determined by the quantization bit-width $b_l$

SE decreases because more fronthaul links add more quantization noise

SE is independent of $L$ and depends only on fronthaul capacity

Correction:

SE saturates at a ceiling determined by the quantization bit-width

b_l

The quantization distortion factor $\alpha_q$ creates an SINR ceiling of $1/\alpha_q$ . Adding more APs increases the pre-quantization SINR but not the post-quantization SINR beyond this ceiling.

Common Mistake: Using Perfect CSI SE as a Design Target

Mistake:

Designing cell-free systems using SE expressions that assume perfect CSI and ignoring the pilot overhead pre-log factor $(\tau_c - \tau_p) / \tau_c$ .

Correction:

With $K = 20$ users and orthogonal pilots ( $\tau_p = 20$ ) in a coherence interval of $\tau_c = 200$ samples, the pre-log factor is $180/200 = 0.9$ — a 10% SE loss just from pilot overhead. With pilot reuse, $\tau_p$ is smaller but pilot contamination reduces the effective SINR. The net SE under imperfect CSI can be 30-50% lower than the perfect-CSI prediction, especially for cell-edge users where estimation quality is poor.

Achievable Rate Expressions

Why Precise Rate Expressions Matter

Definition: Cell-Free Massive MIMO System Model

Definition: MMSE Channel Estimate

Use-and-Then-Forget (UatF) Bound

Theorem: UatF Bound for Uplink Spectral Efficiency

Signal decomposition

Desired signal extraction

UatF bounding technique

SINR expression

Definition: Hardening-Based Bound

Hardening-Based Bound

Example: SE Under Uncorrelated Rayleigh Fading

Channel estimation

Desired signal power

Interference terms

Closed-form SE

Theorem: Downlink SE with Conjugate Beamforming

Received signal

Apply UatF

Impact of Pilot Contamination on SE

Common Mistake: Confusing Coherent and Non-Coherent Combining Gains

Theorem: SE with Centralized MMSE (Level 4)

Stacked system model

MMSE combining

Rate expression

Spectral Efficiency vs Number of APs

Parameters

Quick Check

Example: SE Loss from Pilot Contamination

Without contamination

With contamination from user $j$

Coherent interference

Key Takeaway

Theorem: SE Under Fronthaul Capacity Constraints

Bussgang decomposition

Effective SINR

Practical Fronthaul Bit-Width Selection

Spectral Efficiency (SE)

Large-Scale Fading Decoding (LSFD)

Historical Note: Origins of the Use-and-Then-Forget Bound

Why This Matters: SE Bounds in 5G NR System Design

Quick Check

Common Mistake: Using Perfect CSI SE as a Design Target

Definition:
Cell-Free Massive MIMO System Model

Definition:
MMSE Channel Estimate

Definition:
Hardening-Based Bound