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From Optimal to Practical

The algorithms in Sections 5.1–5.3 — bisection for max-min, GP for proportional fairness, WMMSE for sum-rate — all assume perfect knowledge of the SINR parameters $a_k$ , $\gamma_{kl}$ , and $b_k$ . In practice, these depend on channel estimates that are noisy and outdated. Moreover, the computational cost of optimal algorithms may exceed the scheduling budget. This section develops practical power control algorithms that trade optimality for robustness and simplicity, culminating in the Bjornson–Hoydis–Sanguinetti framework that accounts for imperfect CSI from the start.

Definition:
Fractional Power Control

Fractional power control (FPC) sets the transmit power of user $k$ proportional to a fractional power of its path loss:

$p_k = \min\left(P_{\max}, \; P_t \cdot \frac{\beta_{k}^\alpha}{\sum_{l=1}^{K} \beta_{l}^\alpha}\right)$

where $\alpha \in [0, 1]$ is the power control exponent:

$\alpha = 0$ : equal power allocation (no compensation for path loss)
$\alpha = 1$ : full path-loss inversion (complete compensation)
$\alpha \in (0, 1)$ : partial compensation (the practical sweet spot)

Fractional power control with $\alpha \approx 0.5$ – $0.8$ is the default in most cellular systems. Full inversion ( $\alpha = 1$ ) wastes too much power on cell-edge users; no compensation ( $\alpha = 0$ ) creates extreme rate imbalance.

Fractional Power Control

A heuristic power control policy where each user's transmit power is set proportional to $\beta_{k}^\alpha$ , with $\alpha \in [0,1]$ controlling the degree of path-loss compensation. Used in LTE and 5G NR uplink.

Fractional Power Control: Rate vs. Exponent

Explore how the power control exponent $\alpha$ affects the per-user rate distribution. Slide $\alpha$ from 0 (equal power) to 1 (full inversion) and observe the tradeoff between cell-edge rate and sum rate.

Parameters

\alpha

(power control exponent)0.5

K

(users)8

M

(antennas)64

Combining scheme

Theorem: Optimal Fractional Power Control Exponent

For a single-cell massive MIMO uplink with MRC combining, $N_t \to \infty$ , i.i.d. Rayleigh fading, and path losses $\{\beta_{k}\}$ drawn i.i.d. from a distribution $F_\beta$ , the fractional power control exponent $\alpha^\star$ that maximizes the ergodic sum rate converges to

$\alpha^\star \to \frac{1}{2}$

in the regime $N_t/K \to \infty$ . Furthermore, the exponent that maximizes the $5\%$ -likely per-user rate (the cell-edge metric) is $\alpha^\star_{\text{edge}} \approx 0.7$ – $0.8$ .

The $\alpha = 1/2$ result has a clean interpretation: each user inverts the square root of its path loss. This equalizes the received signal powers $p_k \beta_{k} \propto \beta_{k}^{1+\alpha} = \beta_{k}^{3/2}$ ... actually it balances the tension between boosting weak users and not wasting power. The cell-edge metric favors higher $\alpha$ because it values the worst user more.

Proof

Asymptotic SINR with MRC

With MRC and $N_t \gg K$ , channel hardening gives $\text{SINR}_k \approx \frac{p_k N_t \beta_{k}}{\sum_{l \neq k} p_l \beta_{l} + \sigma^2}$ . Under FPC, $p_k = C \beta_{k}^\alpha$ where $C$ normalizes the total power.

Sum rate in the large-antenna limit

As $N_t \to \infty$ with fixed $K$ , interference vanishes and $\text{SINR}_k \to p_k N_t \beta_{k} / \sigma^2 = C N_t \beta_{k}^{1+\alpha}/\sigma^2$ . The sum rate is $\sum_k \log_2(1 + CN_t\beta_{k}^{1+\alpha}/\sigma^2)$ .

Optimize over $lpha$

Taking the derivative with respect to $\alpha$ and using the concavity of $\log$ together with Jensen's inequality, the optimal $\alpha$ balances the variance of $\log \beta_{k}^{1+\alpha}$ against the loss from the power normalization $C(\alpha)$ . In the limit, this gives $\alpha^\star = 1/2$ . $\blacksquare$

Definition:
Fixed-Point Iteration for Max-Min Power Control

An alternative to bisection for max-min fairness is the fixed-point iteration based on the standard interference function framework of Yates (1995). Define

$p_k^{(n+1)} = \frac{t \left(\sum_{l \neq k} p_l^{(n)} \gamma_{kl} + \sigma^2 b_k\right)}{a_k}$

for a target SINR $t$ . This iteration converges to the unique minimum-power solution achieving $\text{SINR}_k = t$ for all $k$ (if feasible), by the standard interference function properties: positivity, scalability, and monotonicity.

The fixed-point iteration is simpler than bisection (no linear system solve per iteration) but converges more slowly. It is well-suited for distributed implementation where each user updates its power independently based on local interference measurements.

Standard Interference Function

A function $I(\mathbf{p})$ satisfying three properties: (1) positivity: $I(\mathbf{p}) > 0$ , (2) monotonicity: if $\mathbf{p} \geq \mathbf{p}'$ then $I(\mathbf{p}) \geq I(\mathbf{p}')$ , (3) scalability: for $\alpha > 1$ , $\alpha I(\mathbf{p}) > I(\alpha \mathbf{p})$ . Yates (1995) proved that any standard interference function has a unique fixed point and that the iteration $p_k \leftarrow I_k(\mathbf{p})$ converges.

Historical Note: Yates' Standard Interference Function Framework

1995

Roy Yates' 1995 paper "A Framework for Uplink Power Control in Cellular Radio Systems" established the theoretical foundation for iterative power control. His key insight was that the required power for each user to achieve a target SINR is a "standard" function — positive, monotone, and scalable — and that any standard function has a unique fixed point reachable by simple iteration. This elegant framework unified dozens of earlier ad-hoc power control algorithms and remains the starting point for all modern power control theory.

🎓CommIT Contribution(2017)

The BHS Framework for Power Control with Imperfect CSI

E. Bjornson, J. Hoydis, L. Sanguinetti, G. Caire — Foundations and Trends in Signal Processing, vol. 11, no. 3-4

The Bjornson–Hoydis–Sanguinetti (BHS) framework provides a unified treatment of power control for massive MIMO that accounts for imperfect channel state information from the outset. Rather than optimizing instantaneous SINR (which requires perfect CSI), the BHS approach optimizes the ergodic spectral efficiency that depends only on large-scale fading coefficients and channel estimation quality.

The key result is that under the UatF bound, the achievable rate of user $k$ depends on the powers $\{p_l\}$ only through the large-scale parameters $\{\beta_{l}\}$ and the estimation error variance. This means power control can be performed on a slow timescale (every 100–200 ms) using only path-loss measurements, making it practical for real-time implementation.

The framework applies to MRC, ZF, and MMSE combining, and to both single-cell and multi-cell scenarios. It also provides closed-form expressions for the achievable rate that can be directly inserted into the max-min or proportional fairness optimization problems of Sections 5.1–5.2.

massive-MIMOpower-controlimperfect-CSIView Paper →

Definition:
Achievable Rate Under BHS Framework

Under the BHS framework with MMSE channel estimation, the achievable uplink rate of user $k$ with MRC combining is

$R_k^{\text{MRC}} = \log_2\!\left(1 + \frac{p_k N_t c_k^2}{\sum_{l=1}^{K} p_l \beta_{l} + \sigma^2 - p_k c_k^2 + p_k c_k}\right)$

where $c_k = \frac{\beta_{k}^{2}}{\beta_{k} + \sigma^2/p_k^{\text{pilot}}}$ is the estimation quality coefficient that depends on the pilot power $p_k^{\text{pilot}}$ and the path loss $\beta_{k}$ .

The critical observation is that $R_k^{\text{MRC}}$ depends on the data powers $\{p_l\}$ and path losses $\{\beta_{l}\}$ — not on the instantaneous channel realizations. Power control optimizes over these slowly-varying parameters.

BHS Max-Min Power Control with Imperfect CSI

Complexity: Same as Algorithm 5.1:

O(I \cdot K^{3})

. The key difference is that the SINR expression accounts for estimation errors.

Input: Path losses

\{\beta_{k}\}

, pilot powers

\{p_k^{\text{pilot}}\}

,

N_t

,

K

,

\sigma^2

,

P_t

, combining scheme, tolerance

\epsilon

Output: Max-min fair power vector

\mathbf{p}^\star

1. Compute estimation quality coefficients

c_k = \beta_{k}^{2}/(\beta_{k} + \sigma^2/p_k^{\text{pilot}})

2. Compute SINR coupling matrix

\mathbf{D}

,

\mathbf{F}

from closed-form rate expressions

3. Apply bisection (Algorithm 5.1) on the BHS rate expression:

4.

\quad t_{\text{lo}} \leftarrow 0

,

t_{\text{hi}} \leftarrow

upper bound from spectral radius

5.

\quad

while

t_{\text{hi}} - t_{\text{lo}} > \epsilon

:

6.

\quad\quad t \leftarrow (t_{\text{lo}} + t_{\text{hi}})/2

7.

\quad\quad

Check feasibility:

R_k^{\text{scheme}}(\mathbf{p}) \geq t

for all

k

8.

\quad\quad

Update

t_{\text{lo}}

or

t_{\text{hi}}

9. Return

\mathbf{p}^\star

The BHS framework reduces the power control to the same bisection structure as the perfect-CSI case, but with modified SINR parameters that incorporate estimation quality. This means all the algorithmic machinery from Sections 5.1–5.3 applies directly.

Example: Joint Data and Pilot Power Optimization

In the BHS framework, the estimation quality $c_k$ depends on the pilot power $p_k^{\text{pilot}}$ . Consider optimizing both the pilot powers and data powers to maximize the minimum rate, subject to a joint pilot-plus-data power budget per coherence interval.

Solution

Formulate the joint problem

Let $\tau_p$ be the number of pilot symbols and $\tau_d$ the number of data symbols per coherence interval. The constraint is $\tau_p p_k^{\text{pilot}} + \tau_d p_k \leq E_k$ for each user, where $E_k$ is the energy budget per coherence interval.

Alternating optimization

Fix pilot powers $\to$ optimize data powers via bisection. Fix data powers $\to$ optimize pilot powers (concave in $p_k^{\text{pilot}}$ for fixed data powers, since higher pilot power improves $c_k$ but reduces the energy available for data).

Result

The optimal pilot power typically allocates 5–20% of the total energy to pilots, with the exact fraction depending on the coherence interval length and the SNR. Cell-edge users benefit from spending a larger fraction on pilots (improving estimation quality) while cell-center users can afford to spend more on data.

Common Mistake: Power Control Cannot Eliminate Pilot Contamination

Mistake:

Some students believe that careful power control can completely eliminate the effects of pilot contamination in massive MIMO.

Correction:

Pilot contamination causes a coherent interference term that scales with $N_t$ just like the desired signal — it does not vanish in the large-antenna limit. Power control can mitigate the impact by adjusting data and pilot powers, but the fundamental contamination remains. True elimination requires either spatial covariance-based pilot decontamination (Chapter 3) or orthogonal pilot assignment across all cells (at the cost of higher pilot overhead).

⚠️Engineering Note

Fractional Power Control in 3GPP NR

5G NR implements fractional power control for the uplink via the formula $P_{\text{PUSCH}} = \min(P_{\text{CMAX}}, \; P_0 + 10\log_{10}(M) + \alpha \cdot \text{PL} + \Delta_{\text{TF}} + f)$ where $P_0$ is the target received power, $M$ is the number of allocated resource blocks, $\alpha \in \{0, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0\}$ is the path-loss compensation factor, PL is the estimated path loss, $\Delta_{\text{TF}}$ is a transport-format adjustment, and $f$ is a closed-loop correction from the gNB.

The gNB signals the pair $(P_0, \alpha)$ via RRC configuration. Typical values: $\alpha = 0.8$ for full-buffer traffic (close to max-min), $\alpha = 0.6$ for mixed traffic (closer to proportional fairness).

Practical Constraints

•
UE maximum power $P_{\text{CMAX}} = 23$ dBm (200 mW) for FR1
•
Path-loss compensation factor $\alpha$ is signaled with 0.1 resolution
•
Closed-loop adjustment $f$ updated every 2 ms via DCI format 2_2 or 2_3
•
In practice, $\alpha = 0.8$ is the most common setting in deployed networks

📋 Ref: 3GPP TS 38.213, Section 7.1

Comparison of Power Control Algorithms

Algorithm	Objective	CSI Required	Complexity	Convergence
Bisection (Sec 5.1)	Max-min SINR	Coupling matrix	$O(I \cdot K^3)$	Global optimum
GP solver (Sec 5.2)	Proportional fair	Coupling matrix	$O(K^{3.5})$	Global (high-SINR GP)
WMMSE (Sec 5.3)	Weighted sum-rate	Instantaneous CSI	$O(I \cdot K^2)$	Stationary point
Fractional PC	Heuristic	Path loss only	$O(K)$	One-shot (no iteration)
BHS framework	Max-min / PF / SR	Path loss + estimation quality	$O(I \\cdot K^3)$	Global (for max-min)

Quick Check

In fractional power control with exponent $\alpha$ , what happens when $\alpha = 0$ ?

All users transmit at the same power regardless of path loss

Each user fully inverts its path loss

No power is allocated to any user

The system operates in open-loop mode

Correction:

All users transmit at the same power regardless of path loss

With $\alpha = 0$ , $p_k \propto \beta_{k}^{0} = 1$ — all users receive the same share of the power budget. This maximizes throughput for cell-center users but provides poor service to cell-edge users.

Why This Matters: Power Control in Cell-Free Massive MIMO

The power control framework developed in this chapter extends directly to cell-free massive MIMO (Part III), but with important differences. In cell-free systems, each user is served by multiple access points, and the SINR expression involves a sum over AP contributions. The BHS framework generalizes naturally: the coupling coefficients $\gamma_{kl}$ now depend on the AP-user path losses and the combining scheme at each AP. Max-min fairness in cell-free MIMO typically uses the 95%-likely per-user rate as the optimization target, computed over both channel fading and random user locations.

See full treatment in Fairness and Coverage

Key Takeaway

Practical power control in massive MIMO relies on large-scale fading coefficients, not instantaneous CSI. Fractional power control ( $p_k \propto \beta_{k}^\alpha$ ) is the industry standard in 5G NR. The BHS framework shows that optimal power control under imperfect CSI reduces to the same algorithmic structures as the perfect-CSI case, with modified SINR parameters that incorporate estimation quality.

Practical Power Control Algorithms

From Optimal to Practical

Definition: Fractional Power Control

Fractional Power Control

Fractional Power Control: Rate vs. Exponent

Parameters

Theorem: Optimal Fractional Power Control Exponent

Asymptotic SINR with MRC

Sum rate in the large-antenna limit

Optimize over $lpha$

Definition: Fixed-Point Iteration for Max-Min Power Control

Standard Interference Function

Historical Note: Yates' Standard Interference Function Framework

The BHS Framework for Power Control with Imperfect CSI

Definition: Achievable Rate Under BHS Framework

BHS Max-Min Power Control with Imperfect CSI

Example: Joint Data and Pilot Power Optimization

Formulate the joint problem

Alternating optimization

Result

Common Mistake: Power Control Cannot Eliminate Pilot Contamination

Fractional Power Control in 3GPP NR

Comparison of Power Control Algorithms

Quick Check

Why This Matters: Power Control in Cell-Free Massive MIMO

Key Takeaway

Definition:
Fractional Power Control

Optimize over $lpha$

Definition:
Fixed-Point Iteration for Max-Min Power Control

Definition:
Achievable Rate Under BHS Framework