Ferkans — Interactive Telecom Tutor

From Sum Power to Per-Antenna Constraints

All precoding designs so far assume a sum power constraint: $\sum_k p_k \|\mathbf{v}_{k}\|^2 \leq P_t$ . In practice, each antenna element has its own power amplifier (PA) with a maximum output level. A precoder that satisfies the sum power budget may still violate the per-antenna limit, causing PA clipping and nonlinear distortion. This section addresses the per-antenna power constraint (PAPC), which is the more realistic model for hardware-constrained systems.

Definition:
Per-Antenna Power Constraint (PAPC)

The per-antenna power constraint requires that the average power transmitted by the $m$ -th antenna element does not exceed $P_{\max}/N_t$ :

$\left[\sum_{k=1}^{K} p_k \mathbf{v}_{k} \mathbf{v}_{k}^{H}\right]_{mm} \leq \frac{P_{\max}}{N_t}, \qquad m = 1, \ldots, N_t.$

Equivalently, defining the transmit covariance $\mathbf{Q} = \sum_k p_k \mathbf{v}_{k} \mathbf{v}_{k}^{H}$ , the constraint is $\text{diag}(\mathbf{Q}) \preceq (P_{\max}/N_t) \mathbf{1}$ .

The sum power constraint $\text{tr}(\mathbf{Q}) \leq P_t$ allows all the power to concentrate on a few antennas. The PAPC prevents this, ensuring uniform power distribution across the array. This is more conservative: the PAPC feasible set is a strict subset of the sum power feasible set.

Definition:
ZF Precoding Under PAPC

Under PAPC, the ZF precoder design becomes a convex optimisation problem. We seek the precoding matrix that maximises the minimum user rate subject to per-antenna constraints:

$\max_{\{p_k, \mathbf{v}_{k}\}} \min_k \log_2(1 + p_k |\mathbf{h}_k^H \mathbf{v}_{k}|^2 / \sigma^2)$

subject to:

$\mathbf{h}_j^H \mathbf{v}_{k} = 0 \; (j \neq k), \qquad \left[\sum_k p_k \mathbf{v}_{k} \mathbf{v}_{k}^{H}\right]_{mm} \leq \frac{P_{\max}}{N_t}.$

This can be reformulated as a second-order cone program (SOCP) and solved efficiently.

Theorem: PAPC Duality

The downlink MU-MIMO problem with per-antenna power constraints has an uplink dual with a diagonal noise covariance:

$\max_{\text{DL precoders}} R_{\text{sum}}^{\text{DL}}(\text{PAPC}) = \max_{\text{UL powers}, \boldsymbol{\Lambda}} R_{\text{sum}}^{\text{UL}}(\boldsymbol{\Lambda})$

where $\boldsymbol{\Lambda} = \text{diag}(\lambda_1, \ldots, \lambda_{N_t})$ with $\lambda_m \geq 0$ are dual variables corresponding to the per-antenna constraints, satisfying $\sum_m \lambda_m = P_{\max}$ .

The classical MAC-BC duality assumes a single sum power constraint. With PAPC, the duality still holds but the "virtual uplink" has antenna-dependent noise levels $\lambda_m$ , reflecting the fact that heavily loaded antennas are effectively noisier. The dual variables $\lambda_m$ act as prices that redistribute power across antennas.

Proof

Lagrangian relaxation

Introduce Lagrange multipliers $\lambda_m \geq 0$ for each per-antenna constraint. The Lagrangian is

$\mathcal{L} = R_{\text{sum}} - \sum_{m=1}^{N_t} \lambda_m \left([\mathbf{Q}]_{mm} - \frac{P_{\max}}{N_t}\right).$

Rearranging, the penalty term $\sum_m \lambda_m [\mathbf{Q}]_{mm} = \text{tr}(\boldsymbol{\Lambda} \mathbf{Q})$ acts like noise with covariance $\boldsymbol{\Lambda}$ in the dual uplink channel.

Strong duality

Since the downlink problem is convex (the rate region is convex and the PAPC constraints are linear in $\mathbf{Q}$ ), strong duality holds by Slater's condition. The optimal $\boldsymbol{\Lambda}^{\star}$ and uplink powers can be found via a sub-gradient algorithm. $\blacksquare$

Iterative PAPC Precoder Design

Complexity: Each iteration costs

O(N_t^{2}K + N_t^{3})

for the MMSE computation. Typically converges in 10--20 iterations.

Input:

\mathbf{H}

, per-antenna limit

P_{\max}/N_t

, tolerance

\epsilon

1. Initialise dual variables:

\lambda_m = P_{\max}/({N_t}^2)

for all

m

2. Repeat:

a. Form

\boldsymbol{\Lambda} = \text{diag}(\lambda_1, \ldots, \lambda_{N_t})

b. Solve the dual uplink problem: compute MMSE receiver with noise

covariance

\boldsymbol{\Lambda}

c. Transform uplink solution to downlink precoders via MAC-BC transformation

d. Update

\lambda_m \leftarrow \lambda_m \cdot [\mathbf{Q}]_{mm} / (P_{\max}/N_t)

for all

m

(sub-gradient step)

e. Normalise:

\lambda_m \leftarrow \lambda_m \cdot P_{\max} / \sum_m \lambda_m

3. Until

\max_m |[\mathbf{Q}]_{mm} - P_{\max}/N_t| < \epsilon

Output: PAPC-compliant precoding vectors and power allocations

Example: Rate Loss from Per-Antenna Constraints

Compare the sum rate of RZF precoding under (a) sum power constraint $P_t = 10$ dB and (b) per-antenna constraint $P_{\max}/N_t$ (same total budget) for $N_t = 8$ , $K = 4$ , and i.i.d. Rayleigh channels. Quantify the rate loss.

Solution

Sum power RZF

With optimal $\alpha = K\sigma^2/P_t = 4/10 = 0.4$ , Monte Carlo gives $R_{\text{sum}}^{\text{SPC}} \approx 18.2$ bits/s/Hz.

Per-antenna RZF

After the iterative PAPC algorithm, the rate is $R_{\text{sum}}^{\text{PAPC}} \approx 17.4$ bits/s/Hz.

Analysis

The per-antenna constraint causes a rate loss of about 0.8 bits/s/Hz (4.4%). The loss is small here because with $N_t = 8$ antennas, the RZF precoder already distributes power relatively uniformly. The loss increases with $K/N_t$ and with spatially correlated channels that concentrate energy on specific antenna elements.

⚠️Engineering Note

PAPR and Precoding in Practice

The peak-to-average power ratio (PAPR) of the precoded signal determines the PA operating point. Key practical considerations:

PA back-off: If the PAPR of $\mathbf{x} = \mathbf{W}\mathbf{s}$ is high, the PA must operate well below its saturation point, reducing efficiency. MRT tends to have lower PAPR than ZF because ZF can create large peaks when nulling strongly correlated channels.
Constant-envelope precoding: For very low-cost hardware (e.g., 1-bit DACs), the precoded signal is constrained to $|x_m| = \text{const}$ . This is an extreme form of PAPC that requires specialised algorithms (e.g., SQUID).
Digital predistortion (DPD): Modern base stations apply DPD to linearise the PA, relaxing the PAPC somewhat. The precoder design should account for the DPD model for best performance.

Practical Constraints

•
PA output power: typically 2--5 W per element at sub-6 GHz
•
Efficiency drops from ~50% at saturation to ~25% with 6 dB back-off
•
1-bit DAC systems: $|x_m| \in \{+1, -1\}$ (no amplitude modulation)

🔧Engineering Note

Computational Complexity of Precoding

The matrix operations for precoding must execute within the channel coherence time. For 5G NR at 3.5 GHz with 30 kHz SCS, the slot duration is 0.5 ms:

MRT: $O(N_tK)$ per subcarrier — negligible cost.
ZF/RZF: $O(N_tK^{2} + K^{3})$ per subcarrier. With $N_t = 64$ , $K = 16$ , $N_{\text{sc}} = 3276$ (100 MHz BW), this is $\sim 10^9$ operations per slot.
PAPC iterative: 10--20x the ZF cost due to iterations.

Modern baseband ASICs (e.g., Ericsson Silicon) handle this in real time. For cell-free systems with distributed processing, the per-AP cost is much lower because each AP has a small local channel matrix.

Common Mistake: Ignoring Per-Antenna Constraints

Mistake:

Designing the precoder under a sum power constraint and then clipping per-antenna power to satisfy PA limits.

Correction:

Post-hoc clipping destroys the ZF property and introduces distortion that acts as additional interference. The correct approach is to include the PAPC in the precoder optimisation from the start, either via the iterative algorithm or by using the sum power precoder as initialisation and projecting onto the PAPC feasible set.

Quick Check

How does the per-antenna power constraint affect the ZF precoder compared to the sum power constraint?

It always reduces the sum rate

It has no effect when $N_t \gg K$

It always increases the sum rate

It only matters for MRT, not ZF

Correction:

It always reduces the sum rate

The PAPC feasible set is a strict subset of the sum power feasible set (since any PAPC-feasible point also satisfies the sum power constraint $\text{tr}(\mathbf{Q}) \leq P_{\max}$ ). Therefore the maximum achievable rate cannot increase.

Per-Antenna Power Constraint (PAPC)

A constraint requiring each antenna element's average transmit power to not exceed a specified limit, reflecting the hardware reality that each antenna has its own power amplifier with finite maximum output.

Key Takeaway

Per-antenna power constraints reflect the hardware reality of individual power amplifiers. They make the precoder design a convex optimisation problem that can be solved via uplink-downlink duality with antenna-dependent noise levels. The rate loss relative to sum power constraints is typically small (a few percent) but must be accounted for in practical system design.

Per-Antenna Power Constraints

From Sum Power to Per-Antenna Constraints

Definition: Per-Antenna Power Constraint (PAPC)

Definition: ZF Precoding Under PAPC

Theorem: PAPC Duality

Lagrangian relaxation

Strong duality

Iterative PAPC Precoder Design

Example: Rate Loss from Per-Antenna Constraints

Sum power RZF

Per-antenna RZF

Analysis

PAPR and Precoding in Practice

Computational Complexity of Precoding

Common Mistake: Ignoring Per-Antenna Constraints

Quick Check

Per-Antenna Power Constraint (PAPC)

Key Takeaway

Definition:
Per-Antenna Power Constraint (PAPC)

Definition:
ZF Precoding Under PAPC