Iterative PoWER in Cooperative DS-CDMA

• The paper introduces Iterative PoWER as a joint optimization algorithm that minimizes global mean-squared error under quadratic power constraints using adaptive MMSE estimation.
• It alternates between updating linear receive filters and power allocation vectors, effectively managing nonconvex joint resource allocation challenges in cooperative DS-CDMA systems.
• Simulation benchmarks show enhanced capacity with up to 4 dB SNR improvement and up to 50% increase in user support compared to noncooperative schemes.

The Iterative PoWER (Power-and-Weight Efficient Relay) algorithm, as formalized in the JPAIS framework, is a constrained joint optimization procedure designed for cooperative DS-CDMA networks employing amplify-and-forward multi-hop relaying. It enables both global and individual power allocation across relays and optimizes linear receive filters for multiuser interference suppression under explicit power constraints. Iterative PoWER alternates between adaptive linear MMSE estimation of receiver weights and power levels, achieving effective interference mitigation and spectral efficiency, particularly under nonconvex parameter coupling.

1. Constrained Joint Optimization Formulation

Iterative PoWER addresses the fundamental joint resource allocation and receiver filter design in multiuser cooperative CDMA systems with $K$ users, $n_r$ relays, and $(n_r+1)M$-dimensional received vectors $r[i]$. The goal is to minimize the global mean-squared error subject to a quadratic power constraint. For global constraints, the optimization is

$$\min_{W[i],\,a_T[i]}\; E\big\{ \| b[i] - W^H[i] r[i] \|^2 \big\} \qquad \text{s.t.} \quad a_T^H[i]\,a_T[i] = P_T$$

where $W[i] \in \mathbb{C}^{(n_r+1)M \times K}$ is the linear receive matrix, $a_T[i]$ is the relay/user power allocation vector, and $P_T$ is the total transmit power allowed. Analogously, for per-user constraints,

$$\min_{w_k[i],\,a_k[i]}\; E\big\{ | b_k[i] - w_k^H[i] r[i] |^2 \big\} \qquad \text{s.t.} \quad a_k^H[i] a_k[i] = P_{A,k}$$

Lagrange multipliers are introduced, yielding unconstrained Lagrangians whose stationarity via KKT necessitates iterative solution for both filter weights and power vectors.

2. Constrained Linear MMSE Solutions

Explicit MMSE updates for both sets of parameters are derived. With power fixed, the optimal receive matrix is

$W_{\text{opt}} = R^{-1} P_{r b}$

where $R = E\{ r r^H \}$, $P_{r b} = E\{ r b^H \}$. Conversely, with $W$ fixed, the optimal power allocation vector is

$$a_{T,\text{opt}} = (R_{a_T} + \lambda_T I)^{-1} p_{a_T}$$

where $$R_{a_T} = E\{ B_T^H H_T^H C_T^H W W^H C_T H_T B_T \}$$ and $p_{a_T} = E\{ B_T^H H_T^H C_T^H W b \}$. For individual constraints, analogous expressions hold per user. Since $R$ and $R_{a}$ are mutually dependent on both $W$ and $a$, these estimates are alternated iteratively.

3. Iterative Algorithmic Structure

The algorithm proceeds as:

1. Initialization: Select initial $W[0]$, $a_T[0]$ through randomization or matched-filter methods, set adaptation forgetting factor $\alpha$.
2. Adaptive Filter Update: At each symbol/packet index $i$, update $W[i]$ using RLS equations:

$$k_r = \alpha^{-1} \Phi_{r}[i-1] r[i] / (1 + \alpha^{-1} r^H[i] \Phi_{r}[i-1] r[i])$$

$$\Phi_{r}[i] = \alpha^{-1} \Phi_{r}[i-1] - \alpha^{-1} k_r r^H[i] \Phi_{r}[i-1]$$

$W[i] = W[i-1] + k_r (b[i] - W^H[i-1] r[i])^H$

1. Power Allocation Update: Solve $R_{a_T}[i] a_T[i] = p_{a_T}[i]$ using RLS or conjugate gradient, normalize $a_T[i]$ to satisfy $a_T^H[i] a_T[i] = P_T$.
2. Optional Channel Adaptation: Update channel matrices via RLS.
3. Feedback: Relay updated $a_T[i]$ to sources/relays.
4. Iterate until convergence.

This procedure balances filter adaptation, relay power allocation, and channel learning under practical feedback and complexity budgets (Lamare, 2013).

4. Convexity Conditions and Nonconvexity Management

The cost function is generally nonconvex with respect to $(W,a)$. However, convexity can be enforced by setting sufficiently large power constraints:

$$P_T \geq \frac{m^H \Re\{ E[(b^* - W^H t) U_T] \} m} {m^H \Re\{ E[(W^H C H \beta_T) U_T] \} m} \quad \forall m \neq 0$$

This ensures the Hessian of the cost function is positive semi-definite, which avoids local minima and guarantees optimality for alternating updates. Pragmatic power budgets above this threshold permit reliable and efficient convergence, circumventing pitfalls typical to nonconvex joint estimation.

5. Computational Complexity

The per-iteration complexity is given by:

Subsystem	Complexity (Global Constraint)	Complexity (Individual Constraint)
Receive Filter (W)	$O(((n_r+1)M)^2)$	$O(((n_r+1)M)^2)$
Power Vector (a_T)	$O((K(n_r+1))^2)$	$O((n_r+1)^2)$ per user
Channel Matrix	$O((K(n_r+1)L)^2)$	$O(((n_r+1)L)^2)$ per user

RLS and CG are employed for adaptive filter/power/channel updates, which enables subquadratic scaling for modest $K$, $n_r$, and $M$ (Lamare, 2013).

6. Feedback Requirements

Feedback architecture is differentiated by constraint choice:

• Global PoWER (PoWER-GPC): Broadcasts $a_T$ (length $K(n_r+1)$) once per update, feedback cost $n_b K(n_r+1)$ bits per packet (with $n_b$ bits per quantized entry).
• Individual PoWER (PoWER-IPC): Each user sends $a_k$ (length $n_r+1$) to the base station; aggregate cost $n_b(n_r+1)$ bits/user.
• Empirically, $n_b=4$ quantization bits suffice for stability under slow fading, with update periodicity dictated by channel Doppler rate (Lamare, 2013).

7. Performance Benchmarks

Simulation confirms substantial performance and capacity gains for Iterative PoWER relative to classical noncooperative MMSE and equal-power cooperative schemes:

• Under static channels and perfect CSI, PoWER-GPC supports $2$–$3$ more users per chip-spread gain than cooperative interference suppression (CIS) at equal BER, and doubles capacity over noncooperative MMSE.
• At BER $=10^{-3}$, PoWER requires $\approx 2~\mathrm{dB}$ lower SNR than CIS and $\approx 4~\mathrm{dB}$ lower than noncooperative schemes.
• Adaptive RLS-based PoWER reaches $0.5~\mathrm{dB}$ from MMSE upper bound within $\approx 200$ symbols.
• Maintains approximately $1~\mathrm{dB}$ gain for Doppler rates up to $f_dT \approx 10^{-3}$, though higher Doppler necessitates more frequent power vector updates.
• Throughput ($NT=R(1-\text{BER})^P \log_2 M$) exceeds CIS by $10$–$20\%$ at high SNR.
• Robust to feedback errors up to $P_e \approx 10^{-3}$ before cooperative gain vanishes.

This suggests that Iterative PoWER is suitable for dense, interference-limited cooperative networks with stringent adaptability and feedback constraints.

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For technical developments, implementation frameworks, and simulation outcomes, the principal reference is "Joint Iterative Power Allocation and Linear Interference Suppression Algorithms in Cooperative DS-CDMA Networks" (Lamare, 2013).