SVD Water Filling in BD-RIS MIMO Systems

• SVD Water Filling is a method that integrates SVD-based channel diagonalization with water-filling power allocation to decouple MIMO channels into independent subchannels.
• It provides a closed-form, one-shot solution for optimizing RIS phase matrices, enabling precise capacity characterization in high-dimensional wireless scenarios.
• The technique achieves optimal power allocation by dynamically assigning power based on subchannel gains, often matching uniform power allocation performance in large RIS arrays.

SVD Water Filling (SVD WF) refers to the combined strategy of using the singular value decomposition (SVD) to diagonalize a multi-input multi-output (MIMO) channel, followed by water-filling power allocation across the resulting parallel spatial subchannels. In cascaded beyond-diagonal reconfigurable intelligent surface (BD-RIS) assisted MIMO systems, SVD WF achieves capacity-optimal transmission by jointly optimizing the unitary phases of multiple RIS layers and the transmit covariance at the base station. This procedure yields a closed-form, one-shot solution for BD-RIS phase matrices and enables precise capacity characterizations in a range of high-dimensional wireless scenarios, including regimes where simple uniform power allocation approaches optimality.

1. System Model and Channel Architecture

The SVD WF strategy is applied in a MIMO setting where the transmitter (base station, BS) with $M$ antennas communicates with $K$-antenna receivers or $K$ single-antenna users. The system incorporates $L$ cascaded BD-RISs, each with $N_l$ elements. Each RIS layer applies a unitary phase-shift matrix $\Phi_l\in\mathbb{C}^{N_l\times N_l}$, such that $\Phi_l \Phi_l^H = I_{N_l}$.

The composite end-to-end channel, denoted as $H_{\rm cas} \in \mathbb{C}^{K\times M}$, is constructed as

$$H_{\rm cas} = H_L\,\Phi_L\,H_{L-1}\,\Phi_{L-1}\,\cdots\,H_1\,\Phi_1\,H_0,$$

where $H_0$ is the BS-to-first-RIS channel, $H_l$ the matrices between cascaded RISs, and $H_L$ the final RIS-to-user channel.

The BS transmits $x = \sum_{i=1}^R \sqrt{p_i}\,v_i\,s_i$, where $s_i \sim \mathcal{CN}(0,1)$, $\Vert v_i\Vert_2 = 1$, $\sum_i p_i \leq P_t$, and $$R = \operatorname{rank}(H_{\rm cas}) \leq \min(K, M)$$. The received vector is $y = H_{\rm cas}\,x + n$, with $n\sim \mathcal{CN}(0,\sigma_n^2 I_K)$. The instantaneous channel capacity under transmit covariance $Q = \mathbb{E}[x x^H]$ is

$$C = \log_2 \Big|I_K + H_{\rm cas}\, Q\, H_{\rm cas}^H \Big|.$$

2. SVD-Based Channel Diagonalization

The central step in SVD WF is the factorization

$$H_{\rm cas} = U_{\rm cas}\, \Sigma_{\rm cas}\, V_{\rm cas}^H,$$

where $U_{\rm cas}\in\mathbb{C}^{K\times R}$ contains left singular vectors, $V_{\rm cas}\in\mathbb{C}^{M\times R}$ the right singular vectors, and $$\Sigma_{\rm cas} = \mathrm{diag}(\sigma_1,\dots,\sigma_R)$$ lists ordered nonzero singular values.

By selecting precoding directions $v_i = V_{\rm cas}(:,i)$, the MIMO channel is decomposed into $R$ independent subchannels $\tilde y_i = \sigma_i s_i + \tilde n_i$. This diagonalization allows the capacity expression to decouple and simplifies the power allocation task.

3. Water-Filling Power Allocation

Optimal power distribution across modes follows the classical water-filling principle. The problem is to maximize

$$C = \sum_{i=1}^R \log_2\left(1+\frac{p_i \sigma_i^2}{\sigma_n^2}\right)$$

subject to $\sum_{i=1}^R p_i \leq P_t$, $p_i \geq 0$. The solution is

$p_i = (\mu - \tfrac{\sigma_n^2}{\sigma_i^2})^+,$

where $\mu$ is determined such that the power constraint is tight. This assigns more power to subchannels with higher singular value gains $\sigma_i^2/\sigma_n^2$, and may deactivate low-gain modes ($p_i=0$). The water-filling level $\mu$ can be computed by sorting $\sigma_n^2/\sigma_i^2$ and solving a piecewise-linear equation or using iterative index-based search algorithms as in (Xing et al., 2018).

A dynamic algorithm constructs the active set of subchannels and iteratively eliminates those that would receive negative power, ensuring an optimal index assignment with low computational complexity.

4. Closed-Form RIS Phase-Shift Design

With SVD-based diagonalization, the optimization over the RIS phase matrices $\{\Phi_l\}$ admits a closed-form one-shot solution. For each RIS, let

$$H_{l-1} = U_{l-1}\, S_{l-1}\, V_{l-1}^H,\qquad H_{l} = U_{l}\, S_{l}\, V_{l}^H.$$

Then the optimal phase-shift matrix aligns principal directions: $\Phi_l^* = V_l\, U_{l-1}^H, \qquad l=1,\dots,L.$ For $l=1$, use $H_0 \sqrt{Q^*}$ instead of $H_0$, with $$Q^*=V_{\rm cas}\,{\rm diag}(p_1,\dots,p_R)\,V_{\rm cas}^H$$. This alignment maximizes $\operatorname{tr}(H_{\rm cas} Q^* H_{\rm cas}^H)$ and thus the channel determinant, achieving jointly optimal end-to-end transmission.

5. Comparison with Uniform Power Allocation

Uniform power allocation (UPA) serves as a practical alternative to SVD WF. Here, one assigns $Q = (P_t/M) I_M$, and the RIS phase matrices

$\Phi_l^* = V_l\,U_{l-1}^H\qquad (l=1,\dots,L)$

remain optimal. The system capacity simplifies to

$$C_{\rm UPA} = \sum_{i=1}^R \log_2 \Big(1 + \tfrac{P_t}{M} \sigma_i^2\Big).$$

In the limit as $N_l \to \infty$ for all RISs, the singular values $\{\sigma_i\}$ concentrate and become nearly equal. Thus, $C_{\rm SVD-WF} \approx C_{\rm UPA}$ for sufficiently large RIS arrays, indicating that UPA is nearly optimal in this regime and transmitter complexity can be significantly reduced (Manasa et al., 8 Nov 2025).

6. Capacity Characterization and High-SNR Analysis

At high SNR, the capacity under UPA (and approximately under SVD–WF for large RIS arrays) is

$$\log_2 \det \Big(I + \frac{P_t}{M} \Sigma_{\rm cas}^2 \Big) \approx R\log_2 \Big(\frac{P_t}{M}\Big) + \sum_{i=1}^R \log_2( \sigma_i^2).$$

For i.i.d. Rayleigh fading, the ergodic capacity can be approximated via Wishart matrix statistics, with each cascaded RIS introducing a multiplicative array gain. Closed-form approximations reveal that $L$ BD-RISs of size $N$ yield an $O(N^{2L})$ SNR scaling.

The high-SNR ergodic-capacity approximation is

$$\mathbb{E}[C] \approx \min(K,M)\log_2\frac{P_t}{M} + \sum_{l=0}^L \Big[ R_l + \log_2|\Sigma_l| + \frac{1}{\ln 2} \sum_{i=1}^{R_l} \psi\Big( \frac{\nu_l-i+1}{2} \Big) \Big],$$

where $(R_l,\nu_l,\Sigma_l)$ are the Wishart parameters per hop. For large $N$, this reduces to

$$\tilde C = \min(K, M) \log_2 \frac{P_t}{M} + 2\, \min(K, M)\, L \log_2 N + \dots$$

which illuminates the joint impact of system dimensions, SNR, number of RIS layers, and element count on the achievable rate (Manasa et al., 8 Nov 2025).

7. Algorithmic Considerations and Robust Extensions

The index-based water-filling algorithm presented in (Xing et al., 2018) provides a practical, unified method for solving the water-filling optimization. At each iteration, the water level $\mu$ is computed for the current active set of subchannels. Any channel for which $p_i < 0$ is removed from the set, and the process is repeated until non-negativity holds for all allocated powers. The approach directly generalizes to robust water-filling, where effective noise variance may couple power allocation decisions, and only requires derivatives $f_i'(p_i)$ and their monotonic inverses $g_i(\mu)$.

The computational complexity of these unified algorithms is $O(R^2)$, with further improvements possible using sorting or structure in the subchannel gains. The SVD WF method is also adaptable to robust designs under imperfect channel state information (CSI), box constraints, and fairness-driven constraints via simple index adjustment and corresponding updates of effective SNR per subchannel.

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In summary, SVD Water Filling provides the capacity-optimal solution for cascaded BD-RIS assisted MIMO systems by jointly diagonalizing the composite channel and allocating transmit power via the water-filling rule. The approach yields closed-form solutions for both transmit and RIS parameters, admits efficient index-based algorithmic implementation, and establishes a clear relationship between system array size, transmit power, and achievable spectral efficiency. In the asymptotic regime, uniform power allocation is nearly optimal, simplifying hardware and algorithmic requirements without significant loss in performance (Manasa et al., 8 Nov 2025, Xing et al., 2018).