---
title: Multi-Norm Beamforming Techniques
url: https://www.emergentmind.com/topics/multi-norm-beamforming
type: topic
---

# Multi-Norm Beamforming Techniques

Multi-norm beamforming denotes beamforming formulations in which several norms are used to model uncertainty, regularize the beamformer output, or both. In the cited literature, this includes robust adaptive beamforming for a general-rank signal model with matrix induced \(\ell_{p,q}\)-norm uncertainty, downlink 3D-MIMO beamforming with \(\ell_1\)-norm bounded CSI uncertainty and extensions to other vector and mixed norms, and speech enhancement beamforming that minimizes output power together with an \(\ell_1\)-norm penalty under a distortionless constraint [2103.13014] [1812.07492] [2507.18350]. The common structure is that norm choice changes either the geometry of the admissible perturbations or the sparsity profile encouraged at the beamformer output.

## 1. Norms and uncertainty models

For \(X\in\mathbb C^{M\times N}\), the induced matrix \(\ell_{p,q}\)-norm is defined by
\[
\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.
\]
Equivalently,
\[
\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.
\]
In Huang and Vorobyov’s robust adaptive beamforming formulation, this norm is used to constrain matrix perturbations introduced into a factorization of the presumed desired-signal covariance. For the least-squares problem
\[
\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,
\]
the optimal value admits the closed form
\[
\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p
=\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},
\]
with equality obtained by a suitable rank-one construction of the worst-case \(\Delta^\star\) when \(\|Xy-b\|_p\ge\eta\,\|y\|_q\) [2103.13014].

In vector-uncertainty models for downlink beamforming, the same norm logic appears through channel-error sets. For user \(k\), the \(\ell_1\)-bounded uncertainty set is
\[
\mathcal H_k^1=\bigl\{\,h_k=\hat h_k+\delta_k\;\big|\;\|\delta_k\|_1\le\epsilon_k\bigr\}.
\]
The same framework also admits \(\ell_\infty\)-norm uncertainty,
\[
H_k^\infty=\{\hat h_k+\delta_k \mid \|\delta_k\|_\infty \le \epsilon_k\},
\]
mixed \(\ell_1/\ell_2\)-norm uncertainty,
\[
H_k^{1,2}=\{\hat h_k+\delta_k \mid \|\delta_k\|_1 \le \epsilon_k^1,\;\|\delta_k\|_2 \le \epsilon_k^2\},
\]
and intersections of ellipsoids or hyper-rectangles. For a general \(p\)-norm bound \(\|\delta\|_p\le\epsilon\), the dual norm \(q\) satisfies \(1/p+1/q=1\), and the worst-case inner product obeys
\[
\sup_{\|\delta\|_p\le\epsilon} |\delta^H w| = \epsilon\cdot\|w\|_q,
\]
which is the basic device used to translate norm-bounded uncertainty into conic constraints [1812.07492].

## 2. General-rank robust adaptive beamforming

For a general-rank signal model, the presumed signal covariance is written as
\[
\widehat R_s=A^H A,
\]
and the uncertainty is introduced through
\[
A\mapsto A+\Delta,\qquad \|\Delta\|_{2,q}\le\eta_q.
\]
Applying the closed-form least-squares result yields the inner worst-case minimization
\[
\min_{\|\Delta\|_{2,q}\le\eta_q}\;\|(A+\Delta)^H w\|_2
=\max\{\|A^H w\|_2-\eta_q\,\|w\|_q,\;0\}.
\]
The associated worst-case SINR maximization is
\[
\max_w\;\min_{\|\Delta\|_{2,q}\le\eta_q}\;
\frac{\|(A+\Delta)^H w\|_2^2}
{w^H(\widehat R_{i+n}+ \gamma I)\,w},
\]
which is reformulated, after dropping the zero-clamp since the optimum is nonnegative, as
\[
\max_{w}\;\frac{\bigl(\|A^H w\|_2 - \eta_q\,\|w\|_q\bigr)^2}
{w^H(\widehat R_{i+n}+ \gamma I)\,w}
\;\;\approx\;
\max_{w}\;\bigl\{\|A^H w\|_2 - \eta_q\,\|w\|_q\bigr\}
\quad\text{s.t.}\quad
w^H(\widehat R_{i+n}+\gamma I)\,w\le1.
\]
With \(u=A^H w\), the final problem is
\[
\max_{w,t}\;\;t
\quad\text{s.t.}\quad
\|u\|_2 \ge t +\eta_q\,\|w\|_q,
\quad
w^H(\widehat R_{i+n}+\gamma I)\,w\le1.
\]
This recasts worst-case SINR maximization as the maximization of the difference between an \(\ell_2\)-norm function and an \(\ell_q\)-norm function under a convex quadratic constraint [2103.13014].

The same paper studies a generalized RAB problem in which the \(\ell_2\)-term is replaced by an \(\ell_p\)-term and the matrix uncertainty is bounded by \(\|\Delta\|_{p,q}\le\eta_{p,q}\). The inner result becomes
\[
\min_{\|\Delta\|_{p,q}\le\eta_{p,q}}\|(A+\Delta)^H w\|_p
=
\max\{\|A^H w\|_p-\eta_{p,q}\,\|w\|_q,\;0\},
\]
and the corresponding design is
\[
\max_{w}\;\|A^H w\|_p-\eta_{p,q}\,\|w\|_q
\quad\text{s.t.}\quad
w^H(\widehat R_{i+n}+\gamma I)w\le1.
\]
The resulting family of beamformers depends explicitly on the choice of \((p,q)\) rather than only on the conventional Frobenius-norm case \(p=q=2\) [2103.13014].

## 3. Sequential SOCP approximation and norm-pair selection

For any rational \(q\ge1\), \(\|w\|_q\) has an SOC-representable epigraph, and \(\|A^H w\|_2\) is handled through a lower-affine approximation at the current iterate \(w^{(k)}\):
\[
\|A^H w\|_2\ge
\frac{\Re\bigl(w^{(k)H}A\,A^H w\bigr)}{\|A^H w^{(k)}\|_2}
=
\frac{\Re\bigl((A^H w^{(k)})^H\,(A^H w)\bigr)}{\|A^H w^{(k)}\|_2}.
\]
Each SOCP subproblem is then
\[
\max_{w,t}\;\;t
\quad\text{s.t.}\quad
\frac{\Re\bigl((A^H w^{(k)})^H(A^H w)\bigr)}{\|A^H w^{(k)}\|_2}
\ge t + \eta_q\,\|w\|_q,
\qquad
w^H(\widehat R_{i+n}+\gamma I)\,w\le1,
\]
which is fully SOC-representable for any rational \(q\ge1\). The iterative scheme initializes a feasible \(w^{(0)}\), solves the SOCP to obtain \((w^{(k+1)},t^{(k+1)})\), increments \(k\), and stops when
\[
t^{(k)}-t^{(k-1)}\le\varepsilon.
\]
The sequence of objective values \(\{t^{(k)}\}\) is nondecreasing, and under mild boundedness assumptions the iterates converge to a locally stationary point. The stated complexity per iteration is that solving an SOCP in \(O(N)\) variables with \(O(N)\) second-order cones of dimension up to \(O(N)\) costs on the order of \(O(N^3)\) or better using interior-point methods [2103.13014].

The generalized \(\ell_p\)-minus-\(\ell_q\) formulation uses the same sequential-SOCP approach provided \(q\), and \(p\) if one wants to linearize \(\|A^H w\|_p\), are rational. The paper describes a practical norm-pair selection rule: one fixes a small finite candidate set of pairs \((p,q)\), for example \(\{(1,1),(2,1),(\infty,1)\}\), runs the sequential-SOCP algorithm for each pair, computes the actual array-output SINR over held-out snapshots, and selects the \((p,q)\) whose beamformer achieves the largest actual SINR. This suggests that norm choice is treated as a model-selection variable rather than as a fixed convention [2103.13014].

## 4. Downlink 3D-MIMO beamforming under vector and mixed norms

In downlink 3D-MIMO, robust beamforming is posed as minimization of total transmit power under worst-case SINR constraints over the uncertainty set \(\mathcal H_k^1\). With beamformers \(w_1,\dots,w_K\),
\[
\mathrm{SINR}_k(h_k,\{w_j\})=
\frac{|h_k^H w_k|^2}{\sum_{j\neq k}|h_k^H w_j|^2+\sigma_n^2},
\]
and the robust design is
\[
\begin{aligned}
&\min_{\{w_k\}}
&&\sum_{k=1}^K \|w_k\|_2^2 \\
&\text{s.t.}
&&\min_{h_k\in\mathcal H_k^1}\;
\mathrm{SINR}_k(h_k,\{w_j\})\ge\gamma_k,\quad k=1,\dots,K.
\end{aligned}
\]
The reformulation introduces \(p\ge0\), auxiliary variables \(\{t_k\}\), and the constant \(\beta_k=\sqrt{1+1/\gamma_k}\), so that
\[
\sum_{k=1}^K\|w_k\|_2^2\le p,
\qquad
\|[\,h_k^H W,\;\sigma_n\,]\|_2 \le \beta_k\,|h_k^H w_k|.
\]
Using \(\eta:=\max_k\|w_k\|_\infty\), one obtains
\[
\Re(\hat h_k^H w_k)-\epsilon_k\eta \ge t_k/\beta_k,
\qquad
\|w_k\|_\infty\le\eta.
\]
With \(W=[w_1\ \cdots\ w_K]\), row vectors \(v(n)\), and \(\alpha\ge\max_n\|v(n)\|_2\), one further gets
\[
\bigl\|\,[\,\hat h_k^H W,\;\alpha\,\epsilon_k,\;\sigma_n]\,\bigr\|_2 \le t_k,
\qquad
\|v(n)\|_2\le\alpha.
\]
All constraints are linear or second-order-cone constraints, so the final problem is an SOCP solvable efficiently by a standard SOCP solver such as CVX [1812.07492].

The same work makes the multi-norm generalization explicit. For a general \(p\)-norm uncertainty bound \(\|\delta\|_p\le\epsilon\), the “numerator” constraint becomes
\[
\Re(\hat h_k^H w_k)-\epsilon\,\|w_k\|_q\ge t_k/\beta_k,
\]
with the corresponding dual norm \(q\). The paper also notes that \(\|w_k\|_1\) can be represented via linear constraints and a positive slack, \(\|w_k\|_\infty\) is polyhedral and can be recast with auxiliary variables \(\alpha\), and mixed \(\ell_1/\ell_2\) constraints lead to conic combinations of rotated-SOC and SOC constraints. The final result is that, for any desired \(p\)-norm or mixed-norm uncertainty, each worst-case term translates into either a linear, an SOC, or a rotated-SOC constraint, so the robust beamforming problem remains in the class of conic programs solvable in polynomial time [1812.07492].

## 5. Speech enhancement with power-plus-sparsity beamforming

In Qin et al., the beamforming stage follows a dual-path MCLP dereverberation step. At each frame-frequency bin, the early estimate is
\[
\hat{x}(n,\omega)
=
y(n,\omega)-\hat G_t^H(\omega)\,\tilde y_t(n,\omega)-\hat G_f^H(n)\,\tilde y_f(n,\omega)
\in\mathbb C^{M\times1}.
\]
The beamformer seeks \(w\in\mathbb C^{M\times1}\) that minimizes output power together with an \(\ell_1\)-norm penalty while preserving the target direction \(\theta_s\):
\[
\begin{aligned}
\hat w
&=\arg\min_{w\in\mathbb C^M}
\sum_{n=1}^N
\Bigl(
\|\,w^H\,\hat x(n,\omega)\|_2^2
+\lambda_{w}\,\|\,w^H\,\hat x(n,\omega)\|_1
\Bigr) \\
&\quad\text{subject to}\quad
w^H\,a(\theta_s)=1.
\end{aligned}
\]
Here \(\|w^H\hat x\|_2^2\) is the output power term, \(\|w^H\hat x\|_1\) is the sample-wise \(\ell_1\) norm over the complex output, \(\lambda_w>0\) weights the sparsity penalty, and \(a(\theta_s)\) is the steering vector of the desired source. The paper states that speech STFT frames are sparse in magnitude, and that the additional \(\ell_1\) penalty encourages the beamformer output to concentrate energy in a few TF bins, thereby further suppressing diffuse noise and small residual reverberation [2507.18350].

The optimization is handled by ADMM. Auxiliary variables \(z_w(n)\) are introduced as scalar copies of the beamformer output,
\[
z_w(n)=w^H\,\hat x(n,\omega),
\quad n=1,\dots,N,
\]
with dual variables \(\eta_w(n)\in\mathbb C\) for these equalities and \(\eta_1\) for the distortionless constraint. The augmented Lagrangian is
\[
\begin{aligned}
\mathcal{L}(w,\{z_w(n)\},\{\eta_w(n)\},\eta_1)
&=\sum_{n=1}^N
\Bigl\{
\|w^H\hat x(n)\|_2^2
+\lambda_w\,\|z_w(n)\|_1
+\Re\bigl[\eta_w(n)^*(w^H\hat x(n)-z_w(n))\bigr] \\
&\qquad\qquad
+\tfrac1{2\rho_w}\,|w^H\hat x(n)-z_w(n)|^2
\Bigr\}
+\Re\bigl[\eta_1^*(w^Ha(\theta_s)-1)\bigr]
+\tfrac1{2\rho_1}\,|w^Ha(\theta_s)-1|^2 .
\end{aligned}
\]
The \(w\)-update is a small quadratic program with one linear constraint. Defining
\[
R = \sum_{n=1}^N \Bigl(1+\tfrac1{2\rho_w}\Bigr)\,\hat x(n)\,\hat x(n)^H
+\tfrac1{2\rho_1}\,a(\theta_s)\,a(\theta_s)^H,
\]
and
\[
b = \sum_{n=1}^N
\hat x(n)\Bigl(\tfrac1{2\rho_w}\,z_w(n)^*-\tfrac12\,\eta_w(n)^*\Bigr)
+a(\theta_s)\Bigl(\tfrac1{2\rho_1}-\tfrac12\,\eta_1^*\Bigr),
\]
one obtains
\[
w^{(l+1)} = R^{-1}b
\quad\text{followed by a simple rank-1 adjustment to enforce }w^Ha=1.
\]
The \(z\)-update is complex soft-thresholding,
\[
z_w^{(l+1)}(n)=\mathcal S_{\lambda_w/\mu_w}
\Bigl(
w^{(l+1)H}\hat x(n)-\tfrac1{\mu_w}\eta_w^{(l)}(n)
\Bigr),
\qquad
\mu_w=1/\rho_w,
\]
and the dual updates are
\[
\eta_w^{(l+1)}(n)=\eta_w^{(l)}(n)+\gamma_w\bigl(w^{(l+1)H}\hat x(n)-z_w^{(l+1)}(n)\bigr),
\]
\[
\eta_1^{(l+1)}=\eta_1^{(l)}+\gamma_1\bigl(w^{(l+1)H}a(\theta_s)-1\bigr).
\]
Since the objective is convex in \(w\) and the constraints are affine, the paper states that ADMM converges to the global optimum under standard assumptions. In the complete pipeline, Stage 1 is dual-path MCLP, which removes late reverberation by minimizing \(\ell_2+\ell_1\) of the dereverberated multichannel signals via PALM, and Stage 2 is multi-norm beamforming, which removes spatial noise and further sharpens sparsity by minimizing output power plus \(\ell_1\) under a distortionless constraint via ADMM [2507.18350].

## 6. Reported performance, norm trade-offs, and scope

The reported numerical results emphasize that norm choice affects both beamformer quality and computational profile. In the induced \(\ell_{p,q}\)-norm RAB experiments, for \(N=10\), \(T=50\), and an Intel Xeon E5-1620 v3 @3.5 GHz, the \(\ell_{2,q}\)-design with \(q=1\) requires \(\sim 0.12\) s per trial, \(q=2\) requires \(\sim 0.18\) s, and \(q=4\) requires \(\sim 0.35\) s. Under \( \mathrm{INR}=10\) dB and SNR from \(10\) dB to \(40\) dB, the \(q=1\) beamformer outperforms \(q=2\) by \(1\)–\(2\) dB SINR gain across SNR; the generalized design \((p,q)=(\infty,1)\) yields a further \(\approx 0.5\) dB gain over \((2,1)\); and the CPU–SINR trade-off reported is that \((2,1)\) is fastest with second-best SINR, whereas \((\infty,1)\) is slightly slower but gives the best SINR [2103.13014].

In the 3D-MIMO study, the \(\ell_1\)-bounded uncertainty model is reported to consume less beamforming power than the conventional spherical uncertainty under the same SINR thresholds. At \(\epsilon=0.5\) and \(\gamma_k=3\) dB, the \(\ell_1\)-robust design needs approximately \(2.1\) units of power, whereas spherical \(\ell_2\)-robust needs approximately \(3.0\), with the perfect-CSI baseline flat at approximately \(1.6\). For \(\epsilon=0.2\) and SINR target increasing from \(0\) to \(10\) dB, the \(\ell_2\)-robust power grows to approximately \(5.2\) at \(10\) dB, the \(\ell_1\)-robust design to approximately \(3.8\), and the perfect-CSI baseline to approximately \(2.5\) [1812.07492].

In the speech-enhancement setting, the multi-norm beamformer is reported to consistently outperform both the cascade WPE+MVDR and the unified WPD beamformer, which omits the extra \(\ell_1\) term, in PESQ and SI-SNR across a wide range of reverberation times \(T_{60}\) and SNRs, particularly in high reverberation scenarios [2507.18350].

| Setting | Norm mechanism | Reported outcome |
|---|---|---|
| General-rank RAB | \(\ell_2-\ell_q\) and generalized \(\ell_p-\ell_q\) objective under matrix induced norm uncertainty | Actual array-output SINR and CPU-time vary with \((p,q)\) |
| Downlink 3D-MIMO | \(\ell_1\)-bounded CSI uncertainty and extensions to other vector and mixed norms | Lower transmit power than spherical \(\ell_2\)-uncertainty for the same worst-case SINR |
| Speech enhancement | Output power plus \(\ell_1\)-norm penalty under a distortionless constraint | Better PESQ and SI-SNR than WPE+MVDR and WPD |

A common simplification is to equate multi-norm beamforming solely with replacing an \(\ell_2\) or Frobenius uncertainty bound by \(\ell_1\). The cited works show a broader technical scope: multiple norms may appear in an induced matrix error model, in vector or mixed uncertainty sets, or directly in the beamformer objective as a joint power-and-sparsity criterion. The solver class also depends on the formulation rather than on the phrase “multi-norm” itself: sequential SOCP approximation is used for the nonconvex \(\ell_p-\ell_q\) RAB family, direct SOCP reformulation is used for the 3D-MIMO robust design, and ADMM is used for the speech-enhancement beamformer. This suggests that “multi-norm beamforming” is best understood as a family of norm-parameterized beamforming designs rather than as a single optimization template.

Source: https://www.emergentmind.com/topics/multi-norm-beamforming