Minimum Power Distortionless Response Method

• MPDR is a beamforming technique that minimizes the output power while ensuring an undistorted target signal through a closed-form solution derived from spatial covariance inversion.
• Extensions of MPDR, including multi-frame, wideband, and spherical harmonic approaches, enhance its performance in denoising and dereverberation across diverse signal processing scenarios.
• Advanced adaptations leverage sparse priors and iterative reweighting techniques to improve robustness and performance in real-time applications like speech enhancement and acoustic echo suppression.

The Minimum Power Distortionless Response (MPDR) method is a fundamental technique in array signal processing, speech enhancement, and beamforming, offering optimal noise and interference suppression under the strict condition of undistorted target signal preservation. MPDR is rigorously formulated as a constrained quadratic minimization, leading to a closed-form solution via spatial covariance inversion and steering vector normalization. Its importance extends from narrowband arrays to multi-frame, spherical harmonic, and convolutional joint denoising-dereverberation regimes, serving as a core component in modern unified filter designs and neural beamforming pipelines.

1. Mathematical Foundations of the MPDR Beamformer

The MPDR beamformer seeks a linear filter $\mathbf{w} \in \mathbb{C}^M$ that acts on a noisy or reverberant multichannel observation $\mathbf{x}_t = \mathbf{v}\,s_t + \mathbf{n}_t$, where $\mathbf{v}$ is the steering vector for the signal of interest (SOI), $s_t$ is the SOI, and $\mathbf{n}_t$ models noise (and possibly late reverberation). The MPDR criterion is to minimize the output power subject to a distortionless gain requirement: $$\min_{\mathbf{w}} \; \mathbf{w}^{\mathrm{H}} \Phi_x \mathbf{w} \quad \text{s.t.} \;\; \mathbf{w}^{\mathrm{H}} \mathbf{v} = 1,$$ where $$\Phi_x = \sum_t \mathbf{x}_t\mathbf{x}_t^{\mathrm{H}}$$ is the empirical spatial covariance. Applying Lagrange multipliers gives the closed-form solution: $$\mathbf{w}_{\mathrm{MPDR}} = \frac{\Phi_x^{-1}\mathbf{v}}{\mathbf{v}^{\mathrm{H}}\Phi_x^{-1}\mathbf{v}},$$ which generalizes directly to noisy and mixed signal scenarios (Nakatani et al., 2018, Nakatani et al., 2019). This solution guarantees unaltered passage of the SOI and aggressive suppression of noise, interference, and (in extensions) late reverberation components.

2. Extensions: Multi-Frame, Wideband, and Spherical Harmonic Domains

The classic MPDR formulation is extended to handle temporal, spectral, and spatial diversity:

• Multi-Frame MVDR/MPDR: In acoustic echo suppression and speech enhancement, multi-frame data vectors stack temporal STFT frames to exploit inter-frame correlation, yielding a virtual array over time. The corresponding MPDR filter:

$$\mathbf{w}_{\mathrm{MFMVDR}}(k,m) = \frac{R_{mf}^{-1}(k,m)\,y_s(k,m)} {y_s(k,m)^{\mathrm{H}} R_{mf}^{-1}(k,m) y_s(k,m)}$$

allows joint spatial-temporal suppression with distortionless response (Tsai et al., 2022).

• Multi-Frequency Distortionless Restriction (MVMFDR): For digital wideband beamforming, unity gain constraints are imposed across $K$ discrete frequencies, solved via:

$$\mathbf{w}^* = \mathbf{R}^{-1}\mathbf{A} \left(\mathbf{A}^{\mathrm{H}}\mathbf{R}^{-1}\mathbf{A}\right)^{-1}\mathbf{1}_K$$

where $\mathbf{A}$ is the stack of steering vectors at each $f_k$ (Liu et al., 2010).

• Spherical Harmonic MPDR: In spatial audio enhancement, MPDR constraints are formulated per SH mode, incorporating Relative Harmonic Coefficients (ReHCs) for undistorted spatial cue preservation, with closed-form filter:

$$\tilde{\mathbf{w}} = \tilde{\bm{\mathcal{R}}}_{v+u}^{-1}\,\tilde{\mathbf{C}} \left( \tilde{\mathbf{C}}^{\mathrm{H}}\tilde{\bm{\mathcal{R}}}_{v+u}^{-1}\tilde{\mathbf{C}} \right)^{-1} \tilde{\mathbf{b}}$$

ensuring spatial fidelity in reverberant regimes (Zhang et al., 2024).

3. Unified Convolutional Beamforming: Denoising and Dereverberation

The MPDR principle underpins the unified Weighted Power minimization Distortionless response (WPD) convolutional beamformer, which simultaneously integrates denoising (MPDR) and dereverberation (Weighted Prediction Error, WPE) through a temporally-augmented filter vector $\bar{w}$. The WPD optimization criterion generalizes the MPDR form: $$\min_{\bar{w}}\; J(\bar{w}) = \sum_t \left| \bar{w}^{\mathrm{H}}\bar{x}_t \right|^2 / \sigma_t^2 \;\;\;\; \text{s.t.}\;\; \bar{w}^{\mathrm{H}}\bar{v} = 1$$ where $\bar{x}_t$ stacks current and lagged observations, $\sigma_t^2$ estimates SOI power, and $\bar{v}$ is the augmented steering vector. The closed-form WPD filter is

$$\bar{w}_{\mathrm{WPD}} = R^{-1}\bar{v}/(\bar{v}^{\mathrm{H}}R^{-1}\bar{v})$$

with $R$ the power-normalized spatio-temporal covariance (Nakatani et al., 2018). This approach guarantees global optimality by jointly exploiting degrees of freedom for denoising and dereverberation, outperforming traditional cascaded WPE$\to$MPDR pipelines.

4. Sparse Priors, Robustness, and Generalizations

MPDR beamforming is further enhanced by introducing heavy-tailed priors and $\ell_p$-norm penalties:

• Complex Generalized Gaussian Prior (CGGD-MLDR): By weighting covariance estimation according to a super-Gaussian prior on the SOI, the filter update alternates between adaptive power weighting and MPDR minimization:

$$\lambda_s^{i+1}(l) = |w^i{}^{\mathrm{H}}y(l)|^{2-p}$$

$$w^{i+1} = R^{i+1}{}^{-1}h / ( h^{\mathrm{H}}R^{i+1}{}^{-1}h )$$

yielding improved robustness to steering vector mismatch and speech cancellation (Meng et al., 2021).

• $\ell_p$-Norm WPD and IRLS: The convolutional WPD beamformer is generalized to minimize $\sum_t |z_t|^p$ ($0 < p \le 2$), promoting speech sparsity and iterative reweighting. The weight update and filter computation employ IRLS with MPDR-type subproblems, enabling direct control over sparsity-induced robustness (Gode et al., 2021).

5. Implementation Techniques and Algorithmic Variants

MPDR and its extensions require reliable estimates for covariance matrices and steering vectors. Algorithmic details include:

• Covariance Learning: For robot ego-noise, spatial covariance matrices are learned via Principal Component Analysis (PCA) on calibration data, used in real-time MPDR filtering for robust speech enhancement and event detection (Lagacé et al., 2023).
• Inverse Covariance Estimation: Neural parameter estimation of inverse covariance and to-be-preserved vector (steering or IFC vector) directly from input features avoids numerical instability and addresses double-talk distorting sources in acoustic echo suppression (Tsai et al., 2022).
• Run-Time Adaptation: Adaptive neural beamforming pipelines use unsupervised dereverberation methods (WPE, FastMNMF) to generate pseudo ground truth masks for DNN fine-tuning, updating the MPDR/WPD filter online to address changing speech and noise conditions (Fujita et al., 2024).

6. Performance, Bias, and Evaluation Metrics

MPDR beamformers exhibit systematic bias in signal-power estimation due to interference and noise energy leakage, quantified as: $$\mathsf{Bias}_{\text{Capon}} = ( \mathbf{a}^{\mathrm{H}}\mathbf{Q}^{-1} \mathbf{a} )^{-1}$$ This bias is addressed by shrinkage corrections (Capon$^+$), balancing waveform and power estimation accuracy: $$\mathbf{w}_{\text{Capon}^+} = \sqrt{\beta^*}\mathbf{w}_{\text{Capon}}, \qquad \beta^* = \frac{\gamma}{\gamma_{\text{Capon}}}\frac{T}{T+1}$$ where $T$ denotes snapshot count (Ollila et al., 20 Jun 2025). In practical evaluations, WPD beamformers consistently outperform cascaded MPDR/WPE, delivering superior metric improvements in SDR, PESQ, STOI, and reduced ASR WER (Nakatani et al., 2018, Fujita et al., 2024, Gode et al., 2021).

7. Application Areas and Broader Impact

The MPDR methodology is widely applied in:

• Speech enhancement and dereverberation: Real-time ASR preprocessing, noise reduction, and dereverberation in challenging environments.
• Array signal processing: Radar, sonar, wireless localization, direction-of-arrival estimation.
• Sparse reconstruction: Sparse signal estimation mitigating inter-atom interference in coherent dictionaries (SBWMVDR) (Yang et al., 2010).
• Spatial audio and multichannel filtering: Spherical microphone array processing for spatial audio rendering in reverberant scenes (Zhang et al., 2024).
• Adaptive pipelines: Deep neural network beamforming systems with run-time adaptation and pseudo-supervision.

The MPDR framework’s extensibility, optimality, and computational tractability make it foundational for contemporary beamformer design, robust speech enhancement, and array filtering in both classical and deep learning-based signal processing architectures.