⚡ electrical engineering

Phase-optimised linearly-constrained minimum-variance beamformers

This paper proposes a procedure for optimizing the group delay of Linearly-Constrained Minimum-Variance (LCMV) beamformers by minimizing either noise power or processing delay, demonstrating its effectiveness through simulations in VHF communication and UHF bistatic radar applications.

Original authors: Hugh L Kennedy

Published 2026-04-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hugh L Kennedy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are at a crowded, noisy party. You are trying to have a conversation with a specific friend (the signal) who is standing in a specific direction. However, the room is filled with other people shouting (interferers), a loud DJ playing music nearby (jammers), and the general hum of the crowd (noise).

Your goal is to hear your friend clearly while blocking out everyone else. This is exactly what a Beamformer does in radio technology (like radar or cell phones). It uses an array of microphones (antennas) to "listen" in a specific direction and ignore the rest.

This paper introduces a clever new trick to make that listening even better. Here is the breakdown in simple terms:

1. The Problem: The "Echo" Delay

Imagine you are trying to listen to your friend, but the sound takes a weird amount of time to reach your ears because of how you are holding your hands (the antennas).

In traditional systems, engineers assume a "standard" delay. They say, "Okay, we'll wait exactly 4 seconds for the sound to line up."
But what if waiting 4 seconds isn't the best time? What if waiting 3.8 seconds actually blocks out the noise better? Or what if waiting only 0.7 seconds is good enough and saves you time?

The paper argues that we have been too rigid. We are waiting for a fixed time, but we should be optimizing that wait time to get the clearest possible signal.

2. The Solution: Tuning the "Group Delay"

The author, Hugh Kennedy, proposes a method to find the perfect delay (called "group delay") for the beamformer. Think of this delay as the "sweet spot" where the signal from your friend lines up perfectly, while the noise from the DJ and the shouting crowd cancels itself out.

He suggests two ways to find this sweet spot:

Method A (The "Silence" Seeker): Find the delay that makes the background noise as quiet as possible. This gives you the clearest voice, even if it takes a tiny bit longer to hear it.
Method B (The "Speed" Seeker): Find the fastest possible delay that still works well. This is great for real-time applications (like a self-driving car needing instant data), even if the voice is slightly less clear than in Method A.

3. The Magic Trick: "Folding" the Sound

To understand how this works, imagine the sound waves as a piece of fabric.

Old Way: You try to smooth out the fabric in one direction (left to right) to hide the noise.
New Way: The paper shows that by adjusting the timing (the delay), you can "fold" the fabric in both directions (left-to-right AND up-and-down). This creates a much tighter, more complex shape that traps the noise in the creases and leaves a perfect path for your friend's voice.

By changing the delay, the system can create "troughs" (quiet zones) exactly where the noisy people are standing, without hurting the signal from your friend.

4. The Results: Two Real-World Tests

The author tested this idea in two different scenarios:

Scenario 1: The VHF Communication (Like a Walkie-Talkie)
- Goal: Hear a message clearly through static and interference.
- Result: By optimizing the delay, the system heard the message much clearer (about 7-8 dB better than the old method). It was like turning down the volume on the whole party except for your friend.
- Bonus: Even when they chose the "Speed" option (very low delay), the signal was still significantly better than the old standard.
Scenario 2: The UHF Radar (Like a Police Speed Gun or Air Traffic Control)
- Goal: Spot a tiny airplane (the target) against a background of mountains and buildings (clutter).
- Result: The optimized beamformer made the tiny airplane stand out much more clearly against the background noise. It improved the ability to detect the target, which is crucial for safety.

5. Why This Matters

In the past, engineers had to choose between "good performance" and "fast processing."

If you wanted the best signal, you had to accept a long delay.
If you wanted speed, you had to accept a noisy signal.

This paper shows that you don't have to choose blindly anymore. You can calculate the exact right amount of delay for your specific situation.

Need the absolute best quality? Tune for minimum noise.
Need instant reaction? Tune for minimum delay.

The Bottom Line

Think of this paper as giving the beamformer a new pair of glasses. Before, it could only see the noise and the signal as they were. Now, by adjusting the timing (the "group delay"), it can focus so sharply that it makes the noise disappear and the signal pop out, whether the goal is to hear a radio message or spot a distant plane. It's a simple mathematical tweak that makes a massive difference in how well our technology can "listen."

1. Problem Statement

The Linearly-Constrained Minimum-Variance (LCMV) beamformer is a standard adaptive processing technique used in sensor arrays (radar and communications) to suppress interference while maintaining unity gain in a desired "look" direction. However, traditional wideband LCMV designs often assume a fixed group delay (typically the center of the filter, $q_t = (M_t-1)/2$ ) or do not explicitly optimize the temporal phase response.

The paper identifies a critical gap: while spatial phase-center optimization is well-documented, temporal phase-center (group delay) optimization for wideband LCMV beamformers has been largely unexplored. In wideband applications, the choice of group delay affects the filter's ability to shape the spatiotemporal response to suppress interference without distorting the desired signal. The author posits that treating the group delay as a design variable, rather than a fixed parameter, offers a new degree of freedom to either minimize output noise power or reduce processing latency.

2. Methodology

The paper proposes a mathematical procedure to determine the optimal temporal group delay ( $q_t$ ) for an LCMV beamformer.

A. Mathematical Formulation

Objective: Minimize the output noise power $P = \mathbf{h}^\dagger \mathbf{R} \mathbf{h}$ , where $\mathbf{R}$ is the spatiotemporal noise covariance matrix and $\mathbf{h}$ is the weight vector.
Constraints: The beamformer is subject to linear constraints defined by $\mathbf{d} = \mathbf{F}^\dagger \mathbf{h}$ $d = F^{†} h$ . Unlike standard designs that constrain specific frequency bins, this method uses derivative constraints at the DC ( $\omega_t = 0$ $ω_{t} = 0$ ) and Nyquist ( $\omega_t = \pi$ $ω_{t} = π$ ) frequencies.
- $K_{dc}$ constraints at DC to define the passband response and group delay.
- $K_{pi}$ constraints at Nyquist to ensure high-frequency attenuation.
Optimization Variable: The group delay $q_t$ is embedded in the constraint vector $\mathbf{d}$ . Specifically, the $k$ -th derivative constraint at DC is set to $(-i q_t)^k$ .

B. The Optimization Procedure

The paper derives a closed-form expression for the noise power as a function of the group delay, $P(q_t)$ :
$P(q_t) = \mathbf{d}^\dagger(q_t) (\mathbf{F}^\dagger \mathbf{R}^{-1} \mathbf{F})^{-1} \mathbf{d}(q_t)$

Two strategies for selecting the optimal $q_t$ are proposed:

Noise Power Minimization: Solve for the roots of the derivative $P'(q_t) = 0$ . This yields the delay that minimizes the output noise power (maximizing SNR). The solution involves finding roots of a polynomial of degree $2(K_{dc}-1)-1$ .
Latency Minimization: Select the feasible root with the smallest positive real value. This reduces processing delay at the cost of a slight SNR reduction.

If no feasible positive real root exists (due to ill-conditioned matrices), the standard center delay $(M_t-1)/2$ is used as a fallback.

3. Key Contributions

Novel Optimization Framework: The paper introduces a method to analytically determine the optimal temporal group delay for wideband LCMV beamformers, a degree of freedom previously ignored in literature.
Derivative Constraint Strategy: It utilizes high-order derivative constraints at DC and Nyquist to create compact, well-damped impulse responses, avoiding the need for "hard" constraints at discrete frequency bins which can inhibit interference suppression.
Dual-Objective Selection: It provides two distinct criteria for delay selection: one for maximum Signal-to-Noise Ratio (SNR) and one for minimum latency, allowing system designers to trade off performance based on application needs.
Closed-Form Analysis: The derivation allows for the analysis of noise power as a function of delay without needing to solve the full LCMV weight vector equation for every candidate delay.

4. Simulation Results

The proposed method was validated using two distinct simulation scenarios: VHF Communication and UHF Bistatic Radar. Six beamformer variants were compared:

BMF-A: Standard MVDR (no temporal degrees of freedom).
BMF-B: Wideband MVDR using DFT sub-bands.
BMF-C: Wideband LCMV with unity-gain constraints at discrete bins.
BMF-D: Wideband LCMV with derivative constraints (fixed delay).
BMF-E: Wideband LCMV with derivative constraints and Noise-Power Minimized delay.
BMF-F: Wideband LCMV with derivative constraints and Minimum-Latency delay.

Key Findings:

VHF Communication Scenario:
- BMF-E (Optimal Delay for SNR): Achieved the highest SNR (32.27 dB), outperforming the standard LCMV (BMF-D, 31.39 dB) and the fixed-delay MVDR (BMF-B, 25.46 dB).
- BMF-F (Optimal Delay for Latency): Achieved an SNR of 27.61 dB with a group delay of only 0.75 samples (compared to ~4.0 samples for others). This represents a significant latency reduction with only a ~4.6 dB penalty compared to the SNR-optimal solution, but still a >2 dB gain over the non-temporal MVDR.
UHF Radar Scenario:
- BMF-E: Achieved the highest SNR (16.68 dB), improving upon the standard LCMV (16.36 dB).
- BMF-F: Achieved 15.82 dB SNR with a group delay of 0.75 samples.
General Trend: The SNR consistently increased from BMF-A to BMF-E. The use of derivative constraints (BMF-D/E/F) allowed the beamformer response to "fold" more effectively in both angle and frequency domains to place interference in magnitude troughs, a capability limited in BMF-A.

5. Significance

This paper demonstrates that group delay is a critical, tunable parameter in adaptive beamforming, not just a byproduct of filter length.

Performance Gain: Optimizing the delay can yield measurable SNR improvements (approx. 0.3–0.9 dB in simulations) in high-interference environments, which is crucial for detecting weak signals in radar and communication systems.
Latency Control: For real-time applications (e.g., active radar tracking or low-latency comms), the method allows designers to drastically reduce processing delay (from ~4 samples to <1 sample) with a manageable SNR trade-off.
Computational Efficiency: The approach avoids the need for iterative search algorithms to find the optimal delay; instead, it relies on solving a polynomial derived from the covariance matrix structure.

In conclusion, the paper establishes that exploiting the temporal phase-center as an optimization variable significantly enhances the performance of wideband LCMV beamformers, offering a flexible trade-off between detection sensitivity and system latency.