MVDR Beamforming Mathematical Explanation
Minimum Variance → The beamformer minimizes the total output power (signal + noise + interference) coming from all directions.Distortionless Response → It ensures that the signal coming from the desired direction passes through without attenuation or distortion.
1. MVDR Weight Formula
For a desired source at direction 胃k:
wMVDR = (R-1 ak) / (akH R-1 ak)
- ak = steering vector of the desired source
- R = covariance matrix of received signals
Goal: Minimize output power wH R w subject to wH ak = 1.
2. Optimization Problem
minw wH R w s.t. wH ak = 1
wH R w = total output power (signal + noise + interference)
Constraint wH ak = 1 ensures the desired signal passes without attenuation.
3. Solve Using Lagrange Multipliers
Define the Lagrangian:
L = wH R w - 位 (wH ak - 1)
Derivative w.r.t w*:
∂L/∂w* = 2 R w - 位 ak = 0
Solve for w:
w = (位/2) R-1 ak
Use constraint wH ak = 1 to find 位:
位 = 2 / (akH R-1 ak)
Plug back to get MVDR weights:
wMVDR = (R-1 ak) / (akH R-1 ak)
4. Intuition: Why R-1 ak Steers Toward the Source
- R contains all correlations including noise and interference.
- Multiplying by R-1 suppresses directions with high interference/noise power.
- Multiplying by ak aligns the weights with the desired signal’s phase across sensors.
Result:
- Desired source passes unchanged
- Interference and noise are minimized
5. Visualization Example
| Antenna | Signal 1 | Signal 2 | Noise |
|---|---|---|---|
| 1 | phase shift 0° | phase shift 30° | random |
| 2 | phase shift 20° | phase shift 50° | random |
| 3 | phase shift 40° | phase shift 70° | random |
ak encodes the phase pattern of the desired signal. R-1 whitens the noise/interference across antennas. The product R-1 ak combines antennas so the desired phase adds constructively and interference cancels.
6. Summary
The operation R-1 ak aligns the array to the desired source while simultaneously suppressing interference and noise because R-1 acts like a spatial whitening filter.