What is the different between Minimum Variance Distortionless Response (MVDR) and Linearly Constrained Minimum Variance (LCMV) Beamforming Technique?
Two different techniques for beamforming and noise cancellation.
The MVDR Beamformer is a data adaptive beamforming solution whose goal is to minimize the variance of the recorded signal. If the noise and the underlying desired signal are uncorrelated, as is typically the case, then the variance of the recorded signal is the sum of the variances of the desired signal and the noise. Hence, the MVDR solution seeks to minimize this sum, thereby mitigating the effect of the noise. On the other hand the LCMV beamforming requires that the beamformer response to signals from a direction of interest are passed with specified gain and phase delay.
the LCMV beamformer can achieve perfect dereverberation and noise cancellation when the acoustic transfer functions between all sources (including interferences) are known.
The LCMV was originally proposed by Frost in 1972 as an implementation of MVDR in time-domain (thus it is also named as the Frost's beamformer). The original paper can be found in O. L. Frost, "An algorithm for linearly constrained adaptive array processing," in Proceedings of the IEEE, vol. 60, no. 8, pp. 926-935, Aug. 1972. An adaptive least mean squares(LMS) method was corresponding proposed to solve the optimization problem with a constraint.
The idea of MVDR can be traced back to 1969， which is proposed by Capon. Thus it is also known as the Capon beamformer. The original paper can still be found in J. Capon, "High-resolution frequency-wavenumber spectrum analysis," in Proceedings of the IEEE, vol. 57, no. 8, pp. 1408-1418, Aug. 1969.
Thus, the LCMV and MVDR together constructed the fundamental idea of the spatial filter with an array: the signals from the expected directions will well pass the spatial filter without distortions, and the signals coming from Non-expected directions will be attenuated.
When the interference (not only the noises) is further considered in your measurement system, It derived that the MVDR is decomposed into the LCMV and a matched filter (MVDR solution in the absence of interference). This perspective can be found in M. Souden, J. Benesty and S. Affes, "A Study of the LCMV and MVDR Noise Reduction Filters," in IEEE Transactions on Signal Processing, vol. 58, no. 9, pp. 4925-4935, Sept. 2010.
- MVDR beamformer will minimize the signals from all other directions while keeping the desired signal (from known direction) undistorted.
- Only desired signal direction need to be known.
- Sub optimal interference cancellation
- LCMV beamformer will minimize the signals from the undesired signal directions(known directions) and maintains keeping the desired signal (from known direction) undistorted.
- Desired signal and undesired signals direction need to be known
- Effective interference cancellation
The MVDR criterion is actually a special case of LCMV, which captures a larger set of optimization criteria and constraints.
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