Project: An Adaptive NLMS Algorithm for a MVDR Beamformer, design and performance analysis.

Place of Development: University of Buenos Aires - Faculty of Engineering
Date: November 2011
Author: Pablo Manuel Delgado

References:

[1] (Updated )Razia, S; Hossain, T; Matin, M A, "Performance analysis of adaptive beamforming algorithm for smart antenna system", IEEE International Conference on , Informatics, Electronics & Vision (ICIEV), 2012
[2]Kammoun, I.; Jaidane, M., "Exact performances analysis of a selective coefficient adaptive algorithm in acoustic echo cancellation", IEEE International Conference on coustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP '01). 2001
[3] S. Haykin, Adaptive Filter Theory, Third Edition, Ch. 5 and 9, Prentice Hall.
[4] L. Rey Vega, H. Rey, S. Tressens, J. Benesty. Adaptive Filtering: Algorithms and Analysis.

(MATLAB Testing Code soon to be available) 


A Normalized Least Mean Squares adaptive filter algorithm for an antenna array was implemented and analysed through simulation and real time performance for a QPSK modulated, 1 GHz carrier received test signal. The spatial filter response on the converging period was also determined and compared to that of the first steps of the iteration. A SNR, INR performance sensitivity analysis was also carried out through symbol interference and scattering diagrams.

Antennas (and antenna arrays) often operate in dynamic environments, where the signals (both desired and interfering) arrive from changing directions and with varying powers.
These antenna arrays employ an adaptive weighting algorithm, that adapts the weights based on the received signals to improve the performance of the array.
 
The project consisted in analysing incoming far field band limited signals to an array of antennas of arbitrary number.
The following preexisting conditions were taken into account for the design experiment:

• The arbitrary number M of  antennas were separated at least at a half of the wavelength of the signal of interest, this is a direct consequence of the spatial sampling theorem.

• The signal of interest impinges on the array at an angle of \( \theta_0 = \frac{\pi}{2} \) measured from the antennas' base plane (far field signal).
• The environment noise is complex uncorrelated circular Gaussian with a power \(  \sigma^2_v \)
• Moreover, other interfering signals impinge with a known measured angle of \( \theta_1 = \frac{2\pi}{3} \)  
• It was also determined that there are other interfering signals as well, with unknown inference angles. 

The process of adaptive cancellation of the MVDR beamformer proved satisfactory for a known interfering signal angle. This way, initial parameters of the filter design can be identified for the adaptive algorithm initialization, based on the classical wiener beamforming filter. The adaptation process can be evidenced on the filter's spatial response function.
Ultimately, the cancelling and selectivity capability of a well designed filter improves with the number of iterations and with a good INR. 
If an unknown angle of an interfering signal is to be accounted for, the initial conditions cannot be determined clearly and the interfering signal results almost unmodified to the filter, impacting negatively on the symbol detection and received signal estimation.

(Details on the algorithm construction and theoretical background considerations soon to be added.)

Adaptation Scheme for an antennae array

Results and Performance analysis

A number of simulations in MATLAB were carried out as to test the filter performance facing varying factors such as environment noise, array size, INR and computational cost and convergence, i.e. number of iterations for a given resolution specification.

Spatial Response with known (estimated) interference signals

A spatial response was plotted for different antennae quantities and SNR, INR merit figures.
It can be seen that the number of iterations smooths the filter response. The number of antennae set the number of equally spaced zeros that can be detected on the first iterations. The simulation results show that the zeros corresponding to erroneous beamforming positions are smoothed out as the number of iteration increases. The more negative the signal to interference ratio i, the more pronounced these zeros are and the filter must reject interfering signals with more intensity.

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 10 INR = -10 M=10

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 10 INR = -5 M=10

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 10 INR = -1

 M=10

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 20 INR = -5 M=10

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 20 INR = -5 M=6

Spatial response for n=6 iterations (above) and n = 990 (below). SNR = 20 INR = -10 M=6

Scattering Diagrams for a known interference signal

The scattering symbol diagrams for different array sizes, SNR and INR are shown below. It can be seen that, given that the beamforming filter is well applied the INR, i.e. the signal to interference ratio does not impact in a remarkable way on the filter's performance. Nevertheless, thermal noise has an important impact on the scattering of the received symbols. This is clearly due to the filters selectivity onto a single direction of preference, given by the constraint vector. White noise, on the contrary, impinges equally in all directions, giving a suboptimal approach. This concept will be clarified when the filter is utilized with an unknown received signal. 

Scattering Diagram for SNR = 10, INR = -10 and M = 10

Scattering Diagram for SNR = 10, INR = -5 and M = 10

Scattering Diagram for SNR = 10, INR = -1 and M = 10

Scattering Diagram for SNR = 20, INR = -5 and M = 10

Scattering Diagram for SNR = 20, INR = -5 and M =6

Interference of an unknown signal, unestimated angle of arrival.

Spatial Response

Simulations were carried out to show the filter's performance in the presence of an unknown interference signal.  It can be noted that, due to the absence of a zero on the real incidence angle of the interference signal (approx \( \frac{\pi}{4}\) ), the interfering signal is treated as white, uncorrelated noise. This is particularly notable on high interference signal power (i.e. low INR).

Spatial response for n=6 iterations (above) and n=990 (below). Unknown signal interference angle. SNR = 10, INR = -5, M = 10

Spatial response for n=6 iterations (above) and n=990 (below). Unknown signal interference angle. SNR = 30, INR = -10, M = 10

Scattering Diagrams

It can be seen that the incidence of thermal noise does not affect the filter performance on this case. Nevertheless, the fact that the interference signal angle of incidence is not known impacts negatively on the filter's performance regarding low INR. The filter's rejection power is highly compromised and the symbol detection becomes implausible.

Scattering Diagram for SNR=10 INR = -5 and M = 10. Angle of interfering signal is unknown.

Scattering Diagram for SNR=20 INR = -5 and M = 10. Angle of interfering signal is unknown.

Scattering Diagram for SNR=10 INR = -10 and M = 10. Angle of interfering signal is unknown.

Scattering Diagram for SNR=10 INR = -1 and M = 10. Angle of interfering signal is unknown.