The Impact of the Covariance Matrix Sampling on the Angle of Arrival Estimation Accuracy

Abstract

Several works show that the linear Angle of Arrival (AoA) methods such as Projection Matrix (PM) have low computational complexity compared to the subspace methods. Although the PM method is classified as a subspace method, it does not need decomposition of the measured matrix. This work investigates the effect of the sampled columns within the covariance matrix on the projection matrix construction. To the authors' knowledge, this investigation has not been addressed in the literature. Unlike the subspace methods such as Multiple Signal Classification (MUSIC), Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT), Minimum Norm, Propagator, etc., which have to use a specific number of columns, we demonstrate this aspect is not applicable in the PM method. To this end, the projection matrix is formed based on a various number of sampled columns to estimate the arrival angles. A theoretical analysis is accomplished to illustrate the relationship between the number of the sampled columns and the degrees of freedom (DOFs). The analysis shows that with the same aperture size, the DOFs can be increased by increasing only the number of sampled columns in the projection matrix calculation step. An intensive Monte Carlo simulation for different scenarios is presented to validate the theoretical claims. The estimation accuracy of the PM method, based on the proposed selected sampling methodology outperforms all the other techniques with less complexity compared to the Capon and MUSIC methods. The estimation accuracy is evaluated in terms of Root Mean Square Error (RMSE) and the Probability of Successful Detection (PSD). The results are presented and discussed.

Uniform Sampling Methodology to Construct Projection Matrices for Angle-of-Arrival Estimation Applications

“…In a previous works [32,37], the impact on the estimation accuracy and the Probability of Detection (PoD) of AoAs of the number of sampled columns (size of the sampled matrix) that was used in the projection matrix construction has been analysed and investigated. It was demonstrated that an increase in the number of the sampled columns leads to increases in the estimation resolutions and PoD.…”

“…The distribution and locations of these columns under this method can be represented conceptually as shown in Figure 2; the blue lines represent the locations of the selected columns in the CM. The projection matrix can be computed as follows [37]:…”

“…The number of produced nulls using (42) is T, and this number is much more than that obtained by (37). The number of nulls using the USM approach compared the that produced with the conventional methodology can be calculated by dividing (41) on (37) as follows:…”

“…Capon [15] O (M 2 N + M 3 + M 2 J θ ) Min-Norm [41] O (M 2 N + M 3 + MJ θ ) MUSIC [16] O (M 2 N + M 3 + M(M-L)J θ ) PM [37] O (M 2 N + ML + M 2 J θ ) ESPRIT [17] O (M 2 N + M 3 + 3ML 2 + 2L 3 ) Propagator [42] O (M 2 N + M 2 L + M(M-L)J θ ) AV [43] O (M 2 N + M 2 LJ θ ) CG [44] O (M 2 N + M 2 LJ θ ) OGSBI [28] O (M 2 N + max (M (J θ ) 2 , MLJ θ )) per iteration Root SBL [45] O (M 2 N + M J θ ) per iteration MVN [46] O (M 2 N + M 3 + MDJ θ ) SSS [47] O…”

Construction of Projection Matrices Based on Non-Uniform Sampling Distribution for AoA Estimation

“…Alternatively, for complexity reduction purposes, it is possible to utilize L columns from the obtained CM to form such a projection matrix instead of using the complete CM or decomposing it [5]. It has been demonstrated that any increase in the number of sampled columns has a positive impact on the effectiveness of the projection matrix construction through improving the estimation accuracy and Probability of Successful Detection (PSD) [33], [34]. However, the complexity of construction will rise.…”