Overview of Basic DOA Estimation Algorithms |
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•Step 5: Find the d largest peaks of PMUSIC(θ) to obtain DOA estimates.
3.6.3Minimum Norm
The minimum norm method is applicable for linear arrays and can be considered as an improved version of MUSIC in computing DOA estimates. The general expression for the minimum norm method is to search for the locations of the peaks in the power spectrum here:
P (θ ) = PMIN (θ ) = |
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with an array weight w, which is of minimum norm. The weight vector w should have its first element equal to unity and is contained in the noise subspace [15]. The final form of the power spectrum is then:
P (θ ) = PMIN (θ ) = |
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a H (θ )V n V nH WV n V nH a(θ ) |
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with the vector W = p1 p1T where p1 equals the first column of an M × M
identity matrix.
Equation (3.74) can be seen as a squaring of the denominator of the power equation of (3.72) of MUSIC. As the denominator is squared, near zero values should serve to boost the power output to even higher levels. The W matrix is necessary to ensure that the matrix dimensions match mathematically
A simulation was conducted by employing a 6-element ULA with its omnidirectional elements separated by a half wavelength. Three equally powered uncorrelated signals were made to impinge on the array from 5°, 25°, and 45°. In such a case, d = 3, M = 6, θ1 = 5°, θ2 = 25°, and θ2 = 45°. An SNR of 10 dB was assumed. Fifty trials were taken with each trial averaged over 250 snapshots. Figure 3.9 shows the result. As can be seen from the figure, the three peaks are sharp with very good resolutions.
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Introduction to Direction-of-Arrival Estimation |
3.6.4ESPRIT
Due to its simplicity and high resolution capability, ESPRIT has become one of the most popular signal subspace-based DOA estimating schemes. ESPRIT is applicable to array geometries that are composed of two identical subarrays and is restricted to use with array geometries that exhibit invariances. This requirement, however, is not very prohibitive in practical applications since many of the common array geometries used in practice exhibit these invariances [16]. There are three primary steps in any ESPRIT based DOA estimation algorithm:
1.Signal subspace estimation: Computation of a basis matrix for the estimated signal subspace.
2.Solution of the invariance equation: Solution of an (in general) overdetermined system of equations, the invariance equation, derived from the basis matrix.
3.DOA estimation: Computation of the eigenvalues of the solution of the invariance equation formed in step 2.
This algorithm will be explained and described in depth in Chapter 5, as this forms the subject of this book.
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Figure 3.9 DOA estimation with the minimum norm technique.