Multiple Signal Classification (MUSIC) and Its Variants

3.2.1 MUSIC

It can be shown [78] that each column of the manifold matrix in (2.11) must be orthogonal to the noise subspace matrix U_N obtained from (2.19), hence

U^H_Na(θ_l) = 0 (3.1)

for θ_l= θ₁, . . . , θ_L or equivalently

a^H(θ_l)UNU^H_Na(θ_l) = 0 (3.2) where a(θ_l) is defined in (2.12). This is the core idea of the MUSIC estimation method.

In practice, in order to estimate the DOAs, the estimate of the noise subspace matrix ˆUN

obtained from the sample covariance matrix ˆR in (2.28) must be used. Therefore, the following “spectral” function is proposed in [78]

f_MUSIC(θ) = 1 k ˆU^H_Na(θ)k²

= 1

a^H(θ) ˆUNUˆ^H_Na(θ). (3.3) The estimated DOAs ˆθ₁, . . . , ˆθ_L are then obtained as the angles θ corresponding to the L largest maxima of fMUSIC(θ) in (3.3). The denominator of the MUSIC function in (3.3) can be interpreted as the measure of the projection of the array manifold vector onto the noise subspace ˆUN which ideally for the true DOAs is zero. Then, the estimated DOAs are the ones that minimize this projection. To find the DOAs, a scan over the entire field-of-view (FOV) is required and the function f_MUSIC(θ) in (3.3) needs to be evaluated for each θ.

The accuracy of the estimates obtained from the MUSIC spectral function in (3.3) depends on many factors such as:

• the step size ε with which the FOV is being scanned

• SNR

• the number of snapshots N

• the accuracy of the available array manifold vectors a(θ), itself dependent on factors such as the precision of the array and sensor calibration or the exact sensor locations

Some explanation regarding these factors are in order. In order to evaluate the MUSIC function in (3.3), a limited number of scanning points should be selected, hence, a step size should be defined. If the step size is chosen small, then not only the estimation accuracy will increase but also the computational cost of the method. Therefore, some compromise must be made to have a reasonable step size (not too small) and at the same time a reasonable estimation accuracy. The effects of the value of SNR and the number of snapshots on DOA estimation of the MUSIC method will be examined in more detail later in Chapter 4.

Moreover, to reduce the negative effects of these factors on the DOA estimation performance, some methods which are capable of identifying the erroneous estimates will be presented in Chapter 4. The MUSIC algorithm is known to be very sensitive to uncertainties and errors in the array manifold vectors [14], [81], [90]. This makes the exact calibration of the sensor array crucial for DOA estimation. The calibration issue in the arrays, especially large sparse arrays will be addressed in Chapter 5 and some novel techniques to simultaneously estimate the DOAs and to calibrate the sensor array will also be proposed.

3.2.2 Weighted-MUSIC

As it can be observed, in the MUSIC spectral function of (3.3), all the noise eigenvectors are treated equally. The MUSIC method can be extended to include a specific weighting matrix for controlling the effect of each noise eigenvector on the estimates. A proper choice of the weighting matrix will be particularly useful to improve the performance of the estimators in difficult situations such as low number of snapshots and low SNR to overcome some of the shortcomings of the MUSIC method [74], [98]. Toward this end, the following spectrum

function is defined to take into account the different effects of the noise eigenvectors

f_WMUSIC(θ) = 1

a^H(θ) ˆU_NW ˆU^H_Na(θ). (3.4) It is clear that the conventional MUSIC function in (3.3) is a special case of the weighted-MUSIC function in (3.4) with W = I. It should be remarked that the weighted-weighted-MUSIC function in (3.4) plays an important role in constructing the “estimator bank” in Section 4.2.

A useful choice of the weighting matrix is

W = ˆU^H_Ne₁e^T₁Uˆ_N (3.5)

where e₁ is the first column of the M × M identity matrix. The choice of W in (3.5) coincides with the well-known Min-Norm method [31], [32], [74]. In the Min-Norm method, a non-zero vector with minimum norm in the noise subspace, i.e., a linear combination of the noise eigenvectors, is obtained. Then, the orthogonality of this minimum length vector and the array manifold vector is measured similar to the one used for the MUSIC method in (3.3) for the angles in the FOV. The Min-Norm method is known to yield an improved resolution capability of distinguishing two close sources, as compared to the MUSIC method in the ULAs [98].

3.2.3 Root-MUSIC

The root-MUSIC DOA estimation method [7] exploits the Vandermonde structure of the array manifold vector in the ULAs in (2.14) to estimate the DOAs through a search-free algorithm based on polynomial rooting. Defining

ϕ , e−j(2πd/λ) sin θ, (3.6)

the parametric array manifold vector a(θ) becomes

a(ϕ) =1, ϕ, ϕ², · · · , ϕ^{M −1}T

. (3.7)

Furthermore, it is simple to show that

a^H(ϕ) = a^T(1/ϕ). (3.8)

Then, the MUSIC criterion in (3.2) transforms into

a^T(1/ϕ_l)U_NU^H_Na(ϕ_l) = 0 (3.9) where

ϕ_l= e−j(2πd/λ) sin θl (3.10)

for l = 1, . . . , L. Let us define

f_r−ideal(ϕ) , a^T(1/ϕ)U_NU^H_Na(ϕ). (3.11) From (3.8), it can be seen that if ϕ is a root of the polynomial in (3.11), then its conjugate reciprocate 1/ϕ^∗ is also a root. Therefore, from (3.9), the polynomial in (3.11), which is of degree 2M − 2, has 2M − 2 roots with M − 1 roots on/inside the unit-circle and their M − 1 conjugate reciprocate pairs on/outside the unit-circle. In practice, the estimate of the noise subspace matrix, i.e., ˆU_N in (2.28), from the sample covariance matrix ˆR in (2.26) is used and the following polynomial is obtained

f_root−MUSIC(ϕ) = a^T(1/ϕ) ˆU_NUˆ^H_Na(ϕ). (3.12) To estimate the DOAs, the L complex roots of f_root−MUSIC(ϕ), namely ˆϕ₁, . . . , ˆϕ_L, closest to the unit-circle and inside it should be selected and the estimated DOAs can be computed for l = 1, . . . , L from

θˆ_l= sin⁻¹ −λ

2πd]( ˆϕ_l)

(3.13) where ](·) denotes the phase of a complex variable. It has been demonstrated [85], [86] that both MUSIC and root-MUSIC have the same asymptotic performances. From (3.13), one can observe that the estimated DOA ˆθ_l (for l = 1, . . . , L) depends only on the phase of the root ˆϕ_l of the root-MUSIC polynomial in (3.12) and not on the magnitude of ˆϕ_l. Hence, any changes in the magnitude has no effect on the estimated DOAs and the root-MUSIC method

is robust to the radial errors of the estimated roots [98]. Because of this property, the root-MUSIC method enjoys superior performance in comparison to the root-MUSIC method in low SNR and low number of snapshots, although the root-MUSIC method is only applicable to the ULAs and also to the uniform circular arrays (UCAs) [45], and not to any arbitrary array geometry (unlike the MUSIC method). However, there are methods, such as array interpolation [15] and beamspace methods [109], in which the array manifold of an arbitrary array geometry can be approximately transformed into the array manifold of a virtual ULA so that the root-MUSIC method can be implemented.