182 Statistical channel parameter estimation sed on fsic(O)=TF(o)El唱 (7.21) In the r.h.s.term in(7.21),F()is an orthonormal basis of the space spanned by the columns of F().The NAs of the are obtained by minimizing the distance bet n the subspace spanned by the signal originating from single scatterer wing (E )an the two subspac s.It an be shown that this distance is prop al to the Frobenius norm of the projection of one subspace21） space or f the( and other previously published extensions of the standard MUSIC algorithm is given in Subsection 7.1.4. he p USgodelaerof eMUSIC The proposed MUSIC algorithm,which makes use of the pseudo-spectrum(7.21),can be generalized to the scenario by one component (ga scatterer)span a subspace of any thi e,in the case where ispersion of an sDs is characte using a pdf,F()can be obtained by the eigenvalue decomposition of ve inte aces Golub an 6).which all for a co on w the arant of the two subspaces is quivalent to minimizins the norm of (where re tsthe vector containingal principa subspacesnd)isthe operator ise f Thus,in our case ofare the principal angles between the subspace spanned by the columns ofF(and the signal subspace estimated ovaria Thisisa reasonable approach in the ID case where the dimension of the si Btheof the MUSICt proposed in Asztely et a (1997)computes the NA estimates by ce spanned by the columns of F()and the NA es hen the wasint6no The pseudo-spectrum(7.21)can be recast as: )W()FE.} (7.22 where W()is an azimuth-dependent weighting matrix defined as W(@)三F(p)tF(o)F()(F(o)t)日 (7.23)182 Statistical channel parameter estimation We propose a natural extension of the standard MUSIC algorithm for the estimation of the NAs of SDSs based on the 1 st-order GAM model. The extension considers the following generalization of the pseudo-spectrum in (7.20): fMUSIC(φ) = 1 kF˜(φ)HEwk 2 F . (7.21) In the r.h.s. term in (7.21), F˜(φ) is an orthonormal basis of the space spanned by the columns of F(φ). The NAs of the D SDSs are estimated to be the arguments of the pseudo-spectrum corresponding to its D highest peaks. Both the standard MUSIC algorithm and the proposed extension rely on the same principle, i.e. parameter estimates are obtained by minimizing the distance between the subspace spanned by the signal originating from single scatterer and an estimate of this subspace computed from the sample covariance matrix. In the SS case, the signal subspace induced by an SS is spanned by the steering vector c(φ), while in the SDS scenario the subspace induced by an SDS is spanned by the columns of F(φ). In that sense, the latter algorithm is a natural extension of the former one. Following (Edelman et al. 1998, p. 337), the distance between the subspace spanned by the columns of F(φ) and the estimated signal subspace coincides with the Frobenius norm of the difference between the projection matrices of the two subspaces. It can be shown that this distance is proportional to the Frobenius norm of the projection of one subspace onto the null space of the other subspace, i.e. kF˜(φ) HEwk 2 F in our case. Thus, the inverse of the pseudo￾spectrum (7.21) provides with a measure of the distance between the signal subspace spanned by the columns of F(φ) and the estimated signal subspace. A thorough discussion of the relationships between this extended MUSIC algorithm and other previously published extensions of the standard MUSIC algorithm is given in Subsection 7.1.4. Generalization of the Proposed MUSIC Algorithm and its Relation to other Extensions of the Standard MUSIC Algorithm The proposed MUSIC algorithm, which makes use of the pseudo-spectrum (7.21), can be generalized to the scenario where the signals contributed by one component (e.g. a scatterer) span a subspace of any arbitrary dimension. In this case, F˜(φ) is an orthonormal basis of the signal subspace. The argument φ of F˜(φ) may be also multi-dimensional and it is not required that a closed–form expression exists which relates F˜(φ) to φ. For instance, in the case where azimuth dispersion of an SDS is characterized using a pdf, F˜(φ) can be obtained by the eigenvalue decomposition of the covariance matrix calculated using this pdfs. We propose now an alternative interpretation of the proposed MUSIC algorithm using the concept of principal angles between subspaces Golub and Loan (1996), which allows for a comparison with the variant of the MUSIC algorithm published in Christensen et al. (2004). As shown in (Edelman et al. 1998, p. 337), minimizing the distance between two subspaces is equivalent to minimizing the norm of sin(θ), where θ represents the vector containing all principal angles between these two subspaces and sin(·) is the operator computing the element–wise sin of θ. Thus, in our case the NA estimates obtained by maximizing the pseudo-spectrum (7.21) in fact minimize k sin(θ)k where the components of θ are the principal angles between the subspace spanned by the columns of F(φ) and the signal subspace estimated from the sample covariance matrix. This is a reasonable approach in the ID case where the dimension of the signal subspace induced by an SDS is larger than 1. By contrast, the variant of the MUSIC algorithm proposed in Asztély et al. (1997) computes the NA estimates by maximizing the smallest principal angle between the 2-dimensional subspace spanned by the columns of F(φ) and the estimated signal subspace. This maximization is indeed equivalent to the maximization of the objective function λ −1 min(F(φ) HEwEH w F(φ)), with λmin(·) denoting the smallest eigenvalue of the matrix given as argument, described in Asztély et al. (1997) to compute the NA estimates Drmac (2000). The resulting algorithm is applicable when the dimension of the subspace effectively induced by an SDS is equal to one, e.g. in the CD case for which the algorithm was initially designed. The pseudo-spectrum (7.21) can be recast as: fMUSIC(φ)= 1 tr{EH w F(φ)W(φ)F(φ)H Ew} , (7.22) where W(φ) is an azimuth-dependent weighting matrix defined as W(φ) .= F(φ) †F˜(φ)F˜(φ) H(F(φ) † ) H. (7.23) At first glance the representation in (7.22) seems to be similar to the pseudo-spectrum (Krim and Viberg 1996, Eq. (37)) of the weighted MUSIC algorithm. However, the proposed MUSIC algorithm and the standard weighted MUSIC