Non-colocated Time-Reversal MUSIC: High-SNR Distribution of Null Spectrum
D. Ciuonzo P. Salvo Rossi





Non-colocated Time-Reversal MUSIC:
 High-SNR Distribution of Null Spectrum 
    D. Ciuonzo, Senior Member, IEEE and P. Salvo Rossi, Senior Member, IEEEManuscript received 2nd December 2016; accepted 25th January 2017.

D. Ciuonzo is with DIETI, University of Naples “Federico II”,
Naples, Italy. 

P. Salvo Rossi is with the Dept. of Electronics and Telecommunications,
NTNU, Trondheim, Norway. 

E-mail: {domenico.ciuonzo, salvorossi}@ieee.org.


    Received: date / Accepted: date
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We derive the asymptotic distribution of the null spectrum of the
well-known Multiple Signal Classification (MUSIC) in its computational
Time-Reversal (TR) form. The result pertains to a single-frequency
non-colocated multistatic scenario and several TR-MUSIC variants are
here investigated. The analysis builds upon the 1st-order perturbation
of the singular value decomposition and allows a simple characterization
of null-spectrum moments (up to the 2nd order). This enables a comparison
in terms of spectrums stability. Finally, a numerical analysis is
provided to confirm the theoretical findingsNotation - Lower-case (resp. Upper-case) bold letters denote column vectors (resp. matrices), with a_n (resp. a_n,m) being the nth (resp. the (n,m)th) element of a (resp. A); 𝔼{·}, var{·}, (·)^T, (·)^†, (·)^*, Tr[·], vec(·), (·)^-, (·), δ(·), ‖·‖ _F and ‖·‖ denote expectation, variance, transpose, Hermitian, conjugate, matrix trace, vectorization, pseudo-inverse, real part, Kronecker delta, Frobenius and ℓ_2 norm operators, respectively; j denotes the imaginary unit; 0_N× M (resp. I_N) denotes the N× M null (resp. identity) matrix; 0_N (resp. 1_N) denotes the null (resp. ones) column vector of length N; diag(a) denotes the diagonal matrix obtained from the vector a; x_1:M≜[ x_1^T     ⋯ x_M^T ]^T denotes the vector concatenation; 𝒩_ℂ(μ,Σ)  denotes a proper  complex Gaussian pdf with mean vector μ and covariance Σ; 𝒞χ_N^2 denotes a complex chi-square distribution with N (complex) Degrees of Freedom (DOFs); finally the symbol ∼ means “distributed as”..



Time-Reversal (TR), Radar imaging, Null-spectrum, Resolution, TR-MUSIC.





§ INTRODUCTION


Time-Reversal (TR) refers to all those methods which
exploit the invariance of the wave equation (in lossless and stationary
media) by re-transmitting a time-reversed version of the scattered
(or radiated) field measured by an array to focus on a scattering
object (or radiating source), by physical <cit.> or synthetic
<cit.> means. In the latter case (computational
TR), it consists in numerically back-propagating the
field data by using a known Green’s function, representative
of the propagation medium. Since the employed Green function depends
on the object (or source) position, an image is formed by varying
the probed location (this procedure is referred to as “imaging”).
Computational TR has been successfully applied in different contexts
such as subsurface prospecting <cit.>, through-the-wall
<cit.> and microwave imaging <cit.>.

The key entity in TR-imaging is the Multistatic Data Matrix (MDM),
whose entries are the scattered field due to each Transmit-Receive
(Tx-Rx) pair. Two popular methods for TR-imaging are the decomposition
of TR operator (DORT) <cit.> and the TR Multiple Signal
Classification (TR-MUSIC) <cit.>. DORT imaging exploits
the MDM spectrum by back-propagating each eigenvector of the so-called
signal subspace, thus allowing to selectively focus on each
(well-resolved) scatterer. On the other hand, TR-MUSIC imaging is
based on a complementary point of view and relies on the noise
subspace (viz. orthogonal-subspace[Such term underlines that it is orthogonal to the signal subspace.]),
leading to satisfactory performance as long as the data space dimension
exceeds the signal subspace dimension and sufficiently high Signal-to-Noise-Ratio
(SNR) is present. TR-MUSIC was first introduced for a Born Approximated
(BA) linear scattering model <cit.> and, later, successfully
applied to the Foldy-Lax (FL) non-linear model <cit.>.
Also, it became popular mainly due to: (a) algorithmic efficiency;
(b) no need for approximate scattering models; and (c) finer
resolution than the diffraction limits (especially in scenarios with
few scatterers). Recently, TR-MUSIC has been expanded to extended
scatterers in <cit.>.

Though a vast literature on performance analysis of MUSIC <cit.>
for Direction-Of-Arrival (DOA) estimation exists (see <cit.>
for resolution studies and <cit.>
for asymptotic Mean Squared Error (MSE) derivation, with more advanced
studies presented in <cit.>),
such results cannot be directly applied to TR-MUSIC. Indeed, in TR
framework scatterers/sources are generally assumed deterministic and
more importantly a single snapshot is used, whereas MUSIC results
for DOA refer to a different asymptotic condition (i.e. a large number
of snapshots). Also, to our knowledge, no corresponding theoretical
results have been proposed in the literature for TR-MUSIC, except
for <cit.>, providing the asymptotic (high-SNR)
localization MSE for point-like scatterers. Yet, a few works have
tackled achievable theoretical performance both for BA and
FL models via the Cramr-Rao lower-bound <cit.>.

In this letter we provide a null-spectrum[We underline that the MUSIC imaging function is commonly referred
to as “pseudo-spectrum” in DOA literature. Though less used, in
this paper we will instead adopt to the term “null-spectrum” employed
in <cit.>, as the latter work represents the closest counterpart
in DOA estimation to the present study.] analysis of TR-MUSIC for point-like scatterers, via a 1st-order perturbation
of Singular Value Decomposition (SVD) <cit.>, thus having
asymptotic validity (i.e. meaning a high SNR regime). The present
results are based on a homogeneous background assumption and neglecting
mutual coupling, as well as polarization or antenna pattern effects.
Here we build upon <cit.> (tackling the simpler
colocated case) and consider a general non-colocated multistatic
setup with BA/FL models where several TR-MUSIC variants, proposed
in the literature, are here investigated. The obtained results complement
those found in DOA literature <cit.> and allow to obtain
both the mean and the variance of each null-spectrum, as well as to
draw-out its pdf. Also, they highlight performance dependence of null-spectrum
on the scatterers/arrays configurations and compare TR-MUSIC variants
in terms of spectrum stability. We recall that stability property
is important for TR-MUSIC, and has been investigated by numerical
means <cit.> or using compressed-sensing based
approaches <cit.>. Finally, a few numerical examples,
for a 2-D geometry with scalar scattering, are presented to confirm
our findings.

The letter is organized as follows: Sec. <ref>
describes the system model and reviews classic results on SVD perturbation
analysis. Sec. <ref> presents the theoretical
characterization of TR-MUSIC null-spectrum, whereas its validation
is shown in Sec. <ref> via simulations. Finally,
conclusions are in Sec. <ref>.



§ SYSTEM MODEL


We consider localization of M point-like scatterers[The number of scatterers M is assumed to be known, as usually done
in array-processing literature <cit.>.] at unknown positions {x_k}_k=1^M in ℝ^p
with unknown scattering potentials {τ_k}_k=1^M in ℂ.
The Tx (resp. Rx) array consists of N_T (resp. N_R) isotropic
point elements (resp. receivers) located at {r̃_i}_i=1^N_T
in ℝ^p (resp. {r̅_j}_j=1^N_R
in ℝ^p). The illuminators first send signals to the
probed scenario (in a known homogeneous background with wavenumber
κ) and the transducer array records the received signals.
The (single-frequency) measurement model is then <cit.>:

    K_n    =    K(x_1:M,τ)+W
        =    G_r(x_1:M) M(x_1:M,τ) G_t(x_1:M)^T+W

where K_n∈ℂ^N_R× N_T (resp. K(x_1:M,τ))
denotes the measured (resp. noise-free) MDM. Differently W∈ℂ^N_R× N_T
is a noise matrix s.t. vec(W)∼𝒩_ℂ(0_N,σ_w^2 I_N),
where N≜ N_TN_R. Additionally, we have denoted: (i)
the vector of scattering coefficients as τ≜[[ τ_1   ⋯ τ_M ]]^T∈ℂ^M×1; (ii) (b) the Tx (resp. Rx) array matrix as G_t(x_1:M)∈ℂ^N_T× M
(resp. G_r(x_1:M)∈ℂ^N_R× M),
whose (i,j)th entry equals 𝒢(r̃_i,x_j)
(resp. 𝒢(r̅_i,x_j)), where 𝒢(·,·)
denotes the (scalar) background Green function <cit.>.
Also, jth column g_t(x_j) (resp. g_r(x_j))
of G_t(x_1:M) (resp. G_r(x_1:M))
denotes the Tx (resp. Rx) Green's function vector evaluated at x_j.
In Eq. (<ref>) the scattering matrix
M(x_1:M,τ)∈ℂ^M× M equals
M(x_1:M,τ)≜diag(τ)
for BA model <cit.>, while M(x_1:M,τ)≜[diag^-1(τ)-S(x_1:M)]^-1
in the case of FL model <cit.>, where the (m,n)th entry
of S(x_1:M) equals 𝒢(x_m,x_n)
when m≠ n and zero otherwise. We recall that our null-spectrum
analysis of TR-MUSIC is general and can be applied to both
scattering models.

Finally, we define the SNR≜‖K(x_1:M,τ)‖ _F^2/(σ_w^2 N_TN_R)
and, for notational convenience, N_Rdof≜(N_R-M)
and N_Tdof≜(N_T-M) as the dimensions of the
left and right orthogonal subspaces, whereas N_dof≜(N_Rdof+N_Tdof).



 §.§ TR-MUSIC Spatial Spectrum


Several TR-MUSIC variants have been proposed in the literature for
the non co-located setup <cit.>. A first approach consists
in using the so-called Rx mode TR-MUSIC, which evaluates the
null (or spatial) spectrum (assuming M<N_R):

    𝒫_r(x;U_n)≜g̅_r(x)^† P_r,n g̅_r(x)=‖U_n^† g̅_r(x)‖ ^2 ,

where U_n∈ℂ^N_R× N_Rdof
is the matrix of left singular vectors of K_n spanning the
noise subspace, g̅_r(x)≜g_r(x)/‖g_r(x)‖
is the unit-norm Rx Green vector function and P_r,n≜(U_nU_n^†)
(i.e. the “noisy” projector into the left noise subspace). A dual
approach, denoted as Tx mode TR-MUSIC, constructs the null
spectrum (assuming M<N_T):

    𝒫_t(x;V_n)≜g̅_t(x)^T P_t,n g̅_t(x)^*=‖V_n^† g̅_t^*(x)‖ ^2 ,

where V_n∈ℂ^N_T× N_Tdof
is the matrix of right singular vectors of K_n spanning
the noise subspace, g̅_t(x)≜g_t(x)/‖g_t(x)‖
is the unit-norm Tx Green vector function and P_t,n≜(V_nV_n^†)
(i.e. the “noisy” projector into the right noise subspace). Finally,
a combined version of two modes, named generalized TR-MUSIC,
is built as (assuming M<min{N_T,N_R}) <cit.>:

    𝒫_tr(x;U_n,V_n)≜𝒫_t(x;V_n)+𝒫_r(x;U_n).

Usually, the M largest local maxima of 𝒫_r(x;U_n)^-1,
𝒫_t(x;V_n)^-1 and
𝒫_tr(x;U_n,V_n)^-1
are chosen as the estimates {x̂_k}_k=1^M. Indeed,
it can be shown that Eq. (<ref>) (resp. Eq.
(<ref>)) equals zero when x equals one
among {x_k}_k=1^M in the noise-free case, since when
U_n=U_n (resp. V_n=V_n)
this reduces to the eigenvector matrix spanning the left (resp. right)
noise subspace of K(x_1:M,τ) <cit.>.
Similar conclusions hold for 𝒫_tr(x;U_n,V_n)
in a noise-free condition.



 §.§ Review of Results on SVD Perturbation


We consider a rank deficient matrix A∈ℂ^R× T
with rank δ<min{R,T}, whose SVD A=U Σ V^†
is rewritten as:

    A=([ U_s U_n ])([     Σ_s  0_δ×δ̌;  0_δ̅×δ 0_δ̅×δ̌ ])([ V_s^†; V_n^† ]) ,

where δ̅≜(R-δ) and δ̌≜(T-δ),
respectively. Also, U_s∈ℂ^R×δ and
V_s∈ℂ^T×δ (resp. U_n∈ℂ^R×δ̅
and V_n∈ℂ^T×δ̌) denote the
left and right singular vectors of signal (resp. orthogonal) subspaces
in Eq. (<ref>), while Σ_s∈ℝ^δ×δ
collects the (>0) singular values of the signal subspace. Then,
consider A=(A+N), where N is
a perturbing term. Similarly to (<ref>), the SVD A=UΣV^†
is rewritten as

    A=([ U_s U_n ])([    Σ_s 0_δ×δ̌; 0_δ̅×δ    Σ_n ])([ V_s^†; V_n^† ]) ,

showing the effect of N on the spectral representation[Indeed, as opposed to Eq. (<ref>), A
may be full-rank in general.] of A, highlighting the change of the left and
right principal directions. We are here concerned with the perturbations
pertaining to U_n and V_n,
stressed as U_n=U_n+Δ U_n
and V_n=V_n+Δ V_n, where
Δ(·) terms are generally complicated functions of
N. However, when N has a “small magnitude” compared
to A (see <cit.>), a 1st-order perturbation (i.e.
Δ(·) are approximated as linear with N),
will be accurate <cit.>. The key result is that perturbed
orthogonal left subspace U_n (resp. right subspace
V_n) is spanned by U_n+U_sB
(resp. V_n+V_sB̅), where norm (any sub-multiplicative
one, such as ℓ_2 or ‖·‖ _F norm)
of B (resp. B̅) is of the same order of that
of N. Intuitively, a small perturbation is observed
at high-SNR. The expressions for Δ U_n and Δ V_n,
at 1st-order, are[We notice that in obtaining Eq. (<ref>), “in-space”
perturbations (e.g. the contribution to Δ U_n depending
on U_n) are not considered, though they have been shown
to be linear with N (and thus not negligible at first-order)
<cit.>. The reason is that these terms do not affect performance
analysis of TR-MUSIC null-spectrum when evaluated at scatterers positions
{x_k}_k=1^M, due to the null spectrum orthogonality
property.] <cit.>:

    Δ U_n=-(A^-)^† N^† U_n;  Δ V_n=-(A^-) N V_n;

where we have exploited A^-=V_s Σ_s^-1 U_s^†
<cit.>.



§ NULL-SPECTRUM ANALYSIS 


First, we observe that the null spectrums at scatterer positions 𝒫_r(x_k;U_n),
𝒫_t(x_k;V_n) and
𝒫_tr(x_k;U_n,V_n),
k∈{1,…,M}, in Eqs. (<ref>), (<ref>)
and (<ref>) can be simplified, using U_n=U_n+Δ U_n
and V_n=V_n+Δ V_n and exploiting
the properties[Such conditions directly follow from orthogonality between left (resp.
right) signal and orthogonal subspaces U_s and U_n
(resp. V_s and V_n).] U_n^† g̅_r(x_k)=0_N_Rdof
and V_n^†g̅_t^*(x_k)=0_N_Tdof,
as

    𝒫_r(x_k;U_n)=‖ξ_r,k‖ ^2 ,  𝒫_t(x_k;V_n)=‖ξ_t,k‖ ^2,

where ξ_r,k≜Δ U_n^† g̅_r(x_k)∈ℂ^N_Rdof×1
and ξ_t,k≜Δ V_n^† g̅_t^*(x_k)∈ℂ^N_Tdof×1,
respectively. Similarly, 

    𝒫_tr(x_k;U_n,V_n)=‖ξ_t,k‖ ^2+‖ξ_r,k‖ ^2=‖ξ_k‖ ^2,

where ξ_k≜[ ξ_r,k^T ξ_t,k^T ]^T∈ℂ^N_dof×1.
Thus, to characterize 𝒫_r(x_k;U_n),
𝒫_t(x_k;V_n) and
𝒫_tr(x_k;U_n,V_n),
it suffices to study the random vector ξ_k. Indeed, the
marginal pdfs of ξ_r,k and ξ_t,k
are easily drawn from that of ξ_k. As a byproduct, ξ_k
definition also allows an elegant and simpler MSE analysis with respect
to <cit.>, as it can be shown that the position-error
of the estimates with Tx mode (Δx_T,k), Rx mode
(Δx_R,k) and generalized (Δx_TR,k)
TR-MUSIC can be expressed as Δx_T,k≈-Γ_T,k^-1{J_T,k^T V_n ξ_t,k},
Δx_R,k≈-Γ_R,k^-1{J_R,k^† U_n ξ_r,k}
and Δx_TR,k≈-Γ_TR,k^-1{[[ (J_R,k^† U_n) (J_T,k^T V_n) ]]ξ_k}, respectively, where J_T,k, J_R,k, Γ_T,k,
Γ_R,k and Γ_TR,k are suitably defined
known matrices (see <cit.>). Clearly, finding
the exact pdf of ξ_k is hard, as Δ U_n
and Δ V_n are generally complicated functions of the
unknown perturbing matrix W.

However, Δ U_n and Δ V_n assume a (tractable)
closed form with a 1st-order approximation (see Eq. (<ref>)).
This approximation holds tightly at high-SNR, as W will be
statistically “small” compared to noise-free MDM K(x_1:M,τ).
Hence, at high-SNR, ξ_k is (approximately) expressed in
terms of W as:

    ξ_k=[ ξ_r,k; ξ_t,k ]≈[   -U_n^† W t_r,k; -V_n^† W^† t_t,k ] ,

where t_r,k≜K^-(x_1:M,τ) g̅_r(x_k)∈ℂ^N_T×1
and t_t,k≜K^-(x_1:M,τ)^† g̅_t^*(x_k)∈ℂ^N_R×1
are deterministic. Since the vector ξ_k is linear[In the following of the letter we will implicitly mean that the results
hold “approximately” in the high-SNR regime.] with the noise matrix W, it will be Gaussian distributed;
thus we only need to evaluate its moments up to the 2nd order to characterize
it completely. Hereinafter we only sketch the main steps and provide
the detailed proof as supplementary material. First, the mean vector
𝔼{[ ξ_r,k^T ξ_t,k^T ]^T}=0_N_dof,
exploiting 𝔼{W} =0_N_R× N_T.
Secondly, the covariance matrix Ξ_k≜𝔼{ξ_kξ_k^†}
(since 𝔼{ξ_k}=0_N_dof) is
given in closed-form as:

    Ξ_k=[ σ_w^2 ‖t_r,k‖ ^2 I_N_Rdof          0_N_Rdof× N_Tdof;          0_N_Tdof× N_Rdof σ_w^2 ‖t_t,k‖ ^2 I_N_Tdof ] .

The above result is based on circularity of the entries of W,
along with their mutual independence. Thirdly, aiming at completing
the statistical characterization, we evaluate the pseudo-covariance
matrix Ψ_k≜𝔼{ξ_kξ_k^T}
(since 𝔼{ξ_k}=0_N_dof), whose
closed-form is Ψ_k=0_N_dof× N_dof.
The latter result is based on circularity of the entries of W,
along with their mutual independence and exploiting the results V_n^† t_r,k=0_N_Tdof
and U_n^† t_t,k=0_N_Rdof,
arising from subspaces orthogonality V_n^†V_s=0_N_Tdof× M
and U_n^†U_s=0_N_Rdof× M.

Therefore, in summary ξ_k∼𝒩_ℂ(0_N_dof, Ξ_k),
i.e. a proper complex Gaussian vector <cit.>.
Similarly, it is readily inferred that ξ_r,k∼ 𝒩_ℂ(0_N_Rdof, σ_w^2 ‖t_r,k‖ ^2 I_N_Rdof)
and ξ_t,k∼ 𝒩_ℂ(0_N_Tdof, σ_w^2 ‖t_t,k‖ ^2 I_N_Tdof),
respectively, i.e. they are independent proper Gaussian
vectors. Clearly, since ξ_r,k and ξ_t,k
have zero mean and scaled-identity covariance, the corresponding variance-normalized
energies ‖ξ_r,k‖ ^2/(σ_w^2‖t_r,k‖ ^2)∼𝒞χ_N_Rdof^2
and ‖ξ_t,k‖ ^2/(σ_w^2‖t_t,k‖ ^2)∼𝒞χ_N_Tdof^2,
respectively (i.e. they are chi-square distributed). Interestingly
these DOFs coincide with those available for TR-MUSIC localization
through Rx and Tx modes, respectively.

Based on these considerations, the means of the null-spectrum for
Tx and Rx modes are 𝔼{‖ξ_r,k‖ ^2}=σ_w^2 ‖t_r,k‖ ^2 N_Rdof
and 𝔼{‖ξ_t,k‖ ^2}=σ_w^2 ‖t_t,k‖ ^2 N_Tdof,
respectively, whereas for generalized null-spectrum 𝔼{‖ξ_k‖ ^2}=𝔼{‖ξ_r,k‖ ^2}+𝔼{‖ξ_t,k‖ ^2}
(by linearity). By similar reasoning, the variances for Tx and Rx
modes are given by var{‖ξ_r,k‖ ^2}=σ_w^4 ‖t_r,k‖ ^4 N_Rdof
and var{‖ξ_t,k‖ ^2}=σ_w^4 ‖t_t,k‖ ^4 N_Tdof,
respectively, whereas for the generalized null-spectrum var{‖ξ_k‖ ^2}=var{‖ξ_r,k‖ ^2}+var{‖ξ_t,k‖ ^2}
(by independence of ξ_r,k and ξ_t,k).

Hence, once we have obtained the mean and the variance of 𝒫_r(x_k;U_n),
𝒫_t(x_k;V_n) and
𝒫_tr(x_k;U_n,V_n),
respectively, we can consider the Normalized Standard Deviation
(NSD), generically defined as

    NSD_k≜ √(var{𝒫(x_k; ·))} / 𝔼{𝒫(x_k; ·)}.

Clearly, the lower the NSD, the higher the null-spectrum stability
at x_k <cit.>. For Rx and Tx modes
it follows that NSD_r,k=1/√(N_Rdof)
and NSD_t,k=1/√(N_Tdof), respectively.
It is apparent that in both cases the NSD does not depend (at
high SNR) on the scatterers and measurement setup, as well as σ_w^2,
but only on the (complex) DOFs, being equal to N_Rdof
and N_Tdof, respectively. Thus, the NSD becomes (asymptotically)
small only when the number of scatterers is few compared to the Tx
(resp. Rx) elements of the array. Those results are analogous to the
case of MUSIC null-spectrum for DOA, whose NSD depends on the DOFs,
namely the difference between the (Rx) array size and the number of
sources <cit.>. Differently, the NSD for generalized null
spectrum equals 

    NSD_k=√(‖t_r,k‖ ^4N_Rdof+‖t_t,k‖ ^4N_Tdof)/‖t_r,k‖ ^2N_Rdof+‖t_t,k‖ ^2N_Tdof .

Eq. (<ref>) underlines (i) a clear dependence
of generalized null-spectrum NSD on scatterers and measurement setup
and (ii) independence from the noise level σ_w^2. Also,
it is apparent that when ‖t_r,k‖≈0
(resp. ‖t_t,k‖≈0) the
expression reduces to NSD_k≈1/√(N_Tdof)
(resp. NSD_k≈1/√(N_Rdof)), i.e.
the NSD is dominated by Tx (resp. Rx) mode stability. Finally,
the same equation is exploited to obtain the conditions ensuring that
generalized spectrum is “more stable” than Tx and Rx modes (NSD_k≤NSD_t,k
and NSD_k≤NSD_r,k, respectively),
expressed as the pair of inequalities

    1/2[1-N_Rdof/N_Tdof]≤(‖t_t,k‖ /‖t_r,k‖)^2    (Tx)
    1/2[1-N_Tdof/N_Rdof]≤(‖t_r,k‖ /‖t_t,k‖)^2    (Rx)

Clearly, when N_R>N_T (resp. N_T>N_R) the inequality
regarding the Tx (resp. Rx) mode is always verified as the left-hand
side is always negative. Also, in the special case N_T=N_R
the left-hand side is always zero for both inequalities. 







§ NUMERICAL RESULTS


In this section we confirm our findings through simulations, focusing
on 2-D localization, with Green function[We discard the irrelevant constant term j/4.]
being 𝒢(x',x)=H_0^(1)(κ‖x'-x‖).
Here H_n^(1)(·) and κ=2π/λ denote the nth
order Hankel function of the 1st kind and the wavenumber (λ
is the wavelength), respectively. First, we consider a setup with
λ/2-spaced Tx/Rx arrays (N_T=11 and N_R=17, respectively,
see Fig. <ref>). Secondly, to quantify
the level of multiple scattering (as in <cit.>) we define
the index η≜‖K_f(x_1:M,τ)-K_b(x_1:M,τ)‖ _F/‖K_b(x_1:M,τ)‖ _F,
where K_b(x_1:M,τ) and K_f(x_1:M,τ)
denote the MDMs generated via BA and FL models, respectively. Finally,
for simplicity we consider M=2 targets located at (x_1/λ)=[[ -1 -6 ]]^T and (x_2/λ)=[[ +1 -6 ]]^T and having scattering coefficients τ=[[ 3 4 ]]^T; thus η=(0.7445).

Then, we compare the asymptotic NSD (Eq. (<ref>),
solid lines) with the true ones obtained via Monte Carlo (MC) simulation
(dashed lines, 10^5 runs), focusing only on the generalized null-spectrum
for brevity. To this end, Fig. <ref> depicts
the null-spectrum NSD vs. SNR for the two targets being considered,
both for FL and BA models. It is apparent that, as the SNR increases,
the theoretical results tightly approximate the MC-based ones, with
approximations deemed accurate above SNR≈10 dB.
Differently, in Fig. <ref>, we plot the
asymptotic NSD of the three TR-MUSIC variants vs. d, where (x_1/λ)=[[ (-1-d)     -6 ]]^T and (x_2/λ)=[[ (1-d)    -6 ]]^T (i.e. a rigid shift of the two scatterers), in order to investigate
the potentially improved asymptotic stability (viz. NSD) of the generalized
spectrum in comparison to Tx and Rx modes. It is apparent that the
gain is significant when d∈(-5,5), while outside this interval
the NSD expression is either dominated by Tx or Rx mode, which for
the present case NSD_t,k=1/√(11-2)≈0.33
and NSD_r,k=1/√(17-2)≈0.26, with
the generalized NSD never above that of NSD_t,k
(as dictated from Eq. (<ref>)).



§ CONCLUSIONS


We provided an asymptotic (high-SNR) analysis of TR-MUSIC null-spectrum
in a non-colocated multistatic setup, by taking advantage of the 1st-order
perturbation of the SVD of the MDM. Three different variants of TR-MUSIC
were analyzed (i.e. Tx mode, Rx mode and generalized), based on the
characterization of a certain complex-valued Gaussian vector. This
allowed to obtain the asymptotic NSD (a measure of null-spectrum stability)
for all the three imaging procedures. While similar results as the
DOA setup were obtained for Tx and Rx modes, it was shown a clear
dependence of generalized null-spectrum NSD on the scatterer and measurement
setup. Finally, its potential stability advantage was investigated
in comparison to Tx and Rx modes. Future works will analyze mutual
coupling, antenna pattern and polarization effects <cit.>,
and propagation in inhomogeneous (random) media <cit.>.


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