Through-Wall Imaging With UWB Radar System - KEMT FEI TUKE

2.5 Radar Imaging Methods Overview 35
so the elements of the multiple response matrix K can be calculated as:
Knj = ψn (Rj, ω)
. (2.5.50)
VSn (ω)
The main idea of MUSIC is to decompose a selfadjoint matrix into two orthogonal
subspaces which represent the signal space and the noise space. This method
is often called Singular Value Decomposition (SVD) [27]. Since the multistatic
response matrix is complex, symmetric but not Hermitian (i.e. it is not selfadjoint),
the time reversal matrix T is defined from the multistatic response matrix:
T = K ∗ njKnj
(2.5.51)
where ∗ denotes the complex conjugation. The time reversal matrix has the same
range as the multistatic response matrix. Moreover, it is selfadjoint. The time
reversal matrix T can be represented as the direct sum of two orthogonal subspaces.
The first one is spanned by the eigenvectors of T with respect to non-zero
eigenvalues. If noise is present, this subspace is spanned by the eigenvectors with
respect to eigenvalues greater than a certain value, which is specified by the noise
level. This space is referred to as the signal space. Note that its dimension is equal
to the number of the point-like targets. The other subspace, which is spanned by
the eigenvectors with respect to zero eigenvalues (or the eigenvalues smaller than
a certain value in the case of noisy data), is referred to as the noise subspace. The
corresponding eigenvectors are denoted by the eigenvalues:
λ1 ≥ λ2 ≥ ... ≥ λM ≥ λM+1 ≥ ... ≥ λN ≥ 0 (2.5.52)
of the time-reversal matrix T and vn, n = 1, 2, ..., N are the corresponding eigenvectors.
The signal subspace is spanned by v1, ..., vM and the noise subspace is
spanned by the remaining eigenvectors. For detection of the target the vector g p
is calculated from the vector of the Green’s functions:
g p = (G (R1, p) , G (R2, p) , ..., G (RN, p)) ′ . (2.5.53)
The point P is at the location of one of the point-like targets if and only if the
vector gp is perpendicular to all the eigenvectors vn, n = M +1, ...s, N. In practice,
the following pseudo spectrum is usually calculated by:
1
D (P ) =
N�
|(vn, gp )| 2
. (2.5.54)
n=M+1
The point P belongs to the locations of the targets if its pseudo spectrum is very
large (theoretically, in the case of exact data, this value will be infinity).
Simulated time-reversal imaging with MUSIC considering multiple scattering
between the targets is shown in [64]
In this case influence of a wall can be taken into account by applying Green’s
function considering the wall, what is a real challenge itself.