The STFD based techniques exploit both the time-frequency representation of the signals and the spatial diversity provided by the multi-sensor platform. In this article, we consider the discrete form of the spatial pseudo Wigner-Ville distribution (PWVD) matrix using a rectangular window of odd length N that we apply to the perturbed data vector,
(13)
Substituting (5) into (13) we obtain,
(14)
is the STFD data matrix in absence of the sensor error and the additive noise, and D
pp
is the perturbation STFD matrix, while and are the cross STFDs matrices. Under the property of zero mean of noise n(t) and sensor error represented by ΔA, the expectation of the cross-terms vanishes (i,e., and ), and it follows,
(15)
Considering the spatially and temporally white assumptions of the perturbed steering matrix and noise, and using Equations (6) and (7) the above equation becomes,
(16)
Where
(17)
and
(18)
D
ss
(t, f) is a K × K source signal STFD matrix whose elements are given by,
(19)
where the diagonal elements are the auto-TFDs of the source signals, while the off-diagonal elements , are the cross-TFDs. We consider only the t-f points along the actual instantaneous frequency (IF) of each signal. Furthermore, assuming a second-order approximation of the derivative of the phase, we have,
(20)
where the IF f
i
(t) is given by,
(21)
Therefore, it results from (19) that,
(22)
In order to exploit the STFD given by (13) under an eigen-decomposition form, we use an averaging method which consists of averaging the STFD matrix at (t
i
, f
i
) points over the selected sources and over a number of T
o
selected tf-points (T
o
= M - N + 1), where M is the number of snapshots.
(23)
whose expectation is given by,
(24)
where f
k,i
is the IF of the k th signal at the i th time sample. Considering the Equation (14) for the (t
i
, f
k,i
) points and substituting it into (23), we obtain, after a straightforward calculation carried out in Appendix 2, the following expression,
(25)