Block Iterative STMV Algorithm and Its Application in Multi-Targets Detection

STMV beamforming algorithm needs inversion operation of matrix, and its engineering application is limited due to its huge computational cost. This paper proposed block iterative STMV algorithm based on one-phase regressive filter, matrix inversion lemma and inversion of block matrix. The computational cost is reduced approximately as 1/4 M times as original algorithm when array number is M. The simulation results show that this algorithm maintains high azimuth resolution and good performance of detecting mul-ti-targets. Within 1 - 2 dB directional index and higher azimuth discrimination of block iterative STMV algorithm are achieved than STMV algorithm for sea trial data processing. And its good robustness lays the foundation of its engineering application.


Introduction
The bearing spectrum estimation is very important in sonar and radar fields. The original algorithm of bearing spectrum estimation based on arrays is conventional beam-forming (CBF), whose azimuth resolution is restricted by space "Fourier threshold", often termed "Rayleigh threshold" [1] [2]. There are many kinds of high-resolution bearing spectrum estimation algorithms since 1970s and Capon proposed the minimum variance distortion response (MVDR) beam-forming algorithm. MVDR has two manifestations when dealing with wide-band signals. One is incoherent signal-subspace processing method (ISM) proposed by Wax, whose azimuth resolution decreases when dealing with correlated sources. The other one is coherent signal-subspace processing method (CSM) proposed by Wang and Kaveh. Importing the beam-focused idea, this algorithm's estimation

Basic Principles of STMV Algorithm
The basic principle of STMV based on STCM is keeping the response at the steered bearing angle θ as 1 by rotating the beam and minimizing the total energy of output.
Assuming that each array output is ( ) 1 The relationship between STCM and CSDM is also given by the formula.
The best weight vector of beam-former is to keep the response as 1 at the specified direction, meanwhile minimize the power of array output, i.e.

( )
The weight vector can be obtained as Equation (8) Substituting Equation (8) into Equation (4), STCM comprehensively utilizes information of each frequency and the matrix is invertible if the product of frequency points and snapshots is bigger than order of the matrix.

Principles of Iterative STMV Algorithm
In this algorithm, the former data are utilized sufficiently and one-phase regressive filter is used to integrate the covariance matrix ( ) stmv R θ , then STCM ˆ( ) stmv R n estimated by using the nth snapshot can be represented as By matrix inversion lemma, Therefore,

A A A A A A A A A A A A A A A A A A A A A A
Noting that computing cost of multiplication is tens of times of add operation, so this paper only takes computational cost of multiplication into consideration.
Assuming inversion computational cost of m order matrix, then inversion com- Inversion formula and computational cost of n n × block matrix is deduced below.
A are m order Hermitian matrices, 12 A are m order matrix, then ( ) The proof is the same as above, where, 1 E − can be derived by Equation (19).
Illustration: In order to derive inversion formula of n n × block matrix, we can solve 1 11 A − first and then construct matrices D and E. Noting that E is block matrix. By analogy, we can solve 1 A − by applying "combination inversion formula" finally. This process is basement of deriving inversion computational cost of block matrix.
In summary, we consider the general formula of inversion computational cost of n n × block matrix. Inversion computational cost of n n × block matrix is

Principles of Algorithm
Assuming that U is segmented equally into K parts, is Hermitian matrix, then Equation (12) can be represented as ( ) (  ( 1) Applying inversion formula of block matrix equation to solving 3) Applying inversion formula of block matrix to solving 1 E − recursively; 4) Applying "combination inversion formula" to solving

Complexity of Classic STMV Algorithm
Assuming that array number is M, the bandwidth of signal is B and CSDM is estimated by the frequency domain output of N snapshots. The computational cost of STMV is mainly focused on two parts as follows: 2) The following calculation is necessary for each beam while estimating spatial spectrum: a) The STCM is estimated with Equation (6). It should be multiplied In summary, assuming the beam number is L, then the total times of multiplication operation is ( )

Complexity of Iterative STMV Algorithm
Assuming that array number is M, the bandwidth of signal is B. The computational cost of iterative STMV algorithm is mainly focused on two parts as follows: 1) It should be multiplied by In summary, assuming the beam number is L, then the total times of multiplication operation is ( ) ( )

Complexity of Block Iterative STMV Algorithm
Assuming that array number is M, the bandwidth of signal is B and U is segmented equally into K parts, calling i U which length is k M K = , where 1, , i K =  . The computational cost of block iterative STMV algorithm is mainly focused on five parts as follows: 1) It should be multiplied by  In summary, assuming the beam number is L, then the total times of multip- where, the computational cost of k order inverse matrix is ( ) Considering k M K = meanwhile, Equation (31) can be simplified to Assuming that the bandwidth of signal Hz 5 k B = and a beam is formed every two degrees. That's to say, it should estimate spatial spectrum at 91 L = beams. The snapshot is 4 N = , 20 K = . Then, the ratio of computational cost between the STMV algorithm and block iterative algorithm for various array number M are shown in Figure 1.
The computational cost of iterative algorithm proposed in paper [8]

Azimuth Resolution
The simulation conditions are listed as follows. Two sources signals and back- It is seen from the simulation results that compared to STMV algorithm, there is no significant change both on azimuth resolution and main lobe width for block iterative STMV algorithm. The only change is that the side lobe fluctuate slightly. In summary, the azimuth resolution of block iterative STMV algorithm has no significant decrement while the computational cost reduced sharply.

Ability of Detecting Weak Source
The simulation conditions are listed as follows. There is an interference lied at the bearing angle 100˚ with SNR = 5 dB, and weak target source lied at the bear- It is seen from the simulation results that block iterative STMV algorithm inherits the ability to detect weak target source of STMV algorithm while decreasing the computational cost dramatically. Moreover, because the side lobe of spatial spectrum is stationary, the weak target source can be detected clearly. This algorithm can realize detection of weak target source in strong coherent interference.

The Sea Trial Data Analysis
The sea trial data is analyzed to verify the adaptation and stability of the block iterative STMV algorithm. The trial frame is shown in Figure 6, a uniformly spaced linear array of 48 sensors is towed behind the towboat sailing straightly, whose original course is 0˚. The target 2 originally lies at about bearing angle 75˚ and sails in the direction of 0˚. The target 4 sails in the direction of array. During the experiment, shipping is busy and compared to targets 2, 4, targets 1, 3 are far away from the towboat whose radiated noise is weak.
The spatial spectrum estimated with STMV algorithm and the block iterative STMV algorithm after the data filtered with a differential whiten filter is shown in Figure 7, Figure 8.
It is not hard to see that there are five targets from Figure 7, where the target at bearing angle about 20˚ is towboat. For the reason that target (i.e. target 3) at bearing angle about 100˚ is far away from the towboat, its bearing changes slowly. Target (i.e. target 1) at bearing angle about 80˚ whose bearing change fast has bearing crossing with target 3 at about 40 s, but these two targets can be discriminated during the short time before or after the crossing moment, which proves further the high azimuth resolution character of block iterative STMV algorithm.
Besides, from Figure 8, we can see that there are two weak targets at bearing angle about 65˚, whose bearings are first close, then far away.
In order to further illustrate the ability to detect weak target, the spatial spectrum of the 27 th , the 45 th and the 126 st moment are shown as Figures 9-11. Although the SNR of target 2, and target 4 is about 20 dB lower than target 1 and target 3, these two weak targets still can be detected validly.       Multi-targets with strong and weak SNR can be detected with block iterative STMV algorithm in Figures 9-11. The directional index within 1 -2 dB for targets and higher azimuth discrimination are improved than STMV, and the power on background azimuth without target is suppressed about 1dB. It's due to the matrixes are added first in Equation (5) for STMV algorithm and the spectrum are added first in Equation (10) for block iterative STMV algorithm for the continuum sequence, the correlation of target's radial signal is utilized for block iterative STMV algorithm and better performance is achieved. Both of the target signal background noise are Gaussian noise which without any correlation in simulation, so the same results can't be obtained in Figure 4 and Figure 5. sented. The performance of block iterative STMV algorithm which has the directional index within 1 -2 dB for targets, the higher azimuth discrimination and the background azimuth power suppress about 1 dB is improved than STMV algorithm. It also has good robustness when processing sea trial data, which lays the foundation of its engineering application.