Comparative study on PC-SD and MWF Algorithms for STAP RADAR

Abstract—The paramount challenge in the radar system is to alleviate the consequences of cold (homogeneous) clutter, severe dynamic (heterogeneous) hot clutter and jamming interferences while estimating the states of targets under track. To surmount this challenge, Space-Time Adaptive Processing (STAP) intensify the competence of radar systems. Space-time Adaptive Processing is a two-dimensional filtering technique for antenna array with multiple spatially distributed channels. The name 'Space-Time' elucidate the coupling of multifarious spatial channels with pulse-Doppler waveforms. The term “Adaptive processor” signifies that it can employ using a variety of algorithms on many platforms ranging from space satellites to a small low flying unmanned aerial. In order to develop STAP algorithms to operate in adverse environments, where intense environmental interference can reduce STAP proficiency to detect and track ground targets. STAP can effectively suppress these interferences and maximize the signal to interference plus noise ratio (SINR). Methods such as principal component analysis, Multi-stage Weiner Filter (MWF) are applied to STAP system. Rank and Minimum square error are parameters considered for estimating the performance of two stated techniques. Keywords—STAP (space Time Adaptive Processing), homogeneous clutter, heterogeneous clutter, PC-SD, MWF, Rank, MSE.


PROBLEM STATEMENT
Though SAR image tracking has much application appeal and performed admirably, it's not without shortcomings. Disadvantages to tracking with SAR image processing include delays from image synthesizing, overcoming typical image distortion effects and need for classifiers. For example, typical SAR image may need 3 to 5 seconds to be synthesized, therefore all target identification and tracking algorithms needs to wait at least that long. These factors create an environment where critical information is not produced in "real-time" which is crucial for fast decision making. This will explore an alternative -space-time adaptive processing (STAP).
It is capable of detecting target returns with information on target range and direction. It has many desirable traits: first, it is adaptive, meaning that it can function in any environment because it adapts to the environment. Second, STAP is faster than most alternatives such as SAR image tracking because it does not require heavy computations. Third, it is relativity simple meaning that it requires no specialized hardware. Fourth it is versatile to implement, meaning it can work using a variety of algorithms (such as PC and MWF) on many platforms ranging from space satellites to a small low flying unmanned aerial vehicle (UAV).We will provide the descriptions of how Reduced Rank Transformation like PC and MWF impact STAP in practice. Returns from a pulsed -Doppler radar are collected in a coherent processing interval(CPI) which can be represented by a 3-D data cube composed of N elements , J pulses and L range gates. The radar data is then processed at one K×J range gate of interest, which corresponds to a slice of the CPI data cube as shown in Fig  (a). The data is then processed at one range of interest, which corresponds to a slice of the CPI data cube. This slice is a J ×K space-time snapshot whose individual elements correspond to the data from the jth pulse repetition interval (PRI) and the kth sensor element [2,6,7]. Hence this two-dimensional space-time data structure consists of element space information and PRI space-Doppler information. The snapshot is then stacked column-wise to form the KJ ×1 vector x. [6] If a target is present in the range gate of interest, then the return is composed of components due to the target, the interference sources or jammers, clutter, and noise: X = X i + X c + X n (1) If no target is present, then the snapshot consists only of interference, clutter, and white noise. The total input noise vector n is given by n = X i + X c + X n .

III. SIGNAL MODEL
(2) Succinctly stated, most classical STAP algorithms consist of the following steps depicted in Figure (b).
(i) Estimate the parameters interference covariance matrix and target complex amplitude.
(ii) Form a weight vector based on the inverse covariance matrix (iii) Calculate the inner product of the weight vector and the data vector from a cell under test (iv) Compare the squared magnitude of the inner product in step (iii) with a threshold determined according to a specified false alarm probability. [1] The input noise covariance matrix is then defined to be Radar detection is a binary hypothesis problem, where hypothesis H 1 corresponds to target presence and hypothesis H 0 corresponds to target absence. Each of the components of the space-time snapshot vector x are assumed to be independent, complex, multivariate Gaussian. This snapshot, for each of the two hypothesis, is of the form, ∶ ∶ . The KJ× 1-dimensional space-time steering vector v (ϑ1, ω1) is defined as follows: ,  Where b (ω t ) is the J×1 temporal steering vector at the target Doppler frequency ω t and a (ϑ t ) is the K×1 spatial steering vector in the direction provided by the target spatial frequency ϑ . The notion (.)  represents the Kronecker tensor product operator. For convenience in the analysis to follow, the normalized steering vector in the space-time look-direction is defined to be | | is a complex gain whose random phase ϕ is uniformly distributed between 0 and 2 .The random vector x, when conditioned on ϕ , is Gaussian under both hypotheses. The conditional probability densities of x are The likelihood ratio test then takes the form, Where η is some threshold. Using the densities in (8), the test in (9) becomes . is the modified Bessel function of the first kind. The noise covariance matrix R is nonnegative definite and the modified Bessel function is monotonically increasing in its argument. Therefore, the test in (10) reduces to

| |
Where the new threshold η 1 is related to the previous threshold η as follows : The test in (11) was the first STAP detection criterion, developed in the well-known papers by Brennan and Reed [8] and Reed, Mallet and Brennan (RMB) [3].

PRINCIPAL COMPONENT-SIGNAL DEPENDENT :
Principal Component uses the Eigen-value decomposition (EVD) to produce a low rank estimate of the sampled covariance matrix . [4,9,10,11,15,16,21]. This lower rank estimate would still be a good approximation to the original but it would dramatically reduce the required computer processing power. This " speed for accuracy" trade off is widely accepted in the industry due to benefit of reduced cost. Consider the MVDR-SMI beam former as follows: The best reduced r rank approximation of is formed by retaining the r largest eigenvalues and their corresponding eigenvectors and eliminate the rest. Therefore Where is less than N but contains the principal components of or the components with most signal power. Selecting the value for r PC is to find the number of eigen values that are above the noise floor. One assumption from equation (16) is that the eigen value are ordered from highest to lowest. Meaning that the highest eigen value is at i =1, the next highest eigen value is at i=2, and so forth. Principal component signal independent (PC-SI) algorithm is a data dependent signal independent rank reducing algorithm. The block diagram of PC-SI system is shown in figure: 2. Principal component signal independent (PC-SI) algorithm is a data dependent signal independent rank reducing algorithm. It is considered to be data dependent because the data x is considered in weight vector calculation. It is considered signal independent because the steering vector s is not considered. One advantage of algorithm is that it is simple. A disadvantage is that we lose performance by not taking s into account. The PC-SI weight vector would be  In forward recursion ,the filter decomposes the sampled data snapshot x with a sequence of orthogonal projection like B 0 [5].Rank reduction can be accomplished by truncating these decomposition stages to a desired number r MWF .The result is a reduced rank transformation basis that spans the Krylov subspace instead of the eigenvector basis like PC. [12,17].
Since, it tailors its basis selection to the desired steering vector s, the MWF is able to operate in a more compact subspace than PC. [1] After the forward recursion is completed, the MWF computes a series of scalar weights (w 1 , w2, etc) at each stage and subsequently combine them to form the overall MWF weight vector.
⋯ This technique has many desirable properties. First, its main computation operation is the simple vector cross correlation. Second, it does not form a covariance matrix which requires substantial computation work [18]. Last, it doesn't need matrix inversion or eigenvector decomposition, both of which are expensive operations [12,19,20].  (30), we find that each of the Krylov vectors is a weighted sum of the eigenvectors. This is similar to principal components (PC).In fact if all α i =1 the resulting rank compression is same as PC. Since these weight values are the function of both eigenvalue and the cross correlation coefficient then the MWF rank compression will always be better than or equal to its PC counterpart because α i ≤ 1.

RANK COMPRESSION IN MWF
Therefore in Krylov subspace, if r MWF = N then all N Krylov basis vectors are kept and the full Ndimensional space is spanned. But if r MWF < N then the Krylov subspace dimension can be reduced based on low eigen value, low correlation, or a combination of both. In practice, it is observed that environment with low power interferers are well handled by MWF rank compression due to the low product. Environments with closely spaced interference sources are also good candidate for MWF because their close proximity creates a bifurcation into a dominant eigenvector and a weak one. These weaker eigenvectors becomes additional candidates for rank compression by the MWF.
IV. STAP STIMULATION: We examined PC and MWF rank compression for space-time adaptive processing (STAP). As mentioned earlier, STAP environment includes three types of undesirable interference signals: jammers, noise, and clutter. Figure.4: shows the Eigen spectra of two environments. One environment includes 2 randomly placed jammers of 30dB jammer to noise ratio (JNR), 10dB clutter to noise ratio (CNR), noise at 0dB and ICM effects. The second environment is the same as the first minus the jammers.  Table.1.The calculated clutter rank is 17. This is near the simulated result of 18. The added rank could be the result of covariance matrix tapers (CMT). The second curve (red) demonstrates the impact of two 30dB JNR jammers on the eigenspectra. As jammer signals contaminates all channels. From simulation we see that contamination resulted in many more strong eigen values (i.e. rank). Comparing the two curves of Figure 5 we see that adding two jammers have doubled the rank and hence the number of needed adaptive degree of freedoms to cancel out the interference. Therefore it is highly desirable to implement reduce rank transformations (RRT) to lower processing cost. We evaluate the performance of MWF and PC-SD. simulation parameters are defined in Table 1 as follows.    PC-SD performance reaches lowest MSE at rank of 23; this means it needs 23 adaptive degrees of freedoms (ADoF) to suppress the interference to achieve MVDR (Minimum variance Distortion less Response). In contrast, MWF only needs 7 ADoF to accomplish the same. Notice that MWF also offers more flexibility in rank selection. As graph shows, MWF's MSE performance of ranks from 5 to 17 are all well within 3dB range of minimum mean square error (MMSE). This means that the MWF process can stop anywhere within stages 5 to 17 and still yield acceptable result. This type of flexibility is highly desirable. By varying the JNR. While holding CNR at 10dB we decrease the JNR from 50dB (Fig 7 A.) to 20dB (Fig 7  D.). Figure 7 shows the MSE performances. Rank selection for PC-SD seems unaffected by the JNR changes, however MWF shows dramatic changes.  Table.2 shows the rank selections of MWF and PC-SD. MWF adapts to the interference levels and adjusts to its rank selection to received jammer power while PC-SD makes no adjustments. MWF adaptability in this case is desirable given that in practical situations the environment is constantly changing. In addition, MWF rank selections are less than its PC-SD counterparts which means that it could be done faster.  Figure.8 and Table.3 shows rank selection for environments where CNR varies from 40dB (Fig 8 A.) to 10dB (Fig 8 D.) while JNR is constant at 50dB. In this case neither rank selection changes much, however MWF still offers lower rank selection. Figure Table 3 shows the rank selections of MWF and PC-SD. MWF adapts to the interference levels and adjust to its rank selection to received jammer power while PC-SD makes no adjustments. MWF adaptability in this case is desirable given that in practical situations the environment is constantly changing. In addition, MWF rank selections are less than its PC-SD counterparts which means that it could be done faster.   A  50  10  5-17  16-30  B  50  20  5-17  16-24  C  50  30  4-20  16-24  D  50  40  5-19 16-21

CONCLUSION
In this paper we showed that MWF offered superior rank compression than PC-SD, especially in environments where jammer powers are lowered to 30dB. MWF demonstrated that it can adapt its rank selections to the environment but PC-SD did not. MWF did not significantly reduce sample requirements as hoped. In all instances, N×M samples were required to have an adequate estimate of the interference covariance matrix. There are two things we should clarify. First, generating the MSE performance graphs shown are not possible in practice. They are acquired in our simulation because we know exactly what the interference covariance matrix is in our simulated environment, but in practice that would require infinite number of samples which is practically impossible. As a result, optimum rank selection would be more or less "blind". Second, the majority of ranks did not achieve our desired minimum variance distortion less response (MVDR). In both PC-SD and MWF, MSE performance degraded further as ranks increased beyond the optimum rank. For case in Figure 6, only 12 out of 64 possible ranks yielded acceptable results. If we blindly select our process rank, the probability of failure would be 81%.