OPTIMAL radar detection in clutter requires appropriate

Transcription

1 An Efficient CFAR Algorithm for Piecewise Homogeneous Environments. Jeong Hun Kim and Mark R. Bell School of Electrical and Computer Engineering, Purdue University Abstract A new constant false alarm rate (CFAR) detection algorithm operating in non-homogeneous clutter is proposed in this paper. The proposed CFAR algorithm, CUT-inclusive (CI) CFAR, utilizes a goodness-of-fit test for determining and combining homogeneous windows, resulting in a higher detection performance. The CI-CFAR algorithm layout and analytical properties for the ideal case are investigated. In comparison to the traditional Cell-averaging (CA) CFAR and Order-statistic (OS) CFAR, it is demonstrated that CI-CFAR outperforms both in most situations. Also, the relationship between efficient sorting and homogeneity is investigated. Index Terms Anderson-Darling Test, CFAR, Order statistic (OS), CI-CFAR I. INTRODUCTION OPTIMAL radar detection in clutter requires appropriate selection of the detector threshold. One common approach to threshold selection in clutter environments with unknown or changing statistics is to estimate the clutter statistics from resolution cells surrounding the resolution cell under test. The implicit assumptions in this approach are that the resolution cells used to estimate the clutter statistics are target free and the clutter returns from these cells are statistically representative of the clutter statistics of the cell under test. When either of these two assumptions is violated, the threshold computed using this approach may result in a significant degradation of detector performance, either in the form of a higher probability of false alarm or a lower probability of target detection. In radar detection problems, detection thresholds are usually selected to maximize the probability of target detection while holding the false alarm constant. So the standard approach to adaptive threshold selection is to estimate a threshold value based on the radar returns from resolution cells surrounding the cell under test that will yield a constant probability of false alarm. The algorithms used to compute these adaptive thresholds are called Constant False Alarm Rate (CFAR) algorithms. In implementing CFAR algorithms, a decision must be made as to how large a collection of reference cells should be used to estimate the detection threshold. This region is called the reference window. If the clutter statistics are homogeneous across all resolution cells in a neighborhood surrounding the cell under test, then using a larger reference window will result in more accurate estimates of the clutter statistics and a corresponding improvement in detector performance. However, if there is significant spatial variation and hence nonhomogeneous statistical behavior in the neighborhood surrounding the cell under test, selecting too large a reference window will result in inaccurate estimates of the clutter statistics and the corresponding detection threshold will not be well matched to the cell under test. For this reason, selection of the reference window size and region is an important consideration in designing CFAR detectors. Furthermore, because a particular radar scene may include large homogeneous regions as well as regions with significant inhomogeneity, it may not be the case that one reference window size is appropriate across the whole radar scene. For this reason, approaches to the adaptive selection of reference windows are of interest. Multiresolution techniques provide one possible approach to measuring the variation in clutter statistics on different spatial scales, and this information can be used to adaptively select reference windows. In real radar scenes, another issue we must consider is that as the size of the reference window increases, the likelihood that one or more of the reference cells contains a target also increases. Since the presence of interfering targets tends to increase the average radar return from the resolution cells containing these targets, order statistic based techniques can provide robust threshold selection having significant immunity to a small number of interfering targets in the reference window. In this paper, we develop a CFAR algorithm that combines multiresolution and order statistic techniques (in the form of empirical distribution functions) for adaptive reference window selection. One particular benefit of the technique is computational efficiency because the cell under test does not have to be accounted for in setting up the reference window because of the robustness of the order statistic techniques used. For this reason, we call this algorithm Cell-under-test Inclusive CFAR (CI-CFAR.) A. Idea behind CI-CFAR II. CUT-INCLUSIVE(CI) CFAR The main concept of CI-CFAR is to find and group homogeneous regions in a radar scene. One commonly used method for finding and grouping homogeneous samples is image segmentation. However, most detection system require very quick response time from receiving data to declaring whether or not a target was detected and image segmentation methods are generally a very computationally intense process. Hence, using image segmentation in a detection system is not practical. We must find another method that will yield reasonable performance while being computationally efficient.

2 2 One such method is the Empirical Distribution Function (EDF) goodness-of-fit test. The CI-CFAR method in this paper has the following characteristics that distinguish it from other traditional CFAR algorithms: ) The concept of a moving/sliding window is not used. 2) Homogeneous regions are combined into larger reference windows using EDF goodness-of-fit tests. 3) A separate threshold is not calculated for each CUT, but rather a single threshold is calculated for each homogeneous region. B. Statistical Test for Merging (Anderson-Darling test) Goodness-of-Fit tests are a class of hypotheses tests that determine whether a pool of random samples came from a single distribution or not. The χ 2 test is a classic example of such a test. However, it was shown in [2] that goodness-offit tests based on empirical distribution functions (EDF) are generally more powerful. The most well known goodness-offit test based on EDF statistics is probably the Kolmogorov- Smirnov (K-S) test. There are numerous papers out that describe the performance of the K-S test. In [3], Stephens observed that the Anderson-Darling (A-D) test is generally more powerful than the K-S test for testing departures from the true distribution in the tail, particularly when there are many statistical outliers in the sample. This is similar to a typical radar detection problem, therefore the A-D test seems to be an appropriate test to use for CI-CFAR. We have in fact compared the K-S and A-D test in this work and found the A-D test to be superior. Implementation of the A-D goodness-of-fit test is rigorously explained in ([, p ], [4]) and will not be repeated here. C. Description of CI-CFAR Algorithm The CI-CFAR algorithm can be described as follows : ) Given a clutter map, use the Anderson-Darling test to see if whether or not the adjacent q-by-q windows (in one direction only, whether horizontal or vertical) come from the same distribution (i.e. are homogeneous). If they are homogeneous, then combine them into a single window. Repeat for all window pairs. This assumes that each of the q 2 samples in the q-by-q windows are homogeneous. 2) Only for those windows that were determined to be homogeneous and combined into a single window in the previous step, check if the adjacent windows are again homogeneous or not. If they are, merge into a single window. Repeat for all window pairs. 3) Repeat step 2 until no more windows are merged or until a pre-determined number of windows are merged. 4) Including the CUT, use the sorted data (data is already sorted in the previous steps) to find the k-th OS for the entire window. Thus, a single threshold will be used for comparison in deciding whether an outlier is present or not. 5) If the CUT exceeds the assigned threshold for the entire window, declare that a target is present. Otherwise, decide that there is no target. Note, that the single threshold to be used by CI-CFAR is equal to the multiplication of the k-th OS and scalar multiplicative factor γ. Although the use of a single threshold may slightly degrade performance, it will speed up the detection computation because there won t be a need to calculate the threshold over and over for each CUT as in conventional CFAR methods. III. PERFORMANCE ANALYSIS OF CI-CFAR A. Calculating Probability of False Alarm (P fa ) for CI-CFAR For calculating the size of our test, we will first assume that we will be only interested in γ >, which corresponds to the range where P fa is small. (γ : scalar multiplicative factor.) Also, we will assume that the algorithm will be operating in a clutter environment that has exponentially distributed power, which is generally accepted when there are a large number of random scatterers. This also offers ease of comparison with other CFAR algorithms since many CFAR algorithms are usually tested in exponentially powered clutter. Also, the following equality will be used several times in the proof, b b ia (Q ci) ta c a b ( ) b t (t a)! (b t)! (Q ct) for a, b Z and Q, c R. A special form of the above equality will be more useful, b b i0 (Q i) t0 () ( ) b t t! (b t)! (Q t), (2) which can be shown trivially from the original form. For the case of low P fa, we make the following assumptions : ) Threshold multiplicative factor γ is greater than (γ > ) which corresponds to low P fa. 2) The number k in the k-th OS for calculating the threshold is fixed. 3) The number of samples is N and the samples are IID with distribution of Exp(). 4) The CUT is also included in the pool of N samples. 5) Hypothesis H 0 is in effect, i.e. no target present, where, ) Z i, i, 2,..., N : OS of the N samples in increasing order. 2) Z k : k-th OS for calculating the threshold. 3) Z m : m-th OS which is also the CUT with no target. 4) S : Sample space of all possible outcomes. We compute the probability of false alarm as follows : P fa P H0 [H declared] P H0 [Z m γz k ]. Since the sample space can be partitioned as we have S [m < k] [m k] [m > k], P fa P H0 [(Z m γz k ) S] P H0 [(Z m γz k ) (m < k)] + P H0 [(Z m γz k ) (m k)] + P H0 [(Z m γz k ) (m > k)].

3 3 Since γ >, it is obvious that P H0 [(Z m γz k ) (m k)] 0. Thus, P fa P H0 [(Z m γz k ) (m > k)] P H0 [(Z m γz k ) ((m k + ) (m N))] P H0 [(Z m γz k ) (m k + )] + + P H0 [(Z m γz k ) (m N)] P H0 [Z k+ γz k ] + + P H0 [Z N γz k ] N P H0 [Z m γz k ]. mk+ Our problem now is to find P H0 [Z m γz k ] where m k +, k + 2,..., N. For this, it is necessary to find the joint pdf of Z m and Z k. (m > k) Noting that the CUT Z m is fixed at the m-th position, we actually only have N samples to choose from, thus the joint pdf for Z m and Z k, f km (x, y) is, ( )( )( )( )( ) N N k N k N m f km (x, y) k m k N m F k (x)f(x)[f (y) F (x)] m k f(y)[ F (y)] N m where F (x) ( e x ) [0, ] (x), f(x) e x [0, ] (x). Thus we have (N )! f km (x, y) (k )! (m k )! (N m)! ( e x ) k e x [e x e y ] m k e (N m+)y. By using the Binomial theorem to expand the powers and some simple calculus steps, we have P H0 [Z m γz k ] m k s0 k t0 Therefore, N P fa mk+ N mk+ x0 yγx f km (x, y)dydx (N )!( ) m s t (N m)! s! (m k s)! t! (k t)! (N k s)(k + s t + γ(n k s)). P H0 [Z m γz k ] m k s0 k t0 (N )!( ) m s t (N m)! s! (m k s)! t! (k t)! (N k s)(k + s t + γ(n k s)) k i0 N i N i + γ. (3) where γ is the scalar multiplicative factor, N is the number of samples and k is the number corresponding to the OS we have chosen for the threshold (usually the 80-th percentile of N). The last equality can be proven by using (). Similarly, for the case of high P fa or equivalently, when Fig.. CI-CFAR P fa Verification for N9 (a) Low P fa, i.e. γ > (b) High P fa, i.e. 0 γ < 0 γ <, it can be shown that, P fa k 2 i0 N i N i + γ. (4) where γ is the scalar multiplicative factor, N is the number of samples and k is the number corresponding to the OS we have chosen for the threshold (usually the 80-th percentile of N). Fig. demonstrates that the theoretic P fa matches the simulation results. Note the jump in Fig.. This happens at the P fa corresponding to γ and is due to the fact that the CUT is included in deciding the threshold. B. Calculating Probability of Detection ( P d ) for CI-CFAR Calculating P d of CI-CFAR for the ideal case (i.e. target embedded in homogeneous clutter) is very similar to that of finding P fa. As it turns out, there isn t a closed form expression for P d, but it can be simplified to a form where numerical evaluations are simple. For calculating P d, we make the following assumptions : ) Threshold multiplicative factor γ is greater than (γ > ) which corresponds to low P fa. 2) The number k in the k-th OS for calculating the threshold is fixed.

4 4 3) sample (CUT) is distributed with Exp(u) while N samples are Exp(). All samples are independent of each other. There are total of N samples. 4) The CUT is also included in the pool of N samples. 5) Hypothesis H is in effect, i.e. a target is present, where, ) Z i, i, 2,..., N : OS of the N samples in increasing order. 2) Z k : k-th OS for calculating the threshold. 3) Z m : m-th OS which is also the CUT with a target. (Exp(u)) 4) S : Sample space of all possible outcomes. We compute the probability of detection as follows : P d P H [H declared] P H [Z m γz k ] P H [(Z m γz k ) S] P H [(Z m γz k ) (m < k)] + P H [(Z m γz k ) (m k)] + P H [(Z m γz k ) (m > k)]. Since γ >, it is obvious that Thus, P H [(Z m γz k ) (m k)] 0. P d P H [(Z m γz k ) (m > k)] N P H [Z m γz k ]. mk+ Our problem now is to find P H [Z m γz k ] where m k +, k + 2,..., N. For this, it is necessary to find the joint pdf of Z m and Z k when H is in effect. (m > k) Noting that the CUT Z m is fixed at the m-th position with distribution Exp(u), we actually only have N samples of distribution Exp() to choose from, thus the joint pdf for Z m and Z k, f km (x, y) is, ( )( )( )( )( ) N N k N k N m f km (x, y) k m k N m F k (x)f(x)[f (y) F (x)] m k g(y)[ F (y)] N m where F (x) e x [0, ] (x), f(x) e x [0, ] (x) and g(x) u e x u [0, ] (x). Thus we have f km (x, y) (N )! u(k )! (m k )! (N m)! ( e x ) k e x [e x e y ] m k e (N m+ u )y. By using the Binomial theorem to expand the powers and some simple calculus steps, we have P H [Z m γz k ] m k s0 k t0 x0 yγx f km (x, y)dydx (N )! ( ) m s t u(n m)! s! (m k s)! t! (k t)! (N k s + u )(k + s t + γ(n k s + u )). Therefore, N P d mk+ N mk+ P H [Z m γz k ] m k s0 k t0 (N )! ( ) m s t u(n m)! s! (m k s)! t! (k t)! (N k s + u ) (k + s t + γ(n k s + u )). where u is the target mean, γ is the scalar multiplicative factor, N is the number of samples and k is the number corresponding to the OS we have chosen for the threshold (usually the 80-th percentile of N). Fig. 2. CI-CFAR P d Verification for (a) N 9, k 7, u 5 (b) N, k 8, u 3 Fig. 2 demonstrates that the analytic expression for P d is consistent with the simulation results. C. Presentation and interpretation of results Fig. 3 through Fig. 3 are simulation results from a 2-by- 2 exponential (mean of ) clutter map of 44 samples. For our purposes of simulating CI-CFAR, q 3, i.e. the clutter map was pre-divided into 3-by-3 windows where each of the windows are assumed to be homogeneous. Also, in step 3 of the CI-CFAR algorithm, the algorithm was terminated if 4 3-

5 5 A Description Target in homogeneous clutter. Layout B Target is adjacent to an outlier. Target in homogeneous clutter. C Target is adjacent to a window of mean 3. Windows NOT in the direction of A-D test. D Target is adjacent to a window of mean 0. E Windows NOT in the direction of A-D test. Target is in the center of the window. Target is adjacent to a window of mean 0. Windows in the direction of A-D test. F Target is adjacent to a window of mean 0. Windows are in the direction of A-D test. G 6 samples in Target s window has mean 0. Target is not in these 6 samples. H 6 samples in Target s window has mean 3. Target is not in these 6 samples. I 6 samples in Target s window has mean 0. Target is in one of these 6 samples. TABLE I DESCRIPTION OF NON-HOMOGENEOUS CLUTTER SCENARIO Fig. 3. ROC for target, homogeneous clutter. Scenario A Fig. 4. Comparison of CI-CFAR with OS-CFAR and CA-CFAR of window size 3-by-3, 5-by-5 and 7-by-7 by-3 windows were merged by the A-D test. A target of mean, 2, 3, 5, 7, 9 was used and P fa is ranged from 0 6 to 0 4. As explained before, it is assumed that the samples in each 3-by-3 window are homogeneous. Thus there are 6 homogeneous windows with each window having 9 samples. This assumption is needed in this type of CI-CFAR for the following reason. In using the A-D test, we are ultimately determining if a given number of samples are indeed homogeneous through EDF goodness-of-fit tests. It turns out that in most cases, the A-D test will usually determine that 2 samples are indeed homogeneous regardless of their underlying distribution. Another way of saying it is that if sample is added to a pool of 9 samples, then the A-D test will usually determine that the whole pool of 0 samples is homogeneous, even if the added single sample is a huge outlier. Thus, it is somewhat impractical to just start with 2 samples and keep adding single samples to the A-D test because the results will usually hold significant error. That is why the 9 samples in a single window are assumed to be homogeneous. The proposed CI-CFAR algorithm was also tested in a field of clutter where the 9 samples in a single window are NOT homogeneous to see how the performance would be affected in those situations. Fig. 3 corresponds to Scenario A in Table I and represents the case where all of the background clutter is homogeneous, having mean of and only a single target is in the entire clutter map. For this ideal situation, we would expect CA- CFAR to perform best but, as the results show, CI-CFAR has the best performance, CA-CFAR next and OS-CFAR last. The CA-CFAR and OS-CFAR behavior can be explained from the fact that CA-CFAR is optimal in homogeneous clutter but it is at first look somewhat surprising that CI-CFAR performs better than CA-CFAR in this ideal clutter map. This is due to the fact that since the background clutter is homogeneous, the CI-CFAR algorithm will combine 4 windows in most cases, thus instead of having 9 samples to detect targets, it will have 36 samples. Since CA-CFAR and OS-CFAR have a fixed window of 9 samples, it is highly likely that CI-CFAR with 36 samples will have better detection performance. Then it is a good question to ask, what window sizes are required by OS-CFAR and/or CA-CFAR to achieve similar performance to CI-CFAR in homogeneous clutter? Fig. 4 gives the answer to this question. The figure shows the case for a target of mean 3 for illustrative purposes. The results were nearly identical regardless of target mean. While the CI-CFAR algorithm was kept the same as before, the OS-CFAR and CA-CFAR were tested with window sizes of 3-by-3, 5-by-5 and 7-by-7. For the same algorithm, higher performance was achieved for larger window size as expected. It is shown that for 3-by-3 windows, CA-CFAR and OS-CFAR are both inferior to CI- CFAR. However, if the window size is increased to 5-by- 5, OS-CFAR shows almost equivalent performance whereas

6 6 CA-CFAR outperforms CI-CFAR entirely. If the window size is increased to 7-by-7 for OS-CFAR and CA-CFAR, they both outperform CI-CFAR. Thus it can be concluded that given an equal number of samples, OS-CFAR and CA-CFAR both have better performance than CI-CFAR if the clutter is homogeneous. Fig. 5. ROC for 2 adjacent target, homogeneous clutter. Scenario B Fig. 5 is the result of Scenario B, which is identical to Scenario A with the exception of an additional target right next to the previous one. The targets were also placed in the same 3-by-3 window and have the same mean. This demonstrates the case where target may mask the other. In this case, it is known that CA-CFAR will be inferior compared to OS-CFAR as is shown in Fig. 5. Note that for low target mean, CA-CFAR is superior to OS-CFAR because since the target mean is low, it will be virtually the same as having homogeneous clutter around the CUT, thus CA-CFAR will have better performance. As the target mean increases, the clutter of windows will become less homogeneous and thus OS-CFAR will behave better than CA-CFAR. (This is shown in Fig. 5) Even in the 2 target case, CI-CFAR is superior to the other two algorithms because the A-D test will still merge the nearby windows for detection even if there are 2 outliers in one of the windows. Thus, in most cases CI-CFAR will run the detection test with 8 or 36 samples, rather than 9, yielding a better detection even in the case of 2 outliers. Fig. 6. ROC for target, non-homogeneous clutter scenario C Fig. 6 through Fig. 2 are simulation results for the nonhomogeneous clutter scenarios. Fig. 6, Scenario C, is the case where there is a single target in the edge of one of the windows and the window next to the target (which will be adjacent to the target s window) has a different mean of 3, whereas all the other windows have mean. It is also set so that the non-homogeneous windows will not be compared with the target s window by the A-D test for homogeneity. Now, although the entire window has a different mean of 3, the mean is not large enough so that the A-D test will mostly accept that the windows are homogeneous. (It was actually accepted to be homogeneous 53.5% of the time.) Because this non-homogeneous window is accepted half the time as homogeneous by the A-D test, this results in a higher false alarm rate than the intended setting used for calculating the threshold. (This is clearly shown in the log-log plot of Fig. 6) Regardless of this matter, since the non-homogeneous window cannot be merged with the target s window (because it is not in the same direction.) the performance of CI-CFAR is expected to have minimal impact in compared to Fig. 3. This is certainly the case as shown in Fig. 6 and CI-CFAR is again superior to the other algorithms. Fig. 7. ROC for target, non-homogeneous clutter scenario D Scenario D in Fig. 7 is identical to Scenario C except that the non-homogeneous window is set to a higher mean of 0. As can be seen in Fig. 7, the performance of CA- CFAR is drastically reduced due to the high increase of the mean. On the other hand, OS-CFAR and CI-CFAR seem to have only modest degradation in performance. Especially, CI- CFAR seems to have not been affected as much. This is due to the way CI-CFAR is implemented. As explained previously, the A-D test only merges windows in one specific direction that is pre-set in the beginning. This means that if the nonhomogeneous window (no matter how close it is to a target) is in a position that is not consistent with the A-D test direction, it will have very little effect, if not none, on the performance. On the other hand, it also implies that if the non-homogeneous window is indeed in the direction of the A-D test, CI-CFAR will be affected. This will be shown in the next few figures. Fig. 8 and Fig. 9, corresponding to Scenario E and F respectively, are cases where the same non-homogeneous window of mean 0 from Scenario D is moved so that it

7 7 Fig. 8. ROC for target, non-homogeneous clutter scenario E Fig. 0. ROC for target, non-homogeneous clutter scenario G Fig. 9. ROC for target, non-homogeneous clutter scenario F Fig.. ROC for target, non-homogeneous clutter scenario H lies in the direction of the A-D test with the target s window. Scenario E has the target in the center of the window whereas in Scenario F, the target is on the edge of the homogeneous and non-homogeneous window. It can be expected that the performance of CI-CFAR will remain relatively equal for both cases because the non-homogeneous window will have equal effects in both cases and since the target s window is homogeneous, the target location within the window will have no impact on the performance. But CA-CFAR and OS-CFAR will have poorer performance in Scenario F than Scenario E because non-homogeneous samples will be also included in the reference window for Scenario F. This is displayed in Fig. 8 and Fig. 9. Fig. 0, Fig. and Fig. 2, corresponding to Scenario G, H and I respectively, shows how CI-CFAR can fail drastically. In Scenario G and H, there is a single target on the edge of a window in which there are 6 non-homogeneous samples. Scenario G has non-homogeneous clutter of mean 0 while in Scenario H, it has mean 3. Now, because of the way the clutter is laid out, it can be expected that the performance of CA-CFAR and OS-CFAR in Scenario G will be identical to the performance in Scenario F, since the non-homogeneous clutter mean is identical to 0. This can be easily verified from the figures. Now, in both Scenario G and H, it can be noted that the CI-CFAR performs significantly worse than both CA-CFAR and OS-CFAR. This is due to the underlying assumption that Fig. 2. ROC for target, non-homogeneous clutter scenario I the 9-sample window is homogeneous is broken. Because of the underlying assumption, the target is being compared with a non-homogeneous clutter that has higher mean than itself, thus resulting in a lower P d. Scenario I is similar to Scenario G, the only difference being that the target is embedded within one of the non-homogeneous samples. Again, CI-CFAR performs worse than both CA-CFAR and OS-CFAR as shown in Fig. 2. In order to improve performance in the case where the underlying assumption is violated, it is necessary to check the homogeneity of a single window prior to the A-D test.

8 8 Note that it isn t feasible to check the homogeneity of a given window by each sample separately. (For example, the A-D test for this case, i.e. N 9, k 9, results in giving no information regarding the homogeneity of the samples.) Thus, it is still required to divide the window into a smaller sized sub-windows, where the samples in each sub-windows are assumed to be homogeneous, for an EDF test to give any result. This again raises questions such as, what if the smaller sized sub-windows are not homogeneous and does different dividing method have different results? This hasn t been answered yet and is a future research topic. Fig. 3. Probability of detection in terms of SNR Fig. 3 is the plot of P d Vs. SNR for CI-CFAR, CA- CFAR And OS-CFAR with P fa set to 0 6. As can be noted from the figure, in an ideal clutter map, CI-CFAR has better performance because it merged the nearby windows and has more samples to estimate from. Also, it can be seen that CA- CFAR performs better than OS-CFAR as expected in an ideal clutter map. In most cases, CI-CFAR outperforms CA-CFAR and OS- CFAR. It is in those situations where the basic assumption of each window being homogeneous is violated that the performance of CI-CFAR degrades. To overcome this problem, it may be necessary to generate a test that will be able to reasonably determine the homogeneity of 2 samples, or i.e. determine the homogeneity of a single sample with a pool of N samples. Also, it will be worthwhile to expand the CI-CFAR s A-D algorithm so that the direction of the test is not fixed but can be generalized, possibly to all nearby windows. IV. SUMMARY AND CONCLUSION In this paper, a new CFAR algorithm that yields higher performance than CA-CFAR in homogeneous clutter while also robust in non-homogeneous clutter was introduced. The derivation and formula for calculating P fa and P d in homogeneous situations were proven for a simple case of CI- CFAR. A more complex version of CI-CFAR that incorporates the Anderson-Darling test of goodness-of-fit was introduced, which had the benefit of merging homogeneous clutter samples for better estimates and performance. It also decreases computational burden because only a single threshold is calculated for the entire homogeneous window. Also, this enables CI- CFAR to not keep track of the CUT for detection purposes, which makes implementation simpler. (I m still working on this last sentence so that it will sound better.) It has been shown through simulations that CI-CFAR performs better in most clutter map situations than CA-CFAR and OS-CFAR except for the case when the homogeneity of each window assumption is broken. It has also been pointed out that the Anderson-Darling test should be expanded into multi-direction and/or be able to determine the homogeneity of a single sample with respect to a pool of N samples. This will lead to a more general algorithm of CI-CFAR that will yield even better performance and possibly be robust to the inhomogeneity of each window. Also, it might be practical to test the homogeneity of each window before any other steps of the CI-CFAR using the k-sample Anderson-Darling test. Although this will increase the computational burden unnecessarily for the homogeneous clutter map case, it will definitely prevent severe performance loss when the homogeneity of each window assumption is broken. For example, if the homogeneity of each window is determined to be not true, the algorithm may choose to use another CFAR algorithm such as traditional OS-CFAR rather than CI-CFAR to prevent severe performance degrades. ACKNOWLEDGEMENT This work has been funded by the Air Force Office of Scientific Research (AFOSR) MURI Adaptive Waveform Design for Full Spectral Dominance. REFERENCES [] M. 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