arXiv:1603.08982v1 [stat.AP] 29 Mar 2016 uhr f[]ad[,1]hv eie,rsrigt h conc the to of resorting case devised, have concentration 10] numerical [9, stepwise and [4] [6–8 applications of certain authors in unrealistic is assumption ing ovninly rca supinfrteM estimators ML is the noise for the assumption crucial performance threshold a Conventionally, and outs asymptotic an offering the of between advantage (M tradeoff the likelihood have maximum to the known are on criterion based those develope estimation, techniques wireles DOA numerous the and the astronomy Among applic radio wide [1–4]. sonar, found munications radar, has others, which among processing in, array im sensor an is in problem topic estimation (DOA) direction-of-arrival The processing array sensor estimation, ma posteriori estimation, a likelihood maximum process, random variant ntrso efrac vrtecnetoa Lalgorithm. ML conventional the adv a over their performance results show of to simulation terms and in algorithms Numerical proposed es our assess models, context. to non-Gaussian vided radar popular the most in the procescially random of invariant one est spherically DOA assumption, the the for under algorithm iterative problem estimation noi an tion MAP derive the iterative we model an paper to as this accurate well In more often process. is non-Gaussian it a and valid, Gaussia assump not a this times however, follow application, to real-life DOA noise In the sensor tribution. in the estimators assume context ML timation the se Conventionally in direct topic processing. the important ray address an problems, to estimation (MAP) used (DOA) posteriori arrival widely a are maximum techniques and estimation (ML) likelihood maximum The n ope asinpoes aey the namely, process, Gaussian complex rando a and scalar positive a a the is of namely, SIRP root pr cess, square A the the as components: 14–16]. formulated two [12, of process, one Gaussian popular compound and two-scale, notable most t the which mod come among (SIRP) problems, process random such invariant with spherically so-called deal o to context developed noi the been non-Gaussian in have Various e.g., non-stationarity. case, shows wher the clutter [11–13], is radar This high-resolution and/or grazing-angle fulfilled. cond a not the theorem, are when limit scenarios this central certain the in on validity its based immediately is not, or colored self, AIU IEIODADMXMMAPSEIR DIRECTION-OF- POSTERIORI A MAXIMUM AND LIKELIHOOD MAXIMUM h rbe,hwvr sta h asinnieassumption noise Gaussian the that is however, problem, The ne Terms Index ouiomwhite nonuniform ∗ omncto ytm ru,Tcnsh Universit¨at Dar Technische Group, Systems Communication nfrl white uniformly ∗∗ EEE46 nvri´ ai us atreL D´efense, V La Nanterre Ouest Universit´e Paris EA416, LEME — texture ieto-farvletmto,shrclyin- spherically estimation, Direction-of-arrival .INTRODUCTION 1. conigfrtelclpwrchanging, power local the for accounting , and i Zhang Xin ABSTRACT 3 ] eetees hsoversimplify- this Nevertheless, 5]. [3, colored SIAINI H RSNEO IPNOISE SIRP OF PRESENCE THE IN ESTIMATION nieaieM siao o the for estimator ML iterative an , os,respectively. noise, ∗ oamdNblE Korso El Nabil Mohammed , speckle eciigthe describing , ini some- is tion .Tu,the Thus, ]. h radar the e emodels se lhsbe- has el tosfor itions dloses nd 3 5]. 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SIRP the o O siain( ihynnlna rbe) o o si for nor problem), non-linear of highly estimation best parameter (a the estimation current To DOA the in for available gene algorithms [23]. so-called no are model the there variance knowledge, namely, of model, analysis special their multivariate a of to application the restricted Furthermore, is consider, one. they linear problem a The sig is ever, noise. unknown expecta SIRP the the parameter-expanded estimate under parameters a to si algorithm provided received (PX-EM) authors no the maximization the unknown contaminating [18], of and In instead in 17–22], embedded [15, realizations parameters t texture’s estimate to seco and order in of realizations) presence noise-only the (known data assume exclusively almost context SIRP textu its matrix. by covariance characterized speckle fully and is eter(s) SIRP A scattering. local matrix, DOAs, asinpoeswt eoma n nunknown an and mean zero with process Gaussian nwhich in nematrix ance aetasoe hra h etr,dntdby denoted texture, the whereas transpose; gate ⋅ T , . . . , ) T N nti ae,w suetesuc waveforms source the assume we paper, this In h xsigwrsadesn h siainpolm na in problems estimation the addressing works existing The eoe transpose. denotes M × ∗∗ A s ob nnw eemnsi ope eune 3.The [3]. sequences complex deterministic unknown be to , 1 ( θ σ ( ( t n aisPesavento Marius and esrnievector, noise sensor M θ ) ( = Q x t ) sthe is ) [ ( θ θ = < = t ersnsteseke eprlywie complex white, temporally a speckle, the represents 1 1 n = ) [ θ , . . . , θ , . . . , E N ( a t ™ sat amtd,Germany Darmstadt, mstadt, M ) ( arwadfrfil oresgaswt un- with signals source far-field narrowband ) A σ θ = ngeneral in 1 ( ( × .MDLSETUP MODEL 2. ) M » t M θ ) , . . . , 1 ) σ h ra upta the at output array The . τ ] s T etro h orewaveforms, source the of vector ( ledAry France d’Avray, ille H ( t sthe is t ) ( a + ) σ t T ) ( ( ž nacmrhniemne,under manner, comprehensive a in θ t eoe h nphtnme,and number, snapshot the denotes where , n ) M M t , ( )] t ) × t , eoe the denotes ∗ 1 = ( etro nnw signal unknown of vector 1 ⋅ ) T , . . . , = H 1 tnsfrteconju- the for stands T, , . . . , N τ ARRIVAL ( ; t N t esr htre- that sensors hsaso can snapshot th N ) scomposed is , × × M s s. especkle he N algorithm param- re ( literature t inand tion steering ) n terative covari- ralized t , yout ry ndary ( mum how- gnal. tion- t gnal dol- ) on- our nal (1) (2) ise is = of independent, identically distributed (i.i.d.) positive random vari- s t and Q are fixed. We denote this estimate by τˆ t , which has ables at each snapshot. To resolve the ambiguity between the texture the( ) following expression: ( ) and the speckle so as to make the noise parameters uniquely identi- 1 H 1 fiable, we assume that tr Q = N, in which tr ⋅ denotes the trace. τˆ t = x t − A θ s t Q− x t − A θ s t . (8) N In this paper, we mainly{ consider} two kinds of{ texture} distributions ( ) ( ( ) ( ) ( )) ( ( ) ( ) ( )) that are most widely used in the literature, for both of which τ t Meanwhile, by applying Lemma 3.2.2. in [28] to Eq. (7), one can is characterized by two parameters, the shape parameter a and( the) obtain the expression of Qˆ , representing the estimate of Q when θ, scale parameter b. The firstis the gamma distribution, leading to the s t and τ t and are fixed, as: ( ) ( ) K-distributed noise [12, 24], where the pdf of τ t is: T 1 1 H ( ) Qˆ = Q x t − A θ s t x t − A θ s t , (9) 1 1 τ(t) T 1 τ t p τ t ; a,b τ t a− e− b , (3) t= ( ( ) ( ) ( )) ( ( ) ( ) ( )) = a ( ) ( ( ) ) Γ a b ( ) in which replacing τ t by the expression of τˆ t in Eq. (8) leads to ( ) the following iterative( ) expression of Qˆ : ( ) in which Γ ⋅ denotes the gamma function. The second kind of H our considered( ) texture distribution is the inverse gamma distribu- 1 T − − ˆ (i+ ) N x t A θ s t x t A θ s t tion, leading to the t-distributed noise [25, 26], for which Q = Q ( ( ) ( ) ( )) ( ( −)1 ( ) ( )) , T (i) t=1 x t − A θ s t H Qˆ x t − A θ s t a b b −a−1 − ( ( ) ( ) ( )) ‹  ( ( ) ( ) ( )) p τ t ; a,b = τ t e τ(t) . (4) (10) Γ a ( ( ) ) ( ) for which we choose the identity matrix of size N, denoted by IN , ( ) (0) to serve as the initialization matrix Qˆ . Under the assumptions above, the unknown parameter vector of 1 T ˆ (i+ ) our problem is ξ = θT , χT , ζT ,a,b , where χ is a 2NT -element We further need to normalize Q in Eq. (10) to fulfill the (i+1) vector containing the
real and imaginary parts of the elements of assumption that tr Q = N. Let Qˆ denote the normalized esti- 2 n t , t ,...,T N (i+1) { } s = 1 , and ζ is a -element vector containing the real mate Qˆ , which is: and( ) imaginary parts of the entries of the lower triangular part of Q. T T T ˆ (i+1) ˆ (i+1) ˆ (i+1) Let x = x 1 , ..., x T denote the full observation vec- Qn = N Q tr Q . (11)
( ) ( T) tor, and τ = τ 1 ,...,τ T represent the vector of texture real-  › izations at all[ snapshots.( ) The( )] full observation likelihood conditioned Now we consider the estimate of s t when θ, τ t and on τ can be written as: Q are fixed, which, denoted by sˆ t , can( ) be found by( solving) ∂LC ∂s t = 0, as: ( ) 1 H T − ~ ( ) 1 exp τ(t) ρ t ρ t H − H p x τ ; θ, χ, ζ = M Š ( ) ( ) ; (5) sˆ t = A˜ θ A˜ θ A˜ θ x˜ t , (12) ( S ) t=1 πτ t Q ( ) Š ( ) ( ) ( ) ( ) S ( ) S 1 2 1 2 in which A˜ θ = Q− ~ A θ , x˜ t = Q− ~ x t , representing the 1 2 in which ρ t = Q− ~ x t − A θ s t , represents the noise steering matrix( ) and the observation( ) ( at) snapshot t(, both) pre-whitened realization at( ) snapshot t with( ( its) speckle( ) spatially( )) whitened. by the speckle covariance matrix Q, respectively. Eq. (5), multiplied by p τ ; a,b , leads to the joint likelihood of From Eqs. (8), (10) and (12) one can see that the estimates of x and τ : ( ) τ t , Q and s t are mutually dependent, and further dependent on the( ) parameter vector( ) θ. This dependency makes it impossible to ob- p x, τ ; ξ = p x τ ; θ, χ, ζ p τ ; a,b tain a closed-form expression for the LL function concentrated w.r.t. ( ) ( S ) ( ) 1 H T − (6) each of the individual parameters τ t , Q and s t and indepen- exp τ(t) ρ t ρ t = M Š ( ) ( ) p τ t ; a,b . dent of other unknown parameters. To( ) cope with this( ) difficulty, we t=1 πτ t Q ( ( ) ) appeal to the so-called stepwise numerical concentration method in- S ( ) S troduced in [4, 9], and concentrate the LL function iteratively. This 3. ITERATIVE MAXIMUM LIKELIHOOD ESTIMATION can be accomplished by assuming at a particular iteration that, in our case, Qˆ and τˆ t are known and can be used in the computation In our IMLE algorithm we maximize, similarly as in [27], the condi- of sˆ t , which is( then) used in its turn to update Qˆ and τˆ t in the tional likelihood in Eq. (5), instead of the intractable marginal like- next( iteration.) The sequential updating procedure is repea(ted) until +∞ lihood function, ∫0 p x, τ ; ξ dτ , which does not yield a closed- convergence. form expression. In doing( so, we) actually focus on the texture real- Finally, we address the estimation of θ, our parameter of inter- ization τ , which is considered as deterministic, rather than the tex- est, considering the values of Q and τ as fixed and known. Thus, ture process itself. neglecting the constant terms, the conditional LL function in Eq. (7) Let LC denote the conditional log-likelihood (LL) function, can be reformulated as: which arises from Eq. (5), as: T 1 H LC = − Q ρ t ρ t , (13) − − t=1 τ t ( ) ( ) LC = ln p x τ ; θ, χ, ζ = T N ln π T ln Q ( ) ( T S )T S S into which we insert Eq. (12). The resulting expression is then maxi- 1 H (7) − N Q ln τ t − Q ρ t ρ t . mized w.r.t. θ, to obtain the estimate of θ for each iteration, denoted τ t t=1 ( ) t=1 ( ) ( ) θˆ ( ) by , as: T 2 To begin with, we set ∂LC ∂τ t = 0, the solution of which 1 – θˆ = arg min P ˜ t x˜ t , (14) provides an estimate of the parameter~ ( )τ t when the parameters θ, θ Q A(θ) œt=1 τ t [ ( ) ( )[ ¡ ( ) ( ) in which ⋅ denotes the Euclidean norm and P – t I − Solving ∂L ∂τ t 0 leads to the expression of τˆ t when all A˜ (θ) = N J = HY Y −1 H ( ) the remaining unknown~ ( ) parameters are fixed, which is: ( ) A˜ θ A˜ θ A˜ θ A˜ θ , stands for the orthogonal projec- ( )Š ( ) ( ) ( ) ˜ 1 2 2 tion matrix onto the null space of the matrix A θ . ⎧ a − N − 1 b + a − N − 1 b ⎪ 2 Our proposed IMLE algorithm, comprising( three) steps, can be ⎪ Œ ( ) Š ( ) ⎪ 1 summarized as follows: ⎪ + 4b x t − A θ s t H Q− 0 ⎪ ( ) ⎪ Step 1: Initialization. At iteration i = 0, set τˆ t = 1, t = ⎪ ( ( ) ( ) ( ))1 (0) ( ) τˆ t ⎪ 2 (17) ,...,T ˆ = ⎪ ⋅ x t − A θ s t , K-distributed noise, 1 , and Qn = IN . ( ) ⎨ (i) i (i) ⎪ ( ( ) ( ) ( ))  ‘ Step 2: Calculate θˆ from Eq. (14) using τˆ( ) t and Qˆ , then ⎪ n ⎪ 1 H −1 (i) (i) ⎪ x t − A θ s t Q (i) ˆ (i) ˆ ( ) ⎪ + + sˆ t from Eq. (12) using θ , τˆ t and Qn . ⎪ a N 1 ‰ 1 ⎪ ( ( ) ( ) ( )) ( ) ˆ(i) (i) ˆ (i) ( ) ˆ (i+ ) ⎪ ⋅ x t − A θ s t + b , t-distributed noise. Step 3: Use θ , sˆ t and Qn to update Qn from Eqs. (10) ⎪ i i 1 ⎪ ( ( ) ( ) ( )) Ž ˆ( )) (i) ˆ ( + ) ⎩ and (11). Then use θ , sˆ t and the updated matrix Qn to Next we consider the estimation of the texture parameters a 1 find the update τˆ(i+ ) t from( Eq.) (8). Set i = i + 1. and b, denoted by aˆ and ˆb. The latter can be obtained by solving Repeat Step 2 and( Step) 3 until a stop criterion (convergence or ∂LJ ∂b = 0, as: ~ a maximum number of iteration) to obtain the final estimate of θ, T ˆ ∑ 1 τ t denoted by θIMLE. ⎧ t= , K-distributed noise, ⎪ Ta( ) The convergence of our algorithm is guaranteed by the fact that ˆ ⎪ b = ⎪ Ta (18) the value of the objective function in Eq. (14) at each step can either ⎨ , t-distributed noise. ⎪ T 1 improve or maintain but cannot worsen [9]. The same holds true for ⎪ ∑t=1 ⎪ τ(t) the update of Qˆ and τˆ t . In fact, as our simulations will show, the ⎩⎪ convergence can be attained( ) by only two iterations. Thus, the com- Meanwhile, calculation of ∂LJ ∂a results in: ~ putational cost of our algorithm, which lies mainly in the solution of T − − + the highly nonlinear optimization problem in Step 2, is only a few ⎪⎧ T Ψ a T ln b Q ln τ t , K-distributed noise, ∂LJ ⎪ ( ) t=1 ( ) times of that of the conventional ML estimation (CMLE) algorithm, = ⎪ which, incidentally, corresponds to the case in which the noise is ∂a ⎪ T ⎨ − + − uniform white Gaussian, such that Eq. (14) degenerates into: ⎪ T Ψ a T ln b Q ln τ t , t-distributed noise; ⎪ 1 ⎪ ( ) t= ( ) ⎪ (19) T ⎩ – 2 in which Ψ ⋅ stands for the digamma function. It is obvious from θˆCMLE = arg min P A θ x t . (15) θ Q ( ) œt=1 Z ( )Z ¡ Eq. (19) that( )∂LJ ∂a = 0 does not allow an analytical expression of the root. Thus aˆ can~ only be calculated numerically. 4. ITERATIVE MAXIMUM A POSTERIORI ESTIMATION Next, we approach the estimation of the source waveforms and the speckle covariance matrix. By noticing that ∂LJ ∂Q = ~ The IMLE algorithm, presented in Section 3, treats the texture as de- ∂LC ∂Q, and ∂LJ ∂s t = ∂LC ∂s t , it follows immediately ~ ~ ( ) ~ ( ) terministic and thereby ignores information of its statistical proper- that the same expressions of Qˆ and sˆ t in Eqs. (9) and (12), which ties. This has the advantage of easier and faster implementation, and we obtained for the IMLE algorithm,( are) also valid in the case of the is also a natural approach when the texture distribution is either un- IMAPE algorithm. Substituting into Eq. (9) the new expression of known or does not have a closed-form expression, e.g., in the case of τˆ t in Eq. (17) leads to the following expression for Qˆ : Weibull-distributed noise. In general cases, however, such approach ( ) 2 T is suboptimal. Thus, when the texture distribution is available, we ⎧ Q x t − A θ s t ⎪ T have the better choice of exploiting information from the texture’s ⎪ t=1 ( ( ) ( ) ( )) ⎪ prior distribution, i.e., employing the maximum a posteriori (MAP) ⎪ ⋅ x t − A θ s t H ⎪ approach, in designing our estimation procedure. This leads to our ⎪ ( ( ) ( ) ( )) ⎪ −1 IMAPE algorithm that we propose in this section. ⎪ H ˆ (i) ⎪ ⎛ 4b x t − A θ s t Q The MAP estimator maximizes the joint LL function, denoted ⎪ ‚ Œ ( ( ) ( ) ( )) ‹  ⎪ by LJ, which is equal to: ⎪ ⎝ 1 ⎪ 2 ⎪ 2 2 ⎪ ⋅ x t − A θ s t + a − N − 1 b LJ = ln p x, τ ; ξ = ln p x τ ; θ, χ, ζ p τ ; a,b ⎪ ( ) ( ( S ) ( )) ⎪ ( ( ) ( ) ( )) ( ) ‘ T ⎪ + (i+1) ⎪ = LC Q ln p τ t ; a,b Qˆ = ⎪ + − − (20) 1 ⎪ a N 1 b⎞, K-distributed noise, t= ( ( ) ) ⎨ ( ) T ⎪ ⎠ ⎪ T LC − T ln Γ a − Ta ln b + a − 1 ln τ t ⎪ a + N + 1 ⎪⎧ Q ⎪ − ⎪ ( ) ( ) t=1 ( ) ⎪ Q x t A θ s t ⎪ ⎪ T t=1 ⎪ T (16) ⎪ Š ( ( ) ( ) ( )) ⎪ 1 τ t ⎪ ⎪ − ∑t= ⎪ H ⎪ ( ), K-distributed noise, ⎪ ⋅ x t − A θ s t ⎪ b ⎪ = ⎪ ⎪ ( ( ) ( ) ( ))  ⎪ T ⎪ −1 ⎨ L − T ln Γ a + Ta ln b − a + 1 ln τ t ⎪ + − H ˆ (i) ⎪ C Q ⎪ b x t A θ s t Q ⎪ ( ) ( ) t=1 ( ) ⎪ ⎪ ⎪ ‚Œ ( ( ) ( ) ( )) ‹  ⎪ T ⎪ ⎪ 1 ⎪ ⎪ − b Q , t-distributed noise. ⎪ ⋅ x t − A θ s t , t-distributed noise. ⎪ τ t ⎪ ⎪ t=1 ⎪ ( ( ) ( ) ( )) ‘ ⎩⎪ ( ) ⎩⎪ (i+1) which, similar to the expression of Qˆ in Eq. (10) for the IMLE comparison we also plot, in both figures, the MSEs generated by algorithm, needs to be substituted into Eq. (11) to obtain the normal- the CMLE algorithm in Eq. (15), and the deterministic Cram´er-Rao ˆ (i+1) ˆ (i+1) bound (CRB) [13]. From these figures one can clearly see that the ized Q denoted as Qn . Finally, we address the estimation of θ. Adopting the numerical conventional ML algorithm becomes poor under the SIRP noise, and concentration method similar to that in Section 3, we also assume both of our algorithms lead to significantly superior performance. here that Q and τ are known from the previous iteration of the algo- These figures also show that only two iterations are sufficient for rithm. Furthermore, as the estimates of a and b are only dependent both of our algorithms to have a satisfactory performance, in terms on τ , these are also fixed for each iteration. This allows us to drop of a resulting MSE appropriately close to CRB θ , in asymptotic SNR cases. ( ) those terms in the expression of the joint LL function LJ in Eq. (16) that contain only these unknown parameters, and thereby to trans- 0 form it into the same expression as in Eq. (13). This means that θ 10 CMLE can be obtained, also for the IMAPE algorithm, from Eq. (14). IMLE, one iteration The iterative estimation procedure of our IMAPE algorithm also IMLE, two iterations −1 CRB contains three steps, and is summarized as follows: 10 0 Step 1: Initialization. At iteration i = 0, set τˆ( ) t , t = 1,...,T as ( ) the absolute values of independent random numbers from the stan- −2 0 10 1 ˆ ( ) MSE dard normal distribution , and Qn = IN . ˆ(i) (i) ˆ (i) Step 2: Calculate θ from Eq. (14) using τˆ t and Qn , then −3 10 (i) ˆ(i) (i) (ˆ ()i) sˆ t from Eq. (12) using θ , τˆ t and Qn . Meanwhile, substitute( ) Eq. (18) into Eq. (19). First( find) numerically aˆ(i) from −4 (i) ˆ(i) (i) 10 Eq. (19) using τˆ t , then b from Eq. (18) using τˆ t and 0 5 10 15 20 25 aˆ(i). ( ) ( ) SNR [dB] 1 ˆ(i) (i) ˆ (i) (i) ˆ(i) ˆ (i+ ) Step 3: Use θ , sˆ t , Qn , aˆ and b to update Qn from ( ) (i) Fig. 1: MSE vs. SNR under t-distributed noise. Eqs. (20) and (11). Then use θˆ , sˆ(i) t , aˆ(i), ˆb(i) and the updated 1 ˆ (i+ ) (i+(1)) matrix Qn to find the update τˆ t from Eq. (17). Set i = i + 1. ( ) 0 Repeat Step 2 and Step 3 until a stop criterion (convergence or 10 CMLE a maximum number of iteration) to obtain the final θˆ, denoted by IMAPE, one iteration ˆ IMAPE, two iterations θ . −1 CRB IMAPE 10 The remarks at the end of Section 3, upon the convergence and computational cost of our IMLE algorithm, also directly apply toour

−2 IMAPE algorithm. 10 MSE

5. NUMERICAL SIMULATIONS −3 10 In our simulations we consider a uniform linear array comprising

N = 6 omnidirectional sensors with half-wavelength inter-sensor −4 10 spacing. Two equally powered narrowband sources impinge on the 0 5 10 15 20 25 ○ ○ SNR [dB] array with the DOAs θ1 = 30 and θ2 = 60 relative to the broadside. The number of statistically independent snapshots is T = 10. For K-distributed sensor noise we choose a = 1.6 and b = 2; and for Fig. 2: MSE vs. SNR under K-distributed noise. t-distributed sensor noise, a = 1.1 and b = 2. The entries of the 2 speckle covariance matrix Q are generated by [29] Q m,n = σ ⋅ π Sm−nS j 2 (m−n) [ ] 0.9 e , m,n = 1,...,N. The number of Monte-Carlo 6. CONCLUSION trials is 100. The signal-to-noise ratio (SNR) is defined as:

T 2 In this paper we addressed the problem of estimating the DOAs of ∑t=1 s t multiple sources under the SIRP noise, by deriving two new esti- SNR = Y ( )Y , (21) T E τ t tr Q mators belonging respectively to the ML and the MAP family. Our { ( )} { } proposed IMLE and IMAPE algorithms are both based on the step- − in which E τ t is equal to ab for a K-distributed noise and b a wise concentration of the LL function w.r.t. signal and noise param- { ( )} ~( 1 for a t-distributed noise (for a > 1) [30]. eters. As our simulations show, both algorithms require only a few ) In Figs. 1 and 2, we plot the mean square errors (MSEs) of the iterations to attain convergence, and lead to significantly superior estimation of θ under the SIRP noise versus the SNR by implement- performance than the conventional approach. ing our proposed IMLE and IMAPE algorithms, respectively. In Fig. 1 the noise is t-distributed, and in Fig. 2, K-distributed. For

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