A Thresholding Scheme for Target Detection for Noise Radar Systems Based on Random Matrix Theory Yangsoo Kwon∗ , Ram M. Narayanan∗ and Muralidhar Rangaswamy†
∗ Department
of Electrical Engineering, The Pennsylvania State University, PA 16802, Building 620, 2241 Avionics Circle, WPAFB, OH 45433, Email:
[email protected],
[email protected],
[email protected] † AFRL/RYRT
Abstract— Target detection is one of important roles of radar systems. In this paper, we present a threshold method for multiple target detection for of noise radar systems. In order to avoid ambiguous target detection, we find thresholds to guarantee the detection performance with the same receiving antenna elements for a given false alarm probability. The threshold is computed from the largest and smallest eigenvalue distributions based on random matrix theory. Simulations show that the proposed threshold provides exact target detection.
I. I NTRODUCTION
II. S IGNAL MODEL Consider a single input multiple output (SIMO) system with one transmitting antenna and N receiving antennas. The number of targets in the scene is assumed to be K. Denote the transmitted sequence (waveform) at time instant n by x(n). Then the total received signal at the ith receiving antenna element is given as yi (n) =
K
rki x (n − τki (k)) + ηi (n) ,
(1)
k=1
i = 1, 2, · · · , N, k = 1, 2, · · · , K, Noise radar has been widely considered as a promising technique for high-resolution convert detection of multiple targets. One of the most important advantages is low probability of detection since the transmit waveform is constantly varying and never repeats exactly [1]. In general noise radar systems, target detection is accomplished by correlating the reflected signal from the target with a delayed replica of the transmitted random waveform, and identifying the peak in the correlation value. However, if the channel causes non-linear distortions on the transmitted signal, the correlation detector may not successfully detect targets. Further, ambiguously detected targets are sometime considered as non-targets due to unclear imaging results. In order to avoid ambiguous target occurrence, we propose a threshold to guarantee the detection performance with the same receiving antenna elements for a given false alarm probability. The threshold is computed from the largest and smallest eigenvalue distributions based on random matrix theory. Simulations show the proposed detection method can be used for a wide range of SNR environments, and the threshold provides definitive target detection. In this abstract, we proposed a modified total correlation method based on the largest eigenvalue. The proposed method obtains better performance for multiple targets than the conventional total correlation method which is in intermediate SNR regimes operating practical systems. At low SNR the proposed detection method outperforms the conventional method. Moreover, in order to obtain clearer target detection at low SNRs, we propose a thresholding method based on random matrix theory. Simulation results verifies the proposed method achieves clear detection performance.
where rki is the kth target response at the ith receiver. Assuming a Rayleigh distributed fluctuating amplitude and a uniformly distributed random phase, the individual target response is modeled as zero mean complex Gaussian random variables with variance σ 2 [2], [3]. Denote the duration of the transmitted waveform x(k) as M . Since each component of the waveform is a random variable drawn from a Gaussian distribution, the samples are independent and identically distributed (i.i.d.). In (1), τki represents the time delay for the received signal from the kth target at the ith receiver , and ηi denotes complex white Gaussian noise assumed to be i.i.d. with zero mean and variance ση2 at the ith receiver. As shown in (1), the total received signal yi (n) at each receiver element are the superposition of the reflected signals from multiple K targets. For this systems, we assume that the receiving antenna elements are a uniform linear array (ULA) that each element is sufficiently separated for received signal decorrelation. Thus, the spatial correlations among components of target response vectors can be ignored. Further, we assume that the noise vectors are independent. The received signal model given in (1) can be cast into a vector form by considering M consecutive samples as follows, r i (n) = [r1i r2i · · · rki ]T ,
(2) T
x (n) = [x (n) x (n − 1) · · · x (n − M + 1) ] ,
(3)
T
(4)
T
(5)
ηi (n) = [ηi (n) ηi (n − 1) · · · ηi (n − M + 1) ] , yi (n) = [yi (n) yi (n − 1) · · · yi (n − M + 1) ] .
Then we have the received signal at the ith antenna given
by yi (n) = ri T Xi+k + ηi (n) , i
(6)
is the sub-matrix of transmitted waveform matrix, where Xi+k i and X defined by (7). ⎡ ⎢ ⎢ ⎢ X = ⎢ ⎢ ⎣
x (n) x (n − 1) · · · x (n − M + 1) 0 ··· 0 x (n) ··· x (n − M ) x (n − M + 1) · · · . . . 0
. . . 0
. . . 0
.
. . ···
. . . 0
0 0
⎤
⎥ ⎥ ⎥ . ⎥ . (7) ⎥ . . ⎦ . . · · · x (n − M + 1) .
The superscript and subscript of X in (6) are column indices for generating the sub-matrix. For instance, Xi+k is a K × M i matrix generated by X from the ith to the i+kth column. Now, we define a matrix having M × (N + 1) elements composed of x and yi s as T Φ = x y1 y2 · · · yN , (8) and the sample covariance matrix, Σ=
N+1 1 1 ΦΦH = φn φH n, N+1 N + 1 n=1
(9)
where φn is the nth column of Φ. III. M ULTIPLE TARGET T HRESHOLD BASED ON R ANDOM MATRIX THEORY
Based on the signal model in Section II, various target detection schemes have been proposed such as cross correlation or information theory based detection and so on [4]. As we already know, these schemes, however, would provide certain values of a target area which implies that additional efforts should be added to judge targets’ existence on a certain pixel. Further, at low SNR regimes, the correlation value becomes very small, since the noise is i.i.d. In order to improve this performance degradation, we propose a threshold to obtain target existence or absence at a certain pixel. We define the SNR of the received signal of the ith receiver element as ⎞ ⎛ E yi − ηi 2 ⎠ dB, (10) SNRi =10 log10 ⎝ 2 E ηi i = 1, 2, · · · , N. where the elements of the vector, ηi , are drawn from independent Gaussian distribution. We assume that ηi has the same variance ση2 . Thus, SNRi can be generalized as SNR over the receiving antenna elements. In order to apply random matrix theory for target detection, we need to grasp another property of the covariance matrix as derived in (9) with different assumptions. Suppose there is no target in a certain pixel. In this case, only noise is measured by the N receivers since there is no reflected signal from targets. Then, the covariance matrix can be obtained by 1 ΦΦH . (11) N+1 From random matrix theory, the covariance matrix of the received signals, Ση has a Wishart distribution [5] when there is no target in an area, since each element of the sample Ση =
covariance matrix is the sum of random variables drawn from i.i.d. normal distribution. We restate as Theorem 1, the result of [6] as follows. Theorem 1: Suppose A is a n × p matrix, each element of each row is drawn from p-variate IID normal distribution with zero mean and covariance R, Np (0, R). If we define B = AT A, B is said to be a Wishart matrix which has Wishart distribution with n degrees of freedom, Wp (R, n). The joint probability density function (PDF) of B is given as |B|(n−p−1)/2 1 −1 tr R exp − B , (12) n/2 2 2pn/2 Γ (n/2) |R| where Γ(x) is the gamma function of x, and notation tr is trace. Following in Wishart’s footsteps, the density function of eigenvalues (ordered) of Wishart matrix was developed by James in [7]. However, it is complicated to be expressed numerically and there is no known closed form solution. Thus, instead of Wishart matrix analysis, the principal component analysis (PCA) can be utilized for obtaining an adaptive threshold for a pixel by pixel detection. In this section, we focus on the convergence properties of largest and smallest eigenvalues of a sample covariance matrix. The asymptotic distribution of the largest eigenvalue in Wishart matrix have been center and scaling constants [8] [9], and the distribution converges to Tracy-Widom distribution. For the Tracy-Widom distribution, Fβ stands for the limiting cumulative distributions of the largest eigenvalues in three types of ensembles, viz Gaussian orthogonal ensemble, Gaussian unitary ensemble, and Gaussian symplectic ensemble [10], [11]. In the complex signal case, the value of β is two for the Gaussian unitary ensemble (GUE) case which means that the Tracy-Widom distribution with order 2 is defined as ⎛ ∞ ⎞ (13) F2 (s) = exp ⎝− (x − s) q 2 (x) dx⎠ . s
In (13), q(s) is the unique solution to the Painl´eve II d2 q 3 equation given by dx with boundary condition 2 = sq + 2q q (x) ∼ Ai (x) as x → ∞ where Ai (x) denotes the Airy function [10], [11]. Theorem 2: Define the normalized sample covariance as = αΣη with a scaling factor M/σn2 . Assume lim N +1 = Σ M→∞ M α, where a constant α is defined as 0 < α < 1. Also, let μ = 1/3 2 where ζ and ξ denote (ζ + ξ) and ω = (ζ + ξ) ζ1 + 1ξ √ √ M and N + 1, respectively. This setting Tracy is to apply − μ /ω Widom distribution, following which λmax Σ converges to Tracy-Widom distribution of order 2 [8], [9]. σ2 2 This implies that the largest eigenvalue of Ση is ζn (ζ + ξ) as M → ∞. In [12], Bai and Yin found the limit of the smallest eigenvalue of a sample covariance matrix as the following theorem. N +1 Theorem 3: Assume lim M = α, where a conN →∞,M→∞
stant α is defined as 0 < α < 1. With the above condition,
the smallest eigenvalue converges√to a certain value, that is → σ 2 (1 − α)2 [12]. λmin Σ lim n N →∞,M→∞
Pf a =P (λmax > γthr λmin ) 2 σn ˜ ˜ Σ > γ λ λ =P max thr min Σ ζ2 2 ξ σn2 ˜ > γthr σn2 1 − λmax Σ =P ζ2 ζ ⎛ ⎞ ˜ −μ 2 λmax Σ γ (ζ −ξ) −μ thr ⎠ =P ⎝ > ω ω
1
35
0.9 0.8
30
0.7 25 y axis
All eigenvalues of the sample covariance matrix Ση from N x and {yi }i=1 have the same value as σn2 , i.e., λ1 = λ2 = · · · = λN +1 = σn2 . When targets are present, λ1 > λN +1 = 1. Hence, we can determine the target existence in a pixel. To compute a threshold, we start with the predefined false alarm probability Pf a (Pf a = P (λmax > γthr λmin )). From this criterion, the threshold γthr can be obtained based on these two properties of the eigenvalue convergence. The pre-defined false alarm probability can be reformulated as
40
0.6 20 0.5 15 0.4 10
0.3
5
0.2 5
10
15
20 x axis
25
30
35
40
(a) Proposed MI detection scheme.
40
(14)
1 0.9
35
0.8 30 0.7
y axis
where μ and ω are defined in Theorem 2. In (14), since the left hand side of the inequality can be expressed using the complementary CDF of Tracy-Widom distribution of order 2, we have 2 γthr (ζ − ξ) − μ . (15) Pf a = 1−F2 ω
25
0.6
20
0.5 0.4
15 0.3
Taking the inverse function of Tracy-Widom distribution, we have 2
γthr (ζ − ξ) − μ . (16) ω Substituting μ and ω into (16) and simplifying, we obtain the threshold as F2 −1 (1 − Pf a ) =
2
γthr =
(ζ + ξ)
2
(ζ − ξ)
+
(ζξ)
1/3
(ζ − ξ)
2
0.2
5
0.1 5
10
15
20 x axis
25
30
35
40
0
(b) Proposed MI detection scheme with a threshold. Fig. 1. Detection performance comparison at -10 dB SNR.
−2/3
F2−1 (1 − Pf a ) (ζ + ξ)
10
.
(17)
Utilizing the proposed total correlation method, each pixel in a certain area has positive values for the total correlation. Then, applying the above threshold, target existence can be determined. Note that the proposed threshold is a function of M , N , and Pf a . It is of interest to determine a threshold value below which target detection becomes unreliable, especially at low SNRs. However, determining such a threshold value is cumbersome and computationally intensive. In addition, this is scenariospecific, as stated above. IV. S IMULATIONS In order to evaluate the proposed detection algorithm, we conduct simulations for multiple target detection at intermediate and low SNR environments. We assume that three point targets are randomly distributed in an area and there are
one transmitter and 10 receiver antennas (N ) as shown. The transmitted signal is a 1000 pseudo Gaussian random sequence (M ) (i.i.d.), and Pf is set 0.02. The closed-form representation of the inverse of the Tracy-Widom distribution has not been found. However, [8] provides tables for the functions which are various orders of Tracy-Widom distribution . Thus, we utilize the value of the inverse Tracy-Widom distribution to verify detection performance of the proposed scheme. In the previous section, we proposed a threshold for multiple target, γthr . In order to verify the thresholding performance, the threshold is applied to the result of Figure 1(a) to determine target existence in a certain pixel which is based on a multiple target detector using correlation. In this simulation, one and zero for a location indicate the target presence and absence, respectively. As shown in Figure 1(b) shows that the proposed threshold achieves the exact positions of multiple
[6] J. Wishart, “The generalized product moment distribution in samples from a normal multivariate population,” Biometrika, vol. 20A, no. 1-2, pp. 32–52, 1928. [7] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” The Annals of Statistics, vol. 35, no. 2, pp. 475– 501, 1964. [8] M. Johnstone, “On the distribution of the largest eigenvalue in principal components analysis,” Ann. Statist, vol. 29, pp. 295–327, 2001. [9] K. Johansson, “Shape fluctuations and random matrices,” Communications in Mathematical Physics, vol. 209, pp. 437–476, 2000. [10] C. A. Tracy and H. Widom, “On orthogonal and symplectic matrix ensembles,” Communications in Mathematical Physics, vol. 177, no. 3, pp. 727–754, 1996. [11] ——, The Distribution of the Largest Eigenvalue in the Gaussian Ensembles. New York, NY: Springer, 2000. [12] Z. D. Bai and Y. Q. Yin, “Limit of the smallest eigenvalue of a large dimensional sample covariance matrix,” The Annals of Probability, vol. 21, no. 3, pp. 1275–1294, 1993.
1 0.9
Probability of detection
0.8 0.7 0.6 0.5 0.5×γ
0.4
thr
0.25×γ
thr
0.3
γthr
0.2
1.5×γ
0.1
3×γthr
thr
5×γ
thr
0
0
0.2
0.4 0.6 Probability of false alarm
0.8
1
Fig. 2. Receiver operating characteristics curves for different thresholds.
targets. Three target locations have unity and others are zeros. Thus exact target locations can be obtained by the proposed threshold at low SNR environment. In order to test the effectiveness of the threshold value given by (17), we compare the detection probability with manually varied threshold values with coefficients ranging 0.5 to 5. The threshold from (17), γthr , is 1.8117 with parameters, M =500, SNR=20 dB and N =5. We see from Figure 2 that as expected, detection probabilities with lower values (coefficients 0.5 and 0.25) are almost identical to the case of γthr . On the other hand, in the cases of higher thresholds (coefficients 1.5, 3 and 5), the detection probabilities decrease since the system misses targets which have fluctuating target reflection coefficients. Thus, the threshold obtained using (17) yields an upper bound above which the detection probability is degraded. V. C ONCLUSION In this abstract, we have presented a threshold method for detecting multiple targets for noise radar systems. Random matrix theory has been used to set the thresholds for exact target detection. Simulations based on randomly distributed targets have done to verify the method. R EFERENCES [1] R. M. Narayanan and X. Xu, “Principles and applications of coherent random noise radar technology,” in Proc. SPIE Conf. on Noise in Devices and Circuits, vol. 5113, Santa Fe, NM, 2003, pp. 503–514. [2] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial diversity in radars - models and detection performance,” IEEE Transactions on Signal Processing, vol. 54, no. 3, pp. 823–838, March 2006. [3] N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (An introduction),” The Annals of Mathematical Statistics, vol. 34, no. 1, pp. pp. 152–177, 1963. [4] A. Davydov, “Information theoretic approach to noise radar,” in Noise Radar Workshop 2008, Arlington, VA, Nov. 2008. [5] A. M. Tulino and S. Verd´u, “Random matrix theory and wireless communications,” NH: Now Publishers, pp. 66–82, June 2004.
