A closed-form expression for false alarm rate of adaptive MIMO-GLRT detector with distributed MIMO radar

Signal Processing 93 (2013) 2771–2776

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A closed-form expression for false alarm rate of adaptive MIMO-GLRT detector with distributed MIMO radar Jun Liu a, Zi-Jing Zhang a,n, Yunhe Cao a, Shiyong Yang b a b

National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China Department of Computer, Xidian University, Xi’an 710071, China

a r t i c l e i n f o

abstract

Article history: Received 10 December 2012 Received in revised form 10 February 2013 Accepted 1 March 2013 Available online 13 March 2013

In this paper, we consider the detection problem in Gaussian noise using a multiple-inputmultiple-output (MIMO) radar with spatially dispersed antennas. A MIMO version of generalized likelihood ratio test detector is provided, which is referred to as MIMO-GLRT detector. A closed-form expression for the probability of false alarm of this MIMO-GLRT detector is derived. This theoretical result is confirmed with Monte Carlo simulations. & 2013 Elsevier B.V. All rights reserved.

Keywords: Detection False alarm rate Generalized likelihood ratio test Multiple-input–multiple-output (MIMO) radar

1. Introduction The technique of multiple-input multiple-output (MIMO) originated from communications is more and more widely applied to radars [1]. In general, MIMO radar falls into two classes according to the configuration of its antennas: one with colocated antennas [2–4] and the other with widely distributed antennas [5–10]. The latter MIMO radar is also referred to as statistical MIMO radar [11]. We restrict ourselves to this kind of MIMO radar. For brevity, we call it MIMO radar in the following. In the MIMO radar, the transmit–receive antennas are spatially dispersed such that independent observations of a target from multiple different aspects are possible, which brings about the so-called spatial diversity [11]. This spatial diversity can be used to reduce significantly the scintillations of targets’ RCS, and thus improve performance in several aspects, such as detection probability [12,13], and resolutions [5].

n

Corresponding author. Tel.: þ86 2988202312. E-mail addresses: [email protected] (J. Liu), [email protected] (Z.-J. Zhang), [email protected] (Y. Cao), [email protected] (S. Yang). 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.03.001

In [13], several temporal coherent adaptive detectors are proposed to deal with the problems of target detection in the MIMO radar. However, the statistical properties of these detectors have not been analyzed yet, and the probabilities of false alarm and detection are also unknown. Thus, the performance analysis in [13] is conducted with resorting to Monte Carlo simulations. Employing generalized likelihood ratio test (GLRT), Chong et al. developed an adaptive MIMO detector (see [14, formula (4)]), under the conditions that the clutter covariance matrix is unknown and the subarrays of MIMO radar are correlated. When the subarrays are uncorrelated, the MIMO Kelly’s detector is provided (see the second term of Remark 4.1 of [14]). In the Appendix of [15], Chong et al. only offered the false alarm rate of the adaptive MIMO detector (4) in [14] for the correlated case, but did not analyze the performance of the MIMO Kelly’s detector for the uncorrelated case. Notice that this MIMO Kelly’s detector in the uncorrelated case is completely different from the adaptive MIMO detector (4) in [14] in the correlated case, and in fact is equivalent to detector (19) in [13]. Therefore, it deserves to investigate the statistical properties of the MIMO Kelly’s detector in [14] or detector (19) in [13].

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J. Liu et al. / Signal Processing 93 (2013) 2771–2776

In this study, we examine the adaptive detection problem of MIMO radars in the presence of Gaussian noise with unknown covariance matrix. The unknown noise covariance matrix for each transmit–receive pair is estimated from a set of secondary data. We develop a MIMO version of GLRT detector (called MIMO-GLRT detector) which includes the MIMO Kelly’s detector in [14] or detector (19) in [13] as a special case. In particular, a closed-form expression for the probability of false alarm of the proposed MIMO-GLRT detector is derived. The remainder of this paper is organized as follows. Section 2 presents a signal model, and the MIMO-GLRT detector is offered in Section 3. The probability of false alarm of the MIMO-GLRT detector is derived in Section 4. Simulation results are illustrated in Section 5 and finally the paper is summarized in Section 6. 2. Signal model In this section, a signal model for MIMO radar with widely distributed antennas is presented. Suppose that a MIMO radar consists of M transmit antennas and N receive antennas with all the antennas geographically dispersed. The total number of transmit–receive paths available is V ¼MN. Assume that the tth transmitter sends Q t pulses and a target to be detected does not leave the given cell under test during these pulses. We further suppose that all the transmit waveforms are orthogonal to each other, and each receiver uses a bank of M matched filters corresponding to the M orthogonal waveforms. Sampled at the pulse rate via slow-time sampling, the signal received by the rth receive antenna due to the transmission from the tth transmit antenna, which is usually called test data (primary data), can be expressed as a Q t  1 vector, i.e., xr,t ¼ sr,t ar,t þnr,t

ð1Þ

where

 sr,t 2 C 



same, because the statistical properties of noise may be unique for each radar perspective. We further assume that Q t 4 1, t ¼ 1,2, . . . ,M, such that the coherent processing for each test data is possible. Notice that Q t, t ¼ 1,2, . . . ,M are not constrained to be identical, which means that the numbers of the pulses transmitted by different transmit antennas may be distinct. Assume that for each test data vector xr,t , there exists a set of training data (secondary data) free of target signal components, i.e., fyr,t ðkÞ, k ¼ 1,2, . . . ,K r,t 9yr,t ðkÞ  CN ð0,Rr,t Þg. Note that the numbers of secondary data vectors K r,t are not constrained to be the same. Suppose further that these secondary data vectors are independent of each other and of the primary data vectors. Let the null hypothesis (H0) be that the data is free of the target signal and the alternative hypothesis (H1) be that the data contains the target signal. Hence, the detection problem is to decide between the null hypothesis and the alternative one ( xr,t  CN ð0,Rr,t Þ ð2aÞ H0 : yr,t ðkÞ  CN ð0,Rr,t Þ and (

xr,t  CN ðsr,t ar,t ,Rr,t Þ

H1 :

for t ¼ 1,2, . . . ,M, r ¼ 1,2, . . . ,N and k ¼ 1,2, . . . ,K r,t .

3. MIMO-GLRT detector Due to the unknown parameters ar,t and Rr,t , the Neyman–Pearson criterion cannot be employed. According to the GLRT [16], a practical detector can be obtained by replacing all the unknown parameters with their maximum likelihood estimates, i.e., the detector is obtained by maxfar,t ,Rr,t 9t ¼ 1,2,...,M,

Q t 1

denotes a known Q t  1 steering vector for the target relative to the tth transmitter and rth receiver pair [13]. ar,t 2 C is a deterministic but unknown complex scalar accounting for the target reflectivity and the channel propagation effects in the tth transmitter and rth receiver pair. The noise nr,t is assumed to have circularly symmetric, complex Gaussian distribution, i.e., nr,t  CN ð0,Rr,t Þ, where the notation  means ‘‘is distributed as,’’ CN denotes a circularly symmetric, complex Gaussian distribution, and Rr,t is a positive definite covariance matrix of dimension Q t  Q t .

These primary data samples can be assumed independent of each other due to the widely distributed antennas in the MIMO radar. Notice that in the above model (1), these steering vectors sr,t ’s are not necessarily identical even though they describe the same target, since the relative position and velocity of the target with respect to different widely dispersed radars may be distinct. In addition, the covariance matrices Rr,t ’s are also not constrained to be the

ð2bÞ

yr,t ðkÞ  CN ð0,Rr,t Þ

maxfRr,t 9t ¼ 1,2,...,M,

r ¼ 1,2,...,Ng f ðX9H 1 Þ

r ¼ 1,2,...,Ng f ðX9H 0 Þ

H1

¼ _ l0

ð3Þ

H0

where l0 is the detection threshold, f ðÞ denotes probability density function (PDF) and X ¼ ½X1,1 , . . . ,XN,M  with Xr,t ¼ ½xr,t ,yr,t ð1Þ,yr,t ð2Þ, . . . ,yr,t ðK r,t Þ. Due to the independent assumption on these test data vectors, the PDF of X under Hj ðj ¼ 0,1Þ can be represented as f ðX9Hj Þ ¼

N Y M Y

f Xr,t ðXr,t 9Hj Þ , r ¼ 1 t ¼ 1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 9f r,t,j

j ¼ 0,1

ð4Þ

where f r,t,j ¼

1

pQ t JRr,t J 

exp½ðxr,t jsr,t ar,t Þy R1 r,t ðxr,t jsr,t ar,t Þ

K r,t Y

1

k¼1

pQ t JRr,t J

exp½yyr,t ðkÞR1 r,t yr,t ðkÞ

ð5Þ

with J  J and the superscribe ‘‘y’’ denoting the determinant of a matrix and conjugate transpose, respectively Since the random variables Xr,t are independent of each other, the maximization of the left-hand side of (3) can be

J. Liu et al. / Signal Processing 93 (2013) 2771–2776

performed term by term. Using (4), we can write (3) as N Y M max Y far,t ,Rr,t g f Xr,t ðXr,t 9H 1 Þ

maxfRr,t g f Xr,t ðXr,t 9H0 Þ r ¼1t ¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

H1

_ l0 :

ð6Þ

H0

9Ur,t

According to [16] or [17], we can write the ratio Ur,t defined in (6) as  K r,t þ 1 1 Ur,t ¼ ð7Þ 1Fr,t where

Fr,t ¼

1 2 9syr,t R^ r,t xr,t 9

ð8Þ 1 1 ð1 þxyr,t R^ r,t xr,t Þðsyr,t R^ r,t sr,t Þ P with R^ r,t ¼ Kk ¼ 1 yr,t ðkÞyyr,t ðkÞ and 9  9 representing the modulus of a complex number. Inserting (7) into (6) yields K r,t þ 1 H N Y M  Y 1 1 _ l0 : ð9Þ 1 F r,t H 0 r ¼1t ¼1 It is apparent that detector (9) is an extension of the classic GLRT in [16] from a single test data sample to multiple independent test data samples. Thus, this detector here is referred to as MIMO-GLRT detector. In particular, when the numbers of secondary data vectors K r,t are chosen to be all identical, we can take the ðK r,t þ1Þth root of (9), and then the MIMO-GLRT detector is reduced to the MIMO Kelly’s detector in [14] or detector (19) in [13]. 4. False alarm rate

where the operator @d1 þ d2 þ  þ ds =@ed11 @ed22    @eds s ðÞ denotes the mixed ðd1 þ d2 þ    þ ds Þth order partial derivatives of a function with respect to e1 ,e2 , . . . ,es J L ðx,TÞ ¼ xL expðTx1 Þ

where l ¼ ln l0 . Define ar,t ¼ ðK r,t þ1Þ=ðK r,t Q t þ 1Þ, we relabel a1,1 , . . . , a1,M , . . . aN,1 , . . . , aN,M as a1 , a2 , . . . , aV , respectively, where V¼MN. Recall that some of Qt’s may be identical, and so are some of K r,t ’s. Thus, some of ai ’s may be equal. We denote these coefficients with the same value by a new symbol, i.e., a1 , a2 , . . . , aV are denoted by e1 ,e2 , . . . ,es (en aem for nam) where s is the total number of the coefficients with different values, and ei corresponds to some coefficient whose value is repeated di þ 1 times among a1 , a2 , . . . , aV . In addition, di Z0, i ¼ 1,2, . . . ,s and P s þ si ¼ 1 di ¼ V. As derived in Appendix A, the probability of false alarm of the MIMO-GLRT detector can be expressed as " # !1 s s Y X @d1 þ d2 þ  þ ds J V ðen , lÞ ð11Þ di ! P FA ¼ @ed1 @ed2    @eds s n ¼ 1 En i¼1 1

2

ð12Þ

and En ¼ en

s Y

ðen ej Þ:

ð13Þ

j ¼ 1,jan

In particular, the expression (11) bears a simple form for the following two cases. Case I: a1 , a2 , . . . , aV are all the same. At this moment, s¼1, E1 ¼ e1 ¼ a1 and d1 ¼ V1. Then, we have s X JV ðen , lÞ ¼ J V1 ðe1 , lÞ: En n¼1

ð14Þ

Thus, the probability of false alarm for this case is P FA ¼ ¼

1 @V 1 ½J ðe1 , lÞ ðV1Þ! @e1V 1 V 1 V 1 X expðle1 n 1 Þ Cn ðV1nÞ!l en 1 ðV1Þ! n ¼ 0 V1

¼ expðle1 1 Þ

V 1 X n¼0

1 n ðle1 1 Þ n!

ð15Þ

where the second equality is obtained with the lemma introduced in Appendix A. Case II: a1 , a2 , . . . , aV are all distinct. In this case, s ¼V and dk ¼0, ek ¼ ak for k ¼ 1,2, . . . ,V. Therefore, the probability of false alarm for this case can be simplified as P FA ¼

In order to complete the construction of the test in (9), we should provide an approach to set the detection threshold. In this part, we derive a closed-form expression for the probability of false alarm of the MIMO-GLRT detector, which can be employed to compute the detection threshold for any given probability of false alarm. In doing so, we take the logarithm of (9), namely  H N X M X 1 1 L¼ ðK r,t þ 1Þ ln ð10Þ _l 1 F r,t H0 r ¼1t ¼1

2773

¼

V X J V ðen , lÞ En n¼1 V X eVn expðle1 n Þ : E n n¼1

ð16Þ

It is obvious that the probability of false alarm of the MIMO-GLRT does not depend on the noise covariance matrix. Therefore, the MIMO-GLRT exhibits the desirable constant false alarm rate (CFAR) property against the noise covariance matrices. It should be noticed that the classic adaptive matched filter (AMF) [22,23] or Rao test [24] developed in conventional radars can also be extended to the case of distributed MIMO radar. However, it is not easy to analyze the false alarm rate of the MIMO version of AMF, since the expression for the probability of false alarm of the classic AMF includes one-dimensional integral (see [22, Eq. (32)]). Even thought the Rao test has a simple expression for the probability of false alarm as shown in [24], a closed-form expression for the false alarm rate of the MIMO version of Rao test is currently unavailable, which needs further investigations. 5. Simulations results In this section, numerical simulations are conducted to validate the above theoretical analysis. For simplicity, we consider a MIMO radar comprised of two transmit antennas and two receive antennas (i.e., M ¼ N ¼ 2), and each

J. Liu et al. / Signal Processing 93 (2013) 2771–2776

transmit antenna sends eight coherent pulses (i.e., Q 1 ¼ Q 2 ¼ 8). Assume that the Doppler shifts of the target are 0.1, 0.2, 0.3 and 0.4 for the (1,1), (1,2), (2,1) and (2,2) transmit-receive pairs, respectively. The (i,j)th element of the noise covariance matrix is chosen to be ½Ri,j ¼ s2 0:959ij9 , where s2 represents the noise power. Without loss of generality, ar,t ’s are supposed to be the same, and then can be uniformly denoted by a. The signalto-noise ratio (SNR) is defined by 9a9

2

s2

:

ð17Þ

In order to examine the probability of false alarm of the MIMO-GLRT, we consider three cases with different numbers of secondary data: Case (1): K 1,1 ¼ K 1,2 ¼ K 2,1 ¼ K 2,2 ¼ 16, i.e., ai ’s are all identical. Case (2): K 1,1 ¼ 16, K 1,2 ¼ 18, K 2,1 ¼ 24 and K 2,2 ¼ 32, i.e., ai ’s are all distinct. Case (3): K 1,1 ¼ 16, K 1,2 ¼ 28, K 2,1 ¼ 28 and K 2,2 ¼ 32, i.e., a2 ¼ a3 , but a1 , a2 , a4 are all distinct. The expressions for the probability of false alarm in Cases (1) and (2) are (15) and (16), respectively. In Case (3), V¼4, e1 ¼ a1 , e2 ¼ a2 , e3 ¼ a4 and s¼3, d1 ¼ 0, d2 ¼ 1, d3 ¼ 0. According to (11), the probability of false alarm in Case (3) can be denoted by 2 3 3 J 4 ðej , lÞ5 1 @ 4X P FA ¼ 0!1!0! @e2 j ¼ 1 Ej ¼

e31 expðle1 1 Þ 2

ðe1 e2 Þ ðe1 e3 Þ 

þ

100

10−1

10−2

Theoretical results in Case (1) Simulation results in Case (1) Theoretical results in Case (2) Simulation results in Case (2) Theoretical results in Case (3) Simulation results in Case (3)

10−3

e2 ð3e2 þ lÞ expðle1 2 Þ ðe2 e1 Þðe2 e3 Þ

e32 ð2e2 e1 e3 Þ expðle1 2 Þ ðe2 e1 Þ2 ðe2 e3 Þ2

þ

10−4

e33 expðle1 3 Þ ðe3 e1 Þðe3 e2 Þ

: 2

0

5

10

15

20

Detection Threshold (λ)

ð18Þ The probability of false alarm of the MIMO-GLRT as a function of the detection threshold l for the three cases is reported in Fig. 1, where the dashed, solid and dasheddotted lines denote the results obtained from (15), (16) and (18), for Cases (1), (2) and (3), respectively, and the symbols represent the corresponding results obtained from Monte Carlo simulations. The number of independent trials used in each case is 105 . It is shown that the simulation results match the theoretical results pretty well. The detection probability curves of the MIMO-GLRT versus SNR for the three cases are plotted with Monte Carlo simulations in Fig. 2, where the probability of false alarm is fixed to 103 . The dashed, solid and dashed-dotted lines correspond to Cases (1), (2) and (3), respectively. The number of independent trials used to obtain each value of detection probability is 5000. It can be observed that the detection performance is the best in Case (3) with the most secondary data, and is the worst in Case (1) with the least secondary data. This is because more secondary date samples employed in the MIMO-GLRT detector can improve the accuracy in estimating the noise covariance matrix, and hence lead to a gain in detection performance. In what follows, we examine detection performance gains of the MIMO radar with respect to single-input–single-output (SISO) radar and single-input–multiple-output (SIMO) radar.

Fig. 1. Probability of false alarm of MIMO-GLRT detector in three cases. The dashed, solid and dashed-dotted lines indicate the results obtained from (15), (16) and (18) for Cases (1), (2) and (3), respectively. The symbols n, þ and J denote the corresponding results obtained from Monte Carlo simulations.

1 0.9 0.8 Probability of Detection

SNR ¼ 10 log10

To this end, we consider three kinds of radar system, i.e., Case (A): one transmitter and one receiver (i.e., SISO radar). Case (B): one transmitter and two receivers (i.e., SIMO radar). Case (C): two transmitters and two receivers (i.e., MIMO radar). In the above three cases, the numbers of the transmit-receive pairs, are equal to 1, 2, and 4, respectively. For simplicity, the number of secondary data samples is supposed to be 16 in each transmit–receive pair. The detection performance of the MIMO-GLRT for the these three cases is compared in Fig. 3, where P FA ¼ 103 . The dashed-dotted, solid and dashed lines indicate detection probabilities obtained with Monte Carlo simulations for Cases (A), (B) and (C), respectively. The number of independent trials used here is 5000. It is demonstrated that the MIMO radar outperforms the other two cases. This benefit comes at the cost of more transmit-receive pairs.

Probability of False Alarm (PFA)

2774

0.7 0.6 0.5 0.4 0.3 0.2 Case (1) Case (2) Case (3)

0.1 0 −22

−20

−18

−16

−14

−12

−10

−8

SNR(dB)

Fig. 2. Detection probability of MIMO-GLRT detector with fixed P FA ¼ 103 in three cases. The dashed, solid and dashed-dotted lines indicate detection probabilities obtained with Monte Carlo simulations for Cases (1), (2) and (3), respectively.

J. Liu et al. / Signal Processing 93 (2013) 2771–2776

hypothesis H0 is

1

Probability of Detection

2775

0.9

Prðfr,t 4 fr,t 9H0 Þ ¼ ð1fr,t ÞK r,t Q t þ 1 :

0.8 0.7

Then, the cumulative distribution function (CDF) of Fr,t under hypothesis H0 is

0.6

F Fr,t ðfr,t Þ ¼ PrðFr,t o fr,t 9H0 Þ ¼ 1ð1fr,t ÞK r,t Q t þ 1 :

0.5

ð20Þ

Therefore, the PDF of Fr,t under hypothesis H0 can be obtained by taking the derivative of the CDF with respect to fr,t , namely

0.4 0.3 0.2 Case (A) Case (B) Case (C)

0.1 0

ð19Þ

−20

−15

−10

−5

0

f Fr,t ðfr,t Þ ¼

¼ ðK r,t Q t þ1Þð1fr,t ÞK r,t Q t :

5

SNR(dB)

Fig. 3. Detection probability of MIMO-GLRT detector with fixed P FA ¼ 103 in Cases (A)–(C). The dashed-dotted, solid and dashed lines indicate detection probabilities obtained with Monte Carlo simulations for Cases (A), (B) and (C), respectively.

6. Conclusion We consider the problem of detecting a target embedded in Gaussian noise with unknown covariance matrix, employing the MIMO radar with widely distributed antennas. The MIMO-GLRT detector is proposed, which contains the MIMO Kelly’s detector in [14] or detector (19) in [13] as a special case. In particular, the closed-form expression for the probability of false alarm of the MIMO-GLRT detector is derived, which is verified using Monte Carlo simulations. We can use this expression to calculate the detection threshold for any given probability of false alarm. Simulation results reveal that the increase in the number of secondary data samples used in the MIMO-GLRT detector can obtain a gain in detection performance. In addition, the detection performance of the MIMO radar is considerably better than that of the SISO or SIMO radar.

Acknowledgments The authors would like to thank the anonymous reviewers and the Associate Editor for their valuable comments and suggestions which have greatly improved the quality of this paper. This work was supported by the National Natural Science Foundation of China (Nos. 61172137, 10990012 and 61271290). The effort of Shiyong Yang was sponsored by Shaanxi province science and technology research development program (No. 2009K01-40). Appendix A. The derivation of false alarm rate It is obvious that Fr,t defined in (8) corresponds to the test statistic in [18, formula (3)] when the matrix R defined in [18] is degraded into a column vector. According to [18, Eq. (4)], the right-tail probability of Fr,t under

d F F ðf Þ dfr,t r,t r,t ð21Þ

Define Cr,t ¼ 1=ð1Fr,t Þ, and it easily follows from the monotonicity of the function gðxÞ ¼ 1=ð1xÞ that the PDF of Cr,t under hypothesis H0 is [19, p. 200, Theorem 11] ðK r,t Q t þ 2Þ

f Cr,t ðcr,t Þ ¼ ðK r,t Q t þ 1Þcr,t

:

ð22Þ

Let Or,t ¼ ðK r,t Q t þ 1Þln Cr,t , we can obtain that the PDF of Or,t under hypothesis H0 is [19, p. 200, Theorem 11] f Or,t ðor,t Þ ¼ expðor,t Þ

ð23Þ

Recall that Cr,t ¼ 1=ð1Fr,t Þ. Thus, (10) can be written as

L¼

N X M X

H1

ar,t Or,t _ l

ð24Þ

H0

r ¼1t ¼1

where ar,t ¼ ðK r,t þ1Þ=ðK r,t Q t þ1Þ. For ease of notation, we relabel a1,1 , . . . , a1,M , . . . aN,1 , . . . , aN,M as a1 , a2 , . . . , aV , respectively, where V¼MN. Similarly, O1,1 , . . . , ON,M are relabeled as O1 , O2 , . . . , OV , respectively. Then, (24) can be rewritten as

L¼

V X i¼1

H1

ai Oi _ l:

ð25Þ

H0

It is worth noting that under hypothesis H0, the random variable Oi has standard exponential distribution as in (23) and the test statistic L in (24) is exactly a sum of weighted exponential variables Oi . According to Theorem 3 of [20], the probability of false alarm of the MIMO-GLRT detector can be expressed as !1 " # s s Y X @d1 þ d2 þ  þ ds J V ðen , lÞ P FA ¼ di ! ð26Þ @ed1 @ed2    @eds s n ¼ 1 En i¼1 1

2

Q where J L ðx,TÞ9x expðTx1 Þ and En ¼ en sj ¼ 1,jan ðen ej Þ, d d d the operator @d1 þ d2 þ  þ ds =@e11 @e22    @es s ðÞ denotes the mixed (d1 þ d2 þ    þ ds )th order partial derivatives of a function with respect to e1 ,e2 , . . . ,es . Notice that these derivatives of higher order in (26) can be calculated by using the following lemma [21]. L

Lemma. Let J L ðx,TÞ ¼ xL expðTx1 Þ where L is an arbitrary positive integer and T is a positive constant, then the kth partial derivative of J L ðx,TÞ with respect to x for 1r k rL is given by k X @k J L ðx,TÞ ¼ expðTx1 Þ Cpk ðLpÞðkpÞ T p xLkp k @x p¼0

ð27Þ

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J. Liu et al. / Signal Processing 93 (2013) 2771–2776

ðmÞ where Cm ¼ nðn1Þ    n ¼ n!=½m!ðnmÞ!, the notation ðnÞ ð0Þ ðnm þ1Þ for mZ 1 and ðnÞ ¼ 1.

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