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A persymmetric modified-SMI algorithm
Signal Processing 23 (1991) 27-34 Elsevier
27
A persymmetric modified-SMI algorithm* Lujing Cai and Hong Wang Department of Electrical & Computer Engineering, Syracuse University, Syracuse, N Y 13244-1240, USA Received 10 July 1990 Revised 17 October 1990
Abstract. By exploring the covariance structure information to reduce the uncertainty in adaptive processing, this paper presents a Persymmetric Modified Sample Matrix Inversion algorithm (PMSMI), together with the closed-form expressions of probabilities of detection and false larm. The new algorithm, which requires less computation and has a Constant False Alarm Rate (CFAR) feature, can significantly outperform the corresponding unstructured MSMI, especially in a severely non-stationary/non-homogeneous interference environment.
Zusammenfassung. Unter Berficksichtigung der Kenntnisse fiber die Kovarianzstruktur wird in diesem Beitrag zur Verringerung der Unsicherheiten in der adaptiven Verarbeitung der modifizierte empirische Kovarianzmatrix-Inversionsalgorithmus fiir persymmetrische Matrizen vorgestellt. AuSerdem werden geschlossene Ausdriicke ffir die Detektions- und Falschalarmwahrscheinlichkeit angegeben. Der neue Algorithmus, der einen geringeren Rechenaufwand erfordert und eine konstante Falschalarmrate besitzt, kann bedeutend bessere Ergebnisse als der entsprechende empirische Kovarianzmatrix-Inversionsalgorithmus ohne Kovarianzstrukturannahme, insbesondere in einer streng instation~ir/inhomogenen Interferenzumgebung liefern. Rrsumr. Par exploration de l'information de structure de la covariance dans le but de rrduire l'incertitude associre ~t un traitement adaptatif, cet article prrsente un algorithme d'inversion de matrices modifires persymrtriqaes (en anglais Persymmetric Modified Sample Matrix Inversion, PMSMI) et les expressions approchres des probabilit6 de drtection et de fausse alarme. Le nouvel algorithme, qui nrcessite moins de calculs et prrsente un taux constant de fausse alarme (en anglais Constant False Alarm Rate, CFAR), peut avoir des performances significativement meilleures que le MSMI non structurr, ParticuliSrement dans un environnement caractrris6 par une interfrrence srvrrement non stationnaire/non homog~ne. Keywords. Adaptive detection, doubly symmetric covariance.
I. Introduction
In a non-stationary a n d / o r nonhomogen¢ous interference environment of unknown statistics, adaptive detection algorithms, such as the wellknown Sample Matrix Inversion (SMI) [4] and Generalized Likelihood Ratio (GLR) algorithms [2], can suffer severe detection performance degradation, simply due to lack of sufficient amount of data from which the system can 'learn' (estimate) * This work was partially supported by the Rome Air Development Center under Air Force Contract F30602-88-D0027(A-9-1125). The work of L. Cai was also supported by the Syracuse University Graduate School Fellowship. 0165-1684/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
the statistics of the environment. In [5, 6], the multiband SMI and G L R algorithms are proposed and have been shown to be able to deliver a significantly improved detection performance over the corresponding single-band algorithms. The multi-band G L R has a clear advantage over the multiband SMI as it has an embedded CFAR feature. However, the multiband GLR requires more computation and is more difficult to implement. To obtain the desirable CFAR in colored Gaussian interference at a lower cost, [ 1] presents a modified version of the multiband SMI, which is termed the multiband Modified-SMI (MSMI). Those multiband algorithms are quite general in
28
L. Cai, H. Wang / A persymmetric modified-SMI algorithm
the sense that they do not assume that the interference covariance matrix has any special structure in addition to the Hermitian, and they will be called the unstructured algorithms in this paper. To further improve the detection performance, one may exploit the structural information of the interference covariance to reduce the uncertainty in learning. Many applications can result in an interference covariance matrix that has some sort of special structure, e.g., a detection system utilizing symmetrically spaced linear array, or symmetrically spaced pulse train• It is easy to see that the resulting interference covariance is a so-called 'doubly' symmetric matrix, i.e., Hermitian about its principal diagonal and persymmetric about its cross-diagonal. Note that the Toeplitz covariance matrix given by a uniformly spaced pulse train (or array) is a special case of the persymmetry. It is obvious that direct application of the previous multiband algorithms to the above persymmetric case will neglect the available structural formation• The goal of this paper is to incorporate the persymmetric structure into the multiband MSMI to further improve the detection performance in the non-homogeneous interference environment. We note that [3] is among the first to exploit the covariance structure in adaptive processing• This paper is organized as follows• Section 2 describes the multiband data model with the persymmetric covariance. We will present a persymmetric multiband MSMI algorithm and its closedform detection performance in Section 3. The detection performance evaluation and comparison are conducted in Section 4, with the summary given in Section 5.
The primary data set consists of J complex vectors, xj, M x 1, j = 1, 2 , . . . , J, where J is the number of subbands. Under Ho, i.e., the interferenceand-noise-alone hypothesis, we have xj=c~+nj,
j= 1,2,...,J,
where cj and nj represent the interference and noise components and are assumed to be independent, and x j , j = 1, 2 , . . . , J are assumed to be independent and have an identical distribution with zero mean and covariance matrix R. Under H1, i.e., the signal-plus-interference-and-noise hypothesis, we have xj=ajs+cj+nj,
j= 1,2,...,J,
We consider the following model of the received multiband data for signal detection. The whole data set is divided into two sets which will be called the primary data set and the secondary data set, respectively. Signal Processing
(2)
where s is the known signal vector and a~ represents the unknown amplitude of the signal in xj. Under both H0 and H1, the secondary data Xjk, M × 1 , j = 1 , 2 , . . . , J , k = 1,2 . . . . , K, are assumed to have the interference and noise components only, i.e.,
Xjk~-Cjk-~njk,
j=l,2,...,J,
k=l,2,...,K.
(3) Xjk, j = 1 , 2 , . . . , J, k = 1 , 2 , . . . , K, are assumed to be independent and identically distributed random vectors with zero mean and a covariance matrix equal to R. We assume that the secondary data set is independent of the primary data set. We note that the covariance matrix R is totally unknown and the detection of the known signal s is based on the primary and secondary data sets only. For an active system utilizing symmetrically spaced linear array for spatial domain processing, or symmetrically spaced pulse train for temporal domain processing, the resultant covariance matrix R has the persymmetric property, i.e., it satisfies
(4)
R = JR*J,
2. Data modeling
(1)
0
...
0j
0
...
0
where * denotes the complex conjugate and J is a permutation matrix
j=
1
(5)
L. Cai, H. Wang/ A persymmetricmodified-SMl algorithm The signal vector s is also a persymmetric vector satisfying
29
(4) Calculate the test statistics J
77 = X (lY~,jl2+ yorjr)
(5)
s = ds*.
j=l
With the above persymmetric structure imposed on the data, we propose the following structured multiband algorithm.
3. Persymmetric MSMI The Persymmetric multiband Modified-SMI (PMSMI) algorithm is implemented by the following procedures. (1) Perform the covariance matrix estimate using the secondary data J
K
Rp= ~ ~ (XjkX~k+ J(XjkX~)*J)/2, j-1
(7)
k=l
where the superscripts ' H ' and '*' denote the Hermitian transpose and the complex conjugate, respectively. Note that to maintain a nonsingular covariance matrix estimate, we only need JK >1M / 2 instead of JK>! M that is required by the unstructured covariance matrix estimate [5]. (2) Transfer the data from the complex domain to the real domain X e rj
=
½[(I + J ) Re{xj} - (I - J ) Im{x~}],
j= 1,2,...,J,
=
1, 2 , . . . , J,
s, = Re{s}- Im{s},
(9)
( "~r # t r "~r J
f0
dp
(14)
(11)
(12)
Pdlpfp(p) dp,
(15)
where
(10)
rlSr ~1/2"
~0Pflpfp(P) 1
L(p)
_Hal-l_
(13)
12),
1
Pr =
Pa =
where Re(x) and Im(x) denote the real and imaginary parts of x, respectively. (3) Estimate the weight vector I~r =
l=+lWPxo,
with Xerj and Xo,~ being specified by (8) and (9). The test statistic r/ is compared with a predetermined threshold r/o, and the signal presence is accepted if ,7 surpasses the chosen threshold. It should be pointed out that the PGLR is computationally efficient because of its real-domain calculation. We may refer to (7) as a 'forward and backward averaged' covariance matrix estimate, and in fact, Rp/JK is the maximum likelihood estimate of the persymmetric covariance matrix R [3]. If both the primary and secondary data are complex-Gaussian, and if the signal amplitudes at, j - 1, 2 . . . . J, are independent complex-Gaussian variables with zero mean and variance ~2s, the probabilities of false alarm and detection, derived in Appendix A, have the closed-form expressions below
(8)
and /~, = Re(Rp) + d Im(/Ip),
(lWPxe,
j=l
and
Xo,j = ½ [ ( l - J ) Re{xj}+(l+d) Im{xj}), j
J
X
=
-
F(JK + I/2) F ( ( M - 1)/2)F(JK - M / 2 + 1) × (1 __p)(M+l)/2--2pJK-M/2,
(16)
p~p= ~ F ( ( K + I ) J - j - ( M - 1 ) / 2 ) (r/op) J-j x (1 + r/op) (K+I)j-j-(M-I)/2
(17)
Vol, 23, No. I, April 1991
L. Cai, H. Wang / A persymmetric modified-SMl algorithm
30
and
ence variance and Co is an M × M Toeplitz matrix defined by its first row vector
Pdlp = ~ F ( ( K + I ) J - j - ( M - 1 ) / 2 )
r = [1 e -2(~02-i2~f~
j=, ( J - j ) ! F ( J K - ( M - 1 ) / 2 )
. . . e 2('rr~rf(M-1))2-i(M
(rlop)J-j(l+/3p)KJ (v ,)/2 X
j (M , ) / 2 ,
[l+p(~70+/3)](K+l)S
(18) with 2 Hn-I /3 =0.~s ~ s.
(19)
It should be noted that from (14), (16) and (17), the probability of false alarm Pf is affected only by the system dimensional parameters, independent of the interference covariance matrix R. This means that the proposed algorithm possesses the desirable Constant False Alarm Rate (CFAR).
1)2"rrfc],
with < 0.s being the center of the interference spectrum and 0-r the parameter controlling the interference bandwidth. The difference between f~ and ~ will be denoted by Af = - f . . This type of covariance presents a Gaussian-shaped interference spectrum. We note that R such defined satisfies (4), i.e., it is persymmetric. We define the subband signal-to-noise ratio (SNR) and the subband interference-to-noise-ratio (INR) of the multiband system by 0-7 S N R = ~7,
4. Performance evaluation and comparison
...
. . . e -i2~f~ 1 e i2~rf~
ei[(M-l)/i]2wf~]
T,
2,
(23)
0-n
thus the subband signal-to-interference-plusnoise-ratio (SlNR) is 0-~2 SNR SINR = (0-2.+ 0.2) - 1 + I N R '
(24)
For convenience of reference, we list the detection performance of the MSMI below from [1]: Pf= and
(20)
where i = ~ and IZI < 0.5 is a known constant denoting the normalized signal Doppler frequency. Obviously, s satisfies s =ds* as specified by (6); (2) the covariance matrix of the receiver noise nj and njk is given by 0../, 2 where 0..2 represents the noise power and I is an M x M identity matrix; (3) the covariance matrix of the interference cj and ejk is 0.~Co, where 0.~ represents the interfer-
2 0-c
INR-
Pd
Signal Processing
(22)
and
We will evaluate the probability of detection of the PMSMI according to the closed-form expressions given in Section 3, while fixing the probability of false alarm at a certain level by varying the threshold ~7o in (14), and conduct a performance comparison of the PMSMI and MSMI. Before we proceed, some additional condition specifications are set up below for the convenience of numerical evaluation and comparison: (1) the signal vector is assumed to be $ ~---[ e - i [ ( M - 1 ) / 2 ] 2 " u ~
(21)
=
Io' £, )
Pqofp(P) do
(25)
Pdlofp(P) do,
(26)
where (JK)Z
fP(P) = ( M - 2 ) I ( J K - M + 1)! x (1 - p) M-2pJK-M+,,
(27)
Prl,, = ~ [ ( K + l l J - M - j ] ! (hop) J-j × (1 + ~lop) (K+~)j-M-j+~
(28)
L. Cai, H. Wang / A persymmetric modified-SMI algorithm
and
31
1
0.9
Pdlp = ~ [ ( K + I ) J - M - j ] !
Optimum
018
(r/0p) J-j(1 + t i p )
g
KJ - M + ,
X[l+p(,/o+fl)](K+,)~
M ~+,,
0.6
(29) ~6
with/3 being defined as in (19). Figure 1 plots the detection performance curves of PMSMI and unstructured MSMI as a function of SINR. The probability of detection of the optimum processor is also plotted in the same figure for reference. We purposely chose a very small value of K (K = 4 ) , the number of adjacent range cells available, in order to reflect the severeness of environment. As we can see from Fig. 1, for J = 3 , M = 8 , I N R = 3 5 d B and P f = 1 0 -5 , the PMSMI can significantly outperform the unstructured MSMI, although it still has some distance from the optimum. Since K is a key factor for an adaptive detection system in a severely non-homogeneous environment, we plot in Fig. 2 the probability of detection of PMSMI and unstructured MSMI as a function of K. Here J is set equal to 1 to emphasize the effect of K. The advantage of PMSMI shows up when K is small. For K < 12, the unstructured MSMI totally loses detectability, while the PMSMI can still maintain a high value of Pd. We should
1
-
-
, ,,'""-
07
.=
~ .......
0.5 / 0A b
i
M=8 Af = 0.3 Gf = 0.02 pf= 10.5 INR = 35dB SINR=-25 dB
MSMI
0.3 0.2 0,1 0,
8
10
12
14
1'6
Optimum , "
1i
............
0.7 0.7
0.6 /
0.5
PMSM~
0.6
KJ=~
0.5 0,4
M=8
0.3 o,21
,'
",,Optimum
",,
J=3 K=4 M=8 Pf= 105 Af=0.3 INR = 35dB SINR = -25dB
0.4
M=03 Gf = 0.02 pf= 10-5 INR = 35dB
MSMI
24
.. '.,...
09
/
2'2
recall the fact that we only need JK >~M / 2 to maintain a non-singular covariance matrix estimate for PMSMI, while for unstructured MSMI we require JK >>-M instead. Figures 3 and 4 illustrate the probability of detection of both multiband algorithms versus crf and Af, respectively. Here the same conclusion can be drawn for the PMSMI. That is, the PMSMI can deliver a much better detection performance than the unstructured MSMI in a severely nonhomogeneous interference environment.
0.8
g
2(/
Fig. 2. Performance comparison of PMSMI and unstructured MSMl: Pd versus the number of the secondary data vectors, K.
0.9 0.8
18
K
PMSMI
MSMI
02
0.1 01 40
-35
30
-25
-20
-15
SINR
0 0
0.01
0.02
003
004
005
006
0.07
0.08
Gf
Fig. 1. Performance comparison of PMSMI and unstructured MSMI: Pd versus the subband signal-to-interference-plus-noise ratio (SINR).
Fig. 3. Performance comparison of PMSMI and unstructured MSMI: PO versus the interference spectral spread. Vol. 23, No. 1, April 1991
L. Cai, H. W a n g / A persymmetric m o d i f i e d - S M l algorithm
32 1
./_. ....... _
where
Optimur0"
0.9
J=3 K=4 M=8 Pf = 10-5 ~f = 0.02 1NR = 35dB SINR = -25dB
0.8 ,'" PMSMI ~
0.7 0.6 0.5
Xojk = ( X~k -- Jx~ ) / 2.
(A.3)
By the properties of complex Gaussian distribution, it is not difficult to show that
0.3
H
0.2 0.1 0
(A.2)
and
MSM1
0.4
Xe~k = ( Xjk + Jx~ ) / 2
H
E{XejkX~jk} = E{XojkXojk}= R / 2
(A.4)
E{X~jkXo~k}= 0.
(A.5)
. /y"
and 0.05
0.1
0.15
012 Af
0.15
(t'.3
0.35
0.4
Fig. 4. P e r f o r m a n c e c o m p a r i s o n o f P M S M I a n d u n s t r u c t u r e d M S M I : Po v e r s u s t h e f r e q u e n c y s e p a r a t i o n b e t w e e n s i g n a l a n d interference.
Equation (A.5) indicates that Xe~k and Xo~k are independent for allj and k. By some simple algebra A manipulation, Rp becomes
5. Conclusion We proposed the Persymmetric multiband Modified Sample Matrix Inversion algorithm (PMSMI), which can be applied to some systems for further detection performance improvement over the unstructured multiband MSMI while still keeping the desirable CFAR feature in colored Gaussian interference. The probabilities of detection and false alarm of the new algorithm were derived and compared with those of the unstructured multiband MSMI. In a severely nonstationary/non-homogeneous interference environment the comparison shows that the new algorithm can significantly outperform the unstructured one. Furthermore, due to its realdomain operation, the new algorithm needs less computation.
Xjk ~ Xej k -I- Xoj k ,
j= 1,2,...,J, Signal Processing
k = 1 , 2 , . . . , K,
(A.1)
K
j=l
k=l
J
K
= Y X (xoj j=l
k=l
k + XojkXojk).
(A.6)
Introducing the unitary matrix: T = 1[(i + j ) + i ( l - a)],
(A.7)
we consider the linear transformation on Xejk and Xojk
Txejk = ½[(I + J ) Re{Xik} -- ( I -- J ) lm{Xjk}] a
(A.8)
Xerjk ,
TXojk = i½[(l -- J ) ae{xjk} + ( I + J ) lm{Xjk}] A.lXorjk ,
(A.9)
where i=x/-J1. Note that Xeok and xook now become purely real vectors. By applying (A.4) and (A.5), we obtain H
Appendix A. Derivation of Pd and Pf of PMSMI In order to perform the statistical analysis associated with the persymmetric covariance matrix/~p, we decompose the secondary data vectors as follows:
J
H
E{XerjkXeUk} = E { X o o k X o O k } =
TRT"/2
= (Re{R} + J lm{R})/2 & R , / 2 (A.10) and H
E{XeOkXoOk}= 0.
(A.11)
Therefore, Xeok and XoOk, j = 1,2 . . . . . J, k = 1, 2 , . . . , K, are all real independent vectors of
33
L. Cai, H. Wang/ A persymmetric modified-SMl algorithm Gaussian distribution with zero mean and a covariance matrix equal to Rr/2. It then follows that
Substituting it into (A.14) yields H ^ 1 ^ 2 [SrRr (R,+0-~SrSHr)Rrls,]
R, = Re(/~p)+ d lm(/~p) J j-1
s, R,-1 R,R,--1 s~s,H R~--1 s~ sU,#71s, H
~
" (XerjkXerjk
(S.#;,Sr)~
" + XoOkXook).
2
k=l
(A.12)
(s~,#;ls,) ~ P= K=
~
H
1
1)! r/J-' exp(-r//0-2)'
= (o.2)j(j_
(A.13)
~
(A.19)
Sr = O'~sHR-ts.
0-2 = 1 [1 +p/3]. pK
(A.21)
With the use of a similar procedure as in [ 1, 4] except different degrees-of-freedom considerations between the complex and real cases, we can show that p and K are independent and have Beta and X2 distribution, respectively, i.e., their probability density functions are given by K
JK--(M+I)/2
f~(K) -- F ( J K - ( M + ^H
2
H
^
= Wr [Rr + 0-~S,Sr ]Wr.
(A.14)
The conditional probability of detection is thus P ( r / > r/olH1, Wr) =
f++p(r/lH,,
l~r) dr/
o
f
oO
(A.20)
Equation (A.17) becomes
where
-----
(A. 18)
.".H
Since Yerj=WrXe,j and Yo0=WrXo,'j, j = l , 2, . . . J, are linear combinations of the Gaussian vectors, it then follows that they are also Gaussian if conditional on ~ . It is obvious that 71= ~=~ (lye,.jI2+lYo,.jl2) has a X2 distribution with the degrees of freedom equal to 2J, i.e., its probability density function is
Pdl~r
,
s~R71s~
fl = 0-+~SrZu---t~,
p(r/lH,, ~ r )
^
s~ R f R,R~l s,Sr R~l s~
and
0-2sa"~ )/2. ^H
(A.17)
Let
W(2JK, R,./2). Under the signal-plus-interference-and-noise hypothesis H~, by the similar procedure, we may show that Xerj and Xorjj = 1, 2 , . . . , J, are also real independent vectors of Gaussian distribution with zero mean and a covariance matrix equal to (R~ +
s".R;ls,
H ^-1
+ 0-~S~ Rr Sr.
It is obvious t h a t / ~ has a real Wishart distribution,
O.2
A
=
K
r#pr" = Z X
=
,'~_1 (SrH R, s,)
0-' =
1 ) / 2 + 1) e x p ( - K ) ,
0 ~< K ~
(A.22)
and
fp(p) -
F ( J K + I/2) F((M-
1)/2)r(JK
- M/2+
1)
x (1 _p)(,',,4+l)/2-2pJ,,< M/2, 0 ~ p ~ l .
ZJ-I
= J,~o/,4 (J--~l)! exp(z) dz
(A.23) Applying (A.21) to (A.15), we have
= e x p ( - r / . / 0 - 2 ) ~ (r/%'2)J-J j=l
(J-j)!
(A.15) PdI,L = Polyp= exp
(-+) 1 + pfl
From (12), we have
^ $0r :
^ 1 RTSr _H~-I_
xl/2"
( ~ r "Rr "~r)
(A.16)
x ~ (r/oKp'] j-j j=, \ l + p f l / ~(J-J)!" (A.24) Vol. 23, No. I, April 1991
34
L Cai, H. Wang / A persymmetric modified-SMl algorithm
References
It t h e n follows that
fo~
Pd~,= J0 Pdr.of,('¢) dK = ~ F((K+I)J-j-(M-1)/2)
( r / 0 p ) j - j (1 + f l p ) r J - ( M - 1 ) / 2
X[I+p(~O+fl)](K+I)J_j_(M_I)/2
,
(A.25)
which is (18). The final step is a c c o m p l i s h e d by averaging (A.25) over/9 as p e r f o r m e d in (15). The i n t e r f e r e n c e - a n d - n o i s e - a l o n e hypothesis Ho differs from H1 in the sense that the signal comp o n e n t in the p r i m a r y data set is equal to zero. Hence, we o n l y n e e d to set o-2s equal to zero in (18) to arrive at alarm.
SignalProcessing
(17) for
the p r o b a b i l i t y of false
[1] L. Cai and H. Wang, "On adaptive filtering with the CFAR feature and its performance sensitivity to non-Gaussian interference", Proc of the 24th Annual Conference on Information Sciences and Systems, Princeton, NJ, 21-23 March 1990, pp. 558-563. [2] E.J. Kelly, "An adaptive detection algorithm", IEEE Trans. Aerospace Electron. Systems, Vol. AES-22, March 1986, pp. 115-127. [3] R. Nitzberg, "Application of maximum likelihood estimation of persymmetric covariance matrices to adaptive processing", IEEE Trans. Aerospace Electron. Systems, Vol. AES-16, January 1980, pp. 124-127. [4] I.S. Reed, J.D. Mallett and L.E. Brennan, "Rapid convergence rate in adaptive arrays", IEEE Trans. Aerospace Electron. Systems, Vol. AES-10, November 1974, pp. 853863. [5] H. Wang and L. Cai, "On adaptive multiband signal detection with the SMI algorithm", IEEE Trans. Aerospace and Electron. Systems, Vol. AES-26, No. 5, September 1990. [6] H. Wang and L. Cai, "On adaptive multiband signal detection with the GLR algorithm", IEEE Trans. Aerospace and Electron. Systems, Vol. AES-27, No. 1, January 1991.
27
A persymmetric modified-SMI algorithm* Lujing Cai and Hong Wang Department of Electrical & Computer Engineering, Syracuse University, Syracuse, N Y 13244-1240, USA Received 10 July 1990 Revised 17 October 1990
Abstract. By exploring the covariance structure information to reduce the uncertainty in adaptive processing, this paper presents a Persymmetric Modified Sample Matrix Inversion algorithm (PMSMI), together with the closed-form expressions of probabilities of detection and false larm. The new algorithm, which requires less computation and has a Constant False Alarm Rate (CFAR) feature, can significantly outperform the corresponding unstructured MSMI, especially in a severely non-stationary/non-homogeneous interference environment.
Zusammenfassung. Unter Berficksichtigung der Kenntnisse fiber die Kovarianzstruktur wird in diesem Beitrag zur Verringerung der Unsicherheiten in der adaptiven Verarbeitung der modifizierte empirische Kovarianzmatrix-Inversionsalgorithmus fiir persymmetrische Matrizen vorgestellt. AuSerdem werden geschlossene Ausdriicke ffir die Detektions- und Falschalarmwahrscheinlichkeit angegeben. Der neue Algorithmus, der einen geringeren Rechenaufwand erfordert und eine konstante Falschalarmrate besitzt, kann bedeutend bessere Ergebnisse als der entsprechende empirische Kovarianzmatrix-Inversionsalgorithmus ohne Kovarianzstrukturannahme, insbesondere in einer streng instation~ir/inhomogenen Interferenzumgebung liefern. Rrsumr. Par exploration de l'information de structure de la covariance dans le but de rrduire l'incertitude associre ~t un traitement adaptatif, cet article prrsente un algorithme d'inversion de matrices modifires persymrtriqaes (en anglais Persymmetric Modified Sample Matrix Inversion, PMSMI) et les expressions approchres des probabilit6 de drtection et de fausse alarme. Le nouvel algorithme, qui nrcessite moins de calculs et prrsente un taux constant de fausse alarme (en anglais Constant False Alarm Rate, CFAR), peut avoir des performances significativement meilleures que le MSMI non structurr, ParticuliSrement dans un environnement caractrris6 par une interfrrence srvrrement non stationnaire/non homog~ne. Keywords. Adaptive detection, doubly symmetric covariance.
I. Introduction
In a non-stationary a n d / o r nonhomogen¢ous interference environment of unknown statistics, adaptive detection algorithms, such as the wellknown Sample Matrix Inversion (SMI) [4] and Generalized Likelihood Ratio (GLR) algorithms [2], can suffer severe detection performance degradation, simply due to lack of sufficient amount of data from which the system can 'learn' (estimate) * This work was partially supported by the Rome Air Development Center under Air Force Contract F30602-88-D0027(A-9-1125). The work of L. Cai was also supported by the Syracuse University Graduate School Fellowship. 0165-1684/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
the statistics of the environment. In [5, 6], the multiband SMI and G L R algorithms are proposed and have been shown to be able to deliver a significantly improved detection performance over the corresponding single-band algorithms. The multi-band G L R has a clear advantage over the multiband SMI as it has an embedded CFAR feature. However, the multiband GLR requires more computation and is more difficult to implement. To obtain the desirable CFAR in colored Gaussian interference at a lower cost, [ 1] presents a modified version of the multiband SMI, which is termed the multiband Modified-SMI (MSMI). Those multiband algorithms are quite general in
28
L. Cai, H. Wang / A persymmetric modified-SMI algorithm
the sense that they do not assume that the interference covariance matrix has any special structure in addition to the Hermitian, and they will be called the unstructured algorithms in this paper. To further improve the detection performance, one may exploit the structural information of the interference covariance to reduce the uncertainty in learning. Many applications can result in an interference covariance matrix that has some sort of special structure, e.g., a detection system utilizing symmetrically spaced linear array, or symmetrically spaced pulse train• It is easy to see that the resulting interference covariance is a so-called 'doubly' symmetric matrix, i.e., Hermitian about its principal diagonal and persymmetric about its cross-diagonal. Note that the Toeplitz covariance matrix given by a uniformly spaced pulse train (or array) is a special case of the persymmetry. It is obvious that direct application of the previous multiband algorithms to the above persymmetric case will neglect the available structural formation• The goal of this paper is to incorporate the persymmetric structure into the multiband MSMI to further improve the detection performance in the non-homogeneous interference environment. We note that [3] is among the first to exploit the covariance structure in adaptive processing• This paper is organized as follows• Section 2 describes the multiband data model with the persymmetric covariance. We will present a persymmetric multiband MSMI algorithm and its closedform detection performance in Section 3. The detection performance evaluation and comparison are conducted in Section 4, with the summary given in Section 5.
The primary data set consists of J complex vectors, xj, M x 1, j = 1, 2 , . . . , J, where J is the number of subbands. Under Ho, i.e., the interferenceand-noise-alone hypothesis, we have xj=c~+nj,
j= 1,2,...,J,
where cj and nj represent the interference and noise components and are assumed to be independent, and x j , j = 1, 2 , . . . , J are assumed to be independent and have an identical distribution with zero mean and covariance matrix R. Under H1, i.e., the signal-plus-interference-and-noise hypothesis, we have xj=ajs+cj+nj,
j= 1,2,...,J,
We consider the following model of the received multiband data for signal detection. The whole data set is divided into two sets which will be called the primary data set and the secondary data set, respectively. Signal Processing
(2)
where s is the known signal vector and a~ represents the unknown amplitude of the signal in xj. Under both H0 and H1, the secondary data Xjk, M × 1 , j = 1 , 2 , . . . , J , k = 1,2 . . . . , K, are assumed to have the interference and noise components only, i.e.,
Xjk~-Cjk-~njk,
j=l,2,...,J,
k=l,2,...,K.
(3) Xjk, j = 1 , 2 , . . . , J, k = 1 , 2 , . . . , K, are assumed to be independent and identically distributed random vectors with zero mean and a covariance matrix equal to R. We assume that the secondary data set is independent of the primary data set. We note that the covariance matrix R is totally unknown and the detection of the known signal s is based on the primary and secondary data sets only. For an active system utilizing symmetrically spaced linear array for spatial domain processing, or symmetrically spaced pulse train for temporal domain processing, the resultant covariance matrix R has the persymmetric property, i.e., it satisfies
(4)
R = JR*J,
2. Data modeling
(1)
0
...
0j
0
...
0
where * denotes the complex conjugate and J is a permutation matrix
j=
1
(5)
L. Cai, H. Wang/ A persymmetricmodified-SMl algorithm The signal vector s is also a persymmetric vector satisfying
29
(4) Calculate the test statistics J
77 = X (lY~,jl2+ yorjr)
(5)
s = ds*.
j=l
With the above persymmetric structure imposed on the data, we propose the following structured multiband algorithm.
3. Persymmetric MSMI The Persymmetric multiband Modified-SMI (PMSMI) algorithm is implemented by the following procedures. (1) Perform the covariance matrix estimate using the secondary data J
K
Rp= ~ ~ (XjkX~k+ J(XjkX~)*J)/2, j-1
(7)
k=l
where the superscripts ' H ' and '*' denote the Hermitian transpose and the complex conjugate, respectively. Note that to maintain a nonsingular covariance matrix estimate, we only need JK >1M / 2 instead of JK>! M that is required by the unstructured covariance matrix estimate [5]. (2) Transfer the data from the complex domain to the real domain X e rj
=
½[(I + J ) Re{xj} - (I - J ) Im{x~}],
j= 1,2,...,J,
=
1, 2 , . . . , J,
s, = Re{s}- Im{s},
(9)
( "~r # t r "~r J
f0
dp
(14)
(11)
(12)
Pdlpfp(p) dp,
(15)
where
(10)
rlSr ~1/2"
~0Pflpfp(P) 1
L(p)
_Hal-l_
(13)
12),
1
Pr =
Pa =
where Re(x) and Im(x) denote the real and imaginary parts of x, respectively. (3) Estimate the weight vector I~r =
l=+lWPxo,
with Xerj and Xo,~ being specified by (8) and (9). The test statistic r/ is compared with a predetermined threshold r/o, and the signal presence is accepted if ,7 surpasses the chosen threshold. It should be pointed out that the PGLR is computationally efficient because of its real-domain calculation. We may refer to (7) as a 'forward and backward averaged' covariance matrix estimate, and in fact, Rp/JK is the maximum likelihood estimate of the persymmetric covariance matrix R [3]. If both the primary and secondary data are complex-Gaussian, and if the signal amplitudes at, j - 1, 2 . . . . J, are independent complex-Gaussian variables with zero mean and variance ~2s, the probabilities of false alarm and detection, derived in Appendix A, have the closed-form expressions below
(8)
and /~, = Re(Rp) + d Im(/Ip),
(lWPxe,
j=l
and
Xo,j = ½ [ ( l - J ) Re{xj}+(l+d) Im{xj}), j
J
X
=
-
F(JK + I/2) F ( ( M - 1)/2)F(JK - M / 2 + 1) × (1 __p)(M+l)/2--2pJK-M/2,
(16)
p~p= ~ F ( ( K + I ) J - j - ( M - 1 ) / 2 ) (r/op) J-j x (1 + r/op) (K+I)j-j-(M-I)/2
(17)
Vol, 23, No. I, April 1991
L. Cai, H. Wang / A persymmetric modified-SMl algorithm
30
and
ence variance and Co is an M × M Toeplitz matrix defined by its first row vector
Pdlp = ~ F ( ( K + I ) J - j - ( M - 1 ) / 2 )
r = [1 e -2(~02-i2~f~
j=, ( J - j ) ! F ( J K - ( M - 1 ) / 2 )
. . . e 2('rr~rf(M-1))2-i(M
(rlop)J-j(l+/3p)KJ (v ,)/2 X
j (M , ) / 2 ,
[l+p(~70+/3)](K+l)S
(18) with 2 Hn-I /3 =0.~s ~ s.
(19)
It should be noted that from (14), (16) and (17), the probability of false alarm Pf is affected only by the system dimensional parameters, independent of the interference covariance matrix R. This means that the proposed algorithm possesses the desirable Constant False Alarm Rate (CFAR).
1)2"rrfc],
with < 0.s being the center of the interference spectrum and 0-r the parameter controlling the interference bandwidth. The difference between f~ and ~ will be denoted by Af = - f . . This type of covariance presents a Gaussian-shaped interference spectrum. We note that R such defined satisfies (4), i.e., it is persymmetric. We define the subband signal-to-noise ratio (SNR) and the subband interference-to-noise-ratio (INR) of the multiband system by 0-7 S N R = ~7,
4. Performance evaluation and comparison
...
. . . e -i2~f~ 1 e i2~rf~
ei[(M-l)/i]2wf~]
T,
2,
(23)
0-n
thus the subband signal-to-interference-plusnoise-ratio (SlNR) is 0-~2 SNR SINR = (0-2.+ 0.2) - 1 + I N R '
(24)
For convenience of reference, we list the detection performance of the MSMI below from [1]: Pf= and
(20)
where i = ~ and IZI < 0.5 is a known constant denoting the normalized signal Doppler frequency. Obviously, s satisfies s =ds* as specified by (6); (2) the covariance matrix of the receiver noise nj and njk is given by 0../, 2 where 0..2 represents the noise power and I is an M x M identity matrix; (3) the covariance matrix of the interference cj and ejk is 0.~Co, where 0.~ represents the interfer-
2 0-c
INR-
Pd
Signal Processing
(22)
and
We will evaluate the probability of detection of the PMSMI according to the closed-form expressions given in Section 3, while fixing the probability of false alarm at a certain level by varying the threshold ~7o in (14), and conduct a performance comparison of the PMSMI and MSMI. Before we proceed, some additional condition specifications are set up below for the convenience of numerical evaluation and comparison: (1) the signal vector is assumed to be $ ~---[ e - i [ ( M - 1 ) / 2 ] 2 " u ~
(21)
=
Io' £, )
Pqofp(P) do
(25)
Pdlofp(P) do,
(26)
where (JK)Z
fP(P) = ( M - 2 ) I ( J K - M + 1)! x (1 - p) M-2pJK-M+,,
(27)
Prl,, = ~ [ ( K + l l J - M - j ] ! (hop) J-j × (1 + ~lop) (K+~)j-M-j+~
(28)
L. Cai, H. Wang / A persymmetric modified-SMI algorithm
and
31
1
0.9
Pdlp = ~ [ ( K + I ) J - M - j ] !
Optimum
018
(r/0p) J-j(1 + t i p )
g
KJ - M + ,
X[l+p(,/o+fl)](K+,)~
M ~+,,
0.6
(29) ~6
with/3 being defined as in (19). Figure 1 plots the detection performance curves of PMSMI and unstructured MSMI as a function of SINR. The probability of detection of the optimum processor is also plotted in the same figure for reference. We purposely chose a very small value of K (K = 4 ) , the number of adjacent range cells available, in order to reflect the severeness of environment. As we can see from Fig. 1, for J = 3 , M = 8 , I N R = 3 5 d B and P f = 1 0 -5 , the PMSMI can significantly outperform the unstructured MSMI, although it still has some distance from the optimum. Since K is a key factor for an adaptive detection system in a severely non-homogeneous environment, we plot in Fig. 2 the probability of detection of PMSMI and unstructured MSMI as a function of K. Here J is set equal to 1 to emphasize the effect of K. The advantage of PMSMI shows up when K is small. For K < 12, the unstructured MSMI totally loses detectability, while the PMSMI can still maintain a high value of Pd. We should
1
-
-
, ,,'""-
07
.=
~ .......
0.5 / 0A b
i
M=8 Af = 0.3 Gf = 0.02 pf= 10.5 INR = 35dB SINR=-25 dB
MSMI
0.3 0.2 0,1 0,
8
10
12
14
1'6
Optimum , "
1i
............
0.7 0.7
0.6 /
0.5
PMSM~
0.6
KJ=~
0.5 0,4
M=8
0.3 o,21
,'
",,Optimum
",,
J=3 K=4 M=8 Pf= 105 Af=0.3 INR = 35dB SINR = -25dB
0.4
M=03 Gf = 0.02 pf= 10-5 INR = 35dB
MSMI
24
.. '.,...
09
/
2'2
recall the fact that we only need JK >~M / 2 to maintain a non-singular covariance matrix estimate for PMSMI, while for unstructured MSMI we require JK >>-M instead. Figures 3 and 4 illustrate the probability of detection of both multiband algorithms versus crf and Af, respectively. Here the same conclusion can be drawn for the PMSMI. That is, the PMSMI can deliver a much better detection performance than the unstructured MSMI in a severely nonhomogeneous interference environment.
0.8
g
2(/
Fig. 2. Performance comparison of PMSMI and unstructured MSMl: Pd versus the number of the secondary data vectors, K.
0.9 0.8
18
K
PMSMI
MSMI
02
0.1 01 40
-35
30
-25
-20
-15
SINR
0 0
0.01
0.02
003
004
005
006
0.07
0.08
Gf
Fig. 1. Performance comparison of PMSMI and unstructured MSMI: Pd versus the subband signal-to-interference-plus-noise ratio (SINR).
Fig. 3. Performance comparison of PMSMI and unstructured MSMI: PO versus the interference spectral spread. Vol. 23, No. 1, April 1991
L. Cai, H. W a n g / A persymmetric m o d i f i e d - S M l algorithm
32 1
./_. ....... _
where
Optimur0"
0.9
J=3 K=4 M=8 Pf = 10-5 ~f = 0.02 1NR = 35dB SINR = -25dB
0.8 ,'" PMSMI ~
0.7 0.6 0.5
Xojk = ( X~k -- Jx~ ) / 2.
(A.3)
By the properties of complex Gaussian distribution, it is not difficult to show that
0.3
H
0.2 0.1 0
(A.2)
and
MSM1
0.4
Xe~k = ( Xjk + Jx~ ) / 2
H
E{XejkX~jk} = E{XojkXojk}= R / 2
(A.4)
E{X~jkXo~k}= 0.
(A.5)
. /y"
and 0.05
0.1
0.15
012 Af
0.15
(t'.3
0.35
0.4
Fig. 4. P e r f o r m a n c e c o m p a r i s o n o f P M S M I a n d u n s t r u c t u r e d M S M I : Po v e r s u s t h e f r e q u e n c y s e p a r a t i o n b e t w e e n s i g n a l a n d interference.
Equation (A.5) indicates that Xe~k and Xo~k are independent for allj and k. By some simple algebra A manipulation, Rp becomes
5. Conclusion We proposed the Persymmetric multiband Modified Sample Matrix Inversion algorithm (PMSMI), which can be applied to some systems for further detection performance improvement over the unstructured multiband MSMI while still keeping the desirable CFAR feature in colored Gaussian interference. The probabilities of detection and false alarm of the new algorithm were derived and compared with those of the unstructured multiband MSMI. In a severely nonstationary/non-homogeneous interference environment the comparison shows that the new algorithm can significantly outperform the unstructured one. Furthermore, due to its realdomain operation, the new algorithm needs less computation.
Xjk ~ Xej k -I- Xoj k ,
j= 1,2,...,J, Signal Processing
k = 1 , 2 , . . . , K,
(A.1)
K
j=l
k=l
J
K
= Y X (xoj j=l
k=l
k + XojkXojk).
(A.6)
Introducing the unitary matrix: T = 1[(i + j ) + i ( l - a)],
(A.7)
we consider the linear transformation on Xejk and Xojk
Txejk = ½[(I + J ) Re{Xik} -- ( I -- J ) lm{Xjk}] a
(A.8)
Xerjk ,
TXojk = i½[(l -- J ) ae{xjk} + ( I + J ) lm{Xjk}] A.lXorjk ,
(A.9)
where i=x/-J1. Note that Xeok and xook now become purely real vectors. By applying (A.4) and (A.5), we obtain H
Appendix A. Derivation of Pd and Pf of PMSMI In order to perform the statistical analysis associated with the persymmetric covariance matrix/~p, we decompose the secondary data vectors as follows:
J
H
E{XerjkXeUk} = E { X o o k X o O k } =
TRT"/2
= (Re{R} + J lm{R})/2 & R , / 2 (A.10) and H
E{XeOkXoOk}= 0.
(A.11)
Therefore, Xeok and XoOk, j = 1,2 . . . . . J, k = 1, 2 , . . . , K, are all real independent vectors of
33
L. Cai, H. Wang/ A persymmetric modified-SMl algorithm Gaussian distribution with zero mean and a covariance matrix equal to Rr/2. It then follows that
Substituting it into (A.14) yields H ^ 1 ^ 2 [SrRr (R,+0-~SrSHr)Rrls,]
R, = Re(/~p)+ d lm(/~p) J j-1
s, R,-1 R,R,--1 s~s,H R~--1 s~ sU,#71s, H
~
" (XerjkXerjk
(S.#;,Sr)~
" + XoOkXook).
2
k=l
(A.12)
(s~,#;ls,) ~ P= K=
~
H
1
1)! r/J-' exp(-r//0-2)'
= (o.2)j(j_
(A.13)
~
(A.19)
Sr = O'~sHR-ts.
0-2 = 1 [1 +p/3]. pK
(A.21)
With the use of a similar procedure as in [ 1, 4] except different degrees-of-freedom considerations between the complex and real cases, we can show that p and K are independent and have Beta and X2 distribution, respectively, i.e., their probability density functions are given by K
JK--(M+I)/2
f~(K) -- F ( J K - ( M + ^H
2
H
^
= Wr [Rr + 0-~S,Sr ]Wr.
(A.14)
The conditional probability of detection is thus P ( r / > r/olH1, Wr) =
f++p(r/lH,,
l~r) dr/
o
f
oO
(A.20)
Equation (A.17) becomes
where
-----
(A. 18)
.".H
Since Yerj=WrXe,j and Yo0=WrXo,'j, j = l , 2, . . . J, are linear combinations of the Gaussian vectors, it then follows that they are also Gaussian if conditional on ~ . It is obvious that 71= ~=~ (lye,.jI2+lYo,.jl2) has a X2 distribution with the degrees of freedom equal to 2J, i.e., its probability density function is
Pdl~r
,
s~R71s~
fl = 0-+~SrZu---t~,
p(r/lH,, ~ r )
^
s~ R f R,R~l s,Sr R~l s~
and
0-2sa"~ )/2. ^H
(A.17)
Let
W(2JK, R,./2). Under the signal-plus-interference-and-noise hypothesis H~, by the similar procedure, we may show that Xerj and Xorjj = 1, 2 , . . . , J, are also real independent vectors of Gaussian distribution with zero mean and a covariance matrix equal to (R~ +
s".R;ls,
H ^-1
+ 0-~S~ Rr Sr.
It is obvious t h a t / ~ has a real Wishart distribution,
O.2
A
=
K
r#pr" = Z X
=
,'~_1 (SrH R, s,)
0-' =
1 ) / 2 + 1) e x p ( - K ) ,
0 ~< K ~
(A.22)
and
fp(p) -
F ( J K + I/2) F((M-
1)/2)r(JK
- M/2+
1)
x (1 _p)(,',,4+l)/2-2pJ,,< M/2, 0 ~ p ~ l .
ZJ-I
= J,~o/,4 (J--~l)! exp(z) dz
(A.23) Applying (A.21) to (A.15), we have
= e x p ( - r / . / 0 - 2 ) ~ (r/%'2)J-J j=l
(J-j)!
(A.15) PdI,L = Polyp= exp
(-+) 1 + pfl
From (12), we have
^ $0r :
^ 1 RTSr _H~-I_
xl/2"
( ~ r "Rr "~r)
(A.16)
x ~ (r/oKp'] j-j j=, \ l + p f l / ~(J-J)!" (A.24) Vol. 23, No. I, April 1991
34
L Cai, H. Wang / A persymmetric modified-SMl algorithm
References
It t h e n follows that
fo~
Pd~,= J0 Pdr.of,('¢) dK = ~ F((K+I)J-j-(M-1)/2)
( r / 0 p ) j - j (1 + f l p ) r J - ( M - 1 ) / 2
X[I+p(~O+fl)](K+I)J_j_(M_I)/2
,
(A.25)
which is (18). The final step is a c c o m p l i s h e d by averaging (A.25) over/9 as p e r f o r m e d in (15). The i n t e r f e r e n c e - a n d - n o i s e - a l o n e hypothesis Ho differs from H1 in the sense that the signal comp o n e n t in the p r i m a r y data set is equal to zero. Hence, we o n l y n e e d to set o-2s equal to zero in (18) to arrive at alarm.
SignalProcessing
(17) for
the p r o b a b i l i t y of false
[1] L. Cai and H. Wang, "On adaptive filtering with the CFAR feature and its performance sensitivity to non-Gaussian interference", Proc of the 24th Annual Conference on Information Sciences and Systems, Princeton, NJ, 21-23 March 1990, pp. 558-563. [2] E.J. Kelly, "An adaptive detection algorithm", IEEE Trans. Aerospace Electron. Systems, Vol. AES-22, March 1986, pp. 115-127. [3] R. Nitzberg, "Application of maximum likelihood estimation of persymmetric covariance matrices to adaptive processing", IEEE Trans. Aerospace Electron. Systems, Vol. AES-16, January 1980, pp. 124-127. [4] I.S. Reed, J.D. Mallett and L.E. Brennan, "Rapid convergence rate in adaptive arrays", IEEE Trans. Aerospace Electron. Systems, Vol. AES-10, November 1974, pp. 853863. [5] H. Wang and L. Cai, "On adaptive multiband signal detection with the SMI algorithm", IEEE Trans. Aerospace and Electron. Systems, Vol. AES-26, No. 5, September 1990. [6] H. Wang and L. Cai, "On adaptive multiband signal detection with the GLR algorithm", IEEE Trans. Aerospace and Electron. Systems, Vol. AES-27, No. 1, January 1991.