- Email: [email protected]
Predictive adaptive moving target indicator
Signal Processing 10 (1986) 83-97 North-Holland
83
P R E D I C T I V E ADAPTIVE M O V I N G TARGET INDICATOR* S. BARBAROSSA and G. PICARDI INFO-COM Department, Rome University, 00184 Rome, Italy Received 31 July 1984 Revised 18 February 1985
Abstract. In the recent past many attempts have been performed in order to obtain the adaptive suppression of correlated interference (clutter) in coherent radar systems. In this paper the spectral estimation with their modern algorithms is considered. By starting from the Wiener optimum filtering technique, a convenient implementation of the linear prediction algorithm is found to be the lattice filter, in double clutter environment. The performances in search radar applications are analyzed and the Improvement Factor losses, due to clutter estimation error, are evaluated.
Zusammenfassung. In den letzten Jahren wurde viele Arbeit fiber die Probleme der adaptativen Beseitigung der korrelierten Interferenzen in den koh~irenten Radar-Systemen durchgefiihrt. In diesem Artikel handelt es sich um die Spektral-Sch~itzung mit modernen Algorithmen. Von der Theorie der optimalen Wiener-Filter wird ein fiir das Problem der korrelierten Interferenzen angewandter Linear-Prediction Algorithmus zustande gebracht. In dieser Einrichtung treten Gitter-Filter ein. Die Leistungen in Radar Anwendungen und die Sch~itzungsfehler werden zuletzt berechnet. Resumr. Durant les derui~res annres le probl~me de la suppression adaptive des interfrrence correlres (fouillis), dans les systrmes radar cohrrents a 6t6 beaueoup 6tudir. Darts cet article on aborde l'estimation spectrale au moyen des algorithmes modernes. Partant de la throrie du filtrage optimal de Wiener, on prrsente une mise en oeuvre de l'algorithme de prrdiction linraire adaptre au problrme des interfrrences corrrlres. Cette implrmentation fait intervenir des filtre en treillis. Ses performances dans les applications radar et les pertes causres par rerreur d'rstimation sont ensuite 6valures. Keywords. Radar signal processing, adaptive filtering, interference suppression, lattice filters.
1. Introduction
The main problem in radar processors is the detection of a very small target in a disturbed environment. The radar environment is defined by the return echoes from the ground, hill, sea, rain . . . which are unwanted echoes, so called 'clutter'; the small useful targets are, e.g. the echoes of airplanes, missiles and so on. The detection can be performed by estimating and cancelling the clutter; the spatial (the useful targets usually have a limited size with reference to the clutter region) and the spectral (due to the * This work was partially supported by Selenia S.p.A.: C.N.B/22231.1984.
different relative speeds--or Doppler frequency-of the clutter and the useful targets) analyses play an important role to solve this problem. Today in search and tracking radars a standard approach is the use of the Doppler filters banks (as spectrum analyzer) which can be implemented using FFT algorithms; the system is usually called Moving Target Detector (MTD) [1]. This approach is intended to obtain the spectral resolution, thus knowledge of target speeds can be acquired and coherent integration is automatically performed. According to the serial nature of the radar returns, the signals (usually sampled in digital processors) are stored serially in each sweep in a proper memory system and the target history of
0165-1684/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
84
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
each range cell can be analyzed reading the memory in parallel (corner turning approach) and the wanted Doppler informations can be extracted, e.g. using the FFT technique. The spectral analysis is usually performed sequentially on the burst of the sweeps (each analysis is performed on each range cell), so the processor works according to the 'batch' philosophy; the size of the burst is usually one half the pulse number on the 'time on target', because the azimuth target position is unknown in reference to the burst position. The conventional sliding spectral analyses are not used, so that we can use frequency and PRF agility. We can note that the radar 'batch' philosophy is a 'real time' processing technique, according to the conventional pipelined architectures. Therefore the FFT processor must face the following problems connected with the windowing of the data: - High sidelobe level of each filter causes the widening and distorsion of the clutter spectra, and congequent sensitivity losses. We can use a weighting network in order to reduce the sidelobes to avoid the spread of clutter power in the contiguous filters, but this technique causes the widening of the main lobe, reducing both resolution and sensitivity. - Low spectral resolution. These problems can be overcome by an a priori knowledge of the main characteristics of the clutter power density spectrum (e.g. the clutter spectrum can be modelled by a proper order of an autoregressive (AR) model). In this case new spectral analysis methods [2] with their modern algorithms can be used and the problems which must be faced in a modern processor design are: - The choice of a model (in the AR approach the definition of a correct order of regression); - The estimation of the model parameters. This can be performed either by directly processing the input samples or by indirectly using their computed correlation matrix; Signal Processing
- The implementation of the algorithms with a low computational cost, that is with hardware as simple as possible. In the present paper, we focus our attention on search radars, starting from the Wiener optimum filtering technique [3] to obtain coherent integration in a very disturbed environment (e.g. defined by two clutters in the same range cell).
2. The Predictive Adaptive Moving Target Indicator (PAMTI) The problem in search radar is the throughput of the signal processor, so we must look for extremely fast algorithms. In order to define the detection problem, the radar environment is sketched in Fig. 1. range
azimuth
k-th sweep
/"
j - t h range cell
Fig. 1. It is assumed that the useful target echo is present in each one of the N consecutive radar sweeps ( N is smaller--e.g, one half than the pulse number corresponding to the 'time on target') but only in one range cell of NR range cell sets. To detect the target, many range cells (NR) must be used in order to identify the target from the clutter; so the estimation procedure is performed in a range-azimuth (NNR) interval (around the tested range cell) small enough to consider the clutter stationary. The range samples for both clutter and noise are assumed to be statistically independent from each
85
S. Barbarossa, (3. Picardi / Predictive adaptive moving target indicator
other, but they are azimuth correlated. As the radar echoes are only locally stationary the optimum processing leads to an adaptive procedure able to match itself to the different signal and clutter conditions encountered in the space of interest of the radar system. According to the symbols introduced in Fig. 1, the estimation of the target presence in a jth range cell set is performed by estimating the clutter in the continguous NR range cells in the subsequent N sweeps; i . e . N . NR clutter samples are considered. The estimated clutter value can be subtracted from the signal plus clutter value present in the j t h range cell. Since the echoes of a pulse radar are sampled, the useful and disturbing signals may be described in vector form: X(j)
= [xl~, x2) . . . .
x,,~]
Fig, 2. (Note: Characters in this and the following figures with underbar appear boldface in text.)
If the filter to be optimized is a finite impulse response (FIR) filter, the optimum estimate of the sample, in a particular cell belonging to the j t h range cell set, is obtained by minimizing the index of performance
I ,12: Is , -
useful input signal,
C ' ( j ) = [ C~j, C:j . . . . C,,O]
clutter input signal,
NT(j) = In, i, nzi . . . . nN~]
noise input signal
(2)
where
input signal,
s T ( j ) = [Slj, S2j . . . . SN,]
w T ( j ) " X(j)I 2
k = i n d e x of the sweep which must be chosen to obtain the best estimation, ek~ = kth entry of the error vector of the j-range cell set, W ( j ) = weight vector of the FIR filter:
w T ( j ) = [wit, w2j.. •, wNj].
(1) where j = 1, 2 . . . . , N~. represents the subsequent range cells and each entry of the vectors is a complex number, which defines the in phase (I) and quadrature (Q) components of the samples. Therefore the detection can be performed by - an estimation procedure of S ( j ) vector (for each j), whitening the clutter; - a coherent integration (as matched filter of the signal in noise environment) performed by a conventional FFT, with reduced dynamic range and consequently low computational cost (clutter was suppressed by whitening filter). In the following we focus our attention on the Wiener optimum filtering technique [3]. In Fig. 2 such an estimation procedure is sketched. In the radar case the useful signal can be specified as a sinusoid whose frequency (a Doppler frequency) is related to the target motion, generally unknown; the disturbance is the sum of white noise N and clutter C.
(3)
When the clutter power is much higher than the noise power, the minimization of eq. (2) yields Re" Wp-- A
(4)
where Re = covariance matrix of the clutter, Wp=optimum form),
weight
vector
(predictive
A = v e c t o r whose components are crosscorrelations between the estimation sample and the input signal samples. The Rc covariant matrix must be estimated by using N . NR clutter samples. According to the whitening filter approach, we hypothesize the target signal as a white noise (the target Doppler is unknown), so A is a vector with only one element differing from zero, namely that corresponding to the estimation sample. VoL 10, No. 1, January 1986
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S. Barbarossa, G. lh'cardi / Predictive adaptive moving target indicator
From eq. (4) we obtain
wp=
~-'.
B
(5)
where K = complex constant, B = vector whose entry differing from zero is normalized: B r = [ 0 , 0 . . . , 1,...0,0].
(6)
In radar literature it is customary to use the ratio of the input clutter-signal ratio to the output, called Improvement Factor (IF), as a measure of radar performance. When the target is modelled as white noise, the IF is given by W T" W* I F - WT" Roe" W*
(7)
where R~ = matrix of the correlation coefficients of the clutter. The maximum of eq. (7) is found [4] by searching for the minimum eigenvalue of Rcc whose corresponding eigenvector is the optimum weighting vector W; the corresponding filter is referred to as Adaptive Moving Target Indicator (AMTI). By using Wiener or predictive filters (eq. (5)), we obtain, with a lower computational cost, an IF very close to that shown in [4]. The resulting PredictiveAdaptive Moving Target Indicator (PAMTI) decorrelates (whitens) the input correlated clutter. The first problem is the choice of the position of the sample on which the estimation is performed [3]. The central position is the optimum as the neighbouring samples allow to perform a clutter estimation better than that obtained using the side samples; in this last case the lower correlation terms are used and the estimation is obviously worse.
Nevertheless by assuming the side estimation technique (only the first entry of vector B is equal Signal Processing
to 1), eq. (5) can be solved by the Durbin-Levinson recursion or Burg algorithms. So the matrix inversion of eq. (5) can be solved with a computational complexity proportional to o(N 2) operations, while with the classical Gaussian elimination technique we need of o(N 3) operations [8]. Figure 3 shows the improvement factor versus the normalized I spectral width of the clutter, with side position for the estimated sample. The clutter is assumed to have a Gaussian spectrum, so that the correlation coefficient is given by p(~-) = e -~'~(~'o'~
(8)
where cr¢ is the bandwidth of the clutter. Figure 4 shows the IF losses of the side estimation technique with reference to the central technique. The effectiveness of the whitening is demonstrated in [6]. In [3] it is demonstrated that the binomial single and double conventional cancellers are the predictive filters in the absence of clutter Doppler shifts and for highly correlated clutters. The N of Fig. 3 corresponds to the autoregressive (AR) order of the clutter model, so the matching of the AR technique to the clutter modelling problems can be provided by the IF evaluation; another
IFlds
N
70.
50. 30-
10.d2
.d6
.1'
.14'
.1'8 - oo. T
Fig. 3. T is the radar pulse repetition period: 1/T is the pulse repetition frequency.
S. Barbarossa, (3. Picardi / Predictive adaptive moving target indicator
AIFId B
7'
8
5-
6
3 1.
Fig. 4.
way to show the AR modelling validity is referred to in [2] by examining the clutter spectral behaviour. In order to realize the AR filtering, we use the AR model definition 1
X(z)=H(z). E(z)=-• E(z) A(z)
(9)
87
In order to perform the clutter cancellation we can therefore use many approaches and it is necessary to find the optimum trade-off in terms of cost and performances. To reduce the hardware complexity it is convenient to use a minimum value of N, according to the IF requirements (e.g. see Fig. 3). When two or more clutters are simultaneously present, the AR coefficients of the PAMTI tend to match differently the clutter depending on the relative powers, carriers and bandwidths. The optimal choice of N requires the determination of the cancellation effectiveness with respect to all present clutters. A way to solve the multiclutter cancellation problem is to use the ARMA model. This model can be obtained by adding the sequence of each clutter due to the AR modelling technique:
X ( z ) = E l ( z ) +E2(z) + .. . ÷ ~E~(z) Al(z) A2(z) A,(z)"
(10)
where l~(z) =
z-transform of the noise-like sequence,
X(z)
z-transform of the sequence to be modelled,
=
H(z) = FIR filter transfer function (1/H(z) is the whitening filter),
A(z)
=
polynomial analytical expression of the AR model.
3. Double clutter rejection In adaptive radar applications we must account for the presence of two clutter in the same range cell. If the mean frequency (carrier) of one clutter is known, the cancellation can be performed by a classical non adaptive MTI (Moving Target Indicator); the second clutter can be processed by an adaptive filter, performing the estimation either of the central frequency only or of the whole spectrum.
By solving eq. (10), we have an ARMA model and a subsequent ARMA filter. Moreover, by the initial assumptions of our search radar problem, we work with a 'real time' batch technique, so that the input sequence is limited and the ARMA filter, as can be demonstrated, is equivalent to an AR filter. The ARMA architecture will be considered in order to minimize the computational cost of the implemented filter. In order to evaluate the cancellation effectiveness we define [7] the partial improvement factor as follows: W T. W*
I F i - wT " Rcci" W*
(11)
where R¢¢i is the matrix of the correlation coefficients of the ith clutter and W is the weight vector estimated by the whole clutter environment. In the following we consider the two clutter case; the first is (e.g.) ground clutter, the second is rain clutter. Vol. 10, No. 1, January 1986
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
88
The double clutter auto-correlation matrix is given by R~c -
Rc~I + rR~2
(12)
l+r
60
where r is the power ratio of the rain to ground clutter (usually r < 1). Substituting eq. (12) in eq. (7) we find the relation between the overall and partial improvement factor: IF --
20"
l+r 1/IF~ + r/IF2"
(13)
Figure 5 shows the average behaviour of tile partial improvement factor of the ground clutter for a typical applicaffon, versus the normalized bandwidth of the rain clutter. The ordinate axis of Fig. 5 changes by changing the r value, according to the behaviour of Fig. 6. Moreover, due to the whitening action of the predictive filter [6], in the hypothesis of small fractional clutter bandwidth and different carriers of the two clutters, it is possible to write [7]: I F t - ~ c ~ 2 .1F2. ~c~" r
(14)
So we can obtain the IF2 value and also, by eq. (13), the overall IF. IF, I,. J
12 I',4 1
80-
~c1" T=.02 fdl' T-" 0 f,Jz" T =.5 =.01
10 7060" 5040. 30" 20"
.dl
.d2
.ds Fig. 5.
Signal Processing
40-
.i
.3
~i T
i~-° lb ~
lb "~
lb ~
i
I~
~vr
Fig. 6.
We note (with reference to Fig. 3) that a number of zeroes (PAMTI filter order) is needed higher than the sum of those necessary to cancel the two clutter separately. While in a single clutter environment we can use a small order for the PAMTI filter (for N ~ 4 the IF is still satisfying), in a double clutter case at least an order of 8 is necessary. The larger values of this order are a consequence of the fact that the predictor tries to minimize simultaneously the overall power of the two clutters. Moreover, it must be observed that (for a search radar) the number of pulses in the time on target is 10-30. According to the real time batch approach, it is obvious to consider the filter order equal to the sweeps number available in the batch processor. These considerations (considering that the estimation is performed by range samples, as we will see in detail) suggest that the order of prediction must not be chosen by using the usual Akaike technique [8], conventionally used in the modern spectral analysis problems. Moreover in radar problems the estimation technique of the covariance matrix is performed by examining the spatial clutter distribution (see Fig. 1) [2] and it is not convenient to apply the procedure proper to the sequential data.
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
In the next section our attention will be focused on the estimation techniques to evaluate the AR coefficient in order to implement a cancellation filter.
4. AR coefficients estimation In order to solve eq. (4), we must use an adaptive procedure to estimate Rc in N . N R points region of Fig. 1: the region size is limited by the stationarity considerations. Substituting the estimated W vector from eq. (4) into eq. (7), we have
R c. i f
In a similar way we can also compute the variance of (17), but we can demonstrate that the expected value is the most important parameter in order to evaluate the IF losses. By examining eq. (18), "~ is slightly depending on ~cT; Fig. 7 shows the behaviour of (18) per ~rcT~ < 0.1 and can be used for crcT> 0.1 to obtain a conservative design. The behaviour of Fig. 7 is the same for the IF losses for Doppler processing (the Doppler target signal is supposed known) [9]. ~]~B
(15)
lo
The estimated improvement factor (IFA) is a random variable, of which we must evaluate the expected value and variance. In the following we examine different approaches to estimate W.
8-
4-
4.1. M a x i m u m likelihood principle
2-
IF,, =
According to the m a x i m u m likelihood principle, we can use
-~R.i~,
'16
6 4 6
N=2
4
7'0 20
5'o
16o
NR
Fig. 7.
1 NR Rc =
89
X ( j ) " x*T(j).
(16)
The sample covariance matrix Rc is distributed with a complex Wishart distribution, and by following the procedure in [9] we can find the distribution of eq. (15), normalized to IFA value computed by the theoretical knowledge of R¢c by eq. (7): IFA 3' = IFA"
By examining eq. (15), we note that the IFA, and so the IFA loss, is a function of the correlation coefficient estimation, even if we use the correlation parameters. The (16) estimation technique can be used also when the stationarity condition i~not satisfied, but in stationary condition we can reduce the computational cost by using a proper estimation technique.
(17) 4.2. Yule- Walker equations
The expected value of (17) can be computed to be
1 T = - -
NR+ I
tr(R~ I) N R - N + I - - +
lEA
NR+ 1
(18)
This is the expected IFA lOSS if only NR range cells were used to estimate Re.
To solve the Yule-Walker equations (see eqs. (5) and (6) with side sampl-, estimation), the computation of W can be performed by using the autocorrelation coefficients of the clutter estimated by the available input samples. The estimated correlation matrix must be substituted in eq, (5) Vol. 10, No. I, January 1986
90
S. Barbarossa, G. Picardi/ Predictive adaptive mot,ing target indicator
and the Durbin-Levinson recursion [8] provides an efficient solution in order to obtain the weighting vector W 2. The computational cost of the approach includes: the estimation of the autocorrelation coefficients, the algorithm computation. The former is usually very large, unless some simplification is applied on the basis of the specific properties of the signal, as for instance the use of the sign only instead of the value of one of the two samples appearing in the terms of autocorrelation. In [10] and [5] the results of interest in terms of the error variance are discussed in detail. Figure 8 shows the Ns/NRc ratio vs. the normalized clutter bandwidth, where - N~ is the pairs number used in the relay estimator, NRc is the pairs number used in the conventional estimator, in order to obtain the same error variance (and thus the same IF loss). Moreover, we note that the error variance is a function of the correlation of the pairs and increases passing from the uncorrelated to the correlated pairs. This means that some pairs in the correlated case could be discharged without impairing the resulting variance of the error. In the case of the relay estimator this property does not hold. The dashed line of Fig. 8 shows the correlated to uncorrelated pairs ratio (Nc/NR~) needed to obtain the same error variance with conventional estimators. The IF losses due to the correlation coefficient estimation errors were obtained in [11]. By comparing the behaviour of Fig. 7 with those of [11, Fig. 5] (being very similar): we must perform the correlation estimation for different lags by the same pairs;
N~c
20' 105-
-
-
-
-
NRc
.o'2
.o4
Signal Processing
.o'6
.oo
Fig. 8.
the IF losses due to the estimation technique can be evaluated by comparing the error variance (for a small value of the bias); the azimuth contribution is very small, so we can perform in usual cases (very azimuth correlated clutter) the estimation by range averaging; - the behaviour of Fig. 7 can be assumed for IF losses evaluation. Of course, in order to obtain the behaviours of Fig. 7 it is necessary to perform a proper way of the estimation technique. -
-
4.3. Burg algorithm
An effective way to perform clutter cancellation is the use of the Burg algorithm. The Burg approach is based on the lattice structure [2, 8] (see Fig. 9) instead of the FIR filter, to calculate the reflection coefficients directly from the available samples. Therefore the reflection coefficients (Ki of Fig. 9) can be estimated directly by evaluating of the
Z~
~1
2 Rc' in the stationarity hypothesis can be reduced to a Toepliz matrix, so one row only (or column) must be estimated.
" .
E2
ei.1
bl n
%
e 2.,?.
~'2
Fig. 9.
b2 n
"~n
er~.n
b~
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
forward and backward errors [2]: NR
-2y j=l i+1-
5. Whitening and coherent integration filter
N
y.
e,.k(j)b*k-,(j)
k=i+2
NR
N
]~
~ (le,,~(j)12+lb,,k-,(j)l ~)
91
(19)
j=l k=i+2
We may consider eq. (19) as the correlation coefficient of the errors; its evaluation yields to whitening of the clutter (the estimated correlation is referred only to clutter cell sets). In [5], Tables 1 and 2 show the computational cost of Burg and Durbin-Levinson algorithms. Appendix A shows the correlation estimators needed to obtain similar performances for the two algorithms. The little stars in Fig. 7 show the experimental results obtained by computer simulation with the Burg algorithm and the black dots show the results obtained by using the Durbin-Levinson algorithm (for N = 3): in this case the correlation estimators were implemented only in range domain. We notice that from the computational standpoint the Durbin-Levinson and Burg algorithms are very similar [5], but we have selected the clutter cancellation through lattice architecture and eq. (19) on the basis of flexibility, numerical robustness and estimation errors. By changing the correlation estimator (see eq. (19)), we obtain an optimum computational costperformances trade-off.
In order to obtain the gain of the coherent integration, we must estimate the signal in each sweep of the jth range cell. The clutter residues are supposed as white-noise and the optimum filtering technique or coherent integrator or matched filter of the signal in noise environment is the FFT processor. Moreover, the lattice formulation of the whitening filter corresponds to obtaining a decorrelation of the clutter residues, and the output of the adder Zj (,~j) (see Fig. 9) can be used to obtain the estimation of the (i+ 1)st (or ( N - 1 ) s t ) sweep of the jth range cell Set (see Fig. 1). Figure 10 shows a block diagram of the proposed lattice filter followed by FFT integration for a batch of 8 azimuth sweeps (conventional number for a search radar). We can indeed demonstrate that the estimation errors, obtained by the prediction technique, in forward and backward direction are uncorrelated: this concept is of course connected with the autoregressive modelling technique. The small value of the correlation of forward and backward errors can be evaluated. In any case the evaluation of the reflection coefficients for a single clutter with mean frequency equal to zero, leads to K, = (-1)'p'(T)
(20)
with reference to the Gaussian clutter model.
I _1
.c-..
!
v
Fig. 10. Vol. 10, No. 1, January 1986
92
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
ratio; by considering the natural noise inside the input signal it is easy to demonstrate that we have similar amplitude and phase characteristics. Figure 12(a) and (b) show the modified amplitude and phase filtering characteristics obtained with a clutter to noise ratio of 30 dB. In order to obtain the overall IF evaluation of the lattice filter and FFT processor, we need to insert an input sinusoid (the target signal is modelled as a sinusoid with a proper Doppler frequency
By (20) we notice that the reflection coefficients vanish very slowly and Fig. ll(a) and (b) show the amplitude and phase responses, respectively of the filter outputs to the corresponding sections of Fig. 10. We expect that the clutter was cancelled (for single clutter application) after a small section number (3 or 4). Indeed the whitening predictive processor works by modelling the spectrum tails of the clutter distribution in order to obtain an excessive cancellation
A).
(a)
ae IHIf)l~. 28
"4:)c
18
o~.T:.05
t
X,
///
"
f.T
-1.
-28 -38
~f,v,
8
I
.125
.25
.375
.5
[:]gb
.825
(b
368 278 188 915 8
~ f
--[3
i
~c'T=.05
----A I
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¢
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-3GO 8
.125
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.5
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.75
.8?5
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
93
(a
a15 IH (f)ldB 215
oc.T :.05 10 f-T
_,, j / -20
-30
360
I
"
|
.5
.75
.6
(b
qf)
1
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q
%'T =.05 J,
lOO 98
/
15 -90
f.T
-lOg -270 -360
.125
.25
.375
.5
.825
.75
.8?5
Fig. 12.
By considering the matrix Rss of the correlation coefficients of the signal, the IF is given by W x" R,s • W* IFD = W T" Rcc" W*"
The maximum of eq, (21) is achieved if the weight vector is given by Wopt(fa) = KR~-2 • S*
(21)
When the target model is a white noise signal, the R,~ is a unitary diagonal matrix and eq. (21) is the same as eq. (7).
(22)
where K = complex constant, S = signal vector S T= [1
e j2"~fdT
eJ4'rrfdT... ]. VoL 10, No. 1, January 1986
94
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
By changing the target Doppler fd, eq. (22) defines a filter bank and the average IFD is then determined by
_IFDopt =
r
fl/T do
IFDopt(fd) dfd
(23)
where 1/T is the repetition frequency of the radar. The results of eq. (23) obtained for several values of N for a Gaussian clutter are presented in [12, Fig. 2]. The IF D obtained by eqs. (22) and (23) are very similar to the IF of eq. (7) for a small clutter bandwidth (troT<0.1), while for a large bandwidth condition we have some integration gain. Figure 13 shows the optimum IF by [12] and the results of the evaluations obtained by the Fig. 10 block diagram. The very small losses of our proposed solution are referred to the single clutter environment, because only a small number of the lattice stages of Fig. 9 are needed. Moreover, Fig. 14 shows the IFDopt(fd) and the IF obtained by the processor sketched in Fig. 10 behaviours in double clutter environment for typical normalized clutter condition (ground and rain clutter) in search radar applications. The IF of the processor of Fig. 10 was obtained by selecting the FFT output which corresponds to the maximum IF.
6. C o n c l u s i o n s
In order to overcome the spectral distorsion and limitations of conventional adaptive technique, in this paper we have investigated the possibility of realizing an adaptive filter for cancelling the clutter in regions where it may be assumed stationary, by using modem spectral analysis technique. Particular attention was devoted to the architecture based on the generalized Wiener theory and to the recent algorithms of spectral analysis (Durbin-Levinson and Burg). We have analyzed the improvement factor losses due to the use of a limited number of samples and we have evaluated the minimum available value of the losses and the way to obtain this value through the lattice filter. We have considered the design criteria in a double clutter environment and also the possibility of realizing the coherent integration by means of a FFT processor following the whitening lattice filter. We can indeed expect some advantages in the clutter cancellation capability in search radar systems, and the computational cost can be reduced by using a proper correlation estimator (e.g. relay estimator).
-N,, 1
.o13
.;, Fig. 13.
SillnalProcessing
%'T
S. Barbarossa, G. Picardi/ Predictiveadaptive moving target indicator IFIds 56' 48"
•
40"
ac (T =.02
56"
%~T=.05
48"
r
40"
=.01
32"
32-
24"
24-
16"
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Appendix
A.
The
Durbin-Levinson,
Burg
algorithms
comparison
In order to obtain a comparison between the Durbin-Levinson and Burg algorithms, we consider the second order lattice, N = 3 and the correlation is performed by averaging NR range cells (in the real case). The conclusions can be extended to the general cases.
A.1. Durbin- Levinson algorithm Let us consider the following correlation coefficients estimators: NR
NA
E [.lllJ-"l t ' l x -- i = 1
E
xj(i)Xj-l(i)
j=l+l NR
Y i=1
(A.l)
NA
E xj(i) 2 j=l+l
N R
NA
2 ~, ~, xj(i)xj_l(i ) ^
i=l
m:,'J = N.
E
j=l+l
(A.2)
N
E (Xj(i)2+Xj-l(i) 2)
i=1 j=l+l
,,,
NR
NA
~,
~
xj(i)xj-l(i)
i=1 j=/+l
om(l)-((~.
~
• 2\/N" x,(,) 1 / E \ \ i = l j=l+| / \i=l
N )),/2 E ~,-,(0 =
(A.3)
j=/+l
where NA<~ N, and it is easy to demonstrate that the performances (bias and error variance) are very similar for the (A.2) and (A.3) estimators [5]. By following the Durbin-Levinson algorithms [8], we obtain the following reflection coefficients: /~I = --t~(1),
/(2-/~2(I) - :(2) I -:2(1) '
(A.4) Vol, 10, No. I, January 1986
96
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
and the weighting vector
1 -/3(2) )~20= I,
/32(1)-13(2)
)~21= -)3(1) 1-fi2(1)'
~'22 =
(A.5)
1-132(1) "
A.2. Burg algorithm We note from (19) that the Burg algorithm yields reflection coefficients smaller than 1, the~.eby assuring the filtering stability: therefore we can analyze the conditions to obtain similar characteristics to the Durbin-Levinson algorithm. By (19), we have NR
3
2 Y. ~ xj(l)xj_~(l) ^
l=l j=2
gl
(A.6)
Np, 3
=
Y. ~., (xj(l)2+xj-l(l) 2) I=l j = 2
NR 3
y~ y (xj+ £,x~_,)(xj_~+ R,xj_,) R2 = -2 N. ,=,~j=3 +(xj-2+ K,xj-1) ]
+ Kixj-,) I=1 j = 3 NR
=
3
NR
3
NR
3
Z X xjxj_2+R, Z Z (xjxj-,+xj-ixj-2)+£ ~, ~, Y~ xj-i
-2
/=1 j = 3
l=l j=3
/=1 j = 3
"~ NR /=l j=3
"~
/=l j=3
/=l j=3
1311(2) + / ( l ( 0 t + K1/3 ) 1 "~- Ri(Ol~ + £ 1 ~
2
(A.7)
)
where NR
3
Ng
Y~ Y. (x;xj_l + Xj_lx~_2) Ol
--
/~1 j = 3 NR
3
2 2 Y Z (xj+x;_~)
/=1 j = 3
3
2 Y. . Y. X j2_ l '
J~ - - NR
/=l j=3 3 2+
2 Z Y (x; x;_~)
I--I j = 3
To obtain an equal/(2 value using the Burg and Durbin-Levinson algorithms, (A.7) must be equal to the/(2 of (A.4); therefore, using (A.2), / ( , ( a +/(i/3) = -fi2(1),
(A.8)
and by (A.6) and (A.7) we have verified that in this case the two analyzed algorithms (for NA = 3) give equal reflection coefficients and weight vector.
References [1] Muehe, "New technique applied to air traffic control radars", Proc. IEEE, Vol. 62, No. 6, June 1974. Signal Processing
[2] S. Haykin, B.W. Currie and S.B. Kesler, "Maximum entropy spectral analysis of radar clutter", Proc. IEEE, Vol. 70, Sept. 1982, pp. 953-962.
S. l~arbarossa, (3. Picardi / Predictive adaptive moving target indicator [3] T. Bucciarelli, G. Martinelli and G. Picardi, "Clutter cancellation in search radars", AIta Frequenza, Vol. VII, No. 5, Sept. 1983, pp. 389-393. [4] J.K. Hsiao, "On the optimization of MTI clutter rejection", IEEE Trans. Aerospace Electron. Systems, Vol. AES10, Sept. 1974, pp. 622-629. [5] S. Barbarossa and G. Picardi, "A trade off between Durbin-Levinson and Burg algorithms for clutter cancellation in search radars", Int. Conf. on Digital Signal Processing, Florence, 1984. [6] S. Cacopardi, G. Picardi and E. Prestifilippo, "Limits of the clutter cancellation techniques by AR models", ISNCR-84, Tokyo. [7] R. Bicocchi, T. Bucciarelli, S. Cacopardi, P.T. Melacci and G. Picardi, "'Autoregressive spectral estimation: a tool to cancel clutter echoes in modern radars", ASSPSpectrum Estimation Workshop II, Tampa, Nov. 1983, pp. 316-318. [8] S.M. Kay and S.L. Marple, "Spectrum analysis. A modern perspective", Proc. IEEE, Vol. 69, No. 11, Nov. 1981, pp. 1380-1419.
97
[9] I.S. Reed, J.D. Mallett and L.E. Brennan, "Rapid convergence rate in adaptive arrays", IEEE Trans. Aerospace Electron. Systems, Vol. AES-10, No. 6, Nov. 1974, pp. 853-863. [10] S. Cacopardi, "Applicability of the relay correlator to radar signal processing", Electron. Lett., Vol. 19, No. 18, Sept. 1983. [11] T. Bucciarelli, S. Cacopardi, G. Martinelli and G. Orlandi, "Improvement factor losses using spectral AR techniques in search radars", IRSI, Bangalore, 1983, pp. 210-215. [ 12] V.G. H ansen, "Optimum pulse doppler search radar processing and practical approximations", Int. Conf. on Radar, London, 1982, pp. 138-143. [13] J.L. Zolesio, "Anti-clutter autoregressive filtering", Int. Conf. on Radar, Paris, 1984, pp. 115-120. [14] A. Farina and F.A. Studer, "Adaptive implementation of the optimum radar signal processor", Int. Conf. on Radar, Paris, 1984, pp. 93-102.
Vol. 10, No. I, January 1986
83
P R E D I C T I V E ADAPTIVE M O V I N G TARGET INDICATOR* S. BARBAROSSA and G. PICARDI INFO-COM Department, Rome University, 00184 Rome, Italy Received 31 July 1984 Revised 18 February 1985
Abstract. In the recent past many attempts have been performed in order to obtain the adaptive suppression of correlated interference (clutter) in coherent radar systems. In this paper the spectral estimation with their modern algorithms is considered. By starting from the Wiener optimum filtering technique, a convenient implementation of the linear prediction algorithm is found to be the lattice filter, in double clutter environment. The performances in search radar applications are analyzed and the Improvement Factor losses, due to clutter estimation error, are evaluated.
Zusammenfassung. In den letzten Jahren wurde viele Arbeit fiber die Probleme der adaptativen Beseitigung der korrelierten Interferenzen in den koh~irenten Radar-Systemen durchgefiihrt. In diesem Artikel handelt es sich um die Spektral-Sch~itzung mit modernen Algorithmen. Von der Theorie der optimalen Wiener-Filter wird ein fiir das Problem der korrelierten Interferenzen angewandter Linear-Prediction Algorithmus zustande gebracht. In dieser Einrichtung treten Gitter-Filter ein. Die Leistungen in Radar Anwendungen und die Sch~itzungsfehler werden zuletzt berechnet. Resumr. Durant les derui~res annres le probl~me de la suppression adaptive des interfrrence correlres (fouillis), dans les systrmes radar cohrrents a 6t6 beaueoup 6tudir. Darts cet article on aborde l'estimation spectrale au moyen des algorithmes modernes. Partant de la throrie du filtrage optimal de Wiener, on prrsente une mise en oeuvre de l'algorithme de prrdiction linraire adaptre au problrme des interfrrences corrrlres. Cette implrmentation fait intervenir des filtre en treillis. Ses performances dans les applications radar et les pertes causres par rerreur d'rstimation sont ensuite 6valures. Keywords. Radar signal processing, adaptive filtering, interference suppression, lattice filters.
1. Introduction
The main problem in radar processors is the detection of a very small target in a disturbed environment. The radar environment is defined by the return echoes from the ground, hill, sea, rain . . . which are unwanted echoes, so called 'clutter'; the small useful targets are, e.g. the echoes of airplanes, missiles and so on. The detection can be performed by estimating and cancelling the clutter; the spatial (the useful targets usually have a limited size with reference to the clutter region) and the spectral (due to the * This work was partially supported by Selenia S.p.A.: C.N.B/22231.1984.
different relative speeds--or Doppler frequency-of the clutter and the useful targets) analyses play an important role to solve this problem. Today in search and tracking radars a standard approach is the use of the Doppler filters banks (as spectrum analyzer) which can be implemented using FFT algorithms; the system is usually called Moving Target Detector (MTD) [1]. This approach is intended to obtain the spectral resolution, thus knowledge of target speeds can be acquired and coherent integration is automatically performed. According to the serial nature of the radar returns, the signals (usually sampled in digital processors) are stored serially in each sweep in a proper memory system and the target history of
0165-1684/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
84
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
each range cell can be analyzed reading the memory in parallel (corner turning approach) and the wanted Doppler informations can be extracted, e.g. using the FFT technique. The spectral analysis is usually performed sequentially on the burst of the sweeps (each analysis is performed on each range cell), so the processor works according to the 'batch' philosophy; the size of the burst is usually one half the pulse number on the 'time on target', because the azimuth target position is unknown in reference to the burst position. The conventional sliding spectral analyses are not used, so that we can use frequency and PRF agility. We can note that the radar 'batch' philosophy is a 'real time' processing technique, according to the conventional pipelined architectures. Therefore the FFT processor must face the following problems connected with the windowing of the data: - High sidelobe level of each filter causes the widening and distorsion of the clutter spectra, and congequent sensitivity losses. We can use a weighting network in order to reduce the sidelobes to avoid the spread of clutter power in the contiguous filters, but this technique causes the widening of the main lobe, reducing both resolution and sensitivity. - Low spectral resolution. These problems can be overcome by an a priori knowledge of the main characteristics of the clutter power density spectrum (e.g. the clutter spectrum can be modelled by a proper order of an autoregressive (AR) model). In this case new spectral analysis methods [2] with their modern algorithms can be used and the problems which must be faced in a modern processor design are: - The choice of a model (in the AR approach the definition of a correct order of regression); - The estimation of the model parameters. This can be performed either by directly processing the input samples or by indirectly using their computed correlation matrix; Signal Processing
- The implementation of the algorithms with a low computational cost, that is with hardware as simple as possible. In the present paper, we focus our attention on search radars, starting from the Wiener optimum filtering technique [3] to obtain coherent integration in a very disturbed environment (e.g. defined by two clutters in the same range cell).
2. The Predictive Adaptive Moving Target Indicator (PAMTI) The problem in search radar is the throughput of the signal processor, so we must look for extremely fast algorithms. In order to define the detection problem, the radar environment is sketched in Fig. 1. range
azimuth
k-th sweep
/"
j - t h range cell
Fig. 1. It is assumed that the useful target echo is present in each one of the N consecutive radar sweeps ( N is smaller--e.g, one half than the pulse number corresponding to the 'time on target') but only in one range cell of NR range cell sets. To detect the target, many range cells (NR) must be used in order to identify the target from the clutter; so the estimation procedure is performed in a range-azimuth (NNR) interval (around the tested range cell) small enough to consider the clutter stationary. The range samples for both clutter and noise are assumed to be statistically independent from each
85
S. Barbarossa, (3. Picardi / Predictive adaptive moving target indicator
other, but they are azimuth correlated. As the radar echoes are only locally stationary the optimum processing leads to an adaptive procedure able to match itself to the different signal and clutter conditions encountered in the space of interest of the radar system. According to the symbols introduced in Fig. 1, the estimation of the target presence in a jth range cell set is performed by estimating the clutter in the continguous NR range cells in the subsequent N sweeps; i . e . N . NR clutter samples are considered. The estimated clutter value can be subtracted from the signal plus clutter value present in the j t h range cell. Since the echoes of a pulse radar are sampled, the useful and disturbing signals may be described in vector form: X(j)
= [xl~, x2) . . . .
x,,~]
Fig, 2. (Note: Characters in this and the following figures with underbar appear boldface in text.)
If the filter to be optimized is a finite impulse response (FIR) filter, the optimum estimate of the sample, in a particular cell belonging to the j t h range cell set, is obtained by minimizing the index of performance
I ,12: Is , -
useful input signal,
C ' ( j ) = [ C~j, C:j . . . . C,,O]
clutter input signal,
NT(j) = In, i, nzi . . . . nN~]
noise input signal
(2)
where
input signal,
s T ( j ) = [Slj, S2j . . . . SN,]
w T ( j ) " X(j)I 2
k = i n d e x of the sweep which must be chosen to obtain the best estimation, ek~ = kth entry of the error vector of the j-range cell set, W ( j ) = weight vector of the FIR filter:
w T ( j ) = [wit, w2j.. •, wNj].
(1) where j = 1, 2 . . . . , N~. represents the subsequent range cells and each entry of the vectors is a complex number, which defines the in phase (I) and quadrature (Q) components of the samples. Therefore the detection can be performed by - an estimation procedure of S ( j ) vector (for each j), whitening the clutter; - a coherent integration (as matched filter of the signal in noise environment) performed by a conventional FFT, with reduced dynamic range and consequently low computational cost (clutter was suppressed by whitening filter). In the following we focus our attention on the Wiener optimum filtering technique [3]. In Fig. 2 such an estimation procedure is sketched. In the radar case the useful signal can be specified as a sinusoid whose frequency (a Doppler frequency) is related to the target motion, generally unknown; the disturbance is the sum of white noise N and clutter C.
(3)
When the clutter power is much higher than the noise power, the minimization of eq. (2) yields Re" Wp-- A
(4)
where Re = covariance matrix of the clutter, Wp=optimum form),
weight
vector
(predictive
A = v e c t o r whose components are crosscorrelations between the estimation sample and the input signal samples. The Rc covariant matrix must be estimated by using N . NR clutter samples. According to the whitening filter approach, we hypothesize the target signal as a white noise (the target Doppler is unknown), so A is a vector with only one element differing from zero, namely that corresponding to the estimation sample. VoL 10, No. 1, January 1986
86
S. Barbarossa, G. lh'cardi / Predictive adaptive moving target indicator
From eq. (4) we obtain
wp=
~-'.
B
(5)
where K = complex constant, B = vector whose entry differing from zero is normalized: B r = [ 0 , 0 . . . , 1,...0,0].
(6)
In radar literature it is customary to use the ratio of the input clutter-signal ratio to the output, called Improvement Factor (IF), as a measure of radar performance. When the target is modelled as white noise, the IF is given by W T" W* I F - WT" Roe" W*
(7)
where R~ = matrix of the correlation coefficients of the clutter. The maximum of eq. (7) is found [4] by searching for the minimum eigenvalue of Rcc whose corresponding eigenvector is the optimum weighting vector W; the corresponding filter is referred to as Adaptive Moving Target Indicator (AMTI). By using Wiener or predictive filters (eq. (5)), we obtain, with a lower computational cost, an IF very close to that shown in [4]. The resulting PredictiveAdaptive Moving Target Indicator (PAMTI) decorrelates (whitens) the input correlated clutter. The first problem is the choice of the position of the sample on which the estimation is performed [3]. The central position is the optimum as the neighbouring samples allow to perform a clutter estimation better than that obtained using the side samples; in this last case the lower correlation terms are used and the estimation is obviously worse.
Nevertheless by assuming the side estimation technique (only the first entry of vector B is equal Signal Processing
to 1), eq. (5) can be solved by the Durbin-Levinson recursion or Burg algorithms. So the matrix inversion of eq. (5) can be solved with a computational complexity proportional to o(N 2) operations, while with the classical Gaussian elimination technique we need of o(N 3) operations [8]. Figure 3 shows the improvement factor versus the normalized I spectral width of the clutter, with side position for the estimated sample. The clutter is assumed to have a Gaussian spectrum, so that the correlation coefficient is given by p(~-) = e -~'~(~'o'~
(8)
where cr¢ is the bandwidth of the clutter. Figure 4 shows the IF losses of the side estimation technique with reference to the central technique. The effectiveness of the whitening is demonstrated in [6]. In [3] it is demonstrated that the binomial single and double conventional cancellers are the predictive filters in the absence of clutter Doppler shifts and for highly correlated clutters. The N of Fig. 3 corresponds to the autoregressive (AR) order of the clutter model, so the matching of the AR technique to the clutter modelling problems can be provided by the IF evaluation; another
IFlds
N
70.
50. 30-
10.d2
.d6
.1'
.14'
.1'8 - oo. T
Fig. 3. T is the radar pulse repetition period: 1/T is the pulse repetition frequency.
S. Barbarossa, (3. Picardi / Predictive adaptive moving target indicator
AIFId B
7'
8
5-
6
3 1.
Fig. 4.
way to show the AR modelling validity is referred to in [2] by examining the clutter spectral behaviour. In order to realize the AR filtering, we use the AR model definition 1
X(z)=H(z). E(z)=-• E(z) A(z)
(9)
87
In order to perform the clutter cancellation we can therefore use many approaches and it is necessary to find the optimum trade-off in terms of cost and performances. To reduce the hardware complexity it is convenient to use a minimum value of N, according to the IF requirements (e.g. see Fig. 3). When two or more clutters are simultaneously present, the AR coefficients of the PAMTI tend to match differently the clutter depending on the relative powers, carriers and bandwidths. The optimal choice of N requires the determination of the cancellation effectiveness with respect to all present clutters. A way to solve the multiclutter cancellation problem is to use the ARMA model. This model can be obtained by adding the sequence of each clutter due to the AR modelling technique:
X ( z ) = E l ( z ) +E2(z) + .. . ÷ ~E~(z) Al(z) A2(z) A,(z)"
(10)
where l~(z) =
z-transform of the noise-like sequence,
X(z)
z-transform of the sequence to be modelled,
=
H(z) = FIR filter transfer function (1/H(z) is the whitening filter),
A(z)
=
polynomial analytical expression of the AR model.
3. Double clutter rejection In adaptive radar applications we must account for the presence of two clutter in the same range cell. If the mean frequency (carrier) of one clutter is known, the cancellation can be performed by a classical non adaptive MTI (Moving Target Indicator); the second clutter can be processed by an adaptive filter, performing the estimation either of the central frequency only or of the whole spectrum.
By solving eq. (10), we have an ARMA model and a subsequent ARMA filter. Moreover, by the initial assumptions of our search radar problem, we work with a 'real time' batch technique, so that the input sequence is limited and the ARMA filter, as can be demonstrated, is equivalent to an AR filter. The ARMA architecture will be considered in order to minimize the computational cost of the implemented filter. In order to evaluate the cancellation effectiveness we define [7] the partial improvement factor as follows: W T. W*
I F i - wT " Rcci" W*
(11)
where R¢¢i is the matrix of the correlation coefficients of the ith clutter and W is the weight vector estimated by the whole clutter environment. In the following we consider the two clutter case; the first is (e.g.) ground clutter, the second is rain clutter. Vol. 10, No. 1, January 1986
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
88
The double clutter auto-correlation matrix is given by R~c -
Rc~I + rR~2
(12)
l+r
60
where r is the power ratio of the rain to ground clutter (usually r < 1). Substituting eq. (12) in eq. (7) we find the relation between the overall and partial improvement factor: IF --
20"
l+r 1/IF~ + r/IF2"
(13)
Figure 5 shows the average behaviour of tile partial improvement factor of the ground clutter for a typical applicaffon, versus the normalized bandwidth of the rain clutter. The ordinate axis of Fig. 5 changes by changing the r value, according to the behaviour of Fig. 6. Moreover, due to the whitening action of the predictive filter [6], in the hypothesis of small fractional clutter bandwidth and different carriers of the two clutters, it is possible to write [7]: I F t - ~ c ~ 2 .1F2. ~c~" r
(14)
So we can obtain the IF2 value and also, by eq. (13), the overall IF. IF, I,. J
12 I',4 1
80-
~c1" T=.02 fdl' T-" 0 f,Jz" T =.5 =.01
10 7060" 5040. 30" 20"
.dl
.d2
.ds Fig. 5.
Signal Processing
40-
.i
.3
~i T
i~-° lb ~
lb "~
lb ~
i
I~
~vr
Fig. 6.
We note (with reference to Fig. 3) that a number of zeroes (PAMTI filter order) is needed higher than the sum of those necessary to cancel the two clutter separately. While in a single clutter environment we can use a small order for the PAMTI filter (for N ~ 4 the IF is still satisfying), in a double clutter case at least an order of 8 is necessary. The larger values of this order are a consequence of the fact that the predictor tries to minimize simultaneously the overall power of the two clutters. Moreover, it must be observed that (for a search radar) the number of pulses in the time on target is 10-30. According to the real time batch approach, it is obvious to consider the filter order equal to the sweeps number available in the batch processor. These considerations (considering that the estimation is performed by range samples, as we will see in detail) suggest that the order of prediction must not be chosen by using the usual Akaike technique [8], conventionally used in the modern spectral analysis problems. Moreover in radar problems the estimation technique of the covariance matrix is performed by examining the spatial clutter distribution (see Fig. 1) [2] and it is not convenient to apply the procedure proper to the sequential data.
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
In the next section our attention will be focused on the estimation techniques to evaluate the AR coefficient in order to implement a cancellation filter.
4. AR coefficients estimation In order to solve eq. (4), we must use an adaptive procedure to estimate Rc in N . N R points region of Fig. 1: the region size is limited by the stationarity considerations. Substituting the estimated W vector from eq. (4) into eq. (7), we have
R c. i f
In a similar way we can also compute the variance of (17), but we can demonstrate that the expected value is the most important parameter in order to evaluate the IF losses. By examining eq. (18), "~ is slightly depending on ~cT; Fig. 7 shows the behaviour of (18) per ~rcT~ < 0.1 and can be used for crcT> 0.1 to obtain a conservative design. The behaviour of Fig. 7 is the same for the IF losses for Doppler processing (the Doppler target signal is supposed known) [9]. ~]~B
(15)
lo
The estimated improvement factor (IFA) is a random variable, of which we must evaluate the expected value and variance. In the following we examine different approaches to estimate W.
8-
4-
4.1. M a x i m u m likelihood principle
2-
IF,, =
According to the m a x i m u m likelihood principle, we can use
-~R.i~,
'16
6 4 6
N=2
4
7'0 20
5'o
16o
NR
Fig. 7.
1 NR Rc =
89
X ( j ) " x*T(j).
(16)
The sample covariance matrix Rc is distributed with a complex Wishart distribution, and by following the procedure in [9] we can find the distribution of eq. (15), normalized to IFA value computed by the theoretical knowledge of R¢c by eq. (7): IFA 3' = IFA"
By examining eq. (15), we note that the IFA, and so the IFA loss, is a function of the correlation coefficient estimation, even if we use the correlation parameters. The (16) estimation technique can be used also when the stationarity condition i~not satisfied, but in stationary condition we can reduce the computational cost by using a proper estimation technique.
(17) 4.2. Yule- Walker equations
The expected value of (17) can be computed to be
1 T = - -
NR+ I
tr(R~ I) N R - N + I - - +
lEA
NR+ 1
(18)
This is the expected IFA lOSS if only NR range cells were used to estimate Re.
To solve the Yule-Walker equations (see eqs. (5) and (6) with side sampl-, estimation), the computation of W can be performed by using the autocorrelation coefficients of the clutter estimated by the available input samples. The estimated correlation matrix must be substituted in eq, (5) Vol. 10, No. I, January 1986
90
S. Barbarossa, G. Picardi/ Predictive adaptive mot,ing target indicator
and the Durbin-Levinson recursion [8] provides an efficient solution in order to obtain the weighting vector W 2. The computational cost of the approach includes: the estimation of the autocorrelation coefficients, the algorithm computation. The former is usually very large, unless some simplification is applied on the basis of the specific properties of the signal, as for instance the use of the sign only instead of the value of one of the two samples appearing in the terms of autocorrelation. In [10] and [5] the results of interest in terms of the error variance are discussed in detail. Figure 8 shows the Ns/NRc ratio vs. the normalized clutter bandwidth, where - N~ is the pairs number used in the relay estimator, NRc is the pairs number used in the conventional estimator, in order to obtain the same error variance (and thus the same IF loss). Moreover, we note that the error variance is a function of the correlation of the pairs and increases passing from the uncorrelated to the correlated pairs. This means that some pairs in the correlated case could be discharged without impairing the resulting variance of the error. In the case of the relay estimator this property does not hold. The dashed line of Fig. 8 shows the correlated to uncorrelated pairs ratio (Nc/NR~) needed to obtain the same error variance with conventional estimators. The IF losses due to the correlation coefficient estimation errors were obtained in [11]. By comparing the behaviour of Fig. 7 with those of [11, Fig. 5] (being very similar): we must perform the correlation estimation for different lags by the same pairs;
N~c
20' 105-
-
-
-
-
NRc
.o'2
.o4
Signal Processing
.o'6
.oo
Fig. 8.
the IF losses due to the estimation technique can be evaluated by comparing the error variance (for a small value of the bias); the azimuth contribution is very small, so we can perform in usual cases (very azimuth correlated clutter) the estimation by range averaging; - the behaviour of Fig. 7 can be assumed for IF losses evaluation. Of course, in order to obtain the behaviours of Fig. 7 it is necessary to perform a proper way of the estimation technique. -
-
4.3. Burg algorithm
An effective way to perform clutter cancellation is the use of the Burg algorithm. The Burg approach is based on the lattice structure [2, 8] (see Fig. 9) instead of the FIR filter, to calculate the reflection coefficients directly from the available samples. Therefore the reflection coefficients (Ki of Fig. 9) can be estimated directly by evaluating of the
Z~
~1
2 Rc' in the stationarity hypothesis can be reduced to a Toepliz matrix, so one row only (or column) must be estimated.
" .
E2
ei.1
bl n
%
e 2.,?.
~'2
Fig. 9.
b2 n
"~n
er~.n
b~
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
forward and backward errors [2]: NR
-2y j=l i+1-
5. Whitening and coherent integration filter
N
y.
e,.k(j)b*k-,(j)
k=i+2
NR
N
]~
~ (le,,~(j)12+lb,,k-,(j)l ~)
91
(19)
j=l k=i+2
We may consider eq. (19) as the correlation coefficient of the errors; its evaluation yields to whitening of the clutter (the estimated correlation is referred only to clutter cell sets). In [5], Tables 1 and 2 show the computational cost of Burg and Durbin-Levinson algorithms. Appendix A shows the correlation estimators needed to obtain similar performances for the two algorithms. The little stars in Fig. 7 show the experimental results obtained by computer simulation with the Burg algorithm and the black dots show the results obtained by using the Durbin-Levinson algorithm (for N = 3): in this case the correlation estimators were implemented only in range domain. We notice that from the computational standpoint the Durbin-Levinson and Burg algorithms are very similar [5], but we have selected the clutter cancellation through lattice architecture and eq. (19) on the basis of flexibility, numerical robustness and estimation errors. By changing the correlation estimator (see eq. (19)), we obtain an optimum computational costperformances trade-off.
In order to obtain the gain of the coherent integration, we must estimate the signal in each sweep of the jth range cell. The clutter residues are supposed as white-noise and the optimum filtering technique or coherent integrator or matched filter of the signal in noise environment is the FFT processor. Moreover, the lattice formulation of the whitening filter corresponds to obtaining a decorrelation of the clutter residues, and the output of the adder Zj (,~j) (see Fig. 9) can be used to obtain the estimation of the (i+ 1)st (or ( N - 1 ) s t ) sweep of the jth range cell Set (see Fig. 1). Figure 10 shows a block diagram of the proposed lattice filter followed by FFT integration for a batch of 8 azimuth sweeps (conventional number for a search radar). We can indeed demonstrate that the estimation errors, obtained by the prediction technique, in forward and backward direction are uncorrelated: this concept is of course connected with the autoregressive modelling technique. The small value of the correlation of forward and backward errors can be evaluated. In any case the evaluation of the reflection coefficients for a single clutter with mean frequency equal to zero, leads to K, = (-1)'p'(T)
(20)
with reference to the Gaussian clutter model.
I _1
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Fig. 10. Vol. 10, No. 1, January 1986
92
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
ratio; by considering the natural noise inside the input signal it is easy to demonstrate that we have similar amplitude and phase characteristics. Figure 12(a) and (b) show the modified amplitude and phase filtering characteristics obtained with a clutter to noise ratio of 30 dB. In order to obtain the overall IF evaluation of the lattice filter and FFT processor, we need to insert an input sinusoid (the target signal is modelled as a sinusoid with a proper Doppler frequency
By (20) we notice that the reflection coefficients vanish very slowly and Fig. ll(a) and (b) show the amplitude and phase responses, respectively of the filter outputs to the corresponding sections of Fig. 10. We expect that the clutter was cancelled (for single clutter application) after a small section number (3 or 4). Indeed the whitening predictive processor works by modelling the spectrum tails of the clutter distribution in order to obtain an excessive cancellation
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By considering the matrix Rss of the correlation coefficients of the signal, the IF is given by W x" R,s • W* IFD = W T" Rcc" W*"
The maximum of eq, (21) is achieved if the weight vector is given by Wopt(fa) = KR~-2 • S*
(21)
When the target model is a white noise signal, the R,~ is a unitary diagonal matrix and eq. (21) is the same as eq. (7).
(22)
where K = complex constant, S = signal vector S T= [1
e j2"~fdT
eJ4'rrfdT... ]. VoL 10, No. 1, January 1986
94
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
By changing the target Doppler fd, eq. (22) defines a filter bank and the average IFD is then determined by
_IFDopt =
r
fl/T do
IFDopt(fd) dfd
(23)
where 1/T is the repetition frequency of the radar. The results of eq. (23) obtained for several values of N for a Gaussian clutter are presented in [12, Fig. 2]. The IF D obtained by eqs. (22) and (23) are very similar to the IF of eq. (7) for a small clutter bandwidth (troT<0.1), while for a large bandwidth condition we have some integration gain. Figure 13 shows the optimum IF by [12] and the results of the evaluations obtained by the Fig. 10 block diagram. The very small losses of our proposed solution are referred to the single clutter environment, because only a small number of the lattice stages of Fig. 9 are needed. Moreover, Fig. 14 shows the IFDopt(fd) and the IF obtained by the processor sketched in Fig. 10 behaviours in double clutter environment for typical normalized clutter condition (ground and rain clutter) in search radar applications. The IF of the processor of Fig. 10 was obtained by selecting the FFT output which corresponds to the maximum IF.
6. C o n c l u s i o n s
In order to overcome the spectral distorsion and limitations of conventional adaptive technique, in this paper we have investigated the possibility of realizing an adaptive filter for cancelling the clutter in regions where it may be assumed stationary, by using modem spectral analysis technique. Particular attention was devoted to the architecture based on the generalized Wiener theory and to the recent algorithms of spectral analysis (Durbin-Levinson and Burg). We have analyzed the improvement factor losses due to the use of a limited number of samples and we have evaluated the minimum available value of the losses and the way to obtain this value through the lattice filter. We have considered the design criteria in a double clutter environment and also the possibility of realizing the coherent integration by means of a FFT processor following the whitening lattice filter. We can indeed expect some advantages in the clutter cancellation capability in search radar systems, and the computational cost can be reduced by using a proper correlation estimator (e.g. relay estimator).
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Appendix
A.
The
Durbin-Levinson,
Burg
algorithms
comparison
In order to obtain a comparison between the Durbin-Levinson and Burg algorithms, we consider the second order lattice, N = 3 and the correlation is performed by averaging NR range cells (in the real case). The conclusions can be extended to the general cases.
A.1. Durbin- Levinson algorithm Let us consider the following correlation coefficients estimators: NR
NA
E [.lllJ-"l t ' l x -- i = 1
E
xj(i)Xj-l(i)
j=l+l NR
Y i=1
(A.l)
NA
E xj(i) 2 j=l+l
N R
NA
2 ~, ~, xj(i)xj_l(i ) ^
i=l
m:,'J = N.
E
j=l+l
(A.2)
N
E (Xj(i)2+Xj-l(i) 2)
i=1 j=l+l
,,,
NR
NA
~,
~
xj(i)xj-l(i)
i=1 j=/+l
om(l)-((~.
~
• 2\/N" x,(,) 1 / E \ \ i = l j=l+| / \i=l
N )),/2 E ~,-,(0 =
(A.3)
j=/+l
where NA<~ N, and it is easy to demonstrate that the performances (bias and error variance) are very similar for the (A.2) and (A.3) estimators [5]. By following the Durbin-Levinson algorithms [8], we obtain the following reflection coefficients: /~I = --t~(1),
/(2-/~2(I) - :(2) I -:2(1) '
(A.4) Vol, 10, No. I, January 1986
96
S. Barbarossa, G. Picardi / Predictive adaptive moving target indicator
and the weighting vector
1 -/3(2) )~20= I,
/32(1)-13(2)
)~21= -)3(1) 1-fi2(1)'
~'22 =
(A.5)
1-132(1) "
A.2. Burg algorithm We note from (19) that the Burg algorithm yields reflection coefficients smaller than 1, the~.eby assuring the filtering stability: therefore we can analyze the conditions to obtain similar characteristics to the Durbin-Levinson algorithm. By (19), we have NR
3
2 Y. ~ xj(l)xj_~(l) ^
l=l j=2
gl
(A.6)
Np, 3
=
Y. ~., (xj(l)2+xj-l(l) 2) I=l j = 2
NR 3
y~ y (xj+ £,x~_,)(xj_~+ R,xj_,) R2 = -2 N. ,=,~j=3 +(xj-2+ K,xj-1) ]
+ Kixj-,) I=1 j = 3 NR
=
3
NR
3
NR
3
Z X xjxj_2+R, Z Z (xjxj-,+xj-ixj-2)+£ ~, ~, Y~ xj-i
-2
/=1 j = 3
l=l j=3
/=1 j = 3
"~ NR /=l j=3
"~
/=l j=3
/=l j=3
1311(2) + / ( l ( 0 t + K1/3 ) 1 "~- Ri(Ol~ + £ 1 ~
2
(A.7)
)
where NR
3
Ng
Y~ Y. (x;xj_l + Xj_lx~_2) Ol
--
/~1 j = 3 NR
3
2 2 Y Z (xj+x;_~)
/=1 j = 3
3
2 Y. . Y. X j2_ l '
J~ - - NR
/=l j=3 3 2+
2 Z Y (x; x;_~)
I--I j = 3
To obtain an equal/(2 value using the Burg and Durbin-Levinson algorithms, (A.7) must be equal to the/(2 of (A.4); therefore, using (A.2), / ( , ( a +/(i/3) = -fi2(1),
(A.8)
and by (A.6) and (A.7) we have verified that in this case the two analyzed algorithms (for NA = 3) give equal reflection coefficients and weight vector.
References [1] Muehe, "New technique applied to air traffic control radars", Proc. IEEE, Vol. 62, No. 6, June 1974. Signal Processing
[2] S. Haykin, B.W. Currie and S.B. Kesler, "Maximum entropy spectral analysis of radar clutter", Proc. IEEE, Vol. 70, Sept. 1982, pp. 953-962.
S. l~arbarossa, (3. Picardi / Predictive adaptive moving target indicator [3] T. Bucciarelli, G. Martinelli and G. Picardi, "Clutter cancellation in search radars", AIta Frequenza, Vol. VII, No. 5, Sept. 1983, pp. 389-393. [4] J.K. Hsiao, "On the optimization of MTI clutter rejection", IEEE Trans. Aerospace Electron. Systems, Vol. AES10, Sept. 1974, pp. 622-629. [5] S. Barbarossa and G. Picardi, "A trade off between Durbin-Levinson and Burg algorithms for clutter cancellation in search radars", Int. Conf. on Digital Signal Processing, Florence, 1984. [6] S. Cacopardi, G. Picardi and E. Prestifilippo, "Limits of the clutter cancellation techniques by AR models", ISNCR-84, Tokyo. [7] R. Bicocchi, T. Bucciarelli, S. Cacopardi, P.T. Melacci and G. Picardi, "'Autoregressive spectral estimation: a tool to cancel clutter echoes in modern radars", ASSPSpectrum Estimation Workshop II, Tampa, Nov. 1983, pp. 316-318. [8] S.M. Kay and S.L. Marple, "Spectrum analysis. A modern perspective", Proc. IEEE, Vol. 69, No. 11, Nov. 1981, pp. 1380-1419.
97
[9] I.S. Reed, J.D. Mallett and L.E. Brennan, "Rapid convergence rate in adaptive arrays", IEEE Trans. Aerospace Electron. Systems, Vol. AES-10, No. 6, Nov. 1974, pp. 853-863. [10] S. Cacopardi, "Applicability of the relay correlator to radar signal processing", Electron. Lett., Vol. 19, No. 18, Sept. 1983. [11] T. Bucciarelli, S. Cacopardi, G. Martinelli and G. Orlandi, "Improvement factor losses using spectral AR techniques in search radars", IRSI, Bangalore, 1983, pp. 210-215. [ 12] V.G. H ansen, "Optimum pulse doppler search radar processing and practical approximations", Int. Conf. on Radar, London, 1982, pp. 138-143. [13] J.L. Zolesio, "Anti-clutter autoregressive filtering", Int. Conf. on Radar, Paris, 1984, pp. 115-120. [14] A. Farina and F.A. Studer, "Adaptive implementation of the optimum radar signal processor", Int. Conf. on Radar, Paris, 1984, pp. 93-102.
Vol. 10, No. I, January 1986