- Email: [email protected]
Linear and non-linear filters for clutter cancellation in radar systems
SIGNAL
PROCESSING ELSEVIER
Signal Processing 59 (1997) 101-112
Linear and non-linear filters for clutter cancellation in radar systems A. Farina* System Analysis Group, Alenia, Via Tiburtina Km. 12.400, 00131 Rome, Italy Received 24 June 1996: revised 13 December 1996
Abstract The theme of this paper is the cancellation of clutter echoes in modern coherent radar systems. The receiver operating characteristics (ROC) of the following clutter filters are compared: (a) optimum linear filter, (b) the linear filter with Chebyshev tapering, and (c) the non-linear filter which takes the minimum between the outputs of few linear filters which process the same radar signals with properly selected weights. The non-linear filter offers a performance advantage, with respect to the widely used filter (b), in terms of detection capability of a target with Doppler frequency close to the clutter mean Doppler frequency. The non-linear filter requires, in general, five times the computation load of the linear filter(b) but is much less computationally expensive than the optimum linear filter (a). 0 1997 Elsevier Science B.V. Zusammenfassung Das Thema dieser Arbeit ist die Ausliischung von Clutterechos in modernen Kohlrenten Radarsystemen. Es werden die Empfinger Betriebscharakteristiken der folgenden Clutterfilter verglichen: (a) optimales lineares Filter, (b) lineares Filter mit Tschebyscheff-Fenster und (c) dasjenige nichtlineare Filter, das das Minimum der Ausgangssignale mehrerer linearer Filter selektiert, die dieselben Radarsignale mit geeignet gewghlten Gewichten verarbeiten. Das nichtlineare Filter bietet einen Leistungsvorteil gegeniiber dem gemeinhin verwendeten Filter(b), und zwar hinsichtlich der FIhigkeit, ein Ziel mit einer Dopplerfrequenz zu erkennen, die nahe der mittleren Dopplerfrequenz des Clutters liegt. Das nichtlineare Filter benijtigt im allgemeinen die fiinffache Rechenleistung des linearen filters (b), ist jedoch vie1 weniger rechenaufwendig als das optimale lineare Filter (a). 0 1997 Elsevier Science B.V. Rbumi! Le sujet de ce papier est la suppression d%chos brouillts dans les radars cohCrents modernes. On compare les caract&istiques de r&ception (ROC) des filtres suivant: (a) filtre IinCaire optimal, (b) filtre lineaire avec coefficients de Chebyshev, et (c) filtre non-linbaire qui prend le minimum entre les sorties de quelques filtres lintaires qui traitent le meme signal radar avec des poids correctement stlectionnts. Le filtre non-linhire offre un avantage de performances par rapport au filtre (b) largement utilist, en terme de capaciti de dCtection d’une cible avec frkquence de Doppler proche de la frkquence Doppler moyenne du brouillage. Le filtre non-lineaire necessite, en genbral, une charge de calcul cinq fois plus grande que le filtre linkaire (b), mais beaucoup moins que le filtre 1inCaire optimal (a). 0 1997 Elsevier Science B.V. Keywords:
Radar; Clutter cancellation; Non-linear
filter
*Tel.: + 39-6-4150-2279; fax: + 39-6-4150-2665; e.mail: [email protected]. 0165-1684/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO165-1684(97)00040-6
102
A. Farina / Signal Processing
1. Introduction In modern radar systems an adequate level of clutter cancellation is needed to limit the number of false alarms presented to the Operator on the display and to avoid overload of the computer that should initiate and update the target tracks. Assume that the clutter echoes are characterised by a Gaussian probability density function (pdf); the optimum detection of a target with a priori known Doppler frequency is obtained by a linear filter, with proper weights, followed by an envelope detector and a comparison to a certain threshold [ 11. In practical radar systems, the implementation of the optimum detector is impaired by its computational complexity. In fact, the weights of the optimum linear filter depend on the inverse of the clutter covariance matrix which is not known a priori; also, the matrix is varying in an unpredictable way within the radar coverage and during the radar operation time. Thus, the optimum linear filter needs to be adaptive to the received radar data. One of the most efficient way for adaptivity is based on the QR decomposition of the received radar data [4]; nevertheless, the computational complexity still remains a problem. Indicating by N the number of coherent pulses in the train, transmitted and received by the radar, the number of operations per range cell is 12N’. Assume that the range cell duration is 1 ps and N = 16; if the adaptation is every range cell, the required computational power is 3 GFlops which is quite high. Assuming that the clutter is local stationary, the filter weights can be updated every m > 1 range cells: say M = 10; this may reduce the computational load. Nevertheless, these computational issues have greatly limited the application of the optimum linear filter in practical radar systems. A sub-optimum implementation, known as adaptive moving target indicator (AMTI), estimates the mean Doppler frequency of clutter and sets an unadapted filter mask around the estimated frequency. This approach works properly against one clutter source; it may have performance degradation against more clutter sources with a priori unknown Doppler frequency (e.g. sea, ground and/or rain clutter perceived by a ship-borne phased-array radar). In this operational situation
59 (1997) 101-l 12
it is common to cancel the clutter with a filter having a fixed set of weights to which corresponds a very low level of side lobes; the Chebyshev tapering gives the best compromise between the side lobe reduction and the main lobe width degradation. This is the situation when linear filters are used. In the field of non-linear filters new trade-offs can be found which are more advantageous in terms of clutter cancellation and target detection. The non-linear filter, which is proposed in this paper, is made of a set of L linear filters all receiving the same set of radar echoes, 1,2, . . . , L, is the weight vector of the Ith filter. The peak of the filters are coincident with the target Doppler frequency. The output of the non-linear filter is calculated as the minimum of the envelopes of the outputs of the L linear filters; thus, one is using a sort of ‘locally linear’ scheme which is simple and efficient. To understand the rationale of using this non-linear filter consider the case of L = 2, with w1 = constant and w2 = Chebyshev tapering. Assume that the clutter is highly correlated, i.e. the clutter spectrum is narrow. As the clutter Doppler spectrum is well separated by the target Doppler frequency, the non-linear filter selects the output from the Chebyshev filter, thus the clutter is filtered by the side lobes of the Chebyshev filter. When the clutter spectrum moves towards the target, the non-linear filter selects the output from the constant weight filter which has the smallest main lobe width. In practice, the clutter spectrum may be wide and the design of the non-linear filter will be done in a proper way, as explained in the sequel of the paper. Section 2 gives the mathematical details including the clutter model, and the expressions for the linear and non-linear filters. The calculation of the receiver operating characteristics (ROCs) of the filters is also shown. The theme of Section 3 is the design of the linear filters which compose the nonlinear filter. The comparison of the ROCs of the following filters is shown in Section 4: (a) optimum linear filter, (b) linear filter with Chebyshev tapering, (c) non-linear filter, and (d) constant weight filter. The conclusions and the references complete the paper. wJ,
1
=
A. Farina / Signal Processing
2. Mathematical models of the radar echo, clutter filters, and calculation of ROCs The radar receives the N-dimensional signal vector z which collects the N coherent reflected echoes from the transmitted pulse train; under hypothesis Ho : z = d, and under hypothesis HI: z = d + seje S( fd). The N-dimensional signal vector is s(fd) = ~1,ejZnfa/rar, . . . , ,$2dd(N- lVPRF]T where fd is the expected Doppler frequency of the target
and PRF is the pulse repetition frequency of the radar. The unknown constant phase $J is assumed to be evenly distributed in the interval [0,21r]. The signal amplitude s is assumed to be known a priori; the calculations are easily extended to the case of Rayleigh amplitude pdf. The column vector d, having dimension N, consists of the sum of thermal noise and clutter; it has a Gaussian pdf with zero mean and covariance matrix M= E(d*dT). The ijth element (i,j = 42, . . . , N) of A4 is as follows: ~~~ = CNRp(i-j)2e-j2rA(i-I)IPRF + d(i _j), where CNR is the clutter-to-noise power ratio, p is the one lag clutter autocorrelation coefficient, fc is
the mean value of the clutter Doppler spectrum and 6 is the Kroneker operator (6(O) = 1,6(i) = 0 for i # 0). The expression of M derives from the hypothesis that the clutter spectrum is Gaussian shaped. It is assumed that the clutter power is constant pulse-to-pulse (i.e. the clutter process is stationary); the thermal noise is a Gaussian distributed random variable independent from pulse-topulse. The ratio of the signal-to-interference power out of a linear filter to the signal-to-interference power into the filter is the filter improvement factor
c11: S2N
-0.4
(1)
WHMW ’
for a filter with a vector w of N coefficients, where tr[MJ is the trace of the matrix M, s2 N/tr(M) is the input signal-to-interference power ratio, hvTsS(jJ2 is the output signal power, wHMv is the output interference power and H denotes conjugate and transpose. The probability of detection can be
I
-78k
103
59 (1997) 101-112
-0.1 0 Doppler frequencyElf
0.2
0.3
-/ 0.4
Fig. 1. An example of the Doppler frequency response of the optimum linear filter.
0.5
104
A. Farina / Signal Processing
shown to be a function of the output signal-to-noise power ratio: IF s’N/tr(M), and the probability PFA of false alarm [l]:
pD
=
Q
(2)
where Q( *, *) is the Marcum Q function. The filter Doppler frequency response is as follows: G(S) = IwTS(f)12, f~ [ - PRF/2,PRF/2].
(3)
The weight vector of the optimum filter is [l] %pt = PM- ‘F (lid,
(4)
p is a constant selected to have the value of the filter peak equal to one. Fig. 1 portrays the frequency response of the optimum linear filter, with N = 16, against a clutter source having the following parameters: CNR = 40 dB, p = 0.99, and fe = 0.18 PRF; note the null that the filter has set around the Doppler frequency fc. The Doppler frequency of
59 (1997) 101 -I 12
the target has been set at& = 0 Hz; it is immaterial to have the target at zero Doppler instead of at a frequency different from zero; the relevant parameter being the Doppler frequency difference between target and clutter. Indicate by wu and u&by the weight VeCtOrS corresponding t0 the UUiform (i.e. constant value of the filter weights) and the Chebyshev tapers, respectively. Fig. 2 illustrates the superimposed frequency responses of the two filters each having 16 taps, i.e. N = 16; the Chebyshev filter has -60 dB of side lobes with respect to the main lobe peak. Also in this case the Doppler frequency of the main lobe peak is at 0 Hz. The basic scheme of the non-linear filter, considered in this paper, is represented in Fig. 3. The vector z, collecting the N radar echoes T = l/PRF s far apart, is processed by two linear filters having the weights W, and wbt respectively. The corresponding output signals are envelope detected giving & = lZTW,I,
-0.1 0 0.1 Doppler frequency/PRF
0.2
zb = lzTu+,(.
0.3
0.4
(5)
0.5
Fig. 2. Superimposed Doppler frequency response of filters with constant and Chebyshev (with - 60 dB of side lobes) tapering functions.
A. Farina / Signal Processing
105
59 (1997) 101-112
Hl
1
2
5
N
=t=: Hz h
Fig. 3. Scheme of the non-linear Doppler frequency filtering of clutter.
The ROCs of the non-linear filter is calculated as follows:
The output of the non-linear filter, 5 = min(x,, xb),
(6)
is compared to a detection threshold 1. The value of 1 depends not only on the PFA value but also on the linear filter with weight vector Wi (i = a, b) which has provided the minimum envelope value at the output. The mathematical expression for 1 is Cl]
iff 5 = x1.
(9)
In fact, the non-linear filter coincides with the Ith filter which has provided the minimum signal at the output.
0.5
W~MViln+-
1= I
.
(7)
FA I
3. Non-linear filter design
A useful generalisation of the technique involves the calculation of the minimum among L > 2 output signals: 5 = min(x,,xb,
. . . ,xL).
(8)
Features of the non-linear filter are - it uses linear filters which can be designed with standard methods, e.g. using the Eq. (4), _ the weights of the linear filters are pre-calculated, i.e. they are not adaptive to the received radar data.
Consider again Fig. 2; a non-linear filter is sought that should have the side lobe level of the Chebyshev filter and the main lobe of the uniform weight filter. This reduces the minimum Doppler frequency interval between the target and the clutter at which the target can still be discriminated and detected. Taking the minimum of the output values from the uniform and the Chebyshev filters does not completely solve the problem. The first and second highest side lobes of the uniform weight filter have to be lowered at a level as close as
106
A. Farina / Signal Processing
possible to the side lobes of the Chebyshev filter without distorting the main lobe of the uniform weight filter. This result can be achieved by replacing the uniform weight filter with two filters: one having a deep null along the first side lobe peak and the other having a deep null along the second side lobe peak. Fig. 1 shows the Doppler frequency response of the second filter. In summary, the non-linear filter takes the minimum of the following five filters: 1. Chebyshev tapering with -60 dB of side lobe level, 2. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.999,fe = O.l2PRF, 3. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.99,fc = O.l8PRF,
59 (1997) 101-112
4. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.999,fc = -O.l2PRF, and 5. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.99,fc = -0.18PRF. The parameters CNR, p andf, have been selected as a compromise to produce a null in the required Doppler frequency region and to limit the distortions on the filter main lobe. Other parameters can be selected to obtain different trade-offs; for instance, the number L = 5 of filters can be changed: only one filter (for positive Doppler frequencies and another for negative Doppler frequencies) can put a null along the first and second side lobes. The Chebyshev filter with -60 dB of side lobes has been selected to contrast a clutter level of the order of 50-60 dB. If the expected amount of clutter is
f1 m J
4 4
t-j-&” . I . I.1
4
.
.
WZ
-
D2
w: w: . .
-
Z
2 WL
.
-
I.1
M
I.1
. .
I
IDI
N
b
M A X
-
Fig 4. Non-linear clutter filtering applied to a Doppler filter bank.
A. Farina / Signal Processing
different, a proper level for the side lobes of the Chebyshev filter will be chosen. The computational load of the non-linear filter is L times the load of the currently used linear filter with fixed weights. More specifically, the number of complex multiplication per range cell is equal to LN, with L = 5 in our study case. The computational load of the adaptive implementation of the adaptive optimum linear filter is 12N2 [4]. Example for N = 16 and L = 5: computational load of the adaptive optimum linear filter = 3072, computational load of the non-linear filter = 80; thus the non-linear filter is much simpler than the adaptive optimum linear filter. An additional comment in favour of the non-linear filter is the following. The non-linear filter should repeat the multiplication of the same set of received data with different sets of weights; on the contrary, the adaptive optimum linear filter needs to perform a more complex operation on the data, e.g. the QR decomposition of the matrix collecting the N echoes at several range cells [4]. Thus, the implementation of the non-linear filter is not risky from a hardware point of view. Slight modifications to the basic non-linear filter scheme are the following. (a) The system should incorporate a constant false alarm rate (CFAR) threshold device after the operation of the minimum: a cell-averaging technique will allow the estimation of the threshold L of Eq. (7). A CFAR loss will affect the detection performance; this loss is also present in the other linear filters currently used in the today radar. (b) In a monopulse radar, the X (sum) and d (difference) channels can be equipped with the non-linear filters; because the calculation of the target angular co-ordinates is done on the coherent C and A signals, the coherent output values from the filter providing the minimum envelope will be extracted for monopulse processing. (c) When the target Doppler frequency is not known a priori, a bank of filters with the main lobes spanning the Doppler frequency interval [ - PRF/2,PRF/2] is needed [3]. Fig. 4 shows the application of the non-linear clutter filtering to the bank of Doppler filters; fnl,fnZ, . . , f& are the Doppler frequency values of the main lobe peaks which span over the whole Doppler frequency interval. w{ (i = 1,2, . . . , L; j = 1,2, . . . ,K) are the weight vectors of the linear filters. The output
59 (I 997) IOI- I12
107
tj (j = 1,2, . . . , K) of each non-linear filter is processed through a cell-averaging CFAR device (not shown in the figure) to suppress residues due to range-extended clutter which may not have been fully suppressed by the filter. High-resolution clutter maps, not shown in the figure, may also be added to suppress point clutter (i.e. non homogeneous) residues. The results of the CFAR devices are further processed by a suitable detection logic. The scheme of Fig. 4 will not be discussed in details in this paper; the performance evaluation, to follow, will refer only to the scheme of Fig. 3 with L( > 2) filters.
4. Performance evaluation The performance of the following filters are compared: (a) optimum linear filter, (b) filter with Chebyshev tapering at - 60 dB of side lobes down the peak, (c) non-linear filter (see the description in Section 3), and (d) constant weight filter. In the evaluation to follow the considered filters have N = 16. The performance analysis can be easily extended to a different value of N. Fig. 5 shows the power of the interference at the output of the compared filters. The interference at the input is a clutter source with CNR = 60 dB and autocorrelation coefficient p = 0.99. The frequency separation, normalised to the radar PRF, between the Doppler frequency of the target (which is assumed to be 0 Hz) and the mean Doppler frequency fc of the clutter is the varying parameter along the horizontal axis of the figure. The main lobe peaks of the filters occur at f = 0 Hz and coincides with the Doppler frequency of the target. As the mean Doppler frequency of the clutter moves towards the filter main lobe, the residue power grows considerably. It is noted that the optimum linear filter produces the maximum filtering of the disturbance. It is followed by the non-linear filter which outperforms the Chebyshev filter. It happens that the non-linear filter curve follows the main lobe of the uniform weight filter and the side lobes of the Chebyshev filter; this was, in fact, the original design goal. The minimum power level at the
A. Farina / Signal Processing 59 (1997) 101-I 12
108
_ _ _Tapering:constant
-_
‘\
-20
0
I 0.05
I 0.1
‘\.
----___-______
Optimum filter -~-~-~-.-.-.-._._.-._._._._._._,_._._
I , I , , 0.15 0.2 0.25 0.3 0.35 (Doppler freq. separation)/PRF
I 0.4
I 0.45
0.5
Fig. 5. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 60 dB, p = 0.99.
output of the Chebyshev filter is zero because of the exact compensation between the filter side lobes and the input clutter power. Fig. 6 depicts the signal-to-noise power ratio (SNR) per pulse needed to achieve a detection probability Pn = 0.8 for a prefixed value of the probability of false alarm, PFA = lO-‘j; SNR is calculated as SNR = s2/2 s being the amplitude of the steering vector S. The input interference is the same adopted to draw the Fig. 5. The curve labelled with ‘Optimum filter + 3 dB’ takes into account the adaptation loss of the optimum linear filter when the adaptation is performed by using 2N independent range cells [2]. Considering a value of SNR equal to 20 dB the minimum separation between the target and clutter Doppler frequency values are - 0.123PRF for the adaptive optimum linear filter, - 0.160PRF for the non-linear filter, and - 0.197PRF for the Chebyshev filter. This is a measure of the advantage of using the non-linear filter with respect to the Chebyshev
filter. The performance curves of Fig. 6 have been limited to the Doppler frequency interval of [O.lPRF - O.SPRF] because the SNR value would grow up considerably when the Doppler separation between the clutter and the target would be less than O.lPRF. Figs. 7 and 8 refer to the case of CNR = 60 dB and p = 0.999, thus the clutter is more correlated. The interference power at the output of the compared filters is displayed in Fig. 7, while the required SNR per pulse to achieve PD = 0.8 and PFA = lop6 is depicted in Fig 8. Again we note, from Fig. 7, that the non-linear filter follows the side lobes of the Chebyshev filter and the main lobe of the constant weight filter. Considering a value of SNR equal to 20 dB the minimum separation between the target and clutter Doppler frequency values are (see Fig. 8): - 0.05lPRF for the adaptive optimum linear filter, _ O.llOPRF for the non-linear filter, and _ 0.159PRF for the Chebyshev filter.
109
A. Farina /Signal Processing 59 (1997) 101-112
I
,
1
60 .. ‘. . .. __-__
_ ?apering:constant -----__ _---____--_
...Tapering.Chebyshev
* , ‘x ‘k. *. .,......
‘\
_,-._._.-
O-
. . . . . .CPt!murp!wV?d! -.-.-._._ -.- - _.- -.-.-.Optimum filter
I
I
I
-1 8/.l
. . . , . -.-.-.-
0.5
0.4
0.2 (Doppler frequen”~separation)~WIF
Fig. 6. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (normalised to the radar PRF) between clutter and target for the linear and non-linear filters: PD = 0.8, PFA= 10e6, N = 16, CNR = 60 dB, p = 0.99.
40\ 30B 9 5 20B a lo-
\
’ \ \ \ \
\
\
O-
\ 1 ,
\
\ / I
I
’
’
1,
c \
I’
\
I \
I
iI
‘Tapering:Chebyshev
\
\
\
\
\
I 0.05
Tapering:constant ,. \ /-
“‘..:, .
-lO-
-20;
/
\
.-____
I 0.1
Optimum filter --_~-L----------------
I
I
0.3 0.35 0.15 0.2 0.25 (Doppler freq. separation)/PRF
0.4
0.45
0.5
Fig. 7. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 60 dB, p = 0.999.
A. Farina / Signal Processing
110
I
60
59 (1997) 101-112
I
1
Tapering:constant \ -\ \ ’ 1 /-‘\ \ I 1 I /
‘/
‘\,_
*
*
.
.
.
.
-.-._._.-__.
.
.
.
.
.
*
.
.
.
.
.
‘-1
I
\
-.
/
\
\
\
Optimum ?W+? . . . . . . . . .
.
-.-.-.-.-___-__._,-.-.__,_,-._,-.-.-.-.-._.-.-.
O-
Optimum filter
I
I
-1
8’.l
I ‘I 5
0.4
0.2 (Doppler frequen”$separation)/PRF
Fig. 8. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (nonnalised to the radar PRF’) between clutter and target for the linear and non-linear filters: PD = 0.8, PFa = 10-6, N = 16, CNR = 60 dB, p = 0.999.
‘\ ‘\ ‘\
.~-.-.-.-._.-.-._._.-._.-.-.-.-.-.-.-. Optimum filter
-200 I
0.05 ,
0.1 I
0.15 L
0.2 I
0.25 I
0.3 I
0.35
0.4
0.45 ,
(Doppler freq. separation)/PRF Fig. 9. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 50 dB, p = 0.99.
A. Farina / Signal Processing 59 (1997) 101-112 50
I
.-:. >. .,-..._, . .‘..... ii
111
I
I
. ‘.
40
:...,
-..._..
_ ~ap@ig:constant .._ -,-.
-,.
2..
---___
-._._.-
-._ :
-._
-..~_.-~_------_------Optimum filter
I -‘8.t
I
0.2
0.4
0.5
(Doppler frequen0j3separation)/PRF Fig. 10. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (normalised to the radar PRF) between clutter and target for the linear and non-linear filters: Po = 0.8, PFA = 1O-6, N = 16, CNR = 50 dB, p = 0.99.
Comparing Figs. 6 and 8 it is noted that the performance improvement of the nonlinear filter, with respect to the Chebyshev filter, increases as the clutter becomes more correlated. From Figs. 7 and 8 we observe that the curves related to the constant tapering mimic the side lobes of the filter; this is so because of the highly correlated nature of the clutter. Figs. 9 and 10 are related to the case of CNR = 50 dB and p = 0.99. The meaning of the figures is similar to that of the pairs of Figs. 5,6 and 7, 8. From Fig. 9 it is noted that the interference residue at the output of the Chebyshev and nonlinear filter is in the order of - 10 dB because the side lobe level of the filters is -60 dB while the CNR is 50 dB. The non-linear filter compares favourably with the ‘optimum filter + 3 dB’ (see Fig. 10).
5. Conclusions The clutter cancellation in a coherent radar system has been analysed and compared by using
linear and non-linear filters. A simple non-linear filter has been proposed that offers a detection advantage with respect to the conventional low side lobe Chebyshev filter; it also approaches the detection performance of the optimum linear filter without requiring the computational complexity of the latter. The need of adaptivity has been removed in favour of a simple non-linear filtering. Because the non-linear filter does not require homogeneous clutter area, where to estimate the interference covariance matrix (as the adaptive filter needs), it promises performance advantage in non-homogeneous clutter areas, e.g. sea-land interface. It is expected that such non-linear filter will be considered as a suitable candidate for immediate use in modern radar; note that the non-linear filter does not require relevant upgrades to the usual hardware that performs the inner product between a sequence of radar echoes and a set of weights (which is the processing required by the conventional linear filters). Future extensions of the work will be the following. (i) Evaluation of performance degradation due to the limited accuracy of the estimation of
112
A. Farina / Signal Processing
the disturbance residue power at the output of the L linear filters; in fact, there is a probability to select the filter which does not give the minimum power level. Mitigation of this effect requires an adequate number of range cells along which to estimate the residue power level. (ii) Calculation of the performance when the target Doppler frequency is not exactly at the peak of the L filters. If performance degradation will occur, the L linear filters - constituting the non-linear filter - may be designed to have a flat response around the peak in addition to nulls in selected intervals of the Doppler frequency axis. (iii) Calculation of the performance against two clutter sources possibly having different sets of parameters (i.e. CNR,f, and p). (iv) Analysis of the processing scheme of Fig. 4 and comparison (in terms of detection performance and needed computational power) with its adaptive counterpart; see the QRD-MVDR adaptive scheme discussed in [4] and depicted in Fig. 4 of the same reference. Additional possible applications of the non-linear filtering theory are in the areas of arrays of antennas for jammer cancellation and the space-time processing for airborne radar for simultaneous cancellation of clutter and jammer. In summary, this paper demonstrates the practical interest in non-linear filtering for a variety of radar systems.
Notation CNR Z fc
clutter-to-noise power ratio received disturbance vector target Doppler frequency mean value of clutter Doppler spectrum
59 (1997) 101-112
interference covariance matrix number of transmitted and received pulses detection probability PD false alarm probability PFA Qt.1 Marcum function signal amplitude i expected signal vector W filter weight vector received signal vector z i detection threshold one-lag autocorrelation coefficient of clutP ter. M N
Acknowledgements The assistance of Dr. S. Pardini of Alenia and of Dr. P.F. Lombardo, of the University of Rome “La Sapienza”, Dept. InfoCom, in reviewing the manuscript is gratefully acknowledged.
References 111 L.E. Brennan, IS. Reed, Theory of adaptive radar, IEEE Trans. Aerospace Electron. Systems AES-9 (2) (1973) 237-252. c21 IS. Reed, J.D. Mallett, L.E. Brennan, Rapid converge rate in adaptive arrays, IEEE Trans. Aerospace Electron. Systems AES-10 (6) (1974) 853-863. c31W.W. Shrader, V. Gregers-Hansen, MT1 radar, in: M.I. Skolnik (Ed.), Radar Handbook, Chapter 15, McGrawHill, New York, 1990, pp. 15-5, 15-6. 141L. Timmoneri, I.K. Proudler, A. Farina, J.G. McWhirter, QRD-based MVDR algorithm for adaptive multipulse antenna array signal processing, IEE Proc.-Radar, Sonar Navigation 141 (2) (1994) 93-102.
PROCESSING ELSEVIER
Signal Processing 59 (1997) 101-112
Linear and non-linear filters for clutter cancellation in radar systems A. Farina* System Analysis Group, Alenia, Via Tiburtina Km. 12.400, 00131 Rome, Italy Received 24 June 1996: revised 13 December 1996
Abstract The theme of this paper is the cancellation of clutter echoes in modern coherent radar systems. The receiver operating characteristics (ROC) of the following clutter filters are compared: (a) optimum linear filter, (b) the linear filter with Chebyshev tapering, and (c) the non-linear filter which takes the minimum between the outputs of few linear filters which process the same radar signals with properly selected weights. The non-linear filter offers a performance advantage, with respect to the widely used filter (b), in terms of detection capability of a target with Doppler frequency close to the clutter mean Doppler frequency. The non-linear filter requires, in general, five times the computation load of the linear filter(b) but is much less computationally expensive than the optimum linear filter (a). 0 1997 Elsevier Science B.V. Zusammenfassung Das Thema dieser Arbeit ist die Ausliischung von Clutterechos in modernen Kohlrenten Radarsystemen. Es werden die Empfinger Betriebscharakteristiken der folgenden Clutterfilter verglichen: (a) optimales lineares Filter, (b) lineares Filter mit Tschebyscheff-Fenster und (c) dasjenige nichtlineare Filter, das das Minimum der Ausgangssignale mehrerer linearer Filter selektiert, die dieselben Radarsignale mit geeignet gewghlten Gewichten verarbeiten. Das nichtlineare Filter bietet einen Leistungsvorteil gegeniiber dem gemeinhin verwendeten Filter(b), und zwar hinsichtlich der FIhigkeit, ein Ziel mit einer Dopplerfrequenz zu erkennen, die nahe der mittleren Dopplerfrequenz des Clutters liegt. Das nichtlineare Filter benijtigt im allgemeinen die fiinffache Rechenleistung des linearen filters (b), ist jedoch vie1 weniger rechenaufwendig als das optimale lineare Filter (a). 0 1997 Elsevier Science B.V. Rbumi! Le sujet de ce papier est la suppression d%chos brouillts dans les radars cohCrents modernes. On compare les caract&istiques de r&ception (ROC) des filtres suivant: (a) filtre IinCaire optimal, (b) filtre lineaire avec coefficients de Chebyshev, et (c) filtre non-linbaire qui prend le minimum entre les sorties de quelques filtres lintaires qui traitent le meme signal radar avec des poids correctement stlectionnts. Le filtre non-linhire offre un avantage de performances par rapport au filtre (b) largement utilist, en terme de capaciti de dCtection d’une cible avec frkquence de Doppler proche de la frkquence Doppler moyenne du brouillage. Le filtre non-lineaire necessite, en genbral, une charge de calcul cinq fois plus grande que le filtre linkaire (b), mais beaucoup moins que le filtre 1inCaire optimal (a). 0 1997 Elsevier Science B.V. Keywords:
Radar; Clutter cancellation; Non-linear
filter
*Tel.: + 39-6-4150-2279; fax: + 39-6-4150-2665; e.mail: [email protected]. 0165-1684/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO165-1684(97)00040-6
102
A. Farina / Signal Processing
1. Introduction In modern radar systems an adequate level of clutter cancellation is needed to limit the number of false alarms presented to the Operator on the display and to avoid overload of the computer that should initiate and update the target tracks. Assume that the clutter echoes are characterised by a Gaussian probability density function (pdf); the optimum detection of a target with a priori known Doppler frequency is obtained by a linear filter, with proper weights, followed by an envelope detector and a comparison to a certain threshold [ 11. In practical radar systems, the implementation of the optimum detector is impaired by its computational complexity. In fact, the weights of the optimum linear filter depend on the inverse of the clutter covariance matrix which is not known a priori; also, the matrix is varying in an unpredictable way within the radar coverage and during the radar operation time. Thus, the optimum linear filter needs to be adaptive to the received radar data. One of the most efficient way for adaptivity is based on the QR decomposition of the received radar data [4]; nevertheless, the computational complexity still remains a problem. Indicating by N the number of coherent pulses in the train, transmitted and received by the radar, the number of operations per range cell is 12N’. Assume that the range cell duration is 1 ps and N = 16; if the adaptation is every range cell, the required computational power is 3 GFlops which is quite high. Assuming that the clutter is local stationary, the filter weights can be updated every m > 1 range cells: say M = 10; this may reduce the computational load. Nevertheless, these computational issues have greatly limited the application of the optimum linear filter in practical radar systems. A sub-optimum implementation, known as adaptive moving target indicator (AMTI), estimates the mean Doppler frequency of clutter and sets an unadapted filter mask around the estimated frequency. This approach works properly against one clutter source; it may have performance degradation against more clutter sources with a priori unknown Doppler frequency (e.g. sea, ground and/or rain clutter perceived by a ship-borne phased-array radar). In this operational situation
59 (1997) 101-l 12
it is common to cancel the clutter with a filter having a fixed set of weights to which corresponds a very low level of side lobes; the Chebyshev tapering gives the best compromise between the side lobe reduction and the main lobe width degradation. This is the situation when linear filters are used. In the field of non-linear filters new trade-offs can be found which are more advantageous in terms of clutter cancellation and target detection. The non-linear filter, which is proposed in this paper, is made of a set of L linear filters all receiving the same set of radar echoes, 1,2, . . . , L, is the weight vector of the Ith filter. The peak of the filters are coincident with the target Doppler frequency. The output of the non-linear filter is calculated as the minimum of the envelopes of the outputs of the L linear filters; thus, one is using a sort of ‘locally linear’ scheme which is simple and efficient. To understand the rationale of using this non-linear filter consider the case of L = 2, with w1 = constant and w2 = Chebyshev tapering. Assume that the clutter is highly correlated, i.e. the clutter spectrum is narrow. As the clutter Doppler spectrum is well separated by the target Doppler frequency, the non-linear filter selects the output from the Chebyshev filter, thus the clutter is filtered by the side lobes of the Chebyshev filter. When the clutter spectrum moves towards the target, the non-linear filter selects the output from the constant weight filter which has the smallest main lobe width. In practice, the clutter spectrum may be wide and the design of the non-linear filter will be done in a proper way, as explained in the sequel of the paper. Section 2 gives the mathematical details including the clutter model, and the expressions for the linear and non-linear filters. The calculation of the receiver operating characteristics (ROCs) of the filters is also shown. The theme of Section 3 is the design of the linear filters which compose the nonlinear filter. The comparison of the ROCs of the following filters is shown in Section 4: (a) optimum linear filter, (b) linear filter with Chebyshev tapering, (c) non-linear filter, and (d) constant weight filter. The conclusions and the references complete the paper. wJ,
1
=
A. Farina / Signal Processing
2. Mathematical models of the radar echo, clutter filters, and calculation of ROCs The radar receives the N-dimensional signal vector z which collects the N coherent reflected echoes from the transmitted pulse train; under hypothesis Ho : z = d, and under hypothesis HI: z = d + seje S( fd). The N-dimensional signal vector is s(fd) = ~1,ejZnfa/rar, . . . , ,$2dd(N- lVPRF]T where fd is the expected Doppler frequency of the target
and PRF is the pulse repetition frequency of the radar. The unknown constant phase $J is assumed to be evenly distributed in the interval [0,21r]. The signal amplitude s is assumed to be known a priori; the calculations are easily extended to the case of Rayleigh amplitude pdf. The column vector d, having dimension N, consists of the sum of thermal noise and clutter; it has a Gaussian pdf with zero mean and covariance matrix M= E(d*dT). The ijth element (i,j = 42, . . . , N) of A4 is as follows: ~~~ = CNRp(i-j)2e-j2rA(i-I)IPRF + d(i _j), where CNR is the clutter-to-noise power ratio, p is the one lag clutter autocorrelation coefficient, fc is
the mean value of the clutter Doppler spectrum and 6 is the Kroneker operator (6(O) = 1,6(i) = 0 for i # 0). The expression of M derives from the hypothesis that the clutter spectrum is Gaussian shaped. It is assumed that the clutter power is constant pulse-to-pulse (i.e. the clutter process is stationary); the thermal noise is a Gaussian distributed random variable independent from pulse-topulse. The ratio of the signal-to-interference power out of a linear filter to the signal-to-interference power into the filter is the filter improvement factor
c11: S2N
-0.4
(1)
WHMW ’
for a filter with a vector w of N coefficients, where tr[MJ is the trace of the matrix M, s2 N/tr(M) is the input signal-to-interference power ratio, hvTsS(jJ2 is the output signal power, wHMv is the output interference power and H denotes conjugate and transpose. The probability of detection can be
I
-78k
103
59 (1997) 101-112
-0.1 0 Doppler frequencyElf
0.2
0.3
-/ 0.4
Fig. 1. An example of the Doppler frequency response of the optimum linear filter.
0.5
104
A. Farina / Signal Processing
shown to be a function of the output signal-to-noise power ratio: IF s’N/tr(M), and the probability PFA of false alarm [l]:
pD
=
Q
(2)
where Q( *, *) is the Marcum Q function. The filter Doppler frequency response is as follows: G(S) = IwTS(f)12, f~ [ - PRF/2,PRF/2].
(3)
The weight vector of the optimum filter is [l] %pt = PM- ‘F (lid,
(4)
p is a constant selected to have the value of the filter peak equal to one. Fig. 1 portrays the frequency response of the optimum linear filter, with N = 16, against a clutter source having the following parameters: CNR = 40 dB, p = 0.99, and fe = 0.18 PRF; note the null that the filter has set around the Doppler frequency fc. The Doppler frequency of
59 (1997) 101 -I 12
the target has been set at& = 0 Hz; it is immaterial to have the target at zero Doppler instead of at a frequency different from zero; the relevant parameter being the Doppler frequency difference between target and clutter. Indicate by wu and u&by the weight VeCtOrS corresponding t0 the UUiform (i.e. constant value of the filter weights) and the Chebyshev tapers, respectively. Fig. 2 illustrates the superimposed frequency responses of the two filters each having 16 taps, i.e. N = 16; the Chebyshev filter has -60 dB of side lobes with respect to the main lobe peak. Also in this case the Doppler frequency of the main lobe peak is at 0 Hz. The basic scheme of the non-linear filter, considered in this paper, is represented in Fig. 3. The vector z, collecting the N radar echoes T = l/PRF s far apart, is processed by two linear filters having the weights W, and wbt respectively. The corresponding output signals are envelope detected giving & = lZTW,I,
-0.1 0 0.1 Doppler frequency/PRF
0.2
zb = lzTu+,(.
0.3
0.4
(5)
0.5
Fig. 2. Superimposed Doppler frequency response of filters with constant and Chebyshev (with - 60 dB of side lobes) tapering functions.
A. Farina / Signal Processing
105
59 (1997) 101-112
Hl
1
2
5
N
=t=: Hz h
Fig. 3. Scheme of the non-linear Doppler frequency filtering of clutter.
The ROCs of the non-linear filter is calculated as follows:
The output of the non-linear filter, 5 = min(x,, xb),
(6)
is compared to a detection threshold 1. The value of 1 depends not only on the PFA value but also on the linear filter with weight vector Wi (i = a, b) which has provided the minimum envelope value at the output. The mathematical expression for 1 is Cl]
iff 5 = x1.
(9)
In fact, the non-linear filter coincides with the Ith filter which has provided the minimum signal at the output.
0.5
W~MViln+-
1= I
.
(7)
FA I
3. Non-linear filter design
A useful generalisation of the technique involves the calculation of the minimum among L > 2 output signals: 5 = min(x,,xb,
. . . ,xL).
(8)
Features of the non-linear filter are - it uses linear filters which can be designed with standard methods, e.g. using the Eq. (4), _ the weights of the linear filters are pre-calculated, i.e. they are not adaptive to the received radar data.
Consider again Fig. 2; a non-linear filter is sought that should have the side lobe level of the Chebyshev filter and the main lobe of the uniform weight filter. This reduces the minimum Doppler frequency interval between the target and the clutter at which the target can still be discriminated and detected. Taking the minimum of the output values from the uniform and the Chebyshev filters does not completely solve the problem. The first and second highest side lobes of the uniform weight filter have to be lowered at a level as close as
106
A. Farina / Signal Processing
possible to the side lobes of the Chebyshev filter without distorting the main lobe of the uniform weight filter. This result can be achieved by replacing the uniform weight filter with two filters: one having a deep null along the first side lobe peak and the other having a deep null along the second side lobe peak. Fig. 1 shows the Doppler frequency response of the second filter. In summary, the non-linear filter takes the minimum of the following five filters: 1. Chebyshev tapering with -60 dB of side lobe level, 2. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.999,fe = O.l2PRF, 3. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.99,fc = O.l8PRF,
59 (1997) 101-112
4. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.999,fc = -O.l2PRF, and 5. weight vector obtained by Eq. (4) with CNR = 40 dB, p = 0.99,fc = -0.18PRF. The parameters CNR, p andf, have been selected as a compromise to produce a null in the required Doppler frequency region and to limit the distortions on the filter main lobe. Other parameters can be selected to obtain different trade-offs; for instance, the number L = 5 of filters can be changed: only one filter (for positive Doppler frequencies and another for negative Doppler frequencies) can put a null along the first and second side lobes. The Chebyshev filter with -60 dB of side lobes has been selected to contrast a clutter level of the order of 50-60 dB. If the expected amount of clutter is
f1 m J
4 4
t-j-&” . I . I.1
4
.
.
WZ
-
D2
w: w: . .
-
Z
2 WL
.
-
I.1
M
I.1
. .
I
IDI
N
b
M A X
-
Fig 4. Non-linear clutter filtering applied to a Doppler filter bank.
A. Farina / Signal Processing
different, a proper level for the side lobes of the Chebyshev filter will be chosen. The computational load of the non-linear filter is L times the load of the currently used linear filter with fixed weights. More specifically, the number of complex multiplication per range cell is equal to LN, with L = 5 in our study case. The computational load of the adaptive implementation of the adaptive optimum linear filter is 12N2 [4]. Example for N = 16 and L = 5: computational load of the adaptive optimum linear filter = 3072, computational load of the non-linear filter = 80; thus the non-linear filter is much simpler than the adaptive optimum linear filter. An additional comment in favour of the non-linear filter is the following. The non-linear filter should repeat the multiplication of the same set of received data with different sets of weights; on the contrary, the adaptive optimum linear filter needs to perform a more complex operation on the data, e.g. the QR decomposition of the matrix collecting the N echoes at several range cells [4]. Thus, the implementation of the non-linear filter is not risky from a hardware point of view. Slight modifications to the basic non-linear filter scheme are the following. (a) The system should incorporate a constant false alarm rate (CFAR) threshold device after the operation of the minimum: a cell-averaging technique will allow the estimation of the threshold L of Eq. (7). A CFAR loss will affect the detection performance; this loss is also present in the other linear filters currently used in the today radar. (b) In a monopulse radar, the X (sum) and d (difference) channels can be equipped with the non-linear filters; because the calculation of the target angular co-ordinates is done on the coherent C and A signals, the coherent output values from the filter providing the minimum envelope will be extracted for monopulse processing. (c) When the target Doppler frequency is not known a priori, a bank of filters with the main lobes spanning the Doppler frequency interval [ - PRF/2,PRF/2] is needed [3]. Fig. 4 shows the application of the non-linear clutter filtering to the bank of Doppler filters; fnl,fnZ, . . , f& are the Doppler frequency values of the main lobe peaks which span over the whole Doppler frequency interval. w{ (i = 1,2, . . . , L; j = 1,2, . . . ,K) are the weight vectors of the linear filters. The output
59 (I 997) IOI- I12
107
tj (j = 1,2, . . . , K) of each non-linear filter is processed through a cell-averaging CFAR device (not shown in the figure) to suppress residues due to range-extended clutter which may not have been fully suppressed by the filter. High-resolution clutter maps, not shown in the figure, may also be added to suppress point clutter (i.e. non homogeneous) residues. The results of the CFAR devices are further processed by a suitable detection logic. The scheme of Fig. 4 will not be discussed in details in this paper; the performance evaluation, to follow, will refer only to the scheme of Fig. 3 with L( > 2) filters.
4. Performance evaluation The performance of the following filters are compared: (a) optimum linear filter, (b) filter with Chebyshev tapering at - 60 dB of side lobes down the peak, (c) non-linear filter (see the description in Section 3), and (d) constant weight filter. In the evaluation to follow the considered filters have N = 16. The performance analysis can be easily extended to a different value of N. Fig. 5 shows the power of the interference at the output of the compared filters. The interference at the input is a clutter source with CNR = 60 dB and autocorrelation coefficient p = 0.99. The frequency separation, normalised to the radar PRF, between the Doppler frequency of the target (which is assumed to be 0 Hz) and the mean Doppler frequency fc of the clutter is the varying parameter along the horizontal axis of the figure. The main lobe peaks of the filters occur at f = 0 Hz and coincides with the Doppler frequency of the target. As the mean Doppler frequency of the clutter moves towards the filter main lobe, the residue power grows considerably. It is noted that the optimum linear filter produces the maximum filtering of the disturbance. It is followed by the non-linear filter which outperforms the Chebyshev filter. It happens that the non-linear filter curve follows the main lobe of the uniform weight filter and the side lobes of the Chebyshev filter; this was, in fact, the original design goal. The minimum power level at the
A. Farina / Signal Processing 59 (1997) 101-I 12
108
_ _ _Tapering:constant
-_
‘\
-20
0
I 0.05
I 0.1
‘\.
----___-______
Optimum filter -~-~-~-.-.-.-._._.-._._._._._._,_._._
I , I , , 0.15 0.2 0.25 0.3 0.35 (Doppler freq. separation)/PRF
I 0.4
I 0.45
0.5
Fig. 5. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 60 dB, p = 0.99.
output of the Chebyshev filter is zero because of the exact compensation between the filter side lobes and the input clutter power. Fig. 6 depicts the signal-to-noise power ratio (SNR) per pulse needed to achieve a detection probability Pn = 0.8 for a prefixed value of the probability of false alarm, PFA = lO-‘j; SNR is calculated as SNR = s2/2 s being the amplitude of the steering vector S. The input interference is the same adopted to draw the Fig. 5. The curve labelled with ‘Optimum filter + 3 dB’ takes into account the adaptation loss of the optimum linear filter when the adaptation is performed by using 2N independent range cells [2]. Considering a value of SNR equal to 20 dB the minimum separation between the target and clutter Doppler frequency values are - 0.123PRF for the adaptive optimum linear filter, - 0.160PRF for the non-linear filter, and - 0.197PRF for the Chebyshev filter. This is a measure of the advantage of using the non-linear filter with respect to the Chebyshev
filter. The performance curves of Fig. 6 have been limited to the Doppler frequency interval of [O.lPRF - O.SPRF] because the SNR value would grow up considerably when the Doppler separation between the clutter and the target would be less than O.lPRF. Figs. 7 and 8 refer to the case of CNR = 60 dB and p = 0.999, thus the clutter is more correlated. The interference power at the output of the compared filters is displayed in Fig. 7, while the required SNR per pulse to achieve PD = 0.8 and PFA = lop6 is depicted in Fig 8. Again we note, from Fig. 7, that the non-linear filter follows the side lobes of the Chebyshev filter and the main lobe of the constant weight filter. Considering a value of SNR equal to 20 dB the minimum separation between the target and clutter Doppler frequency values are (see Fig. 8): - 0.05lPRF for the adaptive optimum linear filter, _ O.llOPRF for the non-linear filter, and _ 0.159PRF for the Chebyshev filter.
109
A. Farina /Signal Processing 59 (1997) 101-112
I
,
1
60 .. ‘. . .. __-__
_ ?apering:constant -----__ _---____--_
...Tapering.Chebyshev
* , ‘x ‘k. *. .,......
‘\
_,-._._.-
O-
. . . . . .CPt!murp!wV?d! -.-.-._._ -.- - _.- -.-.-.Optimum filter
I
I
I
-1 8/.l
. . . , . -.-.-.-
0.5
0.4
0.2 (Doppler frequen”~separation)~WIF
Fig. 6. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (normalised to the radar PRF) between clutter and target for the linear and non-linear filters: PD = 0.8, PFA= 10e6, N = 16, CNR = 60 dB, p = 0.99.
40\ 30B 9 5 20B a lo-
\
’ \ \ \ \
\
\
O-
\ 1 ,
\
\ / I
I
’
’
1,
c \
I’
\
I \
I
iI
‘Tapering:Chebyshev
\
\
\
\
\
I 0.05
Tapering:constant ,. \ /-
“‘..:, .
-lO-
-20;
/
\
.-____
I 0.1
Optimum filter --_~-L----------------
I
I
0.3 0.35 0.15 0.2 0.25 (Doppler freq. separation)/PRF
0.4
0.45
0.5
Fig. 7. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 60 dB, p = 0.999.
A. Farina / Signal Processing
110
I
60
59 (1997) 101-112
I
1
Tapering:constant \ -\ \ ’ 1 /-‘\ \ I 1 I /
‘/
‘\,_
*
*
.
.
.
.
-.-._._.-__.
.
.
.
.
.
*
.
.
.
.
.
‘-1
I
\
-.
/
\
\
\
Optimum ?W+? . . . . . . . . .
.
-.-.-.-.-___-__._,-.-.__,_,-._,-.-.-.-.-._.-.-.
O-
Optimum filter
I
I
-1
8’.l
I ‘I 5
0.4
0.2 (Doppler frequen”$separation)/PRF
Fig. 8. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (nonnalised to the radar PRF’) between clutter and target for the linear and non-linear filters: PD = 0.8, PFa = 10-6, N = 16, CNR = 60 dB, p = 0.999.
‘\ ‘\ ‘\
.~-.-.-.-._.-.-._._.-._.-.-.-.-.-.-.-. Optimum filter
-200 I
0.05 ,
0.1 I
0.15 L
0.2 I
0.25 I
0.3 I
0.35
0.4
0.45 ,
(Doppler freq. separation)/PRF Fig. 9. Power of the residue disturbance at the output of the linear and non-linear filters versus the Doppler frequency separation (normalised to radar PRF) between clutter and target: N = 16, CNR = 50 dB, p = 0.99.
A. Farina / Signal Processing 59 (1997) 101-112 50
I
.-:. >. .,-..._, . .‘..... ii
111
I
I
. ‘.
40
:...,
-..._..
_ ~ap@ig:constant .._ -,-.
-,.
2..
---___
-._._.-
-._ :
-._
-..~_.-~_------_------Optimum filter
I -‘8.t
I
0.2
0.4
0.5
(Doppler frequen0j3separation)/PRF Fig. 10. Input signal-to-noise power ratio per pulse versus the Doppler frequency separation (normalised to the radar PRF) between clutter and target for the linear and non-linear filters: Po = 0.8, PFA = 1O-6, N = 16, CNR = 50 dB, p = 0.99.
Comparing Figs. 6 and 8 it is noted that the performance improvement of the nonlinear filter, with respect to the Chebyshev filter, increases as the clutter becomes more correlated. From Figs. 7 and 8 we observe that the curves related to the constant tapering mimic the side lobes of the filter; this is so because of the highly correlated nature of the clutter. Figs. 9 and 10 are related to the case of CNR = 50 dB and p = 0.99. The meaning of the figures is similar to that of the pairs of Figs. 5,6 and 7, 8. From Fig. 9 it is noted that the interference residue at the output of the Chebyshev and nonlinear filter is in the order of - 10 dB because the side lobe level of the filters is -60 dB while the CNR is 50 dB. The non-linear filter compares favourably with the ‘optimum filter + 3 dB’ (see Fig. 10).
5. Conclusions The clutter cancellation in a coherent radar system has been analysed and compared by using
linear and non-linear filters. A simple non-linear filter has been proposed that offers a detection advantage with respect to the conventional low side lobe Chebyshev filter; it also approaches the detection performance of the optimum linear filter without requiring the computational complexity of the latter. The need of adaptivity has been removed in favour of a simple non-linear filtering. Because the non-linear filter does not require homogeneous clutter area, where to estimate the interference covariance matrix (as the adaptive filter needs), it promises performance advantage in non-homogeneous clutter areas, e.g. sea-land interface. It is expected that such non-linear filter will be considered as a suitable candidate for immediate use in modern radar; note that the non-linear filter does not require relevant upgrades to the usual hardware that performs the inner product between a sequence of radar echoes and a set of weights (which is the processing required by the conventional linear filters). Future extensions of the work will be the following. (i) Evaluation of performance degradation due to the limited accuracy of the estimation of
112
A. Farina / Signal Processing
the disturbance residue power at the output of the L linear filters; in fact, there is a probability to select the filter which does not give the minimum power level. Mitigation of this effect requires an adequate number of range cells along which to estimate the residue power level. (ii) Calculation of the performance when the target Doppler frequency is not exactly at the peak of the L filters. If performance degradation will occur, the L linear filters - constituting the non-linear filter - may be designed to have a flat response around the peak in addition to nulls in selected intervals of the Doppler frequency axis. (iii) Calculation of the performance against two clutter sources possibly having different sets of parameters (i.e. CNR,f, and p). (iv) Analysis of the processing scheme of Fig. 4 and comparison (in terms of detection performance and needed computational power) with its adaptive counterpart; see the QRD-MVDR adaptive scheme discussed in [4] and depicted in Fig. 4 of the same reference. Additional possible applications of the non-linear filtering theory are in the areas of arrays of antennas for jammer cancellation and the space-time processing for airborne radar for simultaneous cancellation of clutter and jammer. In summary, this paper demonstrates the practical interest in non-linear filtering for a variety of radar systems.
Notation CNR Z fc
clutter-to-noise power ratio received disturbance vector target Doppler frequency mean value of clutter Doppler spectrum
59 (1997) 101-112
interference covariance matrix number of transmitted and received pulses detection probability PD false alarm probability PFA Qt.1 Marcum function signal amplitude i expected signal vector W filter weight vector received signal vector z i detection threshold one-lag autocorrelation coefficient of clutP ter. M N
Acknowledgements The assistance of Dr. S. Pardini of Alenia and of Dr. P.F. Lombardo, of the University of Rome “La Sapienza”, Dept. InfoCom, in reviewing the manuscript is gratefully acknowledged.
References 111 L.E. Brennan, IS. Reed, Theory of adaptive radar, IEEE Trans. Aerospace Electron. Systems AES-9 (2) (1973) 237-252. c21 IS. Reed, J.D. Mallett, L.E. Brennan, Rapid converge rate in adaptive arrays, IEEE Trans. Aerospace Electron. Systems AES-10 (6) (1974) 853-863. c31W.W. Shrader, V. Gregers-Hansen, MT1 radar, in: M.I. Skolnik (Ed.), Radar Handbook, Chapter 15, McGrawHill, New York, 1990, pp. 15-5, 15-6. 141L. Timmoneri, I.K. Proudler, A. Farina, J.G. McWhirter, QRD-based MVDR algorithm for adaptive multipulse antenna array signal processing, IEE Proc.-Radar, Sonar Navigation 141 (2) (1994) 93-102.