Robust direct data domain processing for MTI
2.3 Robust D 3 -STAP
constraints of the form:
h
1 zs;l z2s;l . . . zKs;nl1 . . . zt;l zt;lzs;l . . . zt;lzKs;nl1 . . . zKt;lm1 zKt;lm1zs;l . . . zKt;lm1zKs;ln1
i (2.10)
where:
zs;l¼exp j2pd lsinqs;l
zt;l¼exp j2pfs;l fr
(2.11)
qs;l andfs;l being the DOA and the Doppler frequency values of thelth constraint, and replacing with non-zero entries the corresponding elements of the vectory. As apparent, every additional main beam constraint reduces the number of available DOFs reserved for the interference suppression. In the following section, a robus- tification of D3-STAP (i.e. RD3-STAP) is presented to overcome this problem.
As a final general comment, the D3-STAP approach has been here presented for the uniform linear array (ULA) case. Under this hypothesis, the derivation of the interference only quantities in sxm;nand in s;txm;nis simplified. Nevertheless, a generalization to planar arrays with irregular sub-array structures is possible.
equation of (2.9) (i.e. the look direction constraint) can be conveniently reformu- lated as follows:
jwHcj ¼ jwHðaþeÞj 1 (2.13)
for all vectorscsatisfying (2.12). As already shown in [17], the inequality in (2.13) can be rewritten for smallas follows:
jwHaj kwHk 1 (2.14)
As a consequence, a convenient robust formulation of the D3-STAP problem (viz.
robust D3-STAP, RD3-STAP) is:
minw kFH2wk
subject to:jwHaj kwk 1
(2.15) Several comments are in order. First, it is worth noticing that the RD3-STAP method, as well as D3-STAP, designs an adapted filter vector w of dimensions KmKn1. Second, RD3-STAP reformulates the D3-STAP filter design in terms of convex optimization, [15], which can be easily solved with ready-to-use toolboxes [18,19]. Third, it has to be highlighted that the actual convergence of the RD3- STAP method directly depends upon the value of.
Finally, one has to notice that the RD3-STAP filter design, which has been described here for brevity sole in the FW implementation, can be applied straightforward to the BK and to the FB cases.
Convergence of the RD3-STAP method
We now investigate the admissible values for . For the sake of simplicity, the derivation is confined to the spatial-only case (i.e. only the spatial dimension is considered). Extension to the space–time case is then straightforward. Let us express the worst case error vectore(as in [17]) as:
e¼ w
jjwjjejF (2.16)
whereF¼angleðwHaÞ, and substituting it in (2.13), it can be easily shown that (2.13) is equivalent to (2.14) if and only if:
jwHaj jjwjj ,jwHaj
jjwjj (2.17)
According to [17], (2.17) defines the constraint that ensures (2.13) and (2.14) to be equivalent, so that the convex optimization problem in (2.15) is correctly for- mulated and will give a stable numerical solution. Nevertheless, if no restriction is made on the possible form of the weight vector w, no specific closed form expression for the upper bound error can be derived. Fortunately, for the particular case of jjwjj ¼ jjajj ¼ ffiffiffiffiffiffi
Kn
p , (2.17) can be refined thus deriving a closed form
expression for the upper bound error. To understand this point, one should again consider (2.14) and make explicit the inequality with respect to:
jwHaj jjwjj 1,jwHaj jjwjj 1
jjwjj (2.18)
By comparing (2.17) with (2.18), it is clear that (2.18) poses a stricter requirement on. In particular, recalling that bothwandaareKn1 vectors and applying the Cauchy–Schwartz inequality to (2.18), we can find an upper bound for. That is:
MAX ¼ ffiffiffiffiffiffi Kn
p 1 ffiffiffiffiffiffi Kn
p ¼Kn1 ffiffiffiffiffiffi Kn
p (2.19)
Please note that the upper bound in (2.19) is indirectly a function ofN. In parti- cular, for the spatial-only case, we will choose the maximum number of DOFs, leading toKN ¼ bðNþ1Þ=2c, wherebcis the floor integer operator. To prove that (2.19) is actually an upper bound for, let us recall that for hypothesisjjwjj ¼ ffiffiffiffiffiffi Kn
p and let us consider the following expression for:
¼MAXþp¼Kn1 ffiffiffiffiffiffi Kn
p þp (2.20)
wherepis a real scalar value. Plugging (2.20) into (2.14) we get:
jwHaj 1þ Kn1 ffiffiffiffiffiffi Kn p þp
jjwjj ¼Knþp ffiffiffiffiffiffi Kn
p (2.21)
Depending on the value ofp, (2.21) leads to the following three different cases:
p<0) jwHaj<Kn several solutions are admissible p¼0) jwHaj ¼Kn,w¼a only one solution is admissible p>0) jwHaj>Kn no solutions are admissible 8<
: (2.22)
By observing (2.22), it should be clear thatMAX represents an upper bound for and that (2.19) is a sufficient condition for that ensures the existence of an admissible solution for our convex problem. In addition, in the case limit¼MAX, the only admissible solution isw¼a, which clearly does not have any effect in reducing the output interference power. As a consequence, in practical situations, one will always choose < MAX, in order to maintain some freedom in the selection of the weight vectorwaiming at reducing the interference power.
2.3.1 RD3-STAP with dimension reducing transformations
It has been shown in [1,2] and other publications that certain dimension reducing transformations (DRTs) can minimize the computational workload of adaptive space–time clutter filtering without significant losses in clutter suppression performance. Typical examples for such transforms are illustrated in Figure 2.1.
It shows how a linear equispaced array can be modified by forming equispaced
sub-arrays overlapping in such a way that the distance of phase centres is the same as in the original array. In this way, one obtains an equispaced array whose chan- nels have a signal gain given by the number of sub-array elements. IfPN, this transform results in dramatic saving of the subsequent operations (adaptation, filter weight calculation, filtering).
DRTs with overlapping sub-arrays (as those shown in Figure 2.1b) can be also applied in the RD3-STAP case, as first introduced in [20]. Here the main advan- tages and limitations of RD3-STAP with DRT are described.
The number of elements per sub-array isNSA, whilePis the number of channels after DRT, andNB indicates the spacing between adjacent sub-arrays in element unit distance. The signal from thepth sub-arrayðp¼1;. . .;P¼ððNNSAÞ=NBÞ þ1Þ becomes:
Wp¼XNSA
n¼1
xnþNBðp1Þ (2.23)
so that an interference matrixF2 can be built by weighted subtractions:
Wpz1Wpþ1 p¼1;. . .;P (2.24)
wherezis the target signal phase shift between adjacent sub-arrays. From (2.24), the RD3-STAP filter after DRT can be derived as shown in the preceding para- graphs. Clearly, the application of DRT reduces the dimensions of the interference
1
b1 b2 b3
b1 b2 b3 bN-2 bN-1 bN
bN-2 bN-1 bN
b1 b2 b3 bN-2 bN-1 bN
2 3
2 3
1 2 3 N-2 N-1 N
N-2 N-1 N
1
1 2
∑
∑
∑
P
1 2 . . .P
. . .
. . . . . .
N-2 N-1 N
(a) Linear equispaced array
(c) Summation of central elements
(b) Overlapping sub-arrays
Figure 2.1 Examples of dimension reducing transforms.2006 IET. After [1]
matrixF2in (2.15), and hence also reduces the number of DOFs toQ¼ ðPþ1Þ=2.
As a consequence of this DOF reduction, the RD3-STAP filter pulse response might be unable to synthesize sufficiently deep notches in the near sidelobe region, as already highlighted in [20]. A solution to this resides in the modification of the weighted subtractions (2.24) when DRTs are considered. Specifically, [20]
observes that (2.24) identifies a two-pulse canceller, and therefore it proposes to modify the weighted subtractions to those of a double canceller. The idea behind is that a double canceller is expected to have a sharper and faster transition between the filter notch and the pass band region.
A double canceller RD3-STAP can then be derived starting from the following (interference only) weighted subtractions:
zWp1þ2Wpz1Wpþ1 (2.25)
and following the same RD3-STAP filter design as described above. It is important to notice that in the double canceller case the dimensions of the matrixF2(and of the weight vectorw) are further reduced, so that now the number of available DOFs isQ¼P=2. In the following paragraph (see also [20]), a simulative example is shown which demonstrate how DRTs can be effectively applied to double canceller RD3-STAP filtering.
Single and double RD3-STAP performance metrics
We here consider an ULA ofN¼15 half-wavelength spaced elements.NBis fixed to 1, while we varyNSA from 1 to 13 (only odd integers in this interval). The number of channelsPafter DRT follows directly fromNSA. We assume the useful target to impinge the antenna from broadside with a signal to noise ratio of 13 dB, and one interference to be present with a jammer to noise ratio of 31 dB. Perfor- mance is analysed in terms of normalized SINRout, defined as follows:
SINRout¼ EnjwHsðQÞj2o
EnjwHðgðQÞþrðQÞÞj2o 1
sHs (2.26)
where w is the RD3-STAP filter vector obtained applying (17) in [13] to the reduced dimension vector Wp
p¼1;...;P in (2.23), Efg indicates the expected
value operator, the superscript (Q) indicates that only the firstQelements of the corresponding vector are considered.
In the first analysis, a MonteCarlo simulation with 102 trials has been con- ducted to evaluate the normalized SINRout against the varying interference DOA. In this context, the target DOA has been kept fixed with a mismatch between nominal and true DOA equal to half antenna beamwidth (forN¼15, such mismatch guarantees the existence of a numerical solution for the RD3- STAP problem, as previously discussed). Such a mismatch corresponds a worst case analysis for the more general case of a target DOA uniformly distributed
between the antenna main beam. Obtained results are shown in Figure 2.2, for all different sub-array configurations. Note that for NSA¼1, we have the con- ventional full DOFs case (i.e. one channel per antenna element), hence this can be considered as a benchmark to evaluate the performance of other DRT RD3- STAP cases. As is apparent, all DRT configurations exhibit a wide notch, which is the unavoidable consequence of the robustness to potential target DOA mis- matches. Apart from that, it is clear that the complete performance recovery is faster for higher number of channels, as one could easily foresee. Nevertheless, even for the case P¼15, performance recovery is slow also beyond the DOA region of uncertainty (see e.g. the region between 0.2 and 0.4 rad). This means that the notch in the filter pulse response is too large to place a sufficiently deep null in the near sidelobe region.
Considering the same ULA and the same scenario as before, and varyingNSA from 2 to 12 (only even integers, to account for the changed dimensions ofF2), analogous overlapping sub-array configurations can be obtained for the double canceller. Figure 2.3 shows the performance of the double RD3-STAP after DRT in terms of normalized SINRout against a varying interference DOA. As is apparent, apart from the P¼4 case, all other robust double cancellers exhibit much nicer behaviour in the shoulder region (i.e. between 0.2 and 0.4 rad). This means that DRT can be conveniently applied to robust double RD3-STAP to reduce the number of receiving channels at no performance degradation.
0 –5 –10 –15 –20 –25 SINRout [dB]
–30 –35 –40 –45
–500 0.2 0.4 0.6 0.8
Interference DOA (rad)
1 1.2 1.4
NSA = 1, P = 15 NSA = 3, P = 13 NSA = 5, P = 11 NSA = 7, P = 9 NSA = 11, P = 5
1.6
Figure 2.2 Normalized SINRoutagainst varying interference DOA for single RD3-STAP filter after DRT.2014 VDE VERLAG, Berlin, Offenbach. After [20]