Page 56 - Fall_DTF
P. 56

Localization with Alisssd Sparse Arrays 20 ,, gas, 1: ma s,..p,.,.,.5
to — - - - um
less vulnerable to sourcelnoise cross terms, an example in 5,
Simulation Examples shows that it is not immune to other E3 0 _
types of cross term interference. Chavali (2017) and Liu and E ,5
Buck (2018) provide detailed discussions of cross terms in E: ,m
multiplicative and min processors. .15 V _ _ |— s _ _ \ _ _ _ _ _ _ , _ _
Simulation Examples '25 * _5 LB 17 dB 15 ‘,3
Figure 6 shows two simulations highlighting the competing '30‘ _0‘5 0 0.5 ‘
strengths of the multiplicative and min processors. Both ex- u:c(}5(y)
arnples use the extended 2, 3 coprime design with Dolph- b) “SE2: “mm snlpshms
Chebychev weights. Figure 6a shows the simulation results 20
for Case 1, which contains two high SNR sources near broad- ‘5
side and a low SNR (-5 dB) source at u = -073. Only 100 ‘O
snapshots are awilable for averaging. The plot demonstrates A 5 ’
that both the multiplicative and min processors resolve the E3, 0 '
sources near broadside with the same accuracy as the ULA. E’ ’5 ’
The main difference is that the min processor identifies the vi’ 'l" ’
low SNR source, whereas the multiplicative processor does 'l5 ‘ - - -
not because it is masked by the source/noise cross terms as- '2" ’ l l 
sociated with the broadside sources. If 10,000 more snap- '25 {)dB 8 dB 16 dB
shots were available, the multiplicative processor would reli- '3"_‘ _‘)_5 U 0.5 ‘
ably detect the low-level source. U:C(}S(l1)
Figure 6h shows ‘he sknulation results for Case 2, which has Figure 6. Simulutiwl exumplesfnr twn eases utilizing the extended
_ Dn1ph—Chebychev-shaded enprime urmy. a.- Case 1 illustrates haw a
5‘’“'‘e5 at “ = ‘O-5' 0-65' me 1 ‘”‘“‘ SNRS °f°v 3* me 15 dB’ C1055 term generated by laual snurces rleur broadside masks a law-
Y=SP=CfiVfi1Y» and 10-000 5n8P51'-015 awilablfi 50' averaging‘ 1“ level source in the multiplicative pawet estimate. The min proces-
this scenario, the multiplicative processor outperforms the sor detects the quiet snurce in this case. 1;: Case 2 demonstrates that
mm processor The P101 shows that the mm processor has srlapshlmuveruging mitigates crass terms, resulting in in: accurate
large Peaks 3‘ u = _033 and 0_ in addmon ‘0 ‘he Peaks 3‘ ‘he multiplicative spectrum, whereas the min spectrum contains large
source locations. These false peaks are cross terms due to the false Peaks‘
loud sources at u = 0.66 and 1. To see how the cross term at
u = 0 is gme‘,a‘ed) Consider ‘he bemnpanems in Figure 40 false detections, but time averaging or processor selection
When the array is steered to broadside, the subarray grating (swfiehing between mi“ and m“lfiP]1-Cefive dePe“di“S 0“
lobes align with the sources at u = 0.66 and 1. Thus, the min 5P3fi31 di5‘db“'-l0“ 0f 50“"5e5 and "el“‘iVe SNR5) C3“ mm‘
processor sees large signals coming through both subarrays geie the“ effects‘ when 5e“501'5 “Ye eXPe“5lVev 517“-‘Se “rays
simultaneously, and the result is a peak in the spectrum. The and P"°Ce55i“S elgomhms 3-re 3 Viable ‘3°5t'5eVi“3 ‘heme’
cross terms in the multiplicative processor occur at the same five '50 densely 5P“Ced “mform 3-"eyst
1°‘3e‘i°“5v hm ‘hey are "ed“Ced th"°“3h “Ve“‘3i“S (became Here are a few suggestions for further reading on this subject.
the sources are uncorrelated). Although this article focused on coprime arrays, the same
processing approaches are applicable to nested geometries.
Danclusion snd Suggestions Chavali (2017) compares the performance of the multiplica-
for Further Flssding tive and min processors for coprime and nested arrays using
This article showed how to use aliased subarrays to achieve experimental data from a challenging underwater acoustic
the same resolution as aULA with fewer sensors. Multiplica- environment. The examples in this article use passive (re-
tive and min processors eliminate a.mbiguities due to under- ceive-only) arrays, but similar processing approaches have
sampling, and weights can be designed to control sidelobe been applied to active (transmit/receive) arrays. Hoctor and
levels to facilitate detection of low-level sources. In some Kassam (1990) discuss both active and passive applications,
environments, cross terms can mask sources or generate and Mitra et al. (2010) describe using polynomial factoriza-
54 l AI:uuI:lr:I Tbdsy l H11 2013

   54   55   56   57   58