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Localization with Alissed Sparse Arrays
conventional beamformer shifts the data for each sensor to lb uses uniform weighting). The power pattern characteriz-
time align the wave fronts arriving from the desired direc- es the performance of an array processor in resolving closely
tion and then sums the time-shifted signals. spaced sources of comparable power (main lobe width) and
The remainder of the discussion focuses on narrowband in detecmfg weak “lungs that may be masked by strong”
_ _ sources (sidelobe height).
planewave signals with temporal frequency f and wave-
length X: c/f Fora narrowband signal, beamforming reduc- N0" mhsldet the P°Wet' Pattem (“ghee 1h» 7911 WW9) for
es to applying the appropriate phase shift to the sensor data the ﬁve-sensor array (Figure lb, red triangles), which spans
and summing In pigupe 1c_ the input to the conventional the same aperture as the nine-sensor array using double
beatpformer is a complex vector x Containing narrowband the sensor spacing. The main lobe width for the ﬁve-sensor
data from the sensors. The narrowband data is typically ob- ULA ls al7Pt'°xhhate1Y e‘l“a1 t0 that °tthe hh‘e'5eh5°t' UL-‘tr
tained by Fourier transforming a segment of time data for which is eXPeCted heeahse the)’ 5l7ah the Same aPe‘t‘h'e- The
each sensor and selecting the desired frequency sample. The l7"hha"Y ‘htfetehee hetweeh the Power l7attem5 is that the
beamformer computes a weighted sum of the input data us- hVe'5eh5°" ULA Pattem thehldes tw° ad‘hti°hal Peaks aWaY
ing the weights stored in the vector w. The complex weights fmm the deshed angle These Peaks are called Seating 1°he5
are designed to implement the phase shifts required to align and they are 1°eated at ‘h“ttil71e5 0f A/(5eI1501' 5l7aCh‘g) away
a planewave with arrival angle en Recall from the introduc_ from the look direction 14". The ﬁve-sensor ULA has A spac-
tion that the phase shift is a function of the cosine of the ing» 5° its Stating 1°he5 are heated 11 away {mm "a (“n : 0
arrival angle. Thus, the weight vector is speciﬁed in terms of in this eXa-'hPle)- This a“aY Cannot ‘h5th‘g“i5h a l71aheWaVe
directional cosine un = cos(9n), which deﬁnes the planewave a“'h’h‘g at hmadslde (14 = 0- 9 = 90a) 5'01“ a l71aheWaVe Pt'°l7'
it is designed to detect. In Figure 1c, the output of the con- agatthg d°wh the a"aY (14 = 11- 9 = 0» 1305- Grating 1°he5
ventional beamformer is }/(um), which is an estimate of the are a 5YmPt°m °t 5Patlal ahasthgw and they °ee“’ when the
narrowband signal propagating with directional cosine um 5eh5°t' 5PaCh‘S ls Steatet than M e‘l“a1 t° a hahlwavelehgtht
The ULA processor computes an estimate of the power in _
the si nal b s uarin the beamformer out ut and avera in Interleaved Sparaa Arrays: EB‘-abllshad
S V ‘l S P S S .
the result over time. To obtain the spatial power spectrum NIB‘-'_h°d_a and Recent Innuvatmns
S“ (um), the ULA processor implements the same calculation _I'n_the" seminal Paper‘ Berman and C1ay(1_957) showed that
fo‘;_‘au Possible directional Costnas (_1 g "D g +1)‘ it is possible to use fewer sensors to obtain the same reso-
lution as a densely sampled ULA. They recognized that the
The beampattern characterizes the performance of the bearn- beampattern of a large-aperture ULA can be synthesized by
former. The beampattern B(u) is deﬁned as the output of the multiplying two smaller arrays. Berman and Clay were mo-
beamformer when the input is a unit-amplitude planewave tivated by ideas initially proposed and implemented in radio
with directional cosine u. Essentially, the beampattern is the astronomy, such as the work of Ryle (1952). Although early
frequency response of the spatial ﬁlter. Because the ULA experiments in sonar (Welsby, 1961) and radar (Shaw and
processor computes power, its performance is character- Davies, 1964) implemented product processing with densely
ized by the power pattern P(u), defined as |B(u)|’. Figure sampled ULAs, Davies and Ward (1980) proposed interleav-
lb compares the power patterns for the two ULAs shown in ing a short densely sampled inner ULA with a long under-
Figure la. The first ULA has nine sensors (Figure la, lllue sampled outer ULA and using multiplicative processing to
circles) with half-wavelength spacing (d = M2). The weight mitigate aliasing of narrowband signals. This configuration
vector is designed for the planewave arriving broadside is called a nested array and it is often used for broadband
to the ULA [uﬂ= cos(90’) = 0]. The power pattern for this analysis because it accommodates the change in wavelength
ULA (Figure lb, blue curve) has a peak equal to 1 at 14 = with frequency. After Pal and Vaidyanathan (2010) explor-
un, meaning that it passes the desired signal with unity gain. ing nested arrays, Vaidyanathan and Pal (2011) proposed
The main lobe deﬁnes the passband of the spatial ﬁlter and coprime arrays, which interleave an M-sensor ULA with an
its width is inversely proportional to the ULA aperture (as N-sensor ULA, where M and N are coprime integers (their
quantified by the number of sensors and the sensor spac- greatest common divisor is 1). The M-sensor array has Nd
ing). The sidelobes determine the ﬁlter’s ability to suppress spacing and the N-sensor array has Md spacing. Because of
signals away from the look direction and their height is de- is typically A/2, both subarrays are undersampled. Product
termined by the relative weighting of the sensor data (Figure processing can eliminate the ambiguity due to aliasing be-
an 1 Ai:i:iuII:lr:I Tbday 1 mi 2013
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