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``````from the signal received at the origin (2 = 0). The time shift Sparse Array Processing
is a function of the arrival angle 0; therefore, measuring the Sensors are expensive, and it is not always feasible to fol-
shift allows estimation of the propagation direction. For a low the design rules outlined above. Accordingly, the goal
planewave propagating in the geometry shown in Figure la, of sparse array processing is to achieve a speciﬁed angular
the time shift for a sensor at z is equal to zcos(0)/c, where c resolution using fewer sensors than the sampling theorem
is the sound speed. When the signal arrives broadside to the requires. Sparse arrays typically use nonuniform sensor
array (6 = 90’), it hits all the sensors simultaneously and the spacing. l0hﬂ50ﬂ 3-Dd Dudgeon (1993) and V3“ TEES (2002)
time shift is zero. When the signal arrives from one of the provide useful introductions to the topic of nonuniform ar-
gndﬁfg directions (9 = 0° 0; e =1;-30°), the time shiﬁ is :1/5, rays, including summaries of the relevant literature.
wlllcll ls the lllaxlmlllll Value for any angle‘ This article focuses on a special class of sparse arrays and
Suppose that the goal is to design a linear array to estimate processing techniques for linear apertures, where the non-
planewave arrival angles. Two questions arise: (1) How far uniform sampling is implemented by interleaving two
apart should the sensors be placed? and (2) How many sen- ULAs. To achieve a sparse design, one or both of the ULAs
sors are required? To answer the first question, note that is undersampled (has sensor spacing greater than a half-
the time shift scales with the sensor distance. If the goal is wavelength).
loalislllve planewave? P_l°Pal§al;llfgf at sllllllall Zllgllls’ llll;l_lﬂll Data processing for this type of sparse array typically con-
lll es sense to lllaxlllllze l 6 l elellce lll l 6 llllle S l S sists of two stages. In the first stage, the data for each ULA
associated with the signals. This is accomplished by placing . . .
th f an _H U f rm t 1 th _ is processed separately using a conventional beamformer,
6 sensors as all all as Possl ll‘ ll 0 ml 6 Y‘ ere ls which is alinear ﬁlter. The second stage combines the bearn-
another consideration that limits the recommended distance f . . . .
_ _ _ _ ormer outputs via a nonlinear operation to obtain an es-
between sensors. To use time shifts to identifythe planewave . . .
timate of the spatial power spectrum (power as a function
angle, the processor must ensure that it compares the shifts of angle)‘ The twmstage implememaﬁon has sever al key
between arrival times for a single wave front, which is not advamagey (1) Conventional beamforming is well under
guaranteed when the sensor distance is too large. For iri- ‘. . . .
stood, thus it is straightforward to design a bea.rnformer to
stance, consider a planewave propagating down the array (9 . . . .
achieve specific performance metrics; and (2) conventional
: On)‘ lflllle smsols are sepallalell by one wavelength Ol)‘ llle beainforming for ULAs can exploit the fast Fourier trans-
crest of one wave hits the sensor at z = 0 at the same time that . . . .
form for computational efficiency. Assuming the interleaved
the crest of the next wave hits the sensor at z = X. In this case, . . . .
ULAs are designed appropriately, the nonlinear operation
the observed time shift will be zero, which is the same as the . . . . . . .
eliminates aliasing introduced by undersampling. This ap-
time shift for a broadside signal. This is an example of spa- . . . . .

_ _ _ _ _ proach to sparse array design and processing originated in
llal allaslllg‘ wlllcll Occllls wllllll ll planewave Plopagallllg at seminal acoustics research by Berman and Clay (1957) The
one angle Callllol be dlsllllglllsllllld lllllll ll planewave Plow‘ remainder of this article introduces key concepts illustrates
gallllg al ll dlffemlll angle based 0“ llle lllllll delay between them using simulations, and provides references for further
sensors. To guarantee unique identiﬁcation of angles based reading
on time shifts, the spatial sampling theorem says the sensors A
must be located less than a half-wavelength apart (Johnson Dnnvanﬁonal uniform Line
and Dudgeon, 1993). E y Prncasaing
To answer the second question about the required number To lay the foundation for the discussion of sparse array pro-
of sensors, consider both resolution and spatial aliasing. An- cessing, this section brieﬂy reviews standard ULA process-
gular resolution is a function of the total aperture spanned ing. See the books by Johnson and Dudgeon (1993) and Van
by the sensors: the larger the aperture, the better the reso- Trees (2002) for more details. The block diagram in Figure
lution. To avoid aliasing, the sensors must be less than or 1c shows how to process ULA data to estimate the power
equal to a half-wavelength apart. For a uniform line array arriving at each angle. The first step is conventional bearn-
(ULA), the required resolution determines the aperture and forming. A beamformer combines the received sensor data
the sampling theorem determines how many sensors are re- so that the signal arriving from a desired direction adds con-
quired to span that aperture assuming that the sensors are structively and the signals from all other directions add de-
equally (i.e., uniformly) spaced. structively. Assuming the desired signal is a planewave, the

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