Page 55 - Fall_DTF
P. 55

mm a) Multiplicative 20 mo h) Min 2“
El 15 15
on a log scale to better illuminate the sidelobes. Changing E ml lo lo‘ m
from the 7, 9 design to the 2, 3 design reduces the sidelobes E 5 5
of the multiplicative processor by 10 dB and those of the min §1 02 V 102 0
processor by 23 dB. The cost for this improvement is addi- E .5 .5
tiona.l sensors: the extended array has 39, whereas the unex- ; ms rm ma N
tended array has 15. E '5 ‘E
Z , ,
Further reduction in sidelobes is possible with adjustments mt N ma ,2a
of the weight vectors. A standard approach to lowering the 4 —o.5 o 0.5 1 4 —o.5 o 05 1
sidelobes in a conventional beamformer is to apply a smooth “:°°‘ll) “:°°5W
taper that weights the outer sensors less than the center sen- 20 c) Noise cross terms alter 10000 snapshots
sors (Van Trees, 2002). The cost of tapering is an increase in —MuIupncauve
main lobe width. The weight vectors in Figure 4, a and b, ‘D A
use uniform tapers, which means that each sensor is weight-
ed equally. Figure 4c shows the patterns for the 2, 3 design 3 U _
after a Dolph-Chebychev taper (Van Trees, 2002) has been :3’
applied. The resulting patterns have substantially lower sid- E _m _
elobes: -30 dB and -60 dB for the multiplicative and min B"
processors, respectively The tapered min processor with 39 _20 T 4
sensors achieves essentially the same power pattern as the
tapered ULA processor with 63 sensors. 30 20 dB
4 0.5 0 0.5 I
="°53 T°"'“5 =°'“P“°a"'° ‘"3 pi°""""° Figure 5. Simulutinn illustrutingsaurce/noise crass terms for the M = 2,
Based on the discussion in Cvprime Pr0€=sSins— coprime N = 3 extended cnprime army with Dn1ph—Chel7ychev shading. Time—
multiplicative and min processors can achieve the average spectra fin the multiplicative (u) and min (b) pracessnrs are
resolution as a ULA processor with fewer sensors. Extended shnwn as ufimctian of the number of scans averugeri :: Average spectra
arrays facilitate the design of processors with user-selected f" ‘he "‘"l’iP““"iV9- mi": “"4 UL-AP’”‘955”’5 “f‘9’10-0005"“P5h”‘5-
sidelobe levels. Equivalent performance with fewer sensors  
seem? mo good to be mm. Are there Omar Consequences of The false arrivals are cross terms (Pedinoff and Ksienski,
sparsity to worry about? 1962) which arise when the input contains multiple sources
Figure 5 shows the output of the multiplicative and min Cross terms form when a signal passing through subarray
processors for a planewave source at broadside plus spatially A interacts with a different signal passing through subarray
uncorrelated noise. The signal-to-noise ratio (SNR) at each B. When one or both of the signals in the cross term passes
sensor is 20 dB. The plots show the estimated power spectra through a subarray grating lobe (rather than the main lobe),
obtained by scanning the directional cosines from -1 to +1. the cross term creates a false peak in the spectrum at a lo-
The narrowband input to the processors consists of a series cation away from either of the true signal directions. The
of time snapshots. Figure 5, a and b, shows the average pow- source/noise cross terms shown in Figure 5 occur when the
er spectra for the two processors as a function of the number loud planewave source aligns with one subarray’s grating
of independent snapshots averaged. Figure 5c compares the lobe and interacts with noise processed by the other subar-
final spectra from 10,000 snapshots with the ULA spectrum. ray. This example shows that for  SNR sources, the min
The spectra for the multiplicative and min processors match processor has substantially lower sourcelnoise cross terms
the ULA spectrum for the source near broadside. The main than the multiplicative processor. Averaging reduces the
differences between the coprime and ULA spectra are the multiplicative cross terms when the sources are uncorre-
signals that appear near u = 10.66 and 11. In the multipli- lated, although Chavali et al. (2014) found the decay to be
cative spectrum, these false arrivals decrease with snapshot slow: -5 dB per 10 snapshots. In Figure 5, it takes 10,000
averaging. In the min spectrum, the false arrivals start low snapshots for the multiplicative cross terms to reach the
and do not decrease with averaging. level of the min cross terms. Although the min processor is
Fall 2013 | Acnuulzllsl Tnduy | 53

   53   54   55   56   57