-
Sidelobe control for a MIMO radar virtual array
Micaela Contu, Pierfrancesco Lombardo Department of Information
Engineering, Electronics and Telecommunications (DIET)
University of Rome La Sapienza Rome, Italy
[email protected] , [email protected]
Abstract In this paper we address the control of the sidelobe
level of the virtual array obtained by a coherent MISO (Multiple
Input Single Output) or MIMO (Multiple Input Multiple Output)
radar. This is useful for many practical radar systems that could
exploit the wide virtual apertures provided by coherent MIMO array,
but have a low sidelobe requirement. The solution is obtained by
jointly selecting the taper function for the receiving array and
the transmitter (TX) displacements to provide a final antenna
pattern with the desired properties. Moreover we investigate how
beamwidth and SNR change by varying the number of used TXs and the
number of isotropic elements used for each TX.
I. INTRODUCTION A significant research activity has been carried
out recently
on radar with multiple transmitters [1], that use orthogonal
waveforms, so that the echo corresponding to each one of them can
be separated at the receiving antennas using the appropriate
matched filters. This can yield either Multiple Input Single Output
(MISO) radar or Multiple Input Multiple Output (MIMO) radar,
depending on the number of receiving channels. Using transmitters
or receivers widely separated in angle, and/or waveforms emitted at
different times or frequencies are provided radar echoes without
phase coherence. These types of MISO or MIMO radar, called
statistical MIMO, typically exploit non-coherent combinations of
the echo signal received from the different waveforms. They have
been largely demonstrated to benefit of the diversity gain to
increase target detection capability and/or parameter estimation
accuracy [2].
In contrast, when the angle of view of transmitters and
receivers is narrow and the orthogonal waveforms are emitted at the
same time and frequency band, the coherent MISO and MIMO radar are
obtained [3]. Coherent means that they allow coherent combination
of the waveforms received at each receiving channel from each
individual transmitter. This is well known to provide also the
synthesis of very wide virtual apertures [4] using, for example a
uniform array of receiving elements and a uniform array of largely
separated transmitters, with appropriate separation. The aperture
synthesis property of MISO/MIMO arrays is quite attractive for many
applications, because it provides very desirable narrow antenna
beams, with important improvements in the angle estimation
accuracy.
However, many practical radar applications have strong
requirements in terms of antenna sidelobes and the uniform array,
with its 13 dB Peak to Sidelobe Level (PSL), is often unacceptable.
To make the aperture synthesis approach usable in practical MIMO
radar systems, we address the design of a coherent MISO or MIMO
radar array with controlled PSL. The solution is obtained by
jointly selecting the taper function for the receiving array and
the transmitters displacements to provide a virtual antenna pattern
with the desired properties.
II. EXISTING TECHNIQUES Let us consider an equispaced linear
array of NRX
omnidirectional receiving elements with spacing dR. Assume also
that NTX equispaced transmitters are available, with uniform
spacing dT (see Figure 1a) and that they transmit orthogonal
waveforms. The NRXNTX echoes received by the NRX receivers as the
target reflection, corresponding to the NTX waveforms, have phase
terms that can be interpreted as phase terms of an equivalent
receive virtual array of NRXNTX elements with appropriate steering
vector, [3]-[4]. The relation between transmitting, receiving and
virtual array is described with a convolution. If we set dT=NRXdR a
very special case is obtained in which the convolution provides a
uniform linear virtual array of NRXNTX elements (see Figure 1b). By
defining u= dR/ sin(), being the angle from the antenna broadside,
the global Transmit-Receive antenna pattern shape is
sin(NRXNTXu)/sin(u), which is NTX times narrower than the pattern
of the Receive array only: sin(NRXu)/sin(u).
(a)
(b) Figure 1: (a) Physical array; (b) Uniform virtual array of
NTX NRX elements.
This increase of the equivalent length of the array with
MIMO is well known for untapered receiving arrays, thus
providing untapered virtual arrays, characterized by PSL values of
13.2 dB, very low for many radar applications.
2013 IEEE Radar Conference (RadarCon13)
978-1-4673-5794-4/13/$31.00 2013 IEEE
-
To provide higher PSLvalues, it is possible to apply a taper
function to the synthesized virtual array. Despite this is
certainly effective in controlling the two-way pattern, it shows
two main drawbacks:
The signals received at each receiving element from different
transmitters require different weights. Therefore: (i) each
receiving antenna must be equipped with a down-conversion receiver
and A/D; (ii) all the NTX matched filters are applied to each
element data flow; (iii) only after this stage the weights are
applied to the samples received at the different elements and they
are added together (sketch in Figure 2).
Figure 2: Sketch of a receiver using a different weight for each
waveform.
While the taper function w applied to the whole virtual array
controls the two-way (namely Transmit-Receive) antenna pattern (see
( 1 ) below), it does not guarantee a good PSL for the (one way)
Receive-only pattern (see ( 2 ) below). This is important for radar
operation against hostile environments, that include jamming
signals.
{ }2
00
1
0
*
22
)]()([)()()(
)();(),;(
sws
ww ksH
ks
N
kkH
RX
RXpNTX
psRXTX diagN
GGP
TXTX
=
=
( 1 )
{ } 2002*2
)]()([)()(
)();(
sws
ww ksH
kskHRX
RXsRX diagaN
GP = ( 2 )
where: );( p
NTX
TXG is the transmit antenna pattern, steered toward pointing
direction p, along the generic target DOA ,
)(RXG is the receive element pattern along target DOA , vector
)()()( 0 sss ss = contains all the phase terms
imposed by the two propagation paths transmitter-to-target and
target-to-receiver. As such, it can be written as the Kronecker
product of the two corresponding phase vectors.
vector )()( 0 vv sav = contains both jammer complex samples and
phase terms of the jammer-to-receiver path. It can be written as
Kronecker product of their phase vectors.
As an example for a MIMO array with NTX=3 and NRX=31, the use of
a Taylor taper function yields the desired two way pattern in
Figure 3 with PSL=30 dB. However, the pattern seen by a noise-like
jammer in Figure 4, shows apparently a PSL not better than 14 dB.
As is apparent, the transmit portion of the system is ineffective
for them and only the receiving portion determines the antenna
pattern seen by the jammer.
Figure 3: Two way antenna patterns.
Figure 4: One way antenna patterns.
III. PROPOSED MIMO FOR SIDELOBE CONTROL
The different approach proposed in this paper is based on the
following consideration. If the taper function is selected so that
the same weight is used at each receive element for the samples
corresponding to all transmitters (namely independent of the
received waveform), then:
(i) the receive system is largely simplified, since it is
possible to apply the weights, even at radio-frequency (RF), and
add the samples of the NRX receiving elements before applying the
matched filters to separate the signals received from the different
transmitters. The largely simplified scheme, sketched in Figure 5,
includes possibly a single down-conversion and
-
A/D, after tapering and beamforming at RF (eventually MISO).
Also the number of matched filters to be applied, and consequently
the computational load, is obviously largely reduced with respect
to the original scheme in Figure 2.
(ii) The receive-only antenna pattern seen by the jammer is
directly determined by the taper function assigned to the receive
array, as illustrated in the following.
Figure 5: Sketch of the simplified receiver.
By defining a taper function for the receiving array, so
that
the same weight is always used in the same RX channel
independent of the received waveform (namely independent of the
transmitter from which the signal is initially transmitted), the
two-way and one way antenna patterns can be obtained from ( 1 ) and
( 2 ) respectively, by setting 0wew = , where the NRX1 vector w0 is
the receive taper function and the NTX1 vector e defines the
scaling among the signals received from the different
transmitters:
ww
swsEH
sH
sH
RX
RXpMTX
psRXTX
diag
N
GGP
2
000
222)]()()()()()();(
),;(
=
( 3 )
2
000
22
)]()()()()(
);(
swsww
aEdiag
NG
P sH
Hs
H
RX
RXsRX =
( 4 )
As is apparent from ( 4 ) the second term is not a function of .
Thus, the RX pattern experienced by the jammer is only defined by
the third term
2
000 )]()()( sws diagsH , (other than the wide beam of GRX () )
which is the weighted receive pattern of the RX portion only. This
implies that the shape of the one way RX-only jammer pattern is
independent of the values of vector a including the jammer
samples.
In contrast, the two-way TX-RX pattern in ( 3 ) depends
on the combination of the RX-only pattern 2
000 )]()()( sws diagsH with the transmit pattern, given by 2
)()( E sH . This latter term is affected by both transmitters
locations (vector ) and taper scaling terms (vector e). Obviously,
the virtual taper is given by the
convolution of the taper function with the transmitter weights
in e. Assuming that vector e has elements all equal to one and that
transmitters distance is dT=NRXdR, the resulting two-way TX-RX
antenna pattern is shown in Figure 6. As is apparent, it has the
desired narrow beam width, but the presence of the taper function
imposes an amplitude modulation, that provides higher side lobes
than desired.
Figure 6: Two way antenna pattern for the simplified
receiver
Therefore, using the proposed simplified MISO system, the main
problem remains to control the two-way antenna pattern so to have
low side lobes. We show that this is possible by appropriately
reducing the distance (shift) between adjacent transmitters, which
changes the result of the convolution. Obviously, such PSL
improvement is achieved at the expense of the beam width, that
becomes wider than for the uniform virtual array case. A heuristic
solution can be found by solving the minimization problem:
RXRXTX
RXRXTXshif
optRXTX
PSLPSLthatso
shiftPSLBWBW
=
:
]});([{min 0w ( 5 )
It consists in the search of the optimal shift that provides an
antenna pattern with the narrowest TX-RX beam width, fulfilling the
constraint that the PSLTX-RX is greater than times the PSLRX value
provided by the receive tapering function w0.
In Section IV we show that appreciable results can be
obtained for =1 using only NTX=2 transmitters. Moreover, in
Section V, the use of NTX=3 transmitters and a non-uniform vector e
is introduced to provide extra degrees of freedom for the control
the two way pattern. In other words, we start by setting the taper
function w0 for the desired RX-only (i.e. jammer) pattern and then
we search for optimal TX positions and taper scaling (namely
vectors and e), so that the TX-RX pattern has the desired
properties in terms of width of the TX-RX pattern (-3dB BW) and
PSL.
As apparent, using the scheme in Figure 5, we sacrifice
(NTX-1)xNRX degrees of freedom of a full MIMO scheme, to achieve
the desired patterns sidelobe control.
-
IV. SIDELOBE CONTROL WITH NTX=2 TRANSMITTERS
In this section, we consider the cases of only two transmitters
as: A. omnidirectional transmitters obtained using a single
radiating element and B. transmitters obtained usimg a subarray of
M>1 elements.
A. Heuristic approach for NRX=31, NTX=2 and M=1. If only two
transmitters (NTX=2) at the edge of the array
are used without any scaling (i.e. uniform vector e) we have a
simple but explicative case.
We approached the problem heuristically, trying different tapers
and analyzing the corresponding performance. In this section the
analysis is focused on the case of M=1 and then the generalization
for higher values of the number of elements used to transmit is
made in Section B. The virtual array taper, obtained from the
convolution of uniform TX array and tapered RX array, is used to
interpret the results. As apparent, increasing the shift of the
single transmitter from the edge of the array we obtain better PSL
but the resulting shorter virtual array implies a wider beamwidth.
Therefore, the smallest shift is usually selected that satisfies
our PSL requirements.
Uniform taper is the simplest solution and its performance is
well known in literature. As already mentioned, we have PSLRX=-13.2
dB for jammer and PSLTX-RX =-14.1 dB for target. With the array of
NRX=31 uniformly spaced elements, which is the reference for this
paper, we have that the half power width for jammer is RX-3dB=3.25
while for target it is TX-RX -3dB =1.75 that means a resolution
improvement of a factor 0.53 thanks to the virtual array gain.
Figure 7 compares the heuristic optimization results obtained
with Hamming and Taylor taper functions as the transmitters shift
varies. We note that note that increasing the shift the virtual
array gain decreases. However, it is apparent that Taylor taper
functions allow us to obtain better -3dB values than Hamming, even
though the latter provides better PSL at the small transmitters
shifts.
Figure 7: Comparison between Hamming and Taylor tapering
function
performances. Error! Reference source not found. shows the
numerical
values of the parameters obtained via the heuristic
optimization. We can observe that the Taylor taper functions
family provides the best performance in terms of RX and TXRX
beamwidths, while fulfilling the requirements of PSLRX and PSLTXRX
below assigned values.
TABLE 1.
Tapering
ShiftBeamwidth
(RX) []
Beamwidth (TXRX)
[]
SNR loss [dB]
PSL (RX) [dB]
PSL (TXRX)
[dB] Hamming 8 4.91 3.20 1.45 -42.05 -22.6 Hamming 9 4.91 3.48
1.45 -42.05 -29 Hamming 10 4.91 3.79 1.45 -42.05 -39.5
Taylor 25dB 7 3.90 2.69 0.43 -25.17 -26.07 Taylor 30dB 8 4.15
2.97 0.69 -30.1 -31.3 Taylor 35dB 10 4.37 3.28 0.92 -35 -35
An example is shown in Figure 8 where a 30 dB PSL is
obtained with a shift= 8 elements and a narrowed TX-RX
beamwidth, equal to 0.69 times the RX-only pattern.
a)
b)
c)
Figure 8: 30 dB Taylor tapering functions a) virtual tapering,
b) one way and two way patterns and their zoom in c).
B. Heuristic approach for the full exploitation of all the
elements: NRX=31, NTX=2 and M=15.
An interesting modification, especially interesting for active
phased arrays is related to the use of subarrays to transmit the
assigned waveform, instead of single elements. In this way, the
transmit pattern shape changes from omnidirectional, thus allowing
to concentrate the radiated power around the direction of interest.
This has also the effect to reduce the TXRX side lobe level in
angular directions far away from broadside, thus simplifying the
task of the receive taper function. In facts, the use of a Taylors
function, together with M>1 elements for each transmitter,
allows to reach the same PSL values with lower shift values (see
Figure 9 a and b), so that a narrower TX-RX beamwidth is obtained
than using M=1.
-
a) b) Figure 9: TX, RX and TX-RX patterns obtained with M=1 and
shift=8;
and M=15 with shift=7.
V. SIDELOBE CONTROL WITH NTX=3 TRANSMITTERS
The use of a higher number of transmitters NTX>2 increases
the degrees of freedom of the MIMO array, and potentially also the
length of the global virtual array taper function. In turn, this is
expected to improve the TXRX beamwidth. Since this also implies an
increase of the systems complexity, we consider only the case of
NTX=3.
By recalling from the previous section that the best compromise
between beamwidth and PSL is obtained with Taylors taper functions,
from now on we concentrate on these functions and compare the cases
of NTX=2 and NTX=3. One of the expected advantages of using three
transmitters is the possibility to reduce the shift needed to reach
high PSL values, thus exploiting more virtual arrays elements to
synthesize the two way pattern.
This is shown in Figure 10 (blue, black and green curves) in
term of the best achievable beamwidth as a function of the PSLTXRX,
assuming that the transmitters cannot be located outside the length
of the RX array (namely transmitters obtained from the same
physical array of the receiving aperture). As apparent comparing
the blue and black curves in Figure 10, when the the vector e is
uniform the beamwidth improvement obtained using three transmitters
instead of two is not significant.
Figure 10: -3dB Resolution with three transmitters and Taylor
tapering in
different situations.
To improve the performance, it is possible to consider to allow
non-uniform power emission from the different transmitters. An
example is obtained by transmitting from the central transmitter a
higher power level than from the two external ones. This is easy to
obtain in a passive phased array by using an appropriate three-way
power splitter device.
As shown in Figure 10 green curve, which consider the case of
transmitters positions constrained inside the physical RX antenna
size, the achieved beamwidth is sensibly narrower than in the case
of NTX=2 for high PSL values. On the other side if the PSL level
are under 25dB we have a slightly worse resolution w.r.t. the 2TX
and 3TX without any scaling. This is caused by the constraint of
having all transmitters within the array that does not allow to
increase their distance enough to exploit in the best way the
virtual array.
To explore the case of largely spaced transmitters, we focus at
the study case described in Figure 11 (a) where dT=NRX dR, and
consequently two of the three transmitters are within the physical
array while the third is outside. The shifts refer to the distance
of the phase centers of the external transmitters from the edge of
the array, while the central element is fixed in the last physical
element. Using a uniform power level at the three transmitters
(uniform e), the considerable distance among transmitters
independent of the shifts value, implies an amplitude modulation of
the virtual array elements that prevent the possibility to reach
PSL higher than 20dB.
(a)
Figure 11: (a) Sketch of the transmitters position By using the
non uniform transmitters power, having the
central TX twice the amplitude of the external ones, we obtain
the best performance in terms of resolution as shown by the red
dashed curve in Figure 10. However is clear that the constraint of
having at least one TX outside the array, and then a lower bound
for the distance between transmitters make difficult to
accomplished PSL higher than 30dB.
Even more interesting is the possibility to obtain a -3dB BW
performances increment exploiting at the same time both the
previous enhancements (i.e. more than two transmitters using M
elements each). In fact in this case we have the possibility of
cumulating the advantages of the cases M>1 together with the
advantages of the case NTX>2. The former has been shown to
improve beamwidth, while the latter (despite the limited beamwidth
improvement of the uniform case) is quite important in active
phased arrays to avoid a sensible power loss.
-
If a uniform power distribution is used (uniform vector e), and
we maintain all the transmitters within the physical array length,
it is always possible to guarantee the desired PSL from 25dB to
35dB. It is worth to note that increasing number of transmitters
the maximum number of elements M available for each transmitter is
reduced. In particular, for NTX=3, for NRX=31, and considering
odd-sized sub-arrays, the maximum value is M=9. This is an
interesting case for practical application with an assigned antenna
aperture.
Different situation is obtained if we remove the constraint
of having all the TX within the physical array. As before,
introducing a the scaling among tapering function amplitude the
distance between the transmitter phase centers to obtain an
assigned PSL is bigger than before.
Figure 12 compares the best TXRX beamwidths
achievable with two and three transmitters, using the best
possible value of M, namely the narrowest possible transmit beam.
Obviously for the blue, black, and red curve referring to the case
of transmitters sub-array completely internal to the physical
receive array, the maximum size is bounded by the transmitters
displacement and the assumption that an individual radiating
element can be used only by a single transmit sub-array.
Figure 12: -3dB BW in different cases using the best value for
M.
As apparent, also using transmit subarrays (M>1) instead of
single elements sub-arrays, the TXRX beamwidth improvement
achievable using NTX=3 instead of NTX=2 transmitters is negligible
in the case of uniform transmit power distribution (see green and
blue curves in Fig. 12). Some improvement is instead achievable
using NTX=3, when considering a non-uniform power distribution with
a higher power level transmitted by the central element. Assuming
that the transmit elements are all inside the receive aperture,
this improvement is apparently available only for the high PSL
value (namely PSL =30 dB abd PSL=35 dB). In contrast, a
significant improvement is found in the situation in which we
synthesize the longer virtual array (i.e. with one TX placed
outside the physical array) with the enhancement of using
incremented of M values.
VI. CONCLUSIONS
In this work we have presented a constrained MIMO
radar scheme, which is available to provide at the same time a
high PSL for both the two-way TXRX antenna pattern seen by the
target echo and the one-way RX antenna pattern seen by the jammer.
While the proposed scheme has a reduced flexibility, for an
appropriate design of transmitters distance and size, it has been
shown to be effective in guaranteeing the desired PSL vales.
After a heuristic selection of an appropriate receive taper
function, a design procedure has been proposed that allows us to
select the best parameters for the MIMO array, namely shift and M,
which are directly related to the sub-array distance and size
respectively.
By using only NTX=2 transmitters, a two-way TXRX
antenna pattern beamwidth has been obtained sensibly narrower
than the one-way RX antenna pattern, namely with beamwidth equal to
0.69 the RX beamwidth. This shows that the proposed MIMO scheme can
be used to improve the performance of a radar working with an
assigned antenna aperture and exploiting wide transmitting
beams.
The improvement available by increasing to NTX=3 the
number of the transmitters has been shown to be negligible if
they transmit the same power level. Under the constraint of a high
PSL (30 to 35 dB) an improvement is instead available when the
central transmitter emits a higher power level, even under the
hypothesis that the trasmitters are fully included inside the
receive array physical dimension. When this hypothesis is removed,
quite narrower beamwidths can be achieve for any value of the
PSL.
REFERENCES [1] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D.
Chizhik, R.
Valenzuela, MIMO RADAR: an idea whose time has come, New Jersey
Institute of Technology, Leihigh University, University of
Delaware, Bell Labs, IEEE Radar Conference, pp 71-78, April
2004.
[2] A. M. Haimovich, R. S. Blum, L. J. Cimini, MIMO Radar with
widely separated antennas, IEEE signal processing magazine, vol.
25, pp 116-129, January 2008.
[3] J. Li, P. Stoica, MIMO Radar with colocated antennas, IEEE
signal processing magazine, vol. 24, pp 106-114, September
2007.
[4] C. Yang, Signal Processing Algorithms for MIMO Radar,
California Institute of Technology, June 2009.
/ColorImageDict > /JPEG2000ColorACSImageDict >
/JPEG2000ColorImageDict > /AntiAliasGrayImages false
/CropGrayImages true /GrayImageMinResolution 200
/GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true
/GrayImageDownsampleType /Bicubic /GrayImageResolution 300
/GrayImageDepth -1 /GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true
/GrayImageFilter /DCTEncode /AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict >
/GrayImageDict > /JPEG2000GrayACSImageDict >
/JPEG2000GrayImageDict > /AntiAliasMonoImages false
/CropMonoImages true /MonoImageMinResolution 400
/MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true
/MonoImageDownsampleType /Bicubic /MonoImageResolution 600
/MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167
/EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode
/MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None
] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000
0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ]
/PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier ()
/PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped
/False
/CreateJDFFile false /Description > /Namespace [ (Adobe)
(Common) (1.0) ] /OtherNamespaces [ > /FormElements false
/GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks
false /IncludeInteractive false /IncludeLayers false
/IncludeProfiles true /MultimediaHandling /UseObjectSettings
/Namespace [ (Adobe) (CreativeSuite) (2.0) ]
/PDFXOutputIntentProfileSelector /NA /PreserveEditing false
/UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling
/UseDocumentProfile /UseDocumentBleed false >> ]>>
setdistillerparams> setpagedevice
