VirtualAntennaArrayAnalysisfor MIMOSyntheticApertureRadars · 7/17/2011 · Figure 1: Geometry mode of the MIMO SAR antennas. antennas can form a new virtual antenna array steering

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2012, Article ID 587276, 10 pagesdoi:10.1155/2012/587276

Research Article

Virtual Antenna Array Analysis forMIMO Synthetic Aperture Radars

Wen-Qin Wang

School of Communication and Information Engineering, University of Electronic Science and Technology of China (UESTC),Chengdu 611731, China

Correspondence should be addressed to Wen-Qin Wang, [email protected]

Received 17 July 2011; Revised 9 October 2011; Accepted 18 October 2011

Academic Editor: Wenhua Chen

Copyright © 2012 Wen-Qin Wang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Multiple-input multiple-output (MIMO) synthetic aperture radar (SAR) that employs multiple antennas to transmit orthogonalwaveforms and multiple antennas to receive radar echoes is a recently proposed remote sensing concept. It has been shown thatMIMO SAR can be used to improve remote sensing system performance. Most of the MIMO SAR research so far focused onsignal/data models and corresponding signal processing algorithm. Little work related to MIMO SAR antenna analysis can befound. One of the main advantages of MIMO SAR is that the degrees of freedom can be greatly increased by the concept of virtualantenna array. In this paper, we analyze the virtual antenna array for MIMO SAR high-resolution wide-swath remote sensingapplications. The one-dimensional uniform and nonuniform linear antenna arrays are investigated and their application potentialsin high-resolution wide-swath remote sensing are introduced. The impacts of nonuniform spatial sampling in the virtual antennaarray are analyzed, along with a multichannel filtering-based reconstruction algorithm. Conceptual system and discussions areprovided. It is shown that high operation flexibility and reconfigurability can be obtained by utilizing the virtual antenna arraysprovided by the MIMO SAR systems, thus enabling a satisfactory remote sensing performance.

1. Introduction

Multiple-input multiple-output radar has received muchattention in recent years [1–3]; however, little work aboutMIMO synthetic aperture radar (SAR) has been investigated[4–6]. Note that the MIMO SAR discussed in this paper isdifferent from the general MIMO radars in that aperturesynthesis is employed in the MIMO SAR, but no aperturesynthesis is employed in general MIMO radars [7]. AlthoughSAR is a well-proven remote sensing application whichobtains its high range resolution by utilizing the transmit-ted wide-band waveform and high azimuth resolution byexploiting the relative motion between the imaged targetand the radar platform, current single-antenna SARs cannotprovide some specific remote sensing performance, forexample, simultaneously high-resolution and wide-swath(the width of the ground area covered by the radar beam)imaging [8, 9]. MIMO SAR provides a solution to resolvingthese problems.

MIMO ideas are not new, their origin in control systemscan be traced back to 1970s [10]. The early 1990s saw anemergence of MIMO ideas into the field of communicationsystems. More recently, the ideas of MIMO appears in sensorand radar systems. Given that MIMO SAR is in its infancy,there is no clear definition of what it is. It is generallyassumed that independent signals are transmitted throughdifferent antennas, and these signals, after propagatingthrough the environment, are received by multiple antennas.Unlike conventional phased array radars [11], in MIMOSARs each antenna transmits a unique waveform, orthogonalto the waveforms transmitted by other antennas. In theMIMO SAR receiver, a matched filter-bank is used toextract the orthogonal waveform components. When theorthogonal signals are transmitted from different antennas,the returns of each orthogonal signal will carry independentinformation about the remote sensing targets. The phasedifference caused by different transmitting antennas alongwith the phase differences caused by different receiving

2 International Journal of Antennas and Propagation

Vs

Tx/Rx

(a) single-phase centre multibeam

Vs

Tx Rx

(b) multiple-phase centre multibeam

Figure 1: Geometry mode of the MIMO SAR antennas.

antennas can form a new virtual antenna array steeringvector. With optimally designed antenna array positions,we can create a very long array steering vector with asmall number of antennas [12–14]. More importantly, thisprovides high flexibility and reconfigurability in antennaconfiguration, thus enabling a flexible and reconfigurableSAR remote sensing performance.

Most of the MIMO SAR research so far focused on signal/data models and corresponding signal processing algorithm[15–17]. Little work related to MIMO radar antenna can befound. The antenna effects on a monostatic MIMO radarfor direction estimation were studied in [18] by analyzingthe Cramer-Rao low bound (CRLB). Two different uniformlinear antenna arrays, one narrowband and the anotherwideband, were investigated by exploring the CRLB. Aniterative algorithm was proposed in [19] to design sparseMIMO radar transmit arrays to approximate a desired trans-mit beampattern response. Additionally, several minimumredundancy MIMO radars were proposed by other authors[14, 20–22]. In fact, one of the main advantages of MIMOSAR is that the degrees of freedom can be greatly increasedby the concept of virtual array provided by the multipleantennas. In this paper, we analyze the virtual antenna arraydesign for MIMO SAR high-resolution wide-swath remotesensing, which has not been investigated in the literature.

The remaining sections are organized as follows. Thesystem principle of the MIMO SAR is described in Sec-tion 2. The one-dimensional uniform and nonuniform linearantenna arrays are designed in Section 3. Their applicationpotentials in high-resolution wide-swath remote sensingare also introduced. Next, Section 4 analyzes the impactsof nonuniform spatial sampling in the virtual antennaarrays. Finally, conceptual design system and discussions areprovided in Section 5. This paper is concluded in Section 6.

2. MIMO SAR Virtual Antenna Array

There are two kinds of MIMO SAR configuration, as shownin Figure 1. The operation mode of the single phase cen-tre multibeam (SPCM) MIMO SAR system is shown inFigure 1(a). A distinct channel is associated with each ofthe receive beams, and, hence, the data are split accordingto azimuth angular position or, equivalently, instantaneous

Doppler frequency centre in the azimuth direction. As aresult, given knowledge of the relative squint angles of eachbeam (hence the Doppler center frequency for each beam)and assuming suitable isolation between the beams, eachchannel can be sampled at a Nyquist rate appropriate to thebandwidth covered by each narrow beam, instead of thatcovered by the full beamwidth. This arrangement enablescorrect sampling of the azimuth spectrum with a pulserepetition frequency (PRF) fitting the total antenna azimuthlength, which is significantly smaller than the general PRFrequirement.

The multiple phase centre multibeam (MPCM) MIMOSAR system also synthesizes multiple receive beams in theazimuth direction, as shown in Figure 1(b); however, theoperating mode of this system is quite different from that ofthe previous one. In this case, the system transmits multiplebroad beams and receives the radar returns in multiplebeams which are displaced in the along-track direction.The motivation is that multiple independent sets of targetreturns are obtained for each transmitted pulse if the distancebetween phase centres is suitably set. This method basicallyimplies that we may broaden the azimuth beam from thediffraction-limited width, giving rise to improved resolution,without having to increase the system operating PRF.

As noted previously, one of the main advantages ofMIMO SAR is that the degrees of freedom can be greatlyincreased by the concept of virtual array [23]. Figure 2illustrates a MIMO SAR system. Consider the MIMO SARsystem with a transmit array equipped with M colocatedantennas and a receive array equipped with N colocatedantennas. Suppose both the transmit and receive arrays areclose to each other in space (possibly the same array) so thatthey see targets at same directions.

The MIMO SAR received signal at each receiving antennais the weighted summation of all the transmitted waveform

rn(t) =M∑

m=1

an, msm(t), m ∈ [1, 2, . . . ,M],

n ∈ [1, 2, . . . ,N],

(1)

where rn(t) is the received signal at the nth antenna, sm(t) isthe transmitted waveform at the mth antenna, and an,m is thechannel coefficient with the mth antenna as input and the

International Journal of Antennas and Propagation 3

dt

s1 s2

M

sM

drN

MF MF MF MF

1 21 2 3

Transmitter: antennas Receiver: antennasM N

Figure 2: Illustration of an example MIMO SAR system.

nth antenna as output. When the transmitted waveforms aredesigned to be orthogonal

∫sm(t)s∗m′(t)dt =

{δ(t), m = m′,0, m /=m′,

(2)

where ()∗ denotes a conjugate operator. At each receivingantenna, these orthogonal waveforms can then be extractedby M matched filters. There are a total of M × N extractedsignals. Compared to the traditional phased-array SARwhere the same waveform is used at all the transmittingantennas and a total of N coefficients are obtained for thematched filtering, the MIMO SAR gives more coefficientsand, therefore, provides more degrees of freedom.

Suppose there are K point targets, the received MIMOSAR signals can be written in a vector form

x(t) =K∑

k=1

σk(

aT(θk)s(t))

b(θk) + n(t), (3)

where θk is the target direction, σk is the complex-valuedreflection coefficient of the focal point θk for the kth pointtarget, ()T is a transpose operator, n(t) is the noise vector,a(θk) and b(θk) are the actual transmit and actual receivesteering vectors associated with the direction θk. Without lossof generality, we ignore the noise in the following discussions.The SAR returns due to the mth transmitted waveform canbe extracted by matched filtering the received signal to eachof the waveforms sm(t)

xm =∫

x(t)s∗(t)dt. (4)

The MN × 1 virtual target signal vector can then be writtenas

y = σsa(θs)⊗ b(θs), (5)

where ⊗ and θs denote the Kronker product and the targetdirection, respectively. Note that here perfect waveformorthogonality is assumed. This equation can be representedby

y = σsv(θs), (6)

where

v(θs) = a(θs)⊗ b(θs) (7)

is the MN ×1 steering vector associated with an virtual arrayof MN sensors.

Suppose the transmitter has M antennas, whereas thereceiver has N antennas, (7) means that a virtual antennaarray with utmost number of MN nonoverlapped virtualtransmitting/receiving elements can be obtained to take fulladvantages of the MIMO antenna array. Since differentantenna array configurations have different spatial sam-pling characteristics and signal processing complexity, theMIMO SAR antenna array configuration should be optimallydesigned.

3. Linear Virtual Antenna Array

3.1. Signal Models. Consider a linear transmitting array withM antenna elements and a linear receiving array with Nantenna elements. Without loss of any generality, supposethe transmitting and the receiving arrays are parallel andcolocated. The mth transmitting antenna is located at xT ,m =(λ/2)um and the nth receiving antenna is located at xR,n =(λ/2)vn, where λ is the wavelength. Consider a far-field pointtarget, the transmitter and receiver steering vectors can berepresented, respectively, by

a(θs) =[e ju1π sin θs , e ju2π sin θs , . . . , e juMπ sin θs

]T,

b(θs) =[e jv1π sin θs , e jv2π sin θs , . . . , e jvNπ sin θs

]T.

(8)

From (7), we can get

v(θs) =

⎡⎢⎢⎢⎢⎣

e j(v1+u1)π sin θs e j(v1+u2)π sin θs · · · e j(v1+uM)π sin θs

e j(v2+u1)π sin θs e j(v2+u2)π sin θs · · · e j(v2+uM)π sin θs

...... · · · ...

e j(vN+u1)π sin θs e j(vN+u2)π sin θs · · · e j(vN+uM)π sin θs

⎤⎥⎥⎥⎥⎦.

(9)

Note that the amplitude of the signal reflected by the targethas been normalized to unity. That is, the target response inthemth matched filtering output of the nth receiving antennais expressed as

vm,n(θs) = e j(vn+um)π sin θs . (10)


1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

Transmitting antennasReceiving antennas

(a) transmitting and receiving antennas

Equivalent antennas

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

(b) equivalent virtual antennas

Figure 3: Virtual phase centres of the uniform linear array.

It can be noticed that the phase differences are createdby both the transmitting antenna locations and the receivingantenna locations. The target response expressed in (11) isthe same as the target response received by a receiving arraywith MN antenna elements located at

{xT ,m + xR,m

}, m ∈ [1, 2, . . . ,M], n ∈ [1, 2, . . . ,N].

(11)

The phase differences are created by both transmitting andreceiving antenna locations. This MN-element array is justthe virtual antenna array. An utmost number ofMN-elementvirtual array can be obtained by using only M + N physicalantenna elements. It is as if we have a receiving array of MNelements. The virtual antenna array can be seen as a way tosample the electromagnetic wave in the spatial domain. Thisdegree-of-freedom can greatly increase the design flexibilityof the MIMO SAR systems.

3.2. Effective Phase Centres. To investigate the effective phasecentre caused by the virtual antenna array, in this sectionwe consider several typical linear array configurations forMIMO SAR systems.

3.2.1. Transmitter Is Same to Receiver: M = N = L. If thetransmitting array and the receiving array are uniform lineararrays, we assume that the first element of a(θs) and b(θs),respectively, is the reference element. From (8) and (9), wehave

a(θs) = b(θs) =[

1, e jπ sin θs , e j2π sin θs , . . . , e j(M−1)π sin θs]T

.

(12)

2 4 6 8 10 12 14 16 18 20

0.5

1

1.5

2


(a) transmitting and receiving antennas

0 5 10 15 20

Equivalent antennas

0.5

1

1.5

2

(b) equivalent virtual antennas

Figure 4: Virtual phase centres of the nonuniform linear array.

The equation (11) can then be reexpressed as

v(θs) =

⎡⎢⎢⎢⎢⎣

1 e jπ sin θs · · · e j(L−1)π sin θs

e jπ sin θs e j2π sin θs · · · e jLπ sin θs

...... · · · ...

e j(L−1)π sin θs e jMπ sin θs · · · e j(2L−3)π sin θs

⎤⎥⎥⎥⎥⎦. (13)

In this case, the number of effective virtual phase centres is2L − 1 with the biggest virtual aperture of 2L − 2. SupposeM = N = 4, Figure 3 shows the corresponding virtual arrays.

If the transmitting array and the receiving array arenonuniform linear array, we can express the steering vectoras

a(θs) = b(θs) =[e ju1π sin θs , e ju2π sin θs , . . . , e juMπ sin θs

]T. (14)

In this case, (11) can then be reexpressed as

v(θs) =

⎡⎢⎢⎢⎢⎣

e j(2u1)π sin θs e j(u1+u2)π sin θs · · · e j(u1+uM)π sin θs

e j(u2+u1)π sin θs e j(u2+u2)π sin θs · · · e j(u2+uM)π sin θs

...... · · · ...

e j(uM+u1)π sin θs e j(uM+u2)π sin θs · · · e j(2uM)π sin θs

⎤⎥⎥⎥⎥⎦.

(15)

It can be proved that the utmost number of effective virtualphase centres is L(L+1)/2. Suppose alsoM = N = 4; Figure 4shows the corresponding virtual arrays.

3.2.2. Transmitter and Receiver Have No Overlapped Elements.Suppose M + N = L; the utmost number of effective virtualphase centres can be determined by Lv = N(L − N) ≤ L2/4.For M = 3,N = 4, Figure 5 shows two typical virtual arrays,one is uniform linear array and the other is nonuniformlinear array.


(a)

2 4 6 8 10 12

0.5

1

1.5

2

2 4 6 8 10 12

0.5

1

1.5

2

(b)


Equivalent virtual antennas

(A) Uniform transmitting and/or receiving array

0 5 10 15 20

0.5

1

1.5

2

0.5

1

1.5

2

(a)

2 4 6 8 10 12 14 16 18 20

(b)



(B) Nonuniform transmitting and/or receiving array

Figure 5: The azimuth-variant Doppler characteristics: Case A: using spaceborne transmitter, Case B: using airborne transmitter.

3.2.3. Transmitter and Receiver Have Overlapped Elements.Suppose the transmitter and receiver have Nov overlappedelements, the utmost number of effective virtual phase cen-tres is determined by

Lv ≤ Nov(Nov + 1)2

+ (M −Nov)(N −Nov). (16)

Suppose also M = 4, N = 3; Figure 6 shows the correspond-ing virtual arrays.

Comparing the three cases discussed above, we can con-cluded that the minimum redundant array is obtainedwhen the transmitting array and/or the receiving array arenonuniform linear array.

3.3. System Performance Analysis. Future SAR will be re-quired to produce high-resolution imagery over a widearea of surveillance. However, the minimum antenna areaconstraint makes it a contradiction to simultaneously obtainboth unambiguous high azimuth resolution and wide-swath.As well as consideration of antenna beam-width, the actualachievable resolution and swath for a SAR is subject to anumber of restrictions imposed by various operating factors.The details can be found in [24, 25]. A basic limitation is theminimum antenna area constraint, which can be representedby

Aa ≥ 4vsλRc tanη

c0, (17)

where vs is the velocity of SAR platform, Rc is the slant rangefrom radar to mid-swath, η is the incidence angle, and c0

0 5 10 15 20

0.5

1

1.5

2

0 5 10 15 20

0.5

1

1.5

2



Figure 6: Virtual phase centres of the nonuniform linear array.

is the speed of light. This requirement arises because theilluminated area of the ground must be restricted so that theradar does not receive ambiguous returns in range or/andDoppler. In this respect, a high operating PRF is desired forsuppressing azimuth ambiguity. But the magnitude of theoperating PRF is limited by the range ambiguity require-ment.


a a

b b

c c

Received Shifted

Combined

Figure 7: Azimuthal spectra synthesis for multichannel subsam-pling. Here three channels are assumed. It is for illustration only.

The attainment of wide-swath will become increasinglydifficult if higher spatial resolution is required, due to therequirement of increased PRF. The MIMO SAR can over-come the minimum antenna area. The virtual effective phasecentres enable correct sampling of the azimuth spectrumwith a PRF fitting the total antenna azimuth length, whichis Lv times smaller than the general PRF requirement.Correspondingly, the area of each antenna is restricted by

Aa ≥ 4vsλRc tanη

c0· 1Lv

. (18)

Clearly the minimum antenna area is Lv-times smaller thanthe respective area of a monostatic SAR. Thereafter, thedisplaced phase center antenna (DPCA) technique [26, 27]can be used to gain additional samples along the syntheticaperture which enables an efficient suppression of azimuthambiguities, that is, the multiple beams in azimuth allowfor the division of a broad Doppler spectrum into multiplenarrow-band subspectra with different Doppler centroids.A coherent combination of the subspectra will then yielda broad Doppler spectrum for high azimuth resolution, asshown in Figure 7 [28]. Thus this approach is especiallyattractive for high-resolution SAR imaging that uses a longantenna for unambiguous wide-swath remote sensing.

For a given range and azimuth antenna pattern, the PRFmust be selected such that the total ambiguity noise con-tribution is enough small relative to the signal. Alternately,given a PRF or range of PRFs, the antenna dimensionsshould be enough small such that the ambiguity-to-noiseratio specification is met. Thus, the MIMO SAR systemperformance can be evaluated by the azimuth ambiguity tosignal ratio (AASR), which is defined as [29]

AASR ≈∑∞

m=−∞, m /= 0

∫ 0.5Bd

−0.5BdG2

(f + m · PRF

)d f

∫ 0.5Bd

−0.5BdG2

(f)d f

, (19)

where Bd is the SAR correlator azimuth processing band-width, G( f ) is the equivalent azimuth transmit-receiveantenna pattern, and PRF is the value of PRF.

As an example, we consider only the MIMO SAR config-uration in which the transmitter is same to the receiver, forexample, the configuration illustrated in Figure 3. Supposethe transmitting antennas synchronize perfectly with the

receiving antennas, the kth antenna beam can be representedby

Gk(θ) = sin c2(πLas cos(k · θa)

λsin(θ − k · θa)

), (20)

where Las is the subantenna length, θ is the antenna beam-width in elevation, and θa is the antenna beam-width inazimuth. Note that here the central antenna element isassumed as the reference element. As the 3 dB beam-widthcan be approximately determined by

θ ≈ λ

2vsf , θa = ka

λ

Las(21)

with ka a given constant, we can get

AASRk(PRF)

=⎧⎨⎩

∞∑

m=−∞m /= 0

[∫ (i+0.5)Bds

(i−0.5)Bds

G2k

(f + m · PRF

)d f

+∑

j /= k

∫ (i+0.5)Bds

(i−0.5)Bds

Gk(f + m · PRF)Gj

×(f + m · PRF

)d f

]}

·⎧⎨⎩

∫ (i+0.5)Bds

(i−0.5)Bds

G2k

(f)d f +

∑

j /= k

∫ (i+0.5)Bds

(i−0.5)Bds

Gk(f)Gj

(f)d f

⎫⎬⎭

−1

,

(22)

where Bds is the Doppler bandwidth of each subantenna.

4. Impacts of Nonuniform Spatial Sampling

As investigated previously, different array configurationshave different spatial sampling characteristics and signalprocessing complexity. A uniform array is desired, so thatthe complexity of signal processing can be reduced, and theultimate MIMO SAR image quality can be ensured. To reachthis aim, the optimum PRF must be satisfactory with

PRFuni = 2vsLvda

, (23)

where da is azimuth separation between the virtual arrayelements. This imposes a stringent requirement on thesystem as it states that to ensure equal spacing between allsamples in azimuth the PRF has to be chosen such thatthe SAR platform moves just one half of its antenna lengthbetween subsequent radar pulses. This optimum PRF yieldsa data array equivalent to that of a single-aperture systemoperating with Lv · PRF. In reverse, any deviation fromthe relation will result in a nonequally sampled data arrayalong the synthetic aperture that is no longer equivalent to amonostatic signal and cannot be processed by conventionalmonostatic algorithms without performance degradation.

To analyze the impact of nonuniform displaced phasecentre sampling, we consider the received radar returns

si(t, τ) ≈ σi[h0(t)∗th1,i(t, τ)

], i = 1, 2, . . . ,N , (24)


where t is the range fast time, τ is the azimuth slow time,and ∗t is a convolution operator on the variable t. h0(t) andh1,i(t, τ) denote, respectively, the range reference functionand azimuth reference function

h0(t) = wr(t) · exp(− jπkrt

2), (25)

h1,i(t, τ) = exp{− j

2πλ

[Rc(τ) + Rc

(τ + i

davs

)]}

×wa(τ) · δ[τ − Rc(τ) + Rc(τ + ida/vs)

c0

],

(26)

where Rc(t) is the equivalent slant range and kr is thechirp rate of the transmitted waveforms. Note that perfectorthogonal frequency diversion multiplexing (OFDM) lin-early frequency modulation (LFM) waveforms are assumedin this paper. wr(t) and wa(τ) denote the antenna pattern inrange dimension and azimuth dimension, respectively. Since

Rc(τ) + Rc

(τ + i

davs

)≈ 2Rc

(τ + i

da2vs

), (27)

we then have

si(t, τ) ≈ σi

[h0(t)⊗th1

(t, τ + i

da2vs

)]. (28)

with

h1

(t, τ + i

da2vs

)= wa

(τ + i

da2vs

)· exp

{− j

4πλRc

(τ + i

da2vs

)}

·[τ − 2Rc(τ + i(da/2vs))

c0

].

(29)

Equivalently, the nonuniform PRF can be considered as azi-muth time drift

τer = da2vs

− 1Lv · PRF

. (30)

After matched filtering and range mitigation correction,we can get

si

(k

1Lv · PRF

)

= wa

(k

1Lv · PRF

+ i · τer)

× exp{− j

[2π fd

(1

Lv · PRF+ i · τer

)

+πka

(k

1Lv · PRF

+ i · τer)2

]},

(31)

where k is an integer, fd is the Doppler frequency centroid,and ka is the Doppler chirp rate. It is noticed that thesignals are periodic nonuniform with the period of 1/PRF.This information is particularly importantly for developingnonuniform reconstruction algorithms. The impacts of

sm

m

Lv · PRF

Channel 1

Channel 2

Channel

FFT

FFT

FFT

IFFT

z1[k]

z2[k]

zK [k]

P1( f )

P2( f )

PK ( f )

K

Figure 8: Reconstruction filtering for multichannel subsampling incase of three channels.

nonuniform spectral sampling can be evaluated by the fol-lowing expression [8]:

si(τ) = wa(τ) exp(− jπkaτ

2)

×{J0(2π fdτerbn

)

− j2πkdτerbnJ0(2π fdτerbn

)τ sin

(2πτ

Lv · PRF

)

−2 jJ1(2π fdτerbn

)sin

(2πτ

Lv · PRF

)}.

(32)

The ambiguous Doppler spectrum of a nonuniformlysampled SAR signal can be recovered unambiguously byapplying a system of reconstruction filters. The algorithmillustrated in Figure 8 is based considering the data acqui-sition in the MIMO SAR as a linear system with multiplereceiver channels, each is described by a linear filter. Thereconstruction consists essentially of multiple linear filterswhich are individually applied to the subsampled signals ofthe receiver channels and then combined coherently. Thedetails can be found in [30, 31].

Therefore, the optimal MIMO SAR configuration shouldhave a uniform virtual linear array along the azimuthdimension. Consider a MIMO SAR with an M-element p-spaced transmitting uniform linear array and an N-elementq-spaced receiving uniform linear array. According to theproposition discussed in [32]. The virtual array is an effectiveuniform linear array if and only if 1 ≤ γ ≤ N with the ratiocoefficient γ = p/q or 1 ≤ γ0 ≤ M with γ0 = 1/γ. Moreover,the virtual array is a nonoverlapped MN-element uniformlinear array if and only if γ = N or γ = 1/M.

5. Conceptual System Design and Discussions

To further evaluate the quantitative performance, an exampleMIMO SAR system is considered. The MIMO SAR operatesin X-band with a center frequency of 10 GHz. The geometricground-range and azimuth resolution are set to ρr =0.2m and ρa = 0.2m, respectively. To calculate the systemperformance, an overall loss factor L f = 3 dB, a fixed flyingheight of 30 km, and a receiver noise figure of F = 3 dB areassumed. It is further assumed that the signal bandwidth isadjusted for varying angle of incidence such that the ground-range resolution is constant across the swath. One examplesystem design is provided in Table 1. We can notice that,for the incidence angle given in Table 1 a swath of 18 km


2000 4000 6000 8000 10000 12000 14000−60

−50

−40

−30

−20

−10

0

PRF (Hz)

AA

SR(d

B)

General single-antenna SARMIMO SAR with three colocated antennas

Figure 9: Comparative AASR results between conventional single-antenna SAR as a function of PRF.

dr

dt

TransmitterReceiverEquivalent centre

dt

dr

(b)

(a)

Figure 10: Two general linear antenna configurations for MIMOSAR systems.

and a noise equivalent sigma zero (NESZ) of −48 dB canbe obtained with a total antenna size not larger than that ofcurrent systems. Note that the NESZ is defined as the targetradar cross section for which the final SAR image SNR isequal to one (i.e., SNRimage = 0 dB).

It is also worthwhile to compare its AASR performance toconventional single-aperture SAR. Consider again the systemparameters listed in Table 1; Figure 9 gives the comparativeAASR performance as a function of PRF. In SAR remotesensing applications, AASR is typically specified to be onthe order of −20 dB, but a lower AASR is desired. It canbe noticed from Figure 9 that the AASR is typically below−20 dB with a low operating PRF requirement. This meansthat a wider swath can be obtained.

The equivalent virtual antenna pattern can be impactedby the different antenna configurations, even for the samenumber of equivalent virtual antenna array. Figure 10 shows

−1 −0.5 0 0.5 1−80

−70

−60

−50

−40

−30

−20

−10

0

Normalized frequency

Nor

mal

ized

mag

nit

ude

(dB

)

dt = dr = 1 mdt = dr /2 = 0.5 mdt = 2 · dr = 1 m

Figure 11: Equivalent antenna pattern of the MIMO SAR configu-rations.

Table 1: Performance parameters of an example MIMO SARsystem.

Parameters Variables Values

Mean transmit power Pavg 10 W

Number of transmitting antennas M 3

Number of receiving antennas N 3

Transmit/receive antenna length Las 0.9 m

Platform altitude hs 30 km

Platform velocity vs 500 m/s

Incidence angle η 30◦

Antenna width Ha 0.1 m

Swath width Ws 12 km

Radiometric resolution NESZ −51.16 dB



Swath width Ws 18 km




Swath width Ws 9 km


two general linear antenna configurations, where dt anddr denote the distance separation between two neighboringantennas for the transmitter and the receiver, respectively.Figure 11 shows the comparative equivalent antenna patternsfor the two antenna configurations. It can be noticed thatdifferent equivalent MIMO SAR antenna patterns can beobtained by choosing different array configurations. Thisprovides a potential to develop new GMTI or remote sensingapproaches.


Tx Rx

Equivalent

Figure 12: Example two-dimensional planar virtual array and three-dimensional cylindrical virtual array.

Another note is that, in this paper, we considered only theone-dimensional linear array. Two-dimensional and three-dimensional phantom element array can also be synthesizedmidway between each transmitter-receiver pair. Figure 12shows an example two-dimensional, planar array and athree-dimensional cylindrical array. Note that many otherforms of two-dimensional three-dimensional arrays can beobtained in a similar way.

6. Conclusion

MIMO SAR is a recently proposed remote sensing concept.It has been shown that MIMO SAR can be used toimprove remote sensing system performance. One of themain advantages of MIMO SAR is that the degrees of free-dom can be greatly increased by the concept of virtualantenna array. In this paper, we investigated the virtual linearantenna array for MIMO SAR high-resolution wide-swathremote sensing applications. The impacts of nonuniformspatial sampling in the virtual antenna array are analyzed,along with a multichannel filtering-based reconstructionalgorithm. The virtual high-dimensional antenna arrays arealso investigated. Conceptual design system is provided,along with the system performance. It is shown that highoperation flexibility and reconfigurability can be obtainedby utilizing the virtual antenna arrays provided by theMIMO SAR systems, thus enabling a satisfactory remotesensing performance. High-dimensional virtual antennaarrays including two-dimensional planar array and three-dimensional cylindrical array are also discussed, with an aimfor further investigations. In a subsequent work, we planto further investigate the MIMO SAR sparse antenna arraydesign and the corresponding signal processing algorithmsto resolve the spatial nonuniform sampling problems.

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no. 41101317,the Fundamental Research Funds for the Central Univer-sities under Grant no. ZYGX2010J001, the First Grade of49th Chinese Post-Doctor Research Funds under Grant no.20110490143, and the Open Funds of the State Laboratoryof Remote Sensing Science, Institute of Remote Sensing

Applications, Chinese Academy of Sciences under Grant no.OFSLRSS201011.

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VirtualAntennaArrayAnalysisfor MIMOSyntheticApertureRadars · 7/17/2011 · Figure 1: Geometry mode of the MIMO SAR antennas. antennas can form a new virtual antenna array steering