Research Article Correction of Channel Imbalance for MIMO...

Research ArticleCorrection of Channel Imbalance for MIMO SAR UsingStepped-Frequency Chirps

Xiulian Luo12 Yunkai Deng1 Robert Wang1 Lei Guo12 and Mingjiang Wang12

1 Institute of Electronics Chinese Academy of Sciences Beijing 100190 China2Graduate University of the Chinese Academy of Sciences Beijing 100080 China

Correspondence should be addressed to Xiulian Luo xiaoluo6070126com

Received 20 December 2013 Accepted 19 February 2014 Published 3 April 2014

Academic Editor Wei Xu

Copyright copy 2014 Xiulian Luo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To simultaneously achieve two-dimensional high resolution and wide swath in synthetic aperture radar (SAR) azimuth MIMOstructure combined with stepped-frequency chirp signals was developed via splitting the antenna into 119873 subapertures Duringtransmitting each subaperture transmits a chirp pulse at a different carrier frequency while during receiving every subaperturereceives the119873 scattered pulses at the same time Separating the119873 scattered pulses received by each subaperture and downlinkingthem to the ground yield 1198732 different signal paths Due to the dedicated network in the SAR system the channel imbalance isinevitable To correct the channel imbalance this paper presents an external calibration method where the channel characteristicsare estimated from the peak value of a strong point target for each channel Simulation and real raw data experiments are performedto validate the proposed method

1 Introduction

Resolution and swath width are two key specifications forthe spaceborne stripmap SAR However high azimuth res-olution and wide range swath imaging pose contradictingrequirement on conventional SAR system design In orderto overcome this inherent contradiction the displaced phasecenter antenna (DPCA) techniques [1ndash3] and the multiple-input and multiple-output (MIMO) antenna techniques [4ndash7] are introduced to acquire additional spatial sampled infor-mation On the other hand the range resolution determinedby the pulse bandwidth has to be increased to match theazimuth resolution However transmitting and receivingwideband signals burden the active electronically steeredarray (AESA) antenna and the echo sampled equipment[8 9] thus confining the range resolution Fortunatelya wideband chirp signal can be synthesized by a groupof narrowband subchirps centered at stepped frequencieswhich requires the synthetic bandwidth technique [8ndash10]

As the synthetic bandwidth technique will reduce thenumber of equivalent phase centers (EPCs) the additionalsampled information in DPCA SAR is actually used to

increase the range bandwidth other than the equivalent PRF[11] Therefore this paper focuses on the MIMO SAR usingstepped-frequency chirps (STFC) [4 5] where both theequivalent PRF and the range bandwidth can be 119873 timesincreased However as this MIMO system yields1198732 differentsignal paths the channel imbalance is inevitable For eachsubband the collected data is from a DPCA system wherethe channel imbalance can be regarded as the inner-bandimbalance which results in azimuth ambiguities [12] Theimbalance among different subbands can be regarded as theinterband imbalance which leads to degradation of the rangeperformance after synthetic bandwidth [13]

Therefore in order to acquire high quality images effec-tive correction of the channel imbalance is indispensableThe available correction methods can be classified as twocategories methods on internal calibration [14] and methodson raw data [15 16] However all these correction methodsonly either handled the inner-band imbalance in the DPCAsystemwith single subband [15 16] or concerned themultiplesubbands system with single subaperture [14] In [14] Denget al gave the internal calibration approach where the imbal-ance information is obtained from the internal calibration

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 161294 8 pageshttpdxdoiorg1011552014161294

2 International Journal of Antennas and Propagation

data acquired from the calibration subsystem This methodcan be extended to theMIMO system using STFC Neverthe-less 1198732 internal calibration subsystems are required whichdramatically increases the complexity of the SAR systemsIn addition it cannot correct imbalance introduced by theantennas The subspace projection method proposed in [16]and the azimuth cross-correlation method presented in [15]do not require additional subsystem However they cannotcorrect the interband imbalance as the phase differencecaused by different carrier frequencies is range dependentand hard to compensate

Usually the channel imbalance information (phase andamplitude) does not change within a certain illuminationtime [15] Moreover many artificial corner reflectors havebeen distributed here and there for radiometric calibration[17] Therefore this paper proposes an external calibrationmethod where the channel imbalance information is esti-mated from the peak value of a corner reflector or a strongpoint target in 119873

2 small complex images Afterwards theestimated channel errors are used to correct the channelimbalance when imaging for much larger data

Succeeding sections are organized as follows In Section 2the MIMO SAR using STFC is conceptually designed andthe corresponding image formation processing is brieflysummarized In Section 3 the external correction methodis presented followed by the simulation and real raw dataexperiments in Section 4 to validate the proposed methodFinally the conclusion is drawn in Section 5

2 MIMO SAR UsingStepped-Frequency Chirps

21 Conceptual Design The MIMO SAR simultaneouslytransmits a set of stepped-frequency chirps regarded assubbands via multiple subapertures During receiving everysubaperture receives all the 119873 subbands in one PRI Theschematic diagram and the corresponding timeline of trans-mitted pulses and received echoes are illustrated in Figure 1where the circles represent the location of the equivalentphase centers (EPCs) of the subbands and the number insidethe circles represents the sequence number of the subbands

In order to separate two frequency-adjacent subbandsand absent gaps from the connection of subbands at the sametime the subband bandwidth 119861

119904is required to be equal to

the frequency stepΔ119891The relationship of the time frequencyfor the transmitted pulses is illustrated in Figure 2 where119891

119898

(119898 = 1 2 119873) is the carrier frequency of the119898 th subbandThe baseband echo of the119898 th subband for each subaper-

ture is acquired as follows Demodulate the radio frequency(RF) pulses with the carrier frequency 119891

119898 Then filter the

demodulated signal with an analog low pass filter (LPF)whose bandwidth is greater than 119861

119904 Afterwards convert

the analog signal to a digital signal via an AD convertorwith a sampling rate greater than the cutoff frequency ofthe LPF As illustrated in Figure 3 due to the imperfectfiltering and the Gibbs phenomenon of the signal spectrumsome spectral component of one subband is leaked into theadjacent subband thus resulting in subband interference In

1

s

H

1 11 1 1

2 2 22 2 2

NN N

NN

N

middot middot middot

1 1

2 2

N N

TX1RX1

TX2RX2

TXNRXN

Figure 1 Schematic diagram of the MIMO SAR using STFC


f

fN

f2

f1

ta(n) ta(n + 1)

Ts

PRT PRT

Bs

Bw

sub1

sub2

subN

sub1

sub2

subN

t

Figure 2 Relationship of time frequency of transmitted pulses

order to reduce the interference a digital rectangular windowcan be applied to the sampled data before imaging Thenthe interference only results from the spectral componentoutside the spectrum bandwidth as illustrated by the blue-shaded region in Figure 3 As stated in [13] the influence ofthe subband interference under this situation can be ignoredbecause it results inminus47 dB ambiguities in range In fact whenthe subband bandwidth is larger the ambiguities in range willbe reduced further This is because the relative amplitude ofthe ambiguities is determined by relating the leakage energy

International Journal of Antennas and Propagation 3

LPF

0f2 minus f1

minusfs2

Rectangular window

f

Figure 3 Separation of two frequency-adjacent subbands

to the whole energy [13] Moreover the leakage energy isinversely proportional to the bandwidth-time product

Therefore in this paper for each subaperture adjacentsubbands are assumed to be completely separated that is thesubband interference is not considered

22 Image Formation Processing When the uniform sam-pling condition

1

PRF=119897119886119911

2V119904

sdot 119873 (1)

with PRF the system pulse repetition frequency 119897119886119911the length

of the subaperture and V119904the velocity of the sensor is not

satisfied multichannel reconstruction [1] is required Thereference position during the multichannel reconstructionshould be the center of the whole antenna to make theEPCs to be synthesized coincide spatially Afterwards thefrequency domain bandwidth synthesis method [9 10] isapplied to acquire a large bandwidth pulse followed by theconventional imaging It should be noted that the synthesizedsignal should be firstly converted to the original echo domainif the CS imaging algorithm is adopted The block diagramof the image formation processing for this MIMO SARsystem is presented in Figure 4 Assume that 119873 = 2 andthen the variation of the distribution of EPCs introducedby multichannel reconstruction and bandwidth synthesis isshown in Figure 5 (ignore the first row and the dashed circleswhich will be used in Section 4)

3 Correction of the Channel Imbalance

In this section the correction of both amplitude and phaseimbalance among channels will be presented As this correc-tion method is based on a focused strong point target in 1198732small complex images separating imaging for each subbandand each subaperture is required

The base-band echo for the119898th subband (transmitted bythe119898th subaperture) received by the 119899th subaperture is

119878119898119899

(120591 120578) = 119873119898119899

(120591 120578) + 119860119898119899

sdot exp (119895120593119898119899) sdot rect(

120591 minus 119877119898119899

(120578) 119888

119879119903

)

times expminus1198952120587119891119898sdot (

119877119898119899

(120578)

119888)

sdot exp119895120587119896119903(120591 minus

119877119898119899

(120578)

119888)

2

(2)

with 120591 the range time 120578 the azimuth time 119896119903the chirp rate 119888

the light velocity 119879119903the pulse duration 119860

119898119899the amplitude

characteristic 120593119898119899

the phase characteristic 119873119898119899(120591 120578) the

system noise and the echo of other targets and 119877119898119899(120578)

the instantaneous distance in propagation between thetransmitter and the receiver for the corner reflector (pointtarget) Applying the Taylor series expansion and ignoringthe quadratic phase termquadratic term error during theequivalence of the two-way slant ranges 119877

119898119899(120578) can be

approximated as

119877119898119899

(120578) = 119877 (120578 minus Δ120578119898119899) (3)

where

Δ120578119898119899

=(119898 + 119899 minus 119873 minus 1) 119897

119886119911

2V119904

(4)

is the azimuth time interval between the corresponding EPCand the center of the whole antenna and

119877 (120578) = 2radic1198772

0+ (V119904120578 minus 1199090)2

(5)

represents the two-way slant range from the center of thewhole antenna to the point target In (5) 119877

0and 119909

0indicate

the slant range and azimuth coordinates of the point targetrespectively

First the imaging for the 1198732 echoes is independentlyperformed Due to lower PRF than Nyquist sampling ratestrong azimuth ambiguities of the point target appear in the1198732 images However the ambiguities are far away from the

real point target thus scarcely impacting the estimation Theimaging result can be approximated as [18]

119878119898119899

(120591 120578) = 1198731015840

119898119899(120591 120578) + 119873

1015840

119898119899amb (120591 120578)

+ 119860119896sdot exp (119895120593

119896) sdot 119866 sdot 119901

119903(120591 minus

21198770

119888)

sdot exp minus1198952120587119891119898sdot (21198770

119888) sdot 119901

119886119898119899(120578 minus

1199090

V119904

)

+ 119860119896sdot exp (119895120593

119896) sdot 119901119903(120591 minus

21198770

119888)


Bandwidth synthesis

Multichannel reconstruction


Conventionalimaging

SAR image with high resolution




Subband 1Subaperture 1

Subband 1

Subaperture 1Subband N

Subband N

Subaperture N

Subaperture N

Figure 4 Block diagram of the image formation processing

1PRF1 2 1 2 1 2 1 2

1 1

1 1

2 2

2 2

11 11

2 2 2 2

1

Reference position

MIM

O ec

ho

cons

truc

ted

Mul

ticha

nnel

reco

nstr

uctio

nBa

ndw

idth

synt

hesis

t2 2

t2 1

t1 2 =laz2s

Figure 5 Distribution of the EPCs


119888)

times

119873minus1

sum

119894=minus119873+1 119894 = 0

119866amb119894

sdot 119901119886119898119899

(120578 minus1199090

V119904

+ 119894 sdotPRF119896119886

)

(6)

where 119896119886is the azimuth chirp rate 119866 and 119866amb119894 are the

gain resulting from the imaging of the ldquorealrdquo part and theambiguous part of the point target respectively 1198731015840

119898119899(120591 120578)

and 1198731015840119898119899amb(120591 120578) are the ldquorealrdquo and ambiguous response of

other targets and noise and 119901119886119898119899

and 119901119903are the amplitude

of the azimuth and range impulse responses two sinc-likefunctions [18] According to (3) 119901

119886119898119899(120578) can be rewritten as

119901119886119898119899

(120578) = 119901119886(120578 minus Δ120578

119898119899) (7)

where 119901119886is also a sinc-like function

Considering that the peak of the corner reflector isgenerally much greater than other terms at position (119877

0 1199090)

only the ldquorealrdquo response of the corner reflector is taken intoaccount when deriving the correction method which is

119878119898119899

(120591 120578) = 119860119898119899

sdot exp (119895120593119898119899) sdot 119866 sdot 119901

119903(120591 minus

21198770

119888)

times exp minus1198952120587119891119898sdot (21198770

119888)

sdot 119901119886(120578 minus

1199090

V119904

minus Δ120578119898119899)

(8)

Second the peak position and value of the corner reflectorare estimated from these1198732 small complex images In orderto make the estimation more accurate the 2D-interpolationover the peak point and its surrounding points is performedConsequently the peak position of the corner reflector isacquired to be 120591

119898119899asymp 21198770119888 The amplitude of the peak is

119860119898119899

asymp 119860119898119899

and the phase of the peak is

120601119887119898119899

asymp 120593119898119899

minus 2120587119891119898sdot 120591119898119899 (9)

In fact in order to reduce the influence of the noise 119860119896

and 120601119887119896

are obtained by averaging the values of the pointswithin the two-dimensional 1 dB main lobeThe first channel


(subband 1 and subaperture 1) is set as the reference channelThen the estimated amplitude error matrix is

AE = diag11986011 11986012 119860

1119873 119860

1198731 1198601198732 119860119873119873

11986011

(10)

Third compensate the inherent phase caused by the offsetof the carrier frequencies119891

119898from the center frequency of the

synthesized wideband signal 119891119888 After this the phase of the

peak value is changed as

120601119898119899

= 120601119887119898119899

+ 2120587Δ119891119898sdot 120591119898119899

asymp 120593119898119899

minus 2120587119891119888sdot (21198770

119888) (11)

where

Δ119891119898= (119898 minus

1 + 119873

2)Δ119891 119898 = 1 119873 (12)

Therefore the estimated phase error matrix can be expressedas

ΦE = (diag exp (11989512060111) exp (1198951206011119873)

exp (1198951206011198731) exp (119895120601

119873119873))

times (exp (11989512060111))minus1

(13)

Finally correct the channel imbalance according to (10)and (13) The correction matrix is

C = (AEΦE)minus1

(14)

Therefore the correction is implemented as

S119888= C sdot [119878

11(120591 120578) 119878

1119873(120591 120578)

1198781198731

(120591 120578) 119878119873119873

(120591 120578)]119879

(15)

It should be noted that the data size of 119878119898119899(120591 120578) in

(2) can be much smaller than that in (15) to improve theprocessing efficiency After correction the balanced data S

119888is

used to form the final image with high resolution The wholeprocessing is shown in Figure 6 where the image formationprocessing strategy is presented in Figure 4

4 Experimental Results

41 Simulation Experiment In the simulation the 2D imag-ing of a plane scene containing a letter ldquoArdquo and a 4-time (6 dB)brighter point target is performed and this point target isinside the letter ldquoArdquo The simulation parameters are specifiedin Table 1

Table 1 System parameters

Parameter Symbol ValueCenter frequency (GHz) 119891

1198889685

Slant range of scene center (km) 1198770

30Sensor velocity (ms) V

119904215

Subaperture length (m) 119897az 25System PRF (Hz) PRF 140Transmit pulse length (us) 119879

11990110

Number of subbandssubapertures 119873 2Transmitted pulse bandwidth (MHz) 119861

11990460

Sampling rate in range (MHz) 119865119903

72SNR (dB) SNR 6

The accurate amplitude and phase errors are given in the2nd and 3rd columns of Table 2 respectivelyThefirst channelis set as the reference channel According to the estimationmethod proposed in the last section the amplitude andphase errors are estimated and then presented in the 4thand 5th columns of Table 2 respectively The imaging resultswithout and with correction of the channel imbalance areshown in Figures 7(a) and 7(b) respectively The rangecomparison of the point target and its surroundings is shownin Figure 7(c) From Figure 7 one can observe that theambiguities in azimuth (caused by the inner-band imbalance)and the raising of the side-lobes in range (stemming fromthe interband imbalance) are removed by the correction ofchannel imbalance

From the simulation experiment it is concluded thatthe proposed correction (compensation) method can correctboth the inner-band and the interband imbalance for thesimulated MIMO SAR data

42 Real Raw Data Experiment For the real raw data theimaging scene is much more complicated than the simulateddata thus possibly affecting the estimation accuracy As thereal raw data collected by a MIMO SAR using STFC isnot available in this subsection the proposed correctionmethod is assessed with an X-band airborne SAR using asingle aperture and two STFCs which are transmitted in twoconsecutive PRIs In this SAR both the subband bandwidthand the frequency step are 30MHz the original PRF is1600Hz and the sensor velocity is 215ms The distributionof the original EPCs is illustrated by the first row in Figure 5

To validate the proposed correction method on the realraw data the echo data for the MIMO system have to befirstly constructed As discussed before in theMIMO systemusing STFCs the echo for each subband can be regardedfrom a DPCA system [1] Further in the DPCA system thedata of each channel is equivalent to the counterpart singlechannel data processed by an azimuth time delay functionand then 119873 times subsampling [2] Herein the counterpartsingle channel data is the original subband data and the delayfunction in Doppler domain for the119898th subband received bythe 119899th subaperture is set to

119867119898119899

(119891120578) = exp minus1198952120587119891

120578sdot 119905119898119899 (16)


Imaging

Imaging

Imaging

Imaging

Correction

Imageformation processing

Extract a small size

datacontaining a strong point

target

Calculatethe position

and peak value of the point target

Compensate the phase caused by different carrier

frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN




Figure 6 Block diagram of the channel imbalance correction

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40

minus50minus10 minus5

Range (m)

WithWithout

(c)

Figure 7 Simulation results for OFDM-MIMO SAR (a) Imaging result without correction of channel imbalance (b) Imaging result withcorrection of channel imbalance (c) Range comparison of the amplification of the point target in (a) and (b)

Table 2 Channel errors (CE)

Channel (119898 119899) Exact CE Estimated CE for simulated data Estimated CE for constructed raw data119860119896

120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)

minus10 minus5 0 5 10 15 20Range (m)

0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)

Figure 8 Real raw data results for OFDM-MIMO SAR (a) Without correction (b) With correction (c) Range comparison of theamplification of the corner reflector

with119891120578theDoppler frequency 119905

11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then

the distribution of the EPCs for the four channels can beillustrated by the 2nd 3rd 4th and 5th rows in Figure 5wherethe dashed circles represent the discarded pulses to realizea two-time subsampling With consideration of the channelerrors specified in the 2nd and 3rd columns of Table 2 thefour echo data of the MIMO system using STFCs are wellconstructed

According to the error estimation process presented inFigure 6 the measured channel errors are listed in the 6thand 7th columns of Table 2The final imaging results withoutand with correction of the channel imbalance are shown inFigures 8(a) and 8(b) respectively The corner reflector usedfor external calibration ismarked by ellipse in the two figuresAfter interpolation and amplification the range comparisonof the corner reflector in Figures 8(a) and 8(b) is shown inFigure 8(c) where one can see that the raising of the sidelobes is eliminated by the compensation Additionally in

Figure 8(a) the peak amplitude of the ambiguity marked bydashed circle is minus1589 dB while in Figure 8(b) the ambiguityis not distinguishable from the background

Therefore for the constructed MIMO raw data thepresented correctionmethod is also effective in removing thechannel imbalance

5 Conclusion

MIMO SAR using stepped frequency chirps is of greatpotential for future high-resolution wide-swath SAR mis-sions Channel imbalance is a key problem for this novelSAR mode as it relates to the imaging performance Thispaper proposed a simple and effective method to correctthis channel imbalance which was validated by simulationand real raw data experiments Other problems such as thesystem design and the orbit model are not discussed in thispaper Nevertheless they are also indispensable especiallywhen this novel mode is applied to spaceborne missions


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] NGebert GKrieger andMAMoreira ldquoDigital beamformingon receive techniques and optimization strategies for high-resolution wide-swath SAR imagingrdquo IEEE Transactions onAerospace and Electronic Systems vol 45 no 2 pp 564ndash5922009

[2] W Jing M Xing C-W Qiu Z Bao and T-S Yeo ldquoUnambigu-ous reconstruction and high-resolution imaging for multiple-channel SAR and Airborne experiment resultsrdquo IEEE Geo-science and Remote Sensing Letters vol 6 no 1 pp 102ndash1062009

[3] A Currie and M A Brown ldquoWide-swath SARrdquo IEE Proceed-ings Part F Radar and Signal Processing vol 139 no 2 pp 122ndash135 1992

[4] W-QWang ldquoSpace-time codingMIMO-OFDM SAR for high-resolution imagingrdquo IEEE Transactions on Geoscience andRemote Sensing vol 49 no 8 pp 3094ndash3104 2011

[5] Y Song and R Yang ldquoHigh resolution wide swath SAR usingsub-aperture sub-band techniquerdquo in Proceedings of the CIEInternational Conference on Radar (ICR rsquo06) Shanghai ChinaOctober 2006

[6] G Krieger ldquoMIMO-SAR opportunities and pitfallsrdquo IEEETransactions on Geoscience and Remote Sensing no 99 pp 1ndash18 2013

[7] J HG Ender and J Klare ldquoSystem architectures and algorithmsfor radar imaging by MIMO-SARrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo09) Pasadena Calif USA May2009

[8] A W Doerry SAR Processing with Stepped Chirps and PhasedArray Antennas Sandia Nat Lab Albuquerque NM USA2006

[9] P Berens ldquoSARwith ultra-high range resolution using syntheticbandwidthrdquo in Proceedings of the IEEE International Geoscienceand Remote Sensing Symposium (IGARSS rsquo99) pp 1752ndash1754Hamburg Germany July 1999

[10] W Nel J Tait R Lord and A Wilkinson ldquoThe use ofa frequency domain stepped frequency technique to obtainhigh range resolution on the CSIR X-Band SAR systemrdquo inProceedings of the 6th IEEE AFRICON Conference in Africa pp327ndash332 October 2002

[11] J H G Ender and A R Brenner ldquoPAMIRmdasha wideband phasedarray SARMTI systemrdquo IEE Proceedings Radar Sonar andNavigation vol 150 no 3 pp 165ndash172 2003

[12] C-G Gao Y-K Deng and J Feng ldquoTheoretical analysis on themismatch influence of displaced phase center multiple-beamSAR systemsrdquo Journal of Electronics amp Information Technologyvol 33 no 8 pp 1828ndash1832 2011

[13] X Luo R Wang Y Deng and W Xu ldquoInfluences of channelerrors and interference on the OFDM-MIMOSARrdquo in Proceed-ings of the IEEE Radar Conference pp 1ndash5 Ottawa Canada2013

[14] Y Deng H Zheng R Wang J Feng and Y Liu ldquoInternalcalibration for stepped-frequency chirp SAR imagingrdquo IEEEGeoscience and Remote Sensing Letters vol 8 no 6 pp 1105ndash1109 2011

[15] J Feng C Gao Y Zhang and R Wang ldquoPhase mismatchcalibration of the multichannel SAR based on azimuth cross-correlationrdquo IEEE Geoscience and Remote Sensing Letters vol10 no 4 pp 903ndash907 2013

[16] L Zhang M-D Xing C-W Qiu and Z Bao ldquoAdaptivetwo-step calibration for high-resolution and wide-swath SARimagingrdquo IET Radar Sonar and Navigation vol 4 no 4 pp548ndash559 2010

[17] J C Curlander and R N Mcdonough Synthetic ApertureRadar Systems and Signal Processing John Wiley amp Sons NewYork NY USA 1991

[18] I G Cumming and F H Wong Digital Processing of SyntheticAperture Radar Data Algorithms and Implementation ArtechHouse Norwood Mass USA 2005

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Control Scienceand Engineering

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VLSI Design



Shock and Vibration


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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



data acquired from the calibration subsystem This methodcan be extended to theMIMO system using STFC Neverthe-less 1198732 internal calibration subsystems are required whichdramatically increases the complexity of the SAR systemsIn addition it cannot correct imbalance introduced by theantennas The subspace projection method proposed in [16]and the azimuth cross-correlation method presented in [15]do not require additional subsystem However they cannotcorrect the interband imbalance as the phase differencecaused by different carrier frequencies is range dependentand hard to compensate

Usually the channel imbalance information (phase andamplitude) does not change within a certain illuminationtime [15] Moreover many artificial corner reflectors havebeen distributed here and there for radiometric calibration[17] Therefore this paper proposes an external calibrationmethod where the channel imbalance information is esti-mated from the peak value of a corner reflector or a strongpoint target in 119873

2 small complex images Afterwards theestimated channel errors are used to correct the channelimbalance when imaging for much larger data

Succeeding sections are organized as follows In Section 2the MIMO SAR using STFC is conceptually designed andthe corresponding image formation processing is brieflysummarized In Section 3 the external correction methodis presented followed by the simulation and real raw dataexperiments in Section 4 to validate the proposed methodFinally the conclusion is drawn in Section 5

2 MIMO SAR UsingStepped-Frequency Chirps

21 Conceptual Design The MIMO SAR simultaneouslytransmits a set of stepped-frequency chirps regarded assubbands via multiple subapertures During receiving everysubaperture receives all the 119873 subbands in one PRI Theschematic diagram and the corresponding timeline of trans-mitted pulses and received echoes are illustrated in Figure 1where the circles represent the location of the equivalentphase centers (EPCs) of the subbands and the number insidethe circles represents the sequence number of the subbands

In order to separate two frequency-adjacent subbandsand absent gaps from the connection of subbands at the sametime the subband bandwidth 119861

119904is required to be equal to

the frequency stepΔ119891The relationship of the time frequencyfor the transmitted pulses is illustrated in Figure 2 where119891

119898

(119898 = 1 2 119873) is the carrier frequency of the119898 th subbandThe baseband echo of the119898 th subband for each subaper-

ture is acquired as follows Demodulate the radio frequency(RF) pulses with the carrier frequency 119891

119898 Then filter the

demodulated signal with an analog low pass filter (LPF)whose bandwidth is greater than 119861

119904 Afterwards convert

the analog signal to a digital signal via an AD convertorwith a sampling rate greater than the cutoff frequency ofthe LPF As illustrated in Figure 3 due to the imperfectfiltering and the Gibbs phenomenon of the signal spectrumsome spectral component of one subband is leaked into theadjacent subband thus resulting in subband interference In

1

s

H

1 11 1 1

2 2 22 2 2

NN N

NN

N


1 1

2 2

N N

TX1RX1

TX2RX2

TXNRXN

Figure 1 Schematic diagram of the MIMO SAR using STFC


f

fN

f2

f1

ta(n) ta(n + 1)

Ts

PRT PRT

Bs

Bw

sub1

sub2

subN

sub1

sub2

subN

t

Figure 2 Relationship of time frequency of transmitted pulses

order to reduce the interference a digital rectangular windowcan be applied to the sampled data before imaging Thenthe interference only results from the spectral componentoutside the spectrum bandwidth as illustrated by the blue-shaded region in Figure 3 As stated in [13] the influence ofthe subband interference under this situation can be ignoredbecause it results inminus47 dB ambiguities in range In fact whenthe subband bandwidth is larger the ambiguities in range willbe reduced further This is because the relative amplitude ofthe ambiguities is determined by relating the leakage energy


LPF

0f2 minus f1

minusfs2

Rectangular window

f





1

PRF=119897119886119911

2V119904

sdot 119873 (1)







119878119898119899

(120591 120578) = 119873119898119899

(120591 120578) + 119860119898119899


120591 minus 119877119898119899

(120578) 119888

119879119903

)


119877119898119899

(120578)

119888)

sdot exp119895120587119896119903(120591 minus

119877119898119899

(120578)

119888)

2

(2)








119898119899(120578) can be

approximated as

119877119898119899

(120578) = 119877 (120578 minus Δ120578119898119899) (3)

where

Δ120578119898119899

=(119898 + 119899 minus 119873 minus 1) 119897

119886119911

2V119904

(4)


119877 (120578) = 2radic1198772

0+ (V119904120578 minus 1199090)2

(5)


0and 119909

0indicate




119878119898119899

(120591 120578) = 1198731015840

119898119899(120591 120578) + 119873

1015840

119898119899amb (120591 120578)

+ 119860119896sdot exp (119895120593

119896) sdot 119866 sdot 119901

119903(120591 minus

21198770

119888)


119888) sdot 119901

119886119898119899(120578 minus

1199090

V119904

)

+ 119860119896sdot exp (119895120593

119896) sdot 119901119903(120591 minus

21198770

119888)


Bandwidth synthesis



Conventionalimaging






Subband 1


Subband N

Subaperture N

Subaperture N


1PRF1 2 1 2 1 2 1 2

1 1

1 1

2 2

2 2

11 11

2 2 2 2

1

Reference position

MIM

O ec

ho

cons

truc

ted

Mul

ticha

nnel

reco

nstr

uctio

nBa

ndw

idth

synt

hesis

t2 2

t2 1

t1 2 =laz2s



119888)

times

119873minus1

sum

119894=minus119873+1 119894 = 0

119866amb119894

sdot 119901119886119898119899

(120578 minus1199090

V119904

+ 119894 sdotPRF119896119886

)

(6)



119898119899(120591 120578)






119901119886119898119899

(120578) = 119901119886(120578 minus Δ120578

119898119899) (7)



0 1199090)


119878119898119899

(120591 120578) = 119860119898119899


119903(120591 minus

21198770

119888)


119888)

sdot 119901119886(120578 minus

1199090

V119904

minus Δ120578119898119899)

(8)



119860119898119899

asymp 119860119898119899


120601119887119898119899

asymp 120593119898119899

minus 2120587119891119898sdot 120591119898119899 (9)


and 120601119887119896




AE = diag11986011 11986012 119860

1119873 119860

1198731 1198601198732 119860119873119873

11986011

(10)





120601119898119899

= 120601119887119898119899

+ 2120587Δ119891119898sdot 120591119898119899

asymp 120593119898119899

minus 2120587119891119888sdot (21198770

119888) (11)

where

Δ119891119898= (119898 minus

1 + 119873

2)Δ119891 119898 = 1 119873 (12)


ΦE = (diag exp (11989512060111) exp (1198951206011119873)

exp (1198951206011198731) exp (119895120601

119873119873))

times (exp (11989512060111))minus1

(13)


C = (AEΦE)minus1

(14)


S119888= C sdot [119878

11(120591 120578) 119878

1119873(120591 120578)

1198781198731

(120591 120578) 119878119873119873

(120591 120578)]119879

(15)



119888is






1198889685



119904215


11990110


11990460


72SNR (dB) SNR 6





119867119898119899

(119891120578) = exp minus1198952120587119891

120578sdot 119905119898119899 (16)


Imaging

Imaging

Imaging

Imaging

Correction




target




frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN





minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40


Range (m)

WithWithout

(c)




120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LPF

0f2 minus f1

minusfs2

Rectangular window

f





1

PRF=119897119886119911

2V119904

sdot 119873 (1)







119878119898119899

(120591 120578) = 119873119898119899

(120591 120578) + 119860119898119899


120591 minus 119877119898119899

(120578) 119888

119879119903

)


119877119898119899

(120578)

119888)

sdot exp119895120587119896119903(120591 minus

119877119898119899

(120578)

119888)

2

(2)








119898119899(120578) can be

approximated as

119877119898119899

(120578) = 119877 (120578 minus Δ120578119898119899) (3)

where

Δ120578119898119899

=(119898 + 119899 minus 119873 minus 1) 119897

119886119911

2V119904

(4)


119877 (120578) = 2radic1198772

0+ (V119904120578 minus 1199090)2

(5)


0and 119909

0indicate




119878119898119899

(120591 120578) = 1198731015840

119898119899(120591 120578) + 119873

1015840

119898119899amb (120591 120578)

+ 119860119896sdot exp (119895120593

119896) sdot 119866 sdot 119901

119903(120591 minus

21198770

119888)


119888) sdot 119901

119886119898119899(120578 minus

1199090

V119904

)

+ 119860119896sdot exp (119895120593

119896) sdot 119901119903(120591 minus

21198770

119888)


Bandwidth synthesis



Conventionalimaging






Subband 1


Subband N

Subaperture N

Subaperture N


1PRF1 2 1 2 1 2 1 2

1 1

1 1

2 2

2 2

11 11

2 2 2 2

1

Reference position

MIM

O ec

ho

cons

truc

ted

Mul

ticha

nnel

reco

nstr

uctio

nBa

ndw

idth

synt

hesis

t2 2

t2 1

t1 2 =laz2s



119888)

times

119873minus1

sum

119894=minus119873+1 119894 = 0

119866amb119894

sdot 119901119886119898119899

(120578 minus1199090

V119904

+ 119894 sdotPRF119896119886

)

(6)



119898119899(120591 120578)






119901119886119898119899

(120578) = 119901119886(120578 minus Δ120578

119898119899) (7)



0 1199090)


119878119898119899

(120591 120578) = 119860119898119899


119903(120591 minus

21198770

119888)


119888)

sdot 119901119886(120578 minus

1199090

V119904

minus Δ120578119898119899)

(8)



119860119898119899

asymp 119860119898119899


120601119887119898119899

asymp 120593119898119899

minus 2120587119891119898sdot 120591119898119899 (9)


and 120601119887119896




AE = diag11986011 11986012 119860

1119873 119860

1198731 1198601198732 119860119873119873

11986011

(10)





120601119898119899

= 120601119887119898119899

+ 2120587Δ119891119898sdot 120591119898119899

asymp 120593119898119899

minus 2120587119891119888sdot (21198770

119888) (11)

where

Δ119891119898= (119898 minus

1 + 119873

2)Δ119891 119898 = 1 119873 (12)


ΦE = (diag exp (11989512060111) exp (1198951206011119873)

exp (1198951206011198731) exp (119895120601

119873119873))

times (exp (11989512060111))minus1

(13)


C = (AEΦE)minus1

(14)


S119888= C sdot [119878

11(120591 120578) 119878

1119873(120591 120578)

1198781198731

(120591 120578) 119878119873119873

(120591 120578)]119879

(15)



119888is






1198889685



119904215


11990110


11990460


72SNR (dB) SNR 6





119867119898119899

(119891120578) = exp minus1198952120587119891

120578sdot 119905119898119899 (16)


Imaging

Imaging

Imaging

Imaging

Correction




target




frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN





minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40


Range (m)

WithWithout

(c)




120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bandwidth synthesis



Conventionalimaging






Subband 1


Subband N

Subaperture N

Subaperture N


1PRF1 2 1 2 1 2 1 2

1 1

1 1

2 2

2 2

11 11

2 2 2 2

1

Reference position

MIM

O ec

ho

cons

truc

ted

Mul

ticha

nnel

reco

nstr

uctio

nBa

ndw

idth

synt

hesis

t2 2

t2 1

t1 2 =laz2s



119888)

times

119873minus1

sum

119894=minus119873+1 119894 = 0

119866amb119894

sdot 119901119886119898119899

(120578 minus1199090

V119904

+ 119894 sdotPRF119896119886

)

(6)



119898119899(120591 120578)






119901119886119898119899

(120578) = 119901119886(120578 minus Δ120578

119898119899) (7)



0 1199090)


119878119898119899

(120591 120578) = 119860119898119899


119903(120591 minus

21198770

119888)


119888)

sdot 119901119886(120578 minus

1199090

V119904

minus Δ120578119898119899)

(8)



119860119898119899

asymp 119860119898119899


120601119887119898119899

asymp 120593119898119899

minus 2120587119891119898sdot 120591119898119899 (9)


and 120601119887119896




AE = diag11986011 11986012 119860

1119873 119860

1198731 1198601198732 119860119873119873

11986011

(10)





120601119898119899

= 120601119887119898119899

+ 2120587Δ119891119898sdot 120591119898119899

asymp 120593119898119899

minus 2120587119891119888sdot (21198770

119888) (11)

where

Δ119891119898= (119898 minus

1 + 119873

2)Δ119891 119898 = 1 119873 (12)


ΦE = (diag exp (11989512060111) exp (1198951206011119873)

exp (1198951206011198731) exp (119895120601

119873119873))

times (exp (11989512060111))minus1

(13)


C = (AEΦE)minus1

(14)


S119888= C sdot [119878

11(120591 120578) 119878

1119873(120591 120578)

1198781198731

(120591 120578) 119878119873119873

(120591 120578)]119879

(15)



119888is






1198889685



119904215


11990110


11990460


72SNR (dB) SNR 6





119867119898119899

(119891120578) = exp minus1198952120587119891

120578sdot 119905119898119899 (16)


Imaging

Imaging

Imaging

Imaging

Correction




target




frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN





minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40


Range (m)

WithWithout

(c)




120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











AE = diag11986011 11986012 119860

1119873 119860

1198731 1198601198732 119860119873119873

11986011

(10)





120601119898119899

= 120601119887119898119899

+ 2120587Δ119891119898sdot 120591119898119899

asymp 120593119898119899

minus 2120587119891119888sdot (21198770

119888) (11)

where

Δ119891119898= (119898 minus

1 + 119873

2)Δ119891 119898 = 1 119873 (12)


ΦE = (diag exp (11989512060111) exp (1198951206011119873)

exp (1198951206011198731) exp (119895120601

119873119873))

times (exp (11989512060111))minus1

(13)


C = (AEΦE)minus1

(14)


S119888= C sdot [119878

11(120591 120578) 119878

1119873(120591 120578)

1198781198731

(120591 120578) 119878119873119873

(120591 120578)]119879

(15)



119888is






1198889685



119904215


11990110


11990460


72SNR (dB) SNR 6





119867119898119899

(119891120578) = exp minus1198952120587119891

120578sdot 119905119898119899 (16)


Imaging

Imaging

Imaging

Imaging

Correction




target




frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN





minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40


Range (m)

WithWithout

(c)




120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Imaging

Imaging

Imaging

Imaging

Correction




target




frequencies

Constructthe

correctionmatrix

Echo11

Echo1N

EchoN1

EchoNN





minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(a)

minus200

minus100

0

100

200

Azi

mut

h (m

)

297 298 299 30 301 302

Range (km)

0

minus20

minus40

minus60

minus80

(b)

0 5 10 15

0

Am

plitu

de (d

B) minus10

minus20

minus30

minus40


Range (m)

WithWithout

(c)




120593119896(∘) 119860

119896120593119896(∘) 119860

119896120593119896(∘)

1 1 1 0 1 0 1 01 2 13 25 1296 24308 1253 282012 1 15 30 1500 29649 1468 333252 2 14 45 1401 44426 1352 42720


(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)


0

Am

plitu

de (d

B)

minus10

minus20

minus30

minus40


WithWithout

(c)



11= 0 11990512= 1198971198861199112V119904= (34)sdot

(12PRF) 11990521

= 11990512minus (14PRF) and 119905

22= 11990512+ 11990521 Then





5 Conclusion





References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Correction of Channel Imbalance for MIMO...

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Transcript of Research Article Correction of Channel Imbalance for MIMO...