Research Article Signal Processing for Digital Beamforming...

Research ArticleSignal Processing for Digital Beamforming FMCW SAR

Qin Xin,1 Zhihong Jiang,1 Pu Cheng,1 and Mi He2

1 College of Electronic Science and Engineering, National University of Defense Technology, ChangSha, Hunan 410073, China2 School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China

Correspondence should be addressed to Pu Cheng; [email protected] and Mi He; [email protected]

Received 10 November 2013; Revised 3 March 2014; Accepted 17 March 2014; Published 15 April 2014

Academic Editor: Gianluca Ranzi

Copyright © 2014 Qin Xin et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

According to the limitations of single channel FrequencyModulation ContinuousWave (FMCW) Synthetic Aperture Radar (SAR),Digital Beamforming (DBF) technology is introduced to improve system performance. Combined with multiple receive apertures,DBF FMCWSAR can obtain high resolution in low pulse repetition frequency, which can increase the processing gain and decreasethe sampling frequency. The received signal model of DBF FMCW SAR is derived. The continuous antenna motion which is themain characteristic of FMCW SAR received signal is taken into account in the whole signal processing. The detailed imagingdiagram of DBF FMCW SAR is given. A reference system is also demonstrated in the paper by comparing with a single channelFMCW SAR. The validity of the presented diagram is demonstrated with a point target simulation results.

1. Introduction

The combination of Frequency Modulated ContinuousWave(FMCW) technology and Synthetic Aperture Radar (SAR)paves a way to light-weight, cost-effective, high-resolutionactive microwave remote sensing instrument. This systemis suitable for small platforms such as Unmanned AerialVehicles (UAV). In the last years, several FMCWSAR systemshave been constructed and have been successfully used inice measurement [1], environment monitoring [2], three-dimensional application [3], and so on [4, 5].

However, these systems are currently applied in the shortrange and narrow swath case. According to the single channelFMCWSAR radar equation [6], the most immediate methodfor improving the system performance is to increase thetransmission power. Nevertheless, high power transmissionis not a suitable solution because of the isolation problembetween transmitter and receiver and the increasing systemcomplexity. Moreover, when operating in high resolutionsuch as 0.1mmode, the aperture length of antenna in azimuthis very small. This will decrease the antenna gain of singlechannel FMCW SAR systems, and then it will decrease theSignal Noise Radio (SNR) of SAR images.

In pulsed SAR systems, wide unambiguous swath cov-erage and high azimuth resolution are also contradictingrequirements. Several proposals, such as multiple beam SARsystem operating in a squinted imaging geometry [7], dis-placed phase center antenna technique [8], quad array system[9], and High-Resolution Wide-Swath (HRWS) SAR system[10], have been presented to resolve the aforementioneddilemma. Those systems have their own advantages anddisadvantages. However, their basic principle is ambiguitysuppression using Digital Beamforming (DBF) technologycombined with multiple antennas.

The HRWS SAR concept relies on separated transmitand receive antennas, which is also the characteristic ofFMCWSAR systems. So theHRWS conceptmay be a suitablesolution for improving the system performance of FMCWSAR. Now, multichannel FMCW SAR is becoming a hot areaof miniature SAR research. And many research institutionshave announced their research projects on multichannelFMCW SAR [11–14].

According to the imaging algorithm of pulsed DBF SAR,the basic steps are firstly the azimuth reconstruction and thensingle channel imaging. However, the multichannel azimuthreconstruction algorithm [15] is based on “stop and go”

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 859890, 10 pageshttp://dx.doi.org/10.1155/2014/859890

2 Mathematical Problems in Engineering

hypothesis. Compared with pulsed SAR received signal, themain characteristic of FMCW SAR received signal is that thecontinuous antenna motion must be taken into account; thatis, the “stop and go” hypothesis is not valid for FMCW SAR[16]. So the impact of continuous antenna motion to azimuthreconstruction must be analyzed. Furthermore, there aremany imaging algorithms for single channel FMCW SAR.The best algorithm for DBF FMCW SAR when consideringsome nonideal factors such as sweep frequency nonlinearityshould also be chosen carefully.

The remaining sections are organized as follows. Com-pared with single channel FMCW SAR, the performanceimprovement of DBF FMCW SAR is given in Section 2. Thesignal model, the impact of continuous antenna motion, andthe complete imaging diagram of DBF FMCW SAR are allpresented in Section 3. Next, an example system is designedand the numerical simulation results are given in Section 4.Finally, Section 5 concludes the whole paper.

2. Performance Improvement

The noise equivalent sigma zero (NESZ) is a quantity directlyrelated to SAR imaging performance, which is defined as thetarget Radar Cross Section (RCS) when the SNR of the imageis equal to one [17]. The NESZ of single channel FMCW SARis expressed by [6]

NESZ =(4𝜋)

3𝑟

4 sin𝜙𝐾𝑇0𝐹𝐵𝑁𝐿𝑠

𝑃𝑡𝐺

2𝜆

2𝜌𝑎𝜌𝑟𝐺𝑎𝐺𝑟

, (1)

where 𝑟 is the slant range, 𝜙 is the incidence angle, 𝐾 isthe Boltzmann constant, 𝑇

0is the system noise temperature,

𝐹 is the receiver noise figure, 𝐵𝑁

is the noise bandwidth,𝐿𝑠is the loss factor, 𝑃

𝑡is the transmitted signal power, 𝐺

is the antenna gain, 𝜌𝑎and 𝜌

𝑟are the azimuth and range

resolution, respectively, and 𝐺𝑎and 𝐺

𝑟are the azimuth and

range processing gain, respectively.

2.1. Antenna Gain. From (1), we know that the transmitand receive antenna gains in single channel FMCW SARsystem are the same. However, in DBF FMCW SAR, thereceive antenna gain is not equal to that of the transmitantenna. Assum that𝑀𝑁 receive channels are used,𝑀 is theantenna number in azimuth, and 𝑁 is the antenna numberin elevation. The multiple receive apertures are shown inFigure 1.

The DBF in azimuth is applied to suppress the azimuthambiguity. This does not increase the antenna gain. Inelevation, the channels are coherently combined. Performedby the SCan-On-REceive (SCORE) technique, the DBF inelevation can obtain full array gain over a wide area. Thus,under the same resolution and swath width, the maximumreceive antenna gain of the DBF system is

𝐺re = 𝑁𝐺. (2)

The DBF technique increases the antenna gain by 𝑁 times,compared to the single channel system.

Flight direction

Tx

Rx array

1 2 3 4 1 2 3 4Spatial samples in azimuth

M = 4

N = 3

� · Tr

Figure 1: Illustration of multiple receive apertures.

2.2. Range Processing Gain. Range processing of FMCWSAR is analog with pulsed SAR. They are all based onpulse compressing using matched filter. Theoretically, rangeprocessing gain is about the following:

𝐺𝑟= 𝑇𝑟𝐵𝑟, (3)

where 𝑇𝑟is the pulse width and 𝐵

𝑟is the signal bandwidth.

For FMCW SAR, the value of 𝑇𝑟is equal to 1/𝑓

𝑝, where 𝑓

𝑝

is the pulse repetition frequency (PRF). If we decrease thesystem PRF using𝑀 receive channels in azimuth, we can get𝑀 times signal processing gain comparedwith single channelsystem. The signal processing gain is increased at the cost oflong processing time.

2.3. Sampling Frequency. In order to maintain the low-powerand lightweight characteristics of FMCW SAR, the samplingfrequency should be chosen as lowly as possible. Accordingto the Nyquist theorem, the lowest sampling frequency forFMCW SAR is

𝑓𝑠= 𝑘𝑟⋅

2 (𝑟max − 𝑟min)

𝑐0

= 𝐵𝑟𝑓𝑝⋅

2 (𝑟max − 𝑟min)

𝑐0

, (4)

where 𝑘𝑟is the chirp modulation rate and 𝑟max and 𝑟min are

the maximum and minimum slant range, respectively. Thesampling frequency is also directly proportional to the PRF.That is to say, the sampling frequency ofmultichannel FMCWSAR is 1/𝑀 times of that of single channel FMCW SAR.

TheDBF FMCWSARoperates with a lowPRF to increasethe swath width. To decrease the PRF, the significant andchallenging processing is the azimuth ambiguity suppression.In the next section, we focus on the multichannel signalreconstruction in azimuth.

3. Signal Processing

3.1. Received Signal Model. Let the transmitted signal ofFMCW SAR be written in complex form as follows:

𝑠𝑠𝑇 (𝑛, 𝑡) = exp [𝑗2𝜋 (𝑓

𝑐⋅ 𝑡 +

1

2

𝑘𝑟⋅ (𝑡 − 𝑛𝑇

𝑟)

2)] , (5)

where 𝑓𝑐is the center frequency, 𝑛 is the sweep number,

and 𝑡 is the time variable. Because signal amplitude has littleeffect in SAR signal processing, so for simplicity, antenna

Mathematical Problems in Engineering 3

gain, amplitude effects of propagation on the signal and anyadditional time delays due to atmospheric effects are ignoredin the following discussions.

The received signal is a delayed version of the transmittedone. The 𝑀 channels of received signals in azimuth areprocessed. The𝑚th received signal is

𝑠𝑠𝑚,𝑅 (

𝑛, 𝑡)

= exp[𝑗2𝜋(𝑓𝑐⋅ (𝑡 −

𝑟𝑡,𝑚

𝑐0

) +

1

2

𝑘𝑟

⋅(𝑡 − 𝑛𝑇𝑟−

𝑟𝑡,𝑚

𝑐0

)

2

)] ,

(6)

where 𝑐0is the velocity of light and 𝑟

𝑡,𝑚is the roundtrip

instantaneous slant range which can be written as

𝑟𝑡,𝑚(𝑡; 𝑟0) = 𝑟𝑡𝑥(𝑡; 𝑟0) + 𝑟𝑟𝑥,𝑚

(𝑡; 𝑟0) , (7)

where 𝑟𝑡𝑥(𝑡; 𝑟0) and 𝑟

𝑟𝑥,𝑚(𝑡; 𝑟0) are transmit range and receive

range, respectively. In the single channel case, 𝑟𝑡𝑥(𝑡; 𝑟0) is

equal to 𝑟𝑟𝑥,𝑚

(𝑡; 𝑟0). However, it cannot be valid in the

multichannel case. This is the main difference between themultichannel and the single channel FMCW SAR receivedsignal.

Dechirp on receive is often used in FMCW SAR. Inorder to further decrease the sampling frequency, and then todecrease the data in range dimension, the radar demodulatesthe received signal by mixing it with a replica of the transmit-ted waveform delayed by a time 𝜏ref = 2 ⋅ 𝑟𝑐/𝑐0, where 𝑟𝑐 is theslant range distance of the swath center. The reference signalis

𝑠𝑠ref (𝑛, 𝑡; 𝑟𝑐)

= exp [𝑗2𝜋 (𝑓𝑐⋅ (𝑡 − 𝜏ref) +

1

2

𝑘𝑟⋅ (𝑡 − 𝑛𝑇

𝑟− 𝜏ref)

2)] .

(8)

The intermediate frequency signal that results from mix-ing (8) with (6) is

𝑠𝑠𝑚(𝑛, 𝑡; 𝑟

0)

= exp(−𝑗2𝜋𝑓𝑐

𝑐0

⋅ 𝑟𝑡,𝑚) ⋅ exp[𝑗

𝜋𝑘𝑟

𝑐

2

0

⋅ (𝑟𝑡,𝑚− 2𝑟𝑐)

2]

⋅ exp [−𝑗2𝜋𝑘𝑟

𝑐0

⋅ (𝑟𝑡,𝑚− 2𝑟𝑐) ⋅ (𝑡 − 𝑛𝑇

𝑟−

2𝑟𝑐

𝑐0

)] .

(9)

Using Taylor series, the round-trip range 𝑟𝑡,𝑚(𝑡; 𝑟0) can be

expanded as

𝑟𝑡,𝑚(𝑡; 𝑟0) = √𝑟

2

0+ V2𝑡2 + √𝑟2

0+ (V𝑡 − Δ𝑥

𝑚)

2

≈ √𝑟

2

0+ V2 ⋅ 𝑡2

𝑎+√𝑟

2

0+ (V𝑡𝑎− Δ𝑥𝑚)

2− 𝜆𝑓𝑑𝑡𝑟

≈ 2𝑟0+

Δ𝑥

2

𝑚

4𝑟0

+

(V𝑡𝑎− Δ𝑥𝑚/2)

2

𝑟0

− 𝜆𝑓𝑑𝑡𝑟,

(10)

where 𝑡𝑎= 𝑛𝑇𝑟and 𝑡𝑟= 𝑡 − 𝑛𝑇

𝑟represent azimuth time and

range time, respectively. V is the velocity of the platform, 𝑟0

is the closest distance between antenna and the scatter. Δ𝑥𝑚

is the distance between the transmit antenna and the 𝑚threceive antenna, 𝜆 = 𝑐

0/𝑓𝑐is the wavelength, and 𝑓

𝑑is the

approximateDoppler frequency shift during the transmissionof one sweep as follows:

𝑓𝑑= −

1

𝜆

[

[

[

V2𝑡𝑎

√𝑟

2

0+ V2 ⋅ 𝑡2

𝑎

+

V (V𝑡𝑎− Δ𝑥𝑚)

√𝑟

2

0+ (V𝑡𝑎− Δ𝑥𝑚)

2

]

]

]

. (11)

Different from pulsed SAR, 𝑓𝑑introduced by the contin-

uous antenna motion should be taken into consideration, forthe “stop and go” hypothesis is not valid for FMCWSAR.Thiscan be seen from the simulated results in Figure 5.

After substituting (10) into (9), the received signal ofchannel𝑚 can be rewritten as

𝑠𝑠𝑚(𝑡𝑎, 𝑡𝑟; 𝑟0)

= exp (−𝑗4𝜋𝜆

⋅ 𝑟0) ⋅ exp[−𝑗2𝜋

𝜆

⋅

Δ𝑥

2

𝑚

4𝑟0

]

⋅ exp[−𝑗2𝜋𝜆

⋅

(V𝑡𝑎− Δ𝑥𝑚/2)

2

𝑟0

] ⋅ exp (𝑗2𝜋𝑓𝑑𝑡𝑟)


𝑐0

⋅ (𝑟 (𝑡𝑎; 𝑟0) − 𝑟𝑐) ⋅ (𝑡𝑟−

2𝑟𝑐

𝑐0

)]

⋅ exp[𝑗4𝜋𝑘𝑟

𝑐

2

0

⋅ (𝑟 (𝑡𝑎; 𝑟0) − 𝑟𝑐)

2] ,

(12)

where 𝑟(𝑡𝑎; 𝑟0) is the azimuth dependent distance to the

scatter which can be expressed by

𝑟 (𝑡𝑎; 𝑟0) ≈ √𝑟

2

0+ V2 ⋅ 𝑡2

𝑎.

(13)

In (12), the first to third exponential terms are the azimuthphase history of multichannel FMCW SAR. The fourth termrepresents the range cellmigration due to continuous antennamotion, that is, the Doppler impact in one sweep. Thefifth term represents the range signal, which is a sinusoidalsignal with a constant frequency value corresponding to theazimuth dependent distance to the point target. The sixthexponential term represents the residual video phase (RVP)term.

3.2. The Impact of Continuous Antenna Motion. As men-tioned above, the main characteristic of FMCW SAR is thecontinuous antennamotion.The range cell migration (RCM)introduced by the continuous antenna motion may result inreconstruction mismatch. Even worse, the extraphase termintroduced by the continuous antenna motion may degradethe quality of the focus image.

Multichannel azimuth reconstruction algorithm [15] is asuited approach to process pulsed DBF SAR data in azimuth.


According to the algorithm [15] and the received signalmodelof DBF FMCW SAR described as (12), the reconstruction

filters P(𝑓𝑎) for DBF FMCW SAR can be calculated by the

following equation:

P (𝑓𝑎) = H−1 (𝑓

𝑎) =

[

[

[

[

[

[

[

[

𝑃11(𝑓𝑎) 𝑃12(𝑓𝑎+ 𝑓𝑝) ⋅ ⋅ ⋅ 𝑃

1𝑀(𝑓𝑎+ (𝑀 − 1) 𝑓𝑝

)

𝑃21(𝑓𝑎) 𝑃22(𝑓𝑎+ 𝑓𝑝) ⋅ ⋅ ⋅ 𝑃

2𝑀(𝑓𝑎+ (𝑀 − 1) 𝑓

𝑝)

......

......

𝑃𝑀1(𝑓𝑎) 𝑃𝑀2(𝑓𝑎+ 𝑓𝑝) ⋅ ⋅ ⋅ 𝑃

𝑀𝑀(𝑓𝑎+ (𝑀 − 1) 𝑓

𝑝)

]

]

]

]

]

]

]

]

, (14)

whereH(𝑓𝑎) is

H (𝑓𝑎) =

[

[

[

[

[

[

[

[

[

[

[

[

[

exp(−𝑗𝜋Δ𝑥

2

1

2𝜆𝑟0

) exp(−𝑗𝜋Δ𝑥1V

𝑓𝑎) ⋅ ⋅ ⋅ exp(−𝑗

𝜋Δ𝑥

2

𝑀

2𝜆𝑟0

) exp (−𝑗𝜋Δ𝑥𝑀V

𝑓𝑎)


2

1

2𝜆𝑟0


(𝑓𝑎+ 𝑓𝑝)) ⋅ ⋅ ⋅ exp(−𝑗

𝜋Δ𝑥

2

𝑀

2𝜆𝑟0


(𝑓𝑎+ 𝑓𝑝))

......

...


2

1

2𝜆𝑟0


(𝑓𝑎+ (𝑀 − 1) 𝑓

𝑝)) ⋅ ⋅ ⋅ exp(−𝑗

𝜋Δ𝑥

2

𝑀

2𝜆𝑟0


(𝑓𝑎+ (𝑀 − 1) 𝑓

𝑝))

]

]

]

]

]

]

]

]

]

]

]

]

]

.

(15)

Using basic matrix theory, the system matrix H can bedecomposed as a matrix Q multiplied by a diagonal matrixD, where only the elements of D are dependent on the slantrange 𝑟

0. And the diagonal elements of D can be written as

(D)𝑚,𝑚

= exp[−𝑗𝜋Δ𝑥

2

𝑚

2𝜆𝑟0

] . (16)

In the SAR case, the range history is not a line but spansseveral range cells, which results in mismatch between thesignal and the reconstruction. In FMCW SAR, the additionalmismatch introduced by the continuous antenna motion is

Δ𝜑𝑚=

𝜋Δ𝑥

2

𝑚

2𝜆

⋅

Δ𝑟dfm𝑟0⋅ (𝑟0+ Δ𝑟dfm)

, (17)

in which Δ𝑟dfm represents the maximum RCM introduced bythe continuous antenna motion of channel𝑚. The mismatchcan be derived from (14) to (16).

Figure 2 shows a simulated RCM introduced by thecontinuous antenna motion using the parameter listed inTable 1. In this case, Δ𝑟dfm is about 1m. Calculated by (17),the mismatch introduced by this RCM is about 4 × 10−5 rad,which is negligible. Here, Δ𝑥

𝑚= 5m is assumed. On the

other side, Δ𝑟dfm is about 10 times of the range resolution.This RCM must be compensated according to SAR imagingtheory [17]. In a word, for DBF FMCW SAR, the impact ofcontinuousmotion antenna is negligible duringmultichannelazimuth reconstructing [15]; however, it must be consideredduring RCM correction.

1.5

1

0.5

0

−0.5

−1

−1.5

1000

500

0−20

−10

0

10

20

Appr

oxim

atio

n er

ror (

m)

PRF (Hz)Azimuth time (s)

Figure 2: RCM introduced by the continuous antenna motion.

3.3. Signal Processing Diagram. The first part of signal pro-cessing for DBF FMCW SAR is the azimuth ambiguitysuppression. All the recorded subaperture signals whoseazimuth frequencies are aliasing become an equivalent singlechannel signal without azimuth ambiguity. To focus thesignal, the second part of the diagram is the classic singlechannel SAR imaging.

In the first part, multichannel azimuth reconstructionalgorithm [15] is chosen because the algorithm can relaxthe imposed constraints on system design and the algorithmperformance is rarely affected by the continuous antenna


Table 1: System parameters.

Parameter Multi-ChannelSystem

Single ChannelSystem

RF center frequency 15GHz 15GHzBand Width 1.5 GHz 1.5 GHzOperational Altitude 5000m 5000mOperational Velocity 70m/s 70m/sSwath Width 7 km 7 kmResolution 0.1m × 0.1m 0.1m × 0.1mRadar Losses 2 dB 2 dBIncidence Angle 5–70∘ 5–70∘

Number of Receive Apertures 2 1Transmitted Power 5W 5WTransmit Antenna Gain 19.5 dB 19.5 dBReceive Antenna Gain 19.5 dB 19.5 dBPRF 350Hz 700HzRange Processing Gain 66.3 dB 63.3 dBSampling Frequency 16MHz 32MHz

motion. In the second part, frequency scaling algorithm [16,18–20] is combined with the DBF technology.This algorithmhas not only efficient and accurate processing but also real-time integrated motion compensation [19] and easy sweepfrequency nonlinear correction [16, 20].

The entire signal processing block diagram of DBFFMCW SAR is shown in Figure 3 including the followingsteps.

Step 1. Apply Azimuth Fast Fourier Transform (FFT) to thereceived signal of each channel. Then, the Doppler domainsignal 𝑆𝑠

𝑚(𝑓𝑎, 𝑡𝑟; 𝑟0) is obtained, where 𝑓

𝑎is the azimuth

frequency.

Step 2. Design the azimuth reconstruction filter P(𝑓𝑎) by

using (14) and (15).The reconstruction filter of channel𝑚 canbe written as

𝑃𝑚(𝑓𝑎) = [𝑃𝑚1

(𝑓𝑎) 𝑃𝑚2(𝑓𝑎) ⋅ ⋅ ⋅ 𝑃

𝑚𝑀(𝑓𝑎)] . (18)

Step 3. Multiply each channel signal to the correspondingreconstruction filter. At this time, the signal becomes

𝑆𝑠AR,𝑚 (𝑓𝑎, 𝑡𝑟; 𝑟0) = 𝑆𝑠𝑚 (𝑓𝑎, 𝑡𝑟; 𝑟0) ⋅ 𝑃𝑚 (𝑓𝑎) . (19)

Step 4. Add each channel signal after passing the correspond-ing reconstruction filter. Then, we can obtain the equivalentsingle channel signal 𝑆𝑠(𝑓

𝑎, 𝑡𝑟; 𝑟0), which has nonazimuth

frequency aliasing. And it can be written in convolution [16]as follows:

𝑆𝑠 (𝑓𝑎, 𝑡𝑟; 𝑟0)

= {exp(−𝑗4𝜋𝑟0𝛽

𝜆

) exp (𝑗2𝜋𝑓𝑑𝑡𝑟)


𝑐0

(

𝑟0

𝛽

− 𝑟𝑐)(𝑡𝑟−

2𝑟𝑐

𝑐0

)]

×src (𝑓𝑎, 𝑡𝑟; 𝑟0)}

⊗ exp (−𝑗𝜋𝑘𝑟𝑡

2

𝑟) ,

(20)

where 𝛽 is the RCM factor defined by

𝛽 (𝑓𝑎) =

√

1 −

𝑓

2

𝑎𝜆

2

4V2,

(21)

where src(𝑓𝑎, 𝑡𝑟; 𝑟0) is the secondary range compression as

follows:

src (𝑓𝑎, 𝑡𝑟; 𝑟0) = exp[−𝑗

2𝜋𝑟0𝑘

2

𝑟𝜆

𝑐

2

0

(𝛽

2− 1)

𝛽

3(𝑡𝑟−

2𝑟𝑐

𝑐0

)

2

]

⋅ exp[𝑗2𝜋𝑟0𝑘

3

𝑟𝜆

2

𝑐

3

0

(𝛽

2− 1)

𝛽

5(𝑡𝑟−

2𝑟𝑐

𝑐0

)

3

] .

(22)

Step 5 (frequency scaling transform). Multiplying the signal𝑆𝑠(𝑓𝑎, 𝑡𝑟; 𝑟0) with the frequency scaling factor, we have the

following:

𝐻FS (𝑓𝑎, 𝑡𝑟) = exp [𝑗𝜋𝑘𝑟(1 − 𝛽) 𝑡

2

𝑟] . (23)

Step 6 (continuous antennamotion correction). Multiplyingthe signal with the Doppler frequency correction factor, wehave the following:

𝐻DFC (𝑓𝑎, 𝑡𝑟) = exp (−𝑗2𝜋𝑓𝑑𝛽 ⋅ 𝑡𝑟) . (24)

In actual implementation, Steps 5 and 6 can be completed atthe same time.The signal 𝑆𝑠FS(𝑓𝑎, 𝑡𝑟; 𝑟0) is obtained as follows:

𝑆𝑠FS (𝑓𝑎, 𝑡𝑟; 𝑟0) = 𝑆𝑠 (𝑓𝑎, 𝑡𝑟; 𝑟0) ⋅ 𝐻FS ⋅ 𝐻DFS. (25)

After an equivalent transform, (25) can be expressed as

𝑆𝑠FS (𝑓𝑎, 𝑡𝑟; 𝑟0)

= {exp(−𝑗4𝜋𝑟0𝛽

𝜆

)

× exp [−𝑗4𝜋𝑘𝑟

𝑐0

(

𝑟0

𝛽

− 𝑟𝑐)(𝛽𝑡

𝑟−

2𝑟𝑐

𝑐0

)]

⋅ exp [−𝑗𝜋𝑘𝑟𝛽 (𝛽 − 1) 𝑡

2

𝑟]

× src (𝑓𝑎, 𝛽𝑡𝑟; 𝑟0)} ⊗ exp (−𝑗𝜋𝑘

𝑟𝛽𝑡

2

𝑟) .

(26)


Equivalent single channel signal

Secondary range compression factor

Range FFT

Range FFT

Range IFFT

Focused image

Azimuth IFFT

Doppler frequency correction factor

RVP correction factor

Inverse frequency scaling factor

Bulk range shift factor

Signal of channel 1 Signal of channel M

Azimuth FFT

Reconstruction filter of channel 1

Azimuth FFT

Reconstruction filter of channel M

Frequency scaling factor

Azimuth matched filter factor

· · ·

· · ·

· · ·

Figure 3: Signal processing block diagram for DBF FMCW SAR.

Step 7 (RVP correction). The correction is applied in 2Dfrequency domain. It is to eliminate the last exponentialterm in (26). After FFT in range dimension and multiplyingwith the RVP correction factor𝐻RVPC(𝑓𝑎, 𝑓𝑟), IFFT in rangedomain is performed.The signal 𝑆𝑠RVPC(𝑓𝑎, 𝑡𝑟; 𝑟0) is obtainedas follows:

𝑆𝑠RVPC (𝑓𝑎, 𝑡𝑟; 𝑟0)

= exp(−𝑗4𝜋𝑟0𝛽

𝜆

) exp [−𝑗4𝜋𝑘𝑟

𝑐0

(

𝑟0

𝛽

− 𝑟𝑐)(𝛽𝑡

𝑟−

2𝑟𝑐

𝑐0

)]

⋅ exp [−𝑗𝜋𝑘𝑟𝛽 (𝛽 − 1) 𝑡

2

𝑟] src (𝑓

𝑎, 𝛽𝑡𝑟; 𝑟0) .

(27)

The RVP correction factor is expressed as

𝐻RVPC (𝑓𝑎, 𝑓𝑟) = exp(−𝑗𝜋𝑓

2

𝑟

𝑘𝑟𝛽

) . (28)

Step 8 (inverse frequency scaling transform). Multiply 𝑆𝑠RVPC(𝑓𝑎, 𝑡𝑟; 𝑟0) with the inverse frequency scaling factor 𝐻IFS(𝑓𝑎,

𝑡𝑟) which is

𝐻IFS (𝑓𝑎, 𝑡𝑟) = exp [𝑗𝜋𝑘𝑟𝛽 (𝛽 − 1) 𝑡

2

𝑟] . (29)

Step 9. Secondary range compression: multiply the signalwith the secondary range compression factor𝐻SRC(𝑓𝑎, 𝑡𝑟; 𝑟𝑐)which is described as

𝐻SRC (𝑓𝑎, 𝑡𝑟; 𝑟𝑐) = src(𝑓𝑎, 𝛽𝑡𝑟; 𝑟𝑐)

∗, (30)

where the approximation 𝑟𝑐≈ 𝑟0is required.


−300 −200 −100 0 100 200 3000

5

10

15

20

25

30

35

40

45A

mpl

itude

Doppler frequency (Hz)

(a)

−300 −200 −100 0 100 200 3000

5

10

15

20

25

30

35

40

45

Am

plitu

de

Doppler frequency (Hz)

(b)

Figure 4: The azimuth aliasing spectrum before reconstruction filters (a) and the aliasing-removal spectrum after reconstruction filters (b).

Step 10. Bulk range shift: multiply the signal with the bulkrange shift factor𝐻BV(𝑓𝑎, 𝑡𝑟; 𝑟𝑐) which is described as

𝐻BV (𝑓𝑎, 𝑡𝑟; 𝑟𝑐) = exp [𝑗4𝜋𝑘𝑟𝑟𝑐

𝑐0

(

1

𝛽

− 1)(𝛽𝑡𝑟−

2𝑟𝑐

𝑐0

)] .

(31)

In actual implementation, Step 8 to Step 10 can also becompleted at the same time. We obtain

𝑆𝑠RCMC (𝑓𝑎, 𝑡𝑟; 𝑟0) = 𝑆𝑠RVPC (𝑓𝑎, 𝑡𝑟; 𝑟0) ⋅ 𝐻IFS ⋅ 𝐻SRC ⋅ 𝐻BV

≈ exp(−𝑗4𝜋𝑟0𝛽

𝜆

)

× exp[−𝑗4𝜋𝑘𝑟(𝑟0− 𝑟𝑐)

𝑐0𝛽

(𝛽𝑡𝑟−

2𝑟𝑐

𝑐0

)] .

(32)

Step 11. Apply FFT to 𝑆𝑠RCMC(𝑓𝑎, 𝑡𝑟; 𝑟0) in range dimension.At this time, the range compression is completed as follows:

𝑆𝑠RCMC (𝑓𝑎, 𝑓𝑟; 𝑟0)

=

𝑇

𝛽

exp{−𝑗4𝜋𝑓𝑟𝑟𝑐

𝛽𝑐0

} exp(−𝑗4𝜋𝑟0𝛽

𝜆

)

⋅ sinc[𝜋𝑇𝛽

(𝑓𝑟+

2𝑘𝑟(𝑟0− 𝑟𝑐)

𝑐0

)] .

(33)

Step 12. The 2D focused image of DBF FMCW SAR isobtained after azimuth matched filtering [18].

4. Simulation Results

The parameters of an example FMCW SAR system usingdual channels combined with DBF are listed in Table 1. Thecorresponding parameters of single channel FMCW SAR,

which is very similar to MicroBSAR described in [6], are alsogiven. There are two parts in the simulation. One is imagingsimulation, the other is performance simulation.

4.1. System Parameters Design. To achieve the required res-olution in azimuth, the Doppler bandwidth must be greaterthan 700Hz. The transmit antenna length in azimuth shouldbe less than 0.2m. According to the swath width, the transmitantenna length in elevation can be chosen to be 2.9 cm, whichcorresponds to 40∘ beamwidth. The transmit antenna gain is19.5 dB with the antenna efficiency 0.5.

The receive antenna is the samewith the transmit antennafor single channel FMCW SAR. In the two-channel system,the receive antennas are placed in azimuth. The receiveantenna gain is still the same. However the overall receiveantenna length in azimuth reaches 0.4m.

According to the Nyquist theorem, the PRF of classicFMCW SAR should be greater than the Doppler bandwidth.However, the PRF of DBF FMCW SAR can be chosen tobe 700/2Hz using two receive channels. To ensure a slantrange resolution of 0.1m, a chirp bandwidth of 1.5 GHz isnecessary. The range processing gain 66.3 dB is obtained forDBF FMCW SAR. Nevertheless, that value of single channelFMCW SAR is only 63.3 dB. The sampling frequency of DBFFMCW SAR and single channel system are 16MHz and32MHz, respectively. All the parameters are listed in Table 1.

4.2. Imaging Simulation. The target raw data are simulatedusing the two-channel system parameters listed in Table 1 by(9). To investigate the method of DBF technology and theeffects of the continuous antenna motion, the raw data areprocessed by using the algorithm described in Figure 3.

The azimuth reconstruction is shown in Figure 4.Figure 4(a) shows the azimuth aliasing spectrum of thefirst channel signal in the two-channel system beforereconstruction filter. After filters of 𝑃

1(𝑓𝑎) and 𝑃

2(𝑓𝑎),

respectively, add the first channel signal and the second


10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90

Azimuth frequency (sample)

Rang

e fre

quen

cy (s

ampl

e)

−3dB−9dB

−21dB−27dB

(a)

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90


Rang

e fre

quen

cy (s

ampl

e)

−3 dB−9 dB

−21 dB−27 dB

(b)

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90


Rang

e fre

quen

cy (s

ampl

e)

−3 dB−9 dB

−21 dB−27 dB

(c)

Figure 5: Contour plots of impulse response function of a point target without azimuth reconstruction (a), without Doppler frequencycorrection (b), and with azimuth reconstruction and Doppler frequency correction (c).

channel signal together. Then the azimuth spectrum iscompletely reconstructed. The reconstruction result isshown in Figure 4(b). The DBF coherently combines the twochannel signals in azimuth and reconstructs a wide Dopplerspectrum about 700Hz.

Using the parameters of the two-channel example system,the signal processing presented in Section 3.3 is simulated.Figure 5 plots the 2D contour maps of the impulse responsefunction of a point target. Only the values above −27 dBare plotted. The result of the proposed processing algorithmis plotted at the bottom. Figure 5(a) is the result with-out azimuth reconstruction. The performance is seriouslydegraded because the azimuth spectrum is overlapped as

Figure 4(a) shows. To explore the characteristic of FMCWSAR distinguished with pulse SAR, Figure 5(b) illustrates theresult without Doppler frequency correction factor which isexpressed as (24). In terms of the resolution and shape of theresponse, it is shown that the reconstruction is successful andthe impact of the motion within one sweep in FMCW SARneeds to be compensated.

4.3. Performance Simulation. Using the parameters inTable 1, NESZ for DBF FMCW SAR has been calculated overa range of incidence angles, which is shown in Figure 6. Theresult for single channel system is also shown in Figure 6.When calculating NESZ, the receiver noise figure is chosen


1000 2000 3000 4000 5000 6000 7000 8000 9000−24

−22

−20

−18

−16

−14

−12

−10

−8

−6

−4

Cross track range (m)

NES

Z (d

B)

Classic systemDBF system

Figure 6: NESZ versus cross track range for DBF (solid) and singlechannel FMCW SAR (dotted).

as 3 dB.TheDBF FMCW SAR achieves a NESZ below −10 dBalong the entire swath. 5W of transmitted power is sufficientto image vegetation and farmland. However, the result ofclassic FMCW SAR in the edge of the swath is only −5 dB,which is not sufficient to obtain good far-range imagery. Thereason is that there is higher signal gain for DBF FMCWSAR.

5. Conclusions

A multichannel receive antenna FMCW SAR system com-bined with DBF is introduced. DBF FMCW SAR overcomesthe limitation of high PRF caused by high azimuth resolution.It has higher receive antenna gain and range processinggain than single channel system. The received signal ofDBF FMCW SAR is modeled. The impact of continuousantenna motion which is the main characteristic of FMCWSAR is negligible in multichannel azimuth reconstructionfor DBF FMCW SAR, but it must be considered in theprocess during RCM correction.The whole signal processingdiagram is given. Simulated point target experiments havebeen performed to verify the processing diagram successfully.Future works will include performance analysis thoroughly,moving target indication, and the miniaturization of thesystem.

Conflict of Interests

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

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (nos. 41301397).

References

[1] E. Zaugg, D. Long,M. Edwards et al., “Using theMicroASARonthe NASA SIERRA UAS in the Characterization of Arctic SeaIce Experiment,” in Proceedings of the IEEE International RadarConference (RADAR ’10), pp. 271–276, Washington, DC, USA,May 2010.

[2] M. Edrich, “Ultra-lightweight synthetic aperture radar basedon a 35GHz FMCW sensor concept and online raw datatransmission,” IEE Proceedings: Radar, Sonar and Navigation,vol. 153, no. 2, pp. 129–134, 2006.

[3] M. Weiss, O. Peters, and J. Ender, “A three dimensional SARsystem on an UAV,” in Proceedings of the IEEE InternationalGeoscience and Remote Sensing Symposium (IGARSS ’07), pp.5315–5318, Barcelona, Spain, July 2007.

[4] E. Zaugg, M. Edwards, D. Long, and C. Stringham, “Develop-ments in compact high-performance synthetic aperture radarsystems for use on small unmanned aircraft,” in Proceedings ofthe IEEE Aerospace Conference (AERO ’11), pp. 1–14, Big Sky,Mont, USA, March 2011.

[5] A. Meta, J. J. M. de Wit, and P. Hoogeboom, “Development ofa high resolution airborne millimeter wave FM-CW SAR,” inProceedings of the 1st European Radar Conference (EuRAD ’04),pp. 209–212, Amsterdam, The Netherlands, October 2004.

[6] M. C. Edwards, Design of a continuous-wave synthetic apertureradar system with analog dechirp [M.S. thesis], Brigham YoungUniversity, Provo, Utah, USA, 2009.

[7] B. R. Jean and J. W. Rouse Jr., “A multiple beam syntheticaperture radar design concept for geoscience applications,”IEEE Transactions on Geoscience and Remote Sensing, vol. 21,no. 2, pp. 201–207, 1983.

[8] A. Currie andM.A. Brown, “Wide-swath SAR,” IEE ProceedingsF: Radar and Signal Processing, vol. 139, no. 2, pp. 122–135, 1992.

[9] G. D. Callaghan and I. D. Longstaff, “Wide-swath space-borneSAR using a quad-element array,” IEEE Proceedings: Radar,Sonar and Navigation, vol. 146, no. 3, pp. 159–165, 1999.

[10] M. Suess, B. Grafmueller, and R. Zahn, “A novel high resolution,wide swath SAR system,” in Proceedings of the InternationalGeoscience and Remote Sensing Symposium (IGARRS ’01), pp.1013–1015, Sydney, Australia, July 2001.

[11] W. van Rossum, M. Otten, and P. van Dorp, “MultichannelFMCW SAR,” in Proceedings of the 9th European Conferenceon Synthetic Aperture Radar (EUSAR ’12), pp. 279–282, Nurem-berg, Germany, April 2012.

[12] A. Meta, E. Imbembo, C. Trampuz, A. Coccia, and G. de Luca,“A selection of MetaSensing airborne campaigns at L-, X- andku band,” in Proceedings of the IEEE International Geoscienceand Remote Sensing Symposium (IGARSS ’12), pp. 4571–4574,Munich, Germany, July 2012.

[13] M. C. Edwards and E. C. Zaugg, “Design of a compact, modular,multi-frequency band, multi-mode, multi-channel syntheticaperture radar,” inProceedings of the 9th EuropeanConference onSynthetic Aperture Radar (EUSAR ’12), pp. 44–47, Nuremberg,Germany, April 2012.

[14] X. Qin, W. Zhan, Z.-H. Jiang, and H.-F. Kan, “ImprovingFMCW SAR system performance by digital beamforming,” inProceedings of the IEEE International Geoscience and RemoteSensing Symposium (IGARSS ’12), pp. 4557–4560, Munich,Germany, July 2012.

[15] N.Gebert, G.Krieger, andM.A.Moreira, “Digital beamformingon receive: techniques and optimization strategies for high-resolution wide-swath SAR imaging,” IEEE Transactions on


Aerospace and Electronic Systems, vol. 45, no. 2, pp. 564–592,2009.

[16] Z.-H. Jiang and K. Huang-Fu, “Squint LFMCW SAR dataprocessing using doppler-centroid-dependent frequency scal-ing algorithm,” IEEE Transactions on Geoscience and RemoteSensing, vol. 46, no. 11, pp. 3535–3543, 2008.

[17] J. C. Curlander and R. N. McDonough, Synthetic ApertureRadar: Systems and Signal Processing, John Wiley & Sons, NewYork, NY, USA, 1991.

[18] J. Mittermayer, A. Moreira, and O. Loffeld, “Spotlight SARdata processing using the frequency scaling algorithm,” IEEETransactions on Geoscience and Remote Sensing, vol. 37, no. 5,pp. 2198–2214, 1999.

[19] E. C. Zaugg and D. G. Long, “Theory and application ofmotion compensation for LFM-CW SAR,” IEEE Transactionson Geoscience and Remote Sensing, vol. 46, no. 10, pp. 2990–2998, 2008.

[20] A. Meta, P. Hoogeboom, and L. P. Ligthart, “Signal processingfor FMCW SAR,” IEEE Transactions on Geoscience and RemoteSensing, vol. 45, no. 11, pp. 3519–3532, 2007.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Research Article Signal Processing for Digital Beamforming...

Documents

Transcript of Research Article Signal Processing for Digital Beamforming...