principles of synthetic aperture radar imaging: a system

This article was downloaded by: 10.3.98.104On: 24 Mar 2022Access details: subscription numberPublisher: CRC PressInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK

Principles of Synthetic Aperture Radar Imaging: A SystemSimulation Approach

Kun-Shan Chen

SAR Data and Signal

Publication detailshttps://www.routledgehandbooks.com/doi/10.1201/b19057-4

Kun-Shan ChenPublished online on: 18 Dec 2015

How to cite :- Kun-Shan Chen. 18 Dec 2015, SAR Data and Signal from: Principles of SyntheticAperture Radar Imaging: A System Simulation Approach CRC PressAccessed on: 24 Mar 2022https://www.routledgehandbooks.com/doi/10.1201/b19057-4

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51

3 SAR Data and Signal

3.1 INTRODUCTION

The signal property of synthetic aperture radar (SAR) is one of the core issues to grasp in order to process the raw data into an image. The mapping between the object domain and image domain is persistently bridged and thus controlled by the signal domain. This chapter deals with this aspect in depth. Although in SA, various waveforms, like con-tinuous wave (CW) and phase-coded [1–3], may be used, we will focus our discussion on linear frequency modulation (LFM) (chirp) and frequency-modulated continuous wave (FMCW) for their common use in modern SAR systems.

3.2 CHIRP SIGNAL

3.2.1 Echo Signal in Two DimEnSionS

In Equation 1.54, we indicated that the pulse repetition frequency fp must follow the Nyquist criteria to avoid the range ambiguity within the maximum probing range Rmax. Also, there is a minimum range Rmin corresponding to the near range. The antenna elevation beamwidth βe subtends from the near to the far range (swath) (Figure 3.1). To avoid the air bubbles, SAR is not pointing its boresight at the nadir. The delay time of returns from the near range and far range, τnear, τfar, must be confined within a period. Mathematically, the following two relations should be obeyed [4,5]:

T

R

cp < 2 min

(3.1)

Tf

R

cT

pp= > +1 2 max (3.2)

Remember that for spaceborne SAR, the maximum range is limited by the earth curvature [6,7], depending on the satellite height and elevation angle. If the transmit-ted signal is of the form given by Equation 1.54, the received signal as a function of fast time is the output of the matched filter:

s A s A t s t dt

A p

r t t

t T

t T

r

p

p

( ) ( ) ( ) ( )τ τ τ= ⊗ = −

=

−

+

∫0 0

2

2

0

/

/

ττ π τ π τ−

−

+ −

22

2 22

R

cf

R

ca

R

cc rcos

(3.3)

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52 Principles of Synthetic Aperture Radar Imaging

where A0 is the scatter amplitude [4] and, without loss of generality, is assumed con-stant. Equation 3.3 is indeed also dependent on the slow time because the slant range R is varying with the sensor position within the target exposure time. Referring to Figure 1.6, the slow time-dependent slant range can be expanded about R(ηc), with the beam center crossing time, ηc:

R Ru

R

u

Rcc

cc

c

c

( ) ( )( )

( )cos

( )(,η η η

ηη η

θη

η= + − + −2 2 21

2� ηηc )2 +… (3.4)

where θc is look angle to the scene center and

η θ η θc

c c cR

u

R

u= =0 tan ( )sin

. (3.5)

Due to the time variation of the range, a point target response will continuously appear along the path according to Equation 3.4 within the synthetic aperture length. A coherent sum of these responses during the course of the target exposure time will be out of focus if no range-induced phase variation is corrected. Figure 3.2

RfarRnear

Tp

Tp

Swath

Near FarNadir

βe

θi,near θi, far

τnearτfar

T

TNear Far

Nadir

θi,near θi, far

FIGURE 3.1 Observation geometry defined by pulse and antenna beamwidth.

Azi

mut

h (c

ells)

Slant range (cells)

Synt

hetic

ape

rtur

e le

ngth

Pulse length

Az

Target

R(η) = R0 + u2η22

R(η)sinθcθc

R(η)

R(η)

R(η)

R(η0)R0 = R(η0)

uηc

η = ηc

η = η0

η = ηc

η = η0

η = ηc

η = η0

η

Rg

c

0T

FIGURE 3.2 Mapping from the object domain to the data domain.

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53SAR Data and Signal

schematically illustrates data collection and mapping from the target (object) to the SAR data domain.

In practice, we should take the antenna pattern into the signal reception. The received signal may be written as

s A pR

cT g f

R

cr r p a c( , ) , ( )cos( )τ η τ η π τ η= −

−

0

22

2

+ −

π τ η

aR

cr

22

( )

(3.6)

A typical two-way antenna pattern in the azimuthal direction may be of the form [4]

gaaz

( ). ( )η θ η

β≅

sinc2 0 886 diff (3.7)

where θdiff = θsq − θsq,c, with θsq,c = θsq(ηc) being the squint angle at the scene center. For small squint approximation, Equation 3.7 may be written as

gasq sq c

az

( ). [ ( ) ] .,η

θ η θβ

=−

≅sinc sinc2 20 886 0 8886 1

0βη η

az

cu

Rtan

( )− −

. (3.8)

Figure 3.3 plots a right side-looking SAR (left) to show the antenna pattern illu-minating on a point target with gain variations (middle) along the azimuth direction; on the right is the intensity of the point target within the cell. The antenna gain pat-tern effect along the azimuthal direction is clearly observed.

Azimuth (η)

Range (t)

Target

θdiffθθθθθdidddddd

Received signal strength0.0–1.5–1.0

–0.50.0

Angle at A

zimuth (deg)

0.51.0

1.50.20.40.60.81.0

FIGURE 3.3 A right side-looking SAR (left) to show antenna beam pattern (middle) on a point target response (right).

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3.2.2 DEmoDulaTED Echo Signal

The demodulation is to remove the carrier frequency, which bears no target informa-tion. Following [4], a typical demodulation scheme is illustrated in Figure 3.4, where the echo signal in ① is downconverted by mixing with a reference signal, resulting in ② and ③. By filtering out the upper frequency component and going through an analog-to-digital conversion (ADC) if necessary, the echo signal for postprocessing ⑥ is obtained.

①: cos( ) ( )

cos24 2

2

π τ π η π τ ηf

f R

ca

R

ccc

r− + −

= [[ ( )]2π τ φ τfc +

②: 12

12

4cos[ ( )] cos[ ( )]φ τ π τ φ τ+ +fc

③: 12

12

4sin[ ( )] sin[ ( )]φ τ π φ τ+ +f tc

④: 12

cos[ ( )]φ τ

⑤: 12

sin[ ( )]φ τ

⑥: 12

12

4 2exp{ ( )} exp

( ) ( )j j

f R

cj a t

R

cc

rφ τ π η π η= − + −

2

The final demodulated two-dimensional echo signal is of the form

s A pR

cT g j

f R

cjr r a

c0 0

2 4( , ) , ( )exp

( )τ η τ η π η= −

− + ππ τ η

a tR

cr −

2

2( )

(3.9)

Figure 3.5 plots a demodulated echo signal from a point target, showing both amplitude and phase. Again, the antenna gain pattern effect can be seen on the amplitude response.

Low-pass filter ADC

cos(2πfct)

–sin(2πfct)

Low-pass filter ADC

1

2

3

4

5

6

I, Real

Q, Imaginary

ADC

ADC

4

5

6

I, Real

Q I i

L

c

L

2

3

FIGURE 3.4 Demodulation of the echo signal.

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3.2.3 RangE cEll migRaTion

When the imaging taken from the object domain to the data domain is within the syn-thetic aperture, as illustrated in Figure 3.6, range cell migration (RCM) occurs [4,5,8–10]. This is the energy from a point target response, supposed to be confined at R0, spreading along the azimuth direction when SAR is traveling. This is easily understood from the slow time-varying range R(η), as already illustrated in Figure 3.2. The phase variations associated with slant range, which changes with SAR moving, consist of constant phase, linear phase, quadratic phase, and higher-order terms. The RCM is contributed mainly from the quadratic phase term. This quadratic phase term also determines the depth of focus. For a given synthetic aperture size, the maximum quadratic phase was given in Equations 1.65 and 1.66. Detailed discussion on the phase error and its impact can be found in [4,8,10]. Due to the energy spreading, the effects of the quadratic phase, if not properly compensated, cause image defocusing, lower gain, and higher side lobes in the point target response; that is, in image quality, spatial resolution is degraded with higher fuzziness. Notice that the phase variations, up to the second order, due to the slow-time-dependent range are low-frequency components compared to the phase error in pulse-to-pulse, modulation error and nonlinearity of frequency modulation, which are high frequency since they are induced in the fast-time domain.

The total range migration during the course of synthetic aperture exposure time Ta can be estimated more explicitly from the geometry relation for the cases of low-squint and high-squint angles, as shown in Figure 3.7.

In the low-squint case, the total range of migrations is

∆ totalR RT

RuR

Tc

ac c

a= +

− ≅ +

η η η

2 2 2

2

0

2

( )

(3.10)

Echo signal - amplitude Echo signal - phase (radius)

300

200

100

0

Azi

mut

h (m

)

300

200

100

0A

zim

uth

(m)

Slant range distance (m)1.99 × 104 2.00 × 104 2.01 × 104

Slant range distance (m)1.99 × 104 2.00 × 104 2.01 × 104

0.0 0.2 0.4 0.6 0.8 1.0 –3 –2 –1 0 1 2 3(b)(a)

FIGURE 3.5 (See color insert.) Typical echo signal after demodulation: amplitude (a) and phase (b).

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while for high squint we have

∆ totalRuR

Ta c≅2

02η (3.11)

where the exposure time was given previously as

TR

L uac

a sq c

= 0 886.( )

cos ,

λ ηθ (3.12)

R0

R(η)

SAR path

u

η

τ

Raw data domain

Range cell migration

Object domain

2R0/cR(η)

FIGURE 3.6 Illustration of the object domain and data domain showing the range of cell migration.

Fast time Fast time

Slow time Slow time

ΔRtotal

ΔRtotal

ηc

ηc

ηc + Ta/2 ηc + Ta

2Ta

Ta

Low squint High squint

FIGURE 3.7 Total range migrations during the course of synthetic aperture.

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The Doppler frequency is obviously also a function of the slow time. Measured at the scene center, the Doppler centroid takes the expression

fR u

R

uc

c

c

c

sq cη

η ηλη

ηη

λ ηθ

λ= − ∂

∂= − =

=

2 2 2( )( )

sin , (3.13)

and the Doppler rate is

aR u

Rasq c

cc

= ∂∂

==

2 22

2

2 2

λη

ηθ

λ ηη η

( ) cos

( ),

(3.14)

The Doppler rate as defined is sometimes also called the azimuth FM rate [4] for a stationary target. Note that the Doppler change is only caused by SAR moving along the azimuthal direction. According to Equation 1.61, the total Doppler bandwidth is

B a Tu

df a asq c

ra

= = 0 8862

.cos ,θ

(3.15)

With the total Doppler bandwidth in mind, it is noted that the pulse repetition frequency (PRF) must be selected to meet the Nyquist criteria [11–13], namely,

f Bu

p dfsq c

ra

> = 0 8862

.cos ,θ

(3.16)

Together with Equations 3.1 and 3.2, Equation 3.16 poses constraints on the selec-tion. Under this condition, the selection of PRF must usually be compromised to meet certain requirements [1,2,14,15]. For example, in order to avoid interference from repeated pulse eclipsing, PRF must be such that

( )n

cR T

n

cR Tp p

−

−

< <+

1

2 2near far

PRF (3.17)

where n is the nth pulse, while to avoid the overlapping (indifferentiable) nadir returns, PRF must be chosen by

( )n

cR T

ch

n

cR T

chp p

−

− −

< <+ −

1

2 2 2 2near far

PRF

(3.18)

where h is the sensor height. Equations 3.17 and 3.18 together implicitly state that there is an available zone of PRF to be selected for a given antenna size, imaging swath, incident angle, and sensor height. Also, bear in mind that the PRF determines the azimuth ambiguity-to-signal ratio for a give Doppler bandwidth.

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Other important parameters associated with the two-dimensional raw data domain, such as the range bandwidth, azimuth resolution, and range resolution, can be reexpressed accordingly by taking the antenna gain pattern and squint angle effects into account. Inclined readers are referred to [4,5] for excellent treatment.

3.3 PROPERTIES OF DOPPLER FREQUENCY

3.3.1 SquinT EffEcTS

For a side-looking SAR system, there always exists a squint angle that is the angle between the antenna boresight and flight path minus 90°. The squint angle induces residual Doppler and causes the shift of the Doppler centroid, as shown in Figure 3.8, where for numerical illustration, we set fp = 1000 kHz, T = 4 × 10−6 s, and ηc = 10−6 s = T/4. Accordingly, the shift of the Doppler centroid is fdc = fp/4 Hz.

3.3.2 fm Signal aliaSing

If the sampling frequency is lower than or equal to the PRF, signal aliasing results [11,12]. This aliasing will cause a “ghost image” after azimuth compression. Beyond ΔηPRF, there exist FM signal aliased regions.

∆η ηη η η

PRF = ==

fddf

f

app

ac

(3.19)

Real part (zero squint) Real part (with squint)1.0

0.5

0.0

–0.5

0.05

0.04

0.03

0.02

0.01

Am

plitu

deM

agni

tude

0.05

0.04

0.03

0.02

0.01

Mag

nitu

de

1.0

0.5

0.0

–0.5

Am

plitu

de

–2 × 10–6 –1 × 10–6 1 × 10–6 2 × 10–60Time (s)Spectrum

–2 × 10–6 –1 × 10–6 1 × 10–6 2 × 10–60Time (s)Spectrum

–4 × 107 –2 × 107 2 × 107 4 × 1070Frequency (Hz)

–4 × 107 –2 × 107 2 × 107 4 × 1070Frequency (Hz)

FIGURE 3.8 Squint effects on the Doppler shift.

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For better inspection and to reveal the aliasing problem, we duplicate Figure 5.4 in [4] in Figure 3.9, showing the signal spectrum (real part) of a point target, a two-way antenna beam pattern as a function of θdiff, and the aliases associated with the main compressed target. The decreasing power of the aliased targets (ghosts) is obvious for the weaker gain of the antenna side lobes. The spacing between the true target and those ghosts is about equal to the pulse period (pulse repetition interval [PRI]).

Figure 3.10 is an example of aliasing where target features along the coast repeat-edly appear but with decreasing intensity along the azimuthal direction (indicated by arrows). This is exactly due to azimuth ambiguity.

When there are multiple targets randomly presented in the scene, the Doppler centroid is more uncertain in terms of its width and position. Figure 3.11 shows the resulting Doppler spreading effect, with one target at t = 0 s, five targets at t = 1 × 10–6 s, and two targets at t = 1.5 × 10–6 s, including a small squint effect. The presence of frequency spreading is caused by the mixing of the individual center frequencies corresponding to each target within a Doppler bandwidth. This poses a challenge to focusing a scene with a global Doppler centroid, as illustrated in Figure 3.12. Treatment of Doppler centroid estimation is discussed in Chapter 5.

∆ηPRF

Aliased regionAliased region

~PRI

Compressed target

Aliased targetAliased target

1.0

0.5

0.0

–0.5

–1.0

–10

–20

–30

–40

–50–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

0

–4 × 10–6 –2 × 10–6 2 × 10–6 4 × 10–60

Real part (aliasing)

TimeTwo-way antenna pattern

Am

plitu

deM

agni

tude

(dB)

θdiff (deg)

FIGURE 3.9 FM signal aliasing.

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3.3.3 PlaTfoRm VElociTy

The platform velocity u plays an essential role in SAR system operation. From the radar equation, the slant range as a function of slow time is dependent on the radar velocity, Equation 3.4. As discussed earlier, the Doppler centroid and Doppler rate are both proportional to the squared velocity. Finally, the range cell migration that

FIGURE 3.10 Azimuth ambiguity phenomena caused by signal aliasing (indicated by arrows).

Real part (with squint) Spectrum

642

0–2–4–6

–2 × 10–6 –4 × 107 –2 × 107 2 × 107 4 × 1070–1 × 10–6 1 × 10–6 2 × 10–6 0Time (s) Frequency (Hz)

Am

plitu

de

0.3

0.2

0.1

Mag

nitu

de

FIGURE 3.11 Effect of Doppler spreading.

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needs be corrected is inversely proportional to the squared velocity. Knowing the precise information of u is extremely critical in image focusing. However, it must differentiate the platform velocity and its ground velocity when the antenna beam sweeps across the target within its swath. As explained in [8], for the airborne sys-tem, because of the low flying altitude, the platform velocity is equal to the speed of the antenna beam sweeping across the target on the ground. In the spaceborne system, due to the earth’s curvature effect and rotation and noncircular satellite orbit, the relative velocity is generally not constant and nonlinear. By taking these factors into account, the effective velocity is approximated as [4]

u v vs g≅ (3.20)

Focused with global Doppler centroid Focused with refined Doppler centroid

FIGURE 3.12 Effect of Doppler frequency and rate estimation on image focusing.

Uncompensated Compensated

FIGURE 3.13 Platform velocity variation effect on image focusing.

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where vs is the platform velocity and vg is the ground velocity; both vary with orbital vector, including the position and range (Figure 3.13). Consequently, the squint angle must be scaled from the physical squint angle according to [4]

θ θ θrg

sqs

sq

u

v

v

u= = (3.21)

3.4 FM CONTINUOUS WAVE

3.4.1 Signal PaRamETERS

Referring to Figure 3.14, the transmitted signal is of the form [16,17]

s t j f t att c( ) exp= +

212

2π (3.22)

where a is the frequency change rate within the transmitted bandwidth B. With a, the intermediate frequency (IF) bandwidth Bif determines the range of the minimum and maximum delay times, τmin, τmax.

The maximum unambiguous range is determined by

Rf c

amaxmax=2

(3.23)

where fmax is the corresponding maximum sampling frequency.The received signal in one dimension is given by

s j f t a tr c( ) exp ( ) ( )τ π τ τ= − + −

212

2 (3.24)

Frequency

Time

Slope = a

Bif

τmin

τmax

PRI

B

FIGURE 3.14 FMCW signal parameters.

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The beat signal (intermediate frequency) for a single target takes the form

s t s t s t j f t a t tif t r c( ) ( ) ( ) exp ( ) (= = − − − − +* 212

22 2π τ τ τ ))

exp

+ +

= + −

f t at

j f t at at

c

c

12

212

2

2π τ

(3.25)

The delay time is range dependent; that is, it is subsequently a function of slow time:

τ η= 2R t

c

( , ) (3.26)

R t R u t( , ) ( )η η= + +02 2 2 (3.27)

For short range, the slant range may be approximated to

R t R v t Ru

Rt( , ) ( )η η η

ηη

= + + ≅ +02 2 2

2

(3.28)

with

R R uη η= +02 2( ) (3.29)

Now Equation 3.25 may be written more explicitly, leading to

s t j fR

c

f

ct at

R

c

f

ctif c

d d( , ) expη π λ λη η= −

+ −

2

2 2

−

+ −

2

2

2

22

2

2 2

22

a

cR

a

cR f t

af

ctd

d

η

η λ λ

(3.30)

where the range–Doppler frequency is induced by the time dependence in Equation 3.27 with

fR t

t

u

Rd = − ∂∂

= −2 2 2

λη

λη

η

( , ) (3.31)

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In focusing, it is essential to remove the Doppler frequency term by designing a proper filter [17–21]. The technical aspect is treated in Chapters 5 and 7.

3.4.2 objEcT To DaTa maPPing

The mapping from the object domain to the data domain is illustrated in Figure 3.15; t0 is the near-range time. The range sampling frequency is confined in

f fa

fa

r c c= − +

2 2

, , while the chirp rate is ac

fa

fa

r c c= − +

42 2

π, . Note that fr =

αos,r∣a∣PRI. A simple simulation is carried out using the parameters given in Table 3.1. Two targets are located 750 m apart. The phase and amplitude of the IF signal is plotted in Figure 3.16, where the Fourier transform was performed on the range. Relevant focusing issues are discussed in Chapter 5.

Time

Az

t0 + 3*PRI

t0 + 2*PRI

t0 + 1*PRIt0 + 1*PRI

f r (Rg sampling)

t0

PRI

t0 + 2*PRI

Rg

R

Target

Data domain Object domain(Rg, Az) = (ar , x) = (freq., time)

(N sa

mpl

es)

FIGURE 3.15 Mapping of the object domain and data domain.

TABLE 3.1FMCW Signal Simulation Parameters for Point Targets

Item Parameter Value

Transmitted bandwidth B 100 MHz

Carrier frequency fc 24 GHz

Pulse repeat interval PRI 4.0 ms

Target location R 250, 1000 m

Slant range Rrange 750~2000 m

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REFERENCES

1. Barton, D. K., Modern Radar System Analysis, Artech House, Norwood, MA, 1988. 2. Berkowitz, R. S., ed., Modern Radar: Analysis, Evaluation, and System Design, John

Wiley & Sons, New York, 1965. 3. Cook, C. E., Radar Signals: An Introduction to Theory and Applications, Artech

House, Norwood, MA, 1993. 4. Cumming, I., and Wong, F., Digital Signal Processing of Synthetic Aperture Radar

Data: Algorithms and Implementation, Artech House, Norwood, MA, 2004. 5. Curlander, J. C., and McDonough, R. N., Synthetic Aperture Radar: Systems and Signal

Processing, Wiley-Interscience, New York, 1991. 6. Cantafio, L. J., Space-Based Radar Handbook, Artech House, Norwood, MA, 1989. 7. Pillai, S. U., Li, K. Y., and Himed, B., Space Based Radar: Theory and Applications,

McGraw-Hill, New York, 2008. 8. Carrara, W. G., Majewski, R. M., and Goodman, R. S., Spotlight Synthetic Aperture

Radar: Signal Processing Algorithms, Artech House, Norwood, MA, 1995. 9. Franceschetti, G., and Lanari, R., Synthetic Aperture Radar Processing, CRC Press,

Boca Raton, FL, 1999. 10. Wahl, D. E., Eiche, P. H., Ghiglia, D. C., Thompson, P. A., and Jakowatz, C. V.,

Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Springer, New York, 1996.

11. Brandwood, D., Fourier Transforms in Radar and Signal Processing, 2nd ed., Artech House, Norwood, MA, 2012.

12. Papoulis, A., Fourier Integral and Its Applications, McGraw-Hill, New York, 1962. 13. Soumekh, M., Synthetic Aperture Radar Processing, John Wiley & Sons, New York,

1999. 14. Stimson, G. W., Introduction to Airborne Radar, Scientific Publishing, Mendham, NJ,

1998.

Az

Rg

IF signal - phase IF signal - amplitude1200

1000

800

600

400

200

00 200 400 600 800 1000 1200

Azi

mut

h sa

mpl

e (p

ix)

Slant range sample (pix) Slant range distance (m)

20

10

0

–10

Azi

mut

h (m

)

–20

0 500 1000 1500 2000

–3 –2 –1 0 1 2 3 0.2 0.4 0.6 0.8 1.0

FIGURE 3.16 (See color insert.) Phase and amplitude of the IF signal after Fourier trans-form on the range using the parameters in Table 3.1.

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15. Ulaby, F. T., Moore, R. K., and Fung, A. K., Microwave Remote Sensing: Active and Passive, vol. 2, Radar Remote Sensing and Surface Scattering and Emission Theory, Artech House, Norwood, MA, 1982.

16. Meta, A., Hoogeboom, P., and Ligthart, L., Signal processing for FMCW SAR, IEEE Transactions on Geoscience and Remote Sensing, 45(11): 3519–3532, 2007.

17. Richards, M. A., Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, New York, 2014.

18. Bracewell, R. M., The Fourier Transform and Its Applications, McGraw-Hill, New York, 1999.

19. Ferro-Famil, L., Electromagnetic Imaging Using Synthetic Aperture Radar, Wiley-ISTE, New York, 2014.

20. Gen, S. M., and Huang, F. K., A modified range-Doppler algorithm for de-chirped FM-CW SAR with a squint angle, Modern Radar, 29(11): 49–52, 2007.

21. Skolnik, M. I., Introduction to Radar Systems, McGraw-Hill, Singapore, 1981.

principles of synthetic aperture radar imaging: a system

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