860 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 6, NO. 4, OCTOBER 2009

The Application of the Principle of Chirp Scaling inProcessing Stepped Chirps in Spotlight SAR

Xin Nie, Daiyin Zhu, Member, IEEE, Xinhua Mao, and Zhaoda Zhu, Senior Member, IEEE

Abstract—A new approach to process stepped chirps in spot-light synthetic aperture radar is presented in this letter, which isbased on exploiting the principle of chirp scaling (PCS). In par-ticular, PCS is integrated into the polar format algorithm (PFA),obtaining a more efficient solution compared with the existingpolar interpolation technique. The main contribution of this letteris the implementation of azimuth scaling, in which the bandwidthsynthesis is embedded. The algorithm is developed dedicatedly fordealing with stepped chirps. The signal processing flow is inves-tigated in detail, in which no interpolations but only fast Fouriertransform and complex multiplications are involved. Point-targetsimulation has validated the new approach and indicated thatit is more efficient than the classic interpolation-based one. Theachieved computational gain measured in execution time is around25%–30%.

Index Terms—Azimuth scaling, polar format algorithm (PFA),principle of chirp scaling (PCS), spotlight synthetic aperture radar(SAR), stepped chirps.

I. INTRODUCTION

THE technological problem of very large bandwidthmanagement in high-resolution spotlight-synthetic-

aperture-radar (SAR) systems can be solved by using thesynthetic-bandwidth technique [1]–[4]. The subband signalscan be combined first and then processed with existing standardSAR image-formation algorithms. However, since the distancebetween the SAR sensor and the target keeps changing duringthe coherent integration interval, if the subpulses in a burst arecombined directly, the residual phase error is always inevitable.

To this problem, it has been demonstrated in [5] that theconventional interpolation-based polar format algorithm (PFA)[6]–[9] can be applied to process the stepped chirps via directresampling. In [5], the wavenumber samples for successivepulses are polar formatted in an annular shape according to theircorresponding range wavenumber coordinates, and two tandem1-D interpolations will convert the data into the desired rectan-gular data array to exploit the efficiency of the 2-D fast Fouriertransform (FFT). Thus, the bandwidth synthesis is incorporatedinto the resampling procedure, avoiding the residual phase errorcaused by the separately executed bandwidth synthesis.

Manuscript received June 29, 2008; revised September 27, 2008 andJanuary 16, 2009. First published September 18, 2009; current version pub-lished October 14, 2009. This work was supported by the Aeronautical ScienceFoundation under Contract 20080152004.

The authors are with the College of Information Science and Technology,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China(e-mail: niexingirl@hotmail.com; zhudy@nuaa.edu.cn; xinhua@nuaa.edu.cn;zzdee@nuaa.edu.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LGRS.2009.2027212

As is mostly accepted, the interpolation procedure in PFA isundesired, which would also be problematic when the PFA isapplied to process stepped chirps. Recently, there have beenmore efficient implementations of PFA, using, for example,the chirp z-transform (CZT) [10], [11] and the principle ofchirp scaling (PCS) [12]–[14], in which no interpolations butonly FFTs and complex multiplications are involved. Both ofthem might also provide channels to improve the efficiency inprocessing stepped chirps.

The CZT approach corresponds to a particular implemen-tation of the PCS one. They are strongly related rescalingtools [15]. The PCS approach exploits more the property ofthe linear frequency-modulated (LFM) signal, which also lim-its its validity within the scope of LFM transmission signal.Moreover, this approach requires oversampling of the echosignal to prevent spectrum aliasing during the scaling operation.Fortunately, this request is almost always satisfied in practicalSAR systems. The CZT approach is more flexible because itis totally independent of the nature of the transmitted signal,and it does not require oversampling. However in terms ofcomputation, the CZT approach is, in general, less efficientthan the PCS one (for sensors using LFM chirps) because ofthe mandatory zero augmenting, which requires the FFT lengthto be twice longer than that in the PCS implementation. Due tothis fact, we will use PCS to process stepped chirps in spotlightSAR in this letter.

The presented approach is developed dedicatedly for deal-ing with stepped chirps. Range and azimuth resamplings areimplemented with range and azimuth scalings, respectively.The range scaling could be accomplished as demonstrated in[13], [14]. Therefore, the main contribution in this work is theimplementation of the azimuth scaling procedure, in which thebandwidth synthesis is embedded. Since there is no interpola-tion but only FFTs and complex multiplications are involved,the new algorithm is superior to the existing polar interpolationtechnique for stepped-chirp processing.

In the next section, the signal model of stepped chirps inspotlight SAR is discussed and based on which the range resam-pling procedure is briefly explained. In Section III, the azimuthscaling procedure is presented, and the processing chain isdescribed in detail. The point-target simulation in Section IVvalidates the presented approach. The achieved computationalgain measured in execution time is 25%–30%.

II. RANGE SCALING OF STEPPED CHIRPS

The time–frequency relationship for stepped chirps is shownin Fig. 1. The number of subchirps is N , and the time interval

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NIE et al.: APPLICATION OF THE PRINCIPLE OF CHIRP SCALING 861

Fig. 1. Time–frequency relationship for stepped chirps.

Fig. 2. Spotlight SAR data acquisition geometry.

between two subsequent subchirps is T0. The N subchirpsin one burst represent the decomposition of a single wide-bandwidth chirp with center frequency fc, total bandwidth B,chirp rate γ, and pulse duration Tp. The center frequency of thekth (k = 0, 1, 2, . . . . . . , N − 1) subchirp is

fck = fc + (k + 1/2 − N/2)fstep (1)

where fstep = B/N . Each subchirp is assumed to have a band-width of Bn = fstep for simplicity.

Fig. 2 shows the spotlight SAR imaging geometry, wherethe flying trajectory is parallel to Ox. The closest approachis OP ′, and its projection onto the xy plane is OP . The SARsensor traveling at a fixed velocity v transmits a series of burstsof subchirps defined as above to illuminate the ground targetarea. The along-track coordinate of the aperture center is xc.The instantaneous squint and grazing angles are denoted as θand ψ, respectively. For the use of PCS, the reference grazingangle ψref is defined in the yOz plane, and the reference squintangle θref must be π/2. On this condition, we implement thestabilized-scene polar interpolation technique [8].

Assuming that the burst number is L, the kth transmittedsubchirp in the mth (m ≤ L) burst is

st(τ, t) = st(τ, k;m) = rect(

t − tcTa

)· rect

(τ

Tpn

)

· exp(j2πfckτ) · exp(jπγτ2) (2)

where rect(·) is a rectangular window, t is the azimuth timecentered at tc = xc/v, Ta is the aperture time, τ is the rangetime, and Tpn = Bn/γ is the pulse duration of each subchirp.

Fig. 3. Sampled data geometry in the KxKy plane. (a) Before range resam-pling. (b) After range resampling.

Note that a pair of m and k uniquely determines a specificazimuth time instant t

t = tc −Ta

2+ (m · N + k) · T0,

k = 0, 1, 2, . . . , N − 1; m = 0, . . . , L − 1. (3)

When the dechirp-on-receive approach is employed, the in-put to range resampling would be in the (τ, t) domain. Therange wavenumber coordinate KR of the sample at (τ, t) is

KR(τ, k) =4π

c

[fck + γ

(τ − 2rc(t)

c

)](4)

where c is the light speed and rc(t) is the instantaneous distancebetween the SAR sensor and the scene center. Then, the sam-ples could be polar formatted in an annular shape according totheir range wavenumber coordinates [5].

The ground plane is usually selected as the image displayplane. Let Kx and Ky be the wavenumber variables withrespect to x and y in Fig. 2, respectively. Projection of thepolar-formatted data onto the KxKy plane results in a reason-able 2-D data storage format. Taking N = 3 as an example,Fig. 3(a) shows the samples of six pulses on the polar radii, withKc = (4π/c)fc cos ψref . Each pulse uniquely corresponds toan instantaneous look direction that is represented by the dottedline, while the solid segments identify where the sampled dataare located in the wavenumber domain.

Range resampling could be accomplished by range scaling[13], [14]. As shown in Fig. 3(b), all the output samples ofthe kth subpulses are supposed to distribute in trapezoid Tk

and are horizontally aligned, which are easily accessible toazimuth processing. The intersection of each polar radius andthe horizontal line Ky = Kc is represented by the diamondmark �. The distance between two adjacent diamond marks isa constant, which is denoted as ΔKx

ΔKx =v · T0

r0 · cos ψref· Kc (5)

where r0 is the length of OP ′. The variable ΔKx will be usefulwhen interpreting azimuth scaling, in which the diamond markswill be used as the reference for constructing the output grids.

862 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 6, NO. 4, OCTOBER 2009

Fig. 4. Input and output for the block of k = 2 in the first step of azimuthscaling.

III. AZIMUTH SCALING FOR STEPPED CHIRPS

The range-resampled signal is the input to azimuth scal-ing, which should be performed for each resampled rangewavenumber coordinate to convert the data into the desiredrectangular data array. If a trapezoid is regarded as a block inFig. 3(b), azimuth scaling ought to be implemented block byblock.

Denote the azimuth time in the kth block as t(k). Theformulation of t(k) could be obtained by consulting (3)

t(k) = tc −Ta

2+ (m · N + k) · T0, m = 0, . . . , L − 1

(6)

in which k is a constant and t(k) varies with m. The input dataof the kth block are denoted by y(τ, t(k)). They represent thesamples in Fig. 3(b), where the polar radii for time t(k) inter-sect the horizontal line for each resampled range wavenumbercoordinate

Ky =4π

c

[fck + γ

(τ − 2r0

c

)]cos ψref . (7)

Azimuth scaling is divided into two steps as follows.

A. First Step

This step aims to make the samples in one block distributeuniformly in a rectangle. The appropriate vertical output gridlines will be selected separately for each block. Without lossof generality, Fig. 4 takes the block of k = 2 as an example toillustrate the input and the expected output, while its verticaloutput grid lines are indicated by the broad gray lines. Foran arbitrary block, it contains L subpulses that correspond toL diamond marks. Then, we can consider the vertical linestraversing these L diamond marks as the output grids for thisblock. In order to explain the relationship between the inputand output samples in the azimuth time domain, an arbitraryrow in Fig. 4 is selected, and the input samples are representedby y(τ, t(2)). The azimuth time instants for an arbitrary inputsample marked as point A and the corresponding output samplemarked as point B satisfy

|BP ||AP | =

fc

fc2 + γ(τ − 2r0

c

) . (8)

Fig. 5. Two resampling steps of azimuth scaling for the kth block.

The ratio in (8) features the azimuth resampling procedurewith a scaling factor, according to which the sampling interval(for the azimuth time t(2)) ought to be resized.

In general, for the kth block, the expected output is sup-posed to be y(τ, δat

(k)), and the azimuth scaling factor δa isdefined as

δa = δa(τ, k) =fc

fck + γ(τ − 2·r0

c

) . (9)

Thus, by consulting [13] and [14], the processing chain ofthis step could be shown by the left part of Fig. 5, where

h1

(t(k)

)= exp

[jπγ′

a

(t(k) − tck

)2]

(10)

Φscl(ft) = exp(−jπ

1 − δa

δaγ′a

f2t

)(11)

h2

(t(k)

)= exp

[−jπδaγ

′a

(t(k) − tck

)2]

(12)

Φins(ft) = exp(

jπ1 − δa

δ2aγ

′a

f2t

)exp

(j2πft

δa − 1δa

tck

)

(13)

in which tck is the midpoint for t(k) and ft is the frequencyvariable corresponding to t(k), which is sampled on

ft = −fP /N

2+

l · fP /N

L, l = 0, 1, 2, . . . , L − 1 (14)

where fP is the pulse repetition frequency. In (10)–(13), thevariable γ′

a is defined with the purpose of recovering the chirprate in azimuth [16]. If γa is the Doppler rate at the aperture cen-ter, and Bs is the scene bandwidth after azimuth dechirp, thenγ′

a is defined as γ′a = γa/ξ, where ξ = (γaTa)/(PRF/N −

Bs) is used to prevent the rechirped signal from aliasing in thespectrum domain.

NIE et al.: APPLICATION OF THE PRINCIPLE OF CHIRP SCALING 863

Fig. 6. Sampled data geometry for azimuth scaling. (a) In the first resamplingstep. (b) In the second resampling step.

B. Second Step

The output samples produced in the first step are input to thesecond step. As shown in Fig. 6(a), they are distributed on atuft of parallel lines orthogonal to the Kx axis, and the spacebetween two neighboring lines is ΔKx. Although samplesbelonging to one block distribute uniformly in a rectangle, thesamples of different subpulses belonging to the same burst arenot aligned with respect to their Kx coordinate. Hence, thepurpose of this step is to convert all the samples in differentblocks to distribute uniformly in a rectangle, as shown inFig. 6(b).

Note that, for the kth block, any input sample and its cor-responding output sample of the second step are spaced by aconstant of kΔKx in the Kx dimension of the wavenumber do-main. For a definite resampled range wavenumber coordinate,the output signal is an azimuth time shifted version of the inputsignal. Let Δt denote the amount of azimuth time shift. It willbe derived in the following.

Fig. 6(b) still takes the block of k = 2 as an example toexplain the relationship between the input and output samplesin the azimuth time domain, and the input is represented byy(τ, δat

(2)). The azimuth time instant for an arbitrary inputsample is marked as point C, while that for the correspondingoutput sample is marked as point D. According to the similartrigonometry principle, it could be derived that

|CD|2ΔKx

=r0

4πc ·

[fc2 + γ

(τ − 2·r0

c

)] . (15)

The azimuth time shift from points C to D should be|CD|/v.

Following this idea, Δt for the kth block could be ex-pressed as

Δt(τ, k) =kΔKx · r0

4πc ·

[fck + γ

(τ − 2·r0

c

)]· v

. (16)

Thus, the expected output should be y(τ, δat(k) − Δt). To this

end, we add a complex multiplier before the inverse FFT (IFFT)in Fig. 5

Φ′(ft) = exp[−j2πft

Δt

δa

]. (17)

TABLE IPARAMETERS IN THE SIMULATION

After this resampling step is accomplished for each resam-pled range wavenumber coordinate and block, all the out-put samples are distributed uniformly in a rectangle [referto Fig. 6(b)], accomplishing azimuth scaling. The bandwidthsynthesis is embedded into the scaling procedure, and theoutput is ready for the following 2-D Fourier transform, whichis implemented through independent range and azimuth FFTs.Note that the azimuth FFT will be canceled by the last IFFTin Fig. 5, which means that we can obtain the azimuth focusedimage at point a in Fig. 5.

IV. SIMULATION RESULTS

In this section, point-target simulation is employed to vali-date the presented approach. A spotlight SAR system adoptingstepped chirps is assumed, and the step number is 4. Echo sig-nals with the two echo receiving approaches, i.e., the dechirp-on-receive approach and the direct sampling approach, are bothsimulated. The parameters are listed in Table I.

/tablegraphic_alt> The classic interpolation approach (usingeight-point sinc kernel) and the presented PCS approach areboth used to process the simulated echo signals on the premisethat stabilized-scene polar formatting is applied.

The focused images obtained from the PCS approach areshown in Fig. 7, which clearly indicate the arrangement ofnine simulated targets. The imaged-scene radius is selectedto limit the quadratic phase error induced by range curvaturewithin π/2 [8], [9].

For each algorithm, the impulse response functions (IRFs)in the range and azimuth dimensions of a selected point target,point A on the scene border, are shown in Figs. 8 and 9 for thedechirp and direct sampling cases, respectively.

864 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 6, NO. 4, OCTOBER 2009

Fig. 7. Processing result of the entire simulated scene, using the PCS-basedPFA. (a) For the dechirped signal. (b) For the chirped signal.

Fig. 8. IRF of point A for the dechirped signal.

Fig. 9. IRF of the point A for the chirped signal.

TABLE IIPOINT-TARGET IMPULSE RESPONSE CHARACTERISTICS

FOR POINT A FOR THE DECHIRPED SIGNAL

TABLE IIIPOINT-TARGET IMPULSE RESPONSE CHARACTERISTICS

FOR POINT A FOR THE CHIRPED SIGNAL

The measured resolution, peak sidelobe ratio, and inte-grated sidelobe ratio in both dimensions are summarized inTables II and III.

Comparing the results, we can claim that the theoretical res-olution is always achieved in both dimensions. The differencebetween the IRFs obtained via these two approaches is trivial,which suggests that, in terms of precision, both approaches arealmost equivalent. We also get the conclusion that, with respectto the conventional interpolation-based approach, the presentedmethodology could save at least 25% of the execution time.

V. CONCLUSION

In this letter, PCS was applied in processing stepped chirpsin spotlight SAR imaging, obtaining a more efficient solutioncompared with the classic interpolation-based technique. Thenew algorithm might be preferable in real-time processing inpractical SAR systems since there is no interpolation but onlyFFTs and complex multiplications involved. In particular, whenimplemented in parallel digital signal processing devices, thenew algorithm is cost effective since the FFT operation is moresuitable for parallel processing than the interpolation approach.Finally, note that our algorithm is reduced to the one proposedin [13] and [14] when the step number N is chosen to be one.

REFERENCES

[1] R. T. Lord and M. R. Inggs, “High resolution SAR processing usingstepped-frequencies,” in Proc. IEEE IGARSS. Remote Sensing, 1997,pp. 490–492.

[2] A. J. Wilkinson, R. T. Lord, and M. R. Inggs, “Stepped-frequency process-ing reconstruction of target reflectivity spectrum,” in Proc. COMSIG,Sep. 1998, pp. 101–104.

[3] A. Wahlen, H. Essen, and T. Brehm, “High resolution millimeterwaveSAR,” in Proc. Eur. Radar Conf., Amsterdam, The Netherlands, 2004,pp. 217–220.

[4] H. Schimpf, A. Wahlen, and H. Essen, “High range resolution by meansof synthetic bandwidth generated by frequency-stepped chirps,” Electron.Lett., vol. 39, no. 18, pp. 1346–1348, Sep. 2003.

[5] A. W. Doerry, “SAR processing with stepped chirps and phased arrayantennas,” Sandia Nat. Lab., Albuquerque, NM, Sep. 2006.

[6] J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans.Aerosp. Electron. Syst., vol. AES-16, no. 1, pp. 23–52, Jan. 1980.

[7] D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio,“Development in radar imaging,” IEEE Trans. Aerosp. Electron. Syst.,vol. AES-20, no. 4, pp. 363–400, Jul. 1984.

[8] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Syn-thetic Aperture Radar: Signal Processing Algorithms. Norwood, MA:Artech House, 1995.

[9] C. V. Jokowartz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, andP. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A SignalProcessing Approach. Norwell, MA: Kluwer, 1996.

[10] D. Zhu and Z. Zhu, “Range resampling in the polar format algorithm forspotlight SAR image formation using the chirp z-transform,” IEEE Trans.Signal Process., vol. 55, no. 3, pp. 1011–1023, Mar. 2007.

[11] Y. Tang, M. Xing, and Z. Bao, “The polar format imaging algorithm basedon double chirp-Z transforms,” IEEE Geosci. Remote Sens. Lett., vol. 5,no. 4, pp. 610–614, Oct. 2008.

[12] A. Papoulis, Systems and Transforms With Application in Optics.New York: McGraw-Hill, 1968, pp. 203–204.

[13] D. Zhu, “A novel approach to residual video phase removal in spot-light SAR image formation,” in Proc. IET Int. Conf. Radar Syst.,Edinburgh, U.K., Oct. 2007, pp. 1–4.

[14] D. Zhu, S. Ye, and Z. Zhu, “Polar format algorithm using chirp scalingfor spotlight SAR image formation,” IEEE Trans. Aerosp. Electron. Syst.,vol. 44, no. 4, pp. 1433–1448, Oct. 2008.

[15] R. Lanari and G. Fornaro, “A short discussion on the exact com-pensation of the SAR range-dependent range cell migration effect,”IEEE Trans. Geosci. Remote Sens., vol. 35, no. 6, pp. 1446–1452,Nov. 1997.

[16] D. Zhu, Y. Li, and Z. Zhu, “A keystone transform without interpolation forSAR ground moving target imaging,” IEEE Geosci. Remote Sens. Lett.,vol. 4, no. 1, pp. 18–22, Jan. 2007.