The Application of the Principle of Chirp Scaling in Processing Stepped Chirps in Spotlight SAR Xin Nie Daiyin Zhu Xinhua Mao Ling Wang Zhaoda Zhu College of Information Science and Technology， Nanjing University of Aeronautics & Astronautics, 210016, Nanjing, China Email:
[email protected] Abstract—A new approach for processing stepped chirps in spotlight SAR is presented in this manuscript, which is based on exploiting the principle of chirp scaling (PCS). In particular the PCS is integrated in a Polar Format Algorithm (PFA), obtaining a more efficient solution compared with the existing interpolation based technique. The main contribution is the implementation of the azimuth scaling with the bandwidth synthesis embedded in, and it is developed dedicatedly for dealing with stepped chirps. The signal processing flow is investigated in detail, with no interpolations but only FFT’s and complex multiplications involved, and point target simulation has validated the new approach based on PCS is feasible and more efficient than the existing interpolation based approach.
I.
INTRODUCTION
The technological problem of wide bandwidth management in high resolution spotlight Synthetic Aperture Radar (SAR) systems can be solved by adopting stepped chirps, and the most common imaging approach is using the synthetic bandwidth technique[1-4] to realize the combination of sub-bands first, and then processing the synthesized signal with existing standard SAR image formation algorithms. Nevertheless, SAR sensor keeps in a moving state, the distance between which and targets is always changing between different pulses. If the sub-pulses in a burst are combined directly, that would lead to residual phase errors, which is an inevitable disadvantage as long as the bandwidth synthesis is accomplished separately .
the efficiency of the 2-D FFT. Thus the bandwidth synthesis is merged into the resampling procedure, avoiding the separate bandwidth synthesis procedure. As mostly accepted, although PFA prevails in its efficient nonplanar motion compensation capability and the compatibility with autofocus, the conventional interpolation based approach leads to great computational burden, which would also be a big problem when PFA is applied to process stepped chirps. Fortunately, some other more efficient implementations of the PFA totally free of interpolation, like the chirp z-transform (CZT) based PFA [10] and the principle of chirp scaling (PCS) [11] based PFA [12], have been addressed, with only FFT’s and complex vector multiplications involved, offering other effective channel to raise the processing efficiency for stepped chirps. The CZT approach and the PCS one are strongly related “rescaling tools” with some differences in the related implementations, but the FFT length in PCS is shorter than that in the CZT [13]. Therefore, for the sake of efficiency improvement, it is desirable to exploit PCS to implement the resampling in processing stepped chirps as well. Limited in SAR systems adopting LFM signal with only one carrier frequency, the original PCS based PFA [12] could not be used directly to process stepped chirps.
In this literature, a new approach particularly for processing stepped chirps is presented, still based on exploiting PCS and by integrating PCS in PFA. This approach is newly developed dedicatedly for dealing with stepped To this problem, it has been demonstrated in [5] that the chirps. The range and azimuth resampling are implemented conventional interpolation based Polar Format Algorithm with the range and azimuth scaling, respectively. Since the (PFA) [6-9] can be applied to process the stepped chirps via range scaling could be accomplished as [12] demonstrates, the direct resampling. The wavenumber samples for successive key point is the implementation of the azimuth scaling pulses can still be placed in an annular shape according to procedure in which the bandwidth synthesis is embedded. their corresponding range wavenumber via the polar format Besides the scaling factor, the appropriate selection of the approach, followed by two tandem 1-D interpolation to coefficients has to be made to prevent aliasing. There is no convert the data into desired rectangular data array to exploit interpolation but only FFT’s and complex multiplications involved during the processing of our method, so the new Supported by National Nature Science Foundation of China algorithm is superior to the classic interpolation based (No.60502030),and Aeronautical Science Foundation of China (No. approach for the complete avoiding of interpolation.
05D52027)
978-1-4244-2871-7/09/$25.00 ©2009 IEEE
In the next section, the waveform model of stepped chirps in spotlight SAR is given, and the range resampling procedure is explained briefly. In section III, the azimuth scaling procedure for stepped chirps is demonstrated, actually with two steps of resampling involved, and the signal processing flows are deduced in detail. In section IV, point target simulation has validated the presented approach is feasible and more efficient than the existing techniques. II.
RANGE SCALING FOR STEPPED CHIRPS
Assume that the step number is N, the time interval between two subsequent sub-chirps is PRI, the pulse repetition frequency PRF=1/PRI, and the N sub-chirps in one burst are used to construct a single wide-bandwidth chirp with centre frequency f c , total bandwidth B , chirp rate γ and pulse duration T p . Then the centre frequency of the sub-chirps should be stepped by a constant increment f step to cover the whole synthesized bandwidth, where f step = B / N .Therefore, the
center frequency of the kth (k =0, 1，2 ……N-1) sub-chirp is f c (k ) = f c + (k + 1/ 2 - N / 2) f step .
(1)
⎛ τ - 2r / c ⎞ ⎡ 4π ⎛ t −t ⎞ ⎤ sde (τ , t ) =rect ⎜ c ⎟ rect ⎜ fc (k ) ( r − rc ) ⎥ ⎟ exp ⎢− j ⎜ ⎟ ⎦ ⎝ Ta ⎠ ⎝ Tpn ⎠ ⎣ c ⎡ 4πγ ⎡ 4πγ ⎤ ⎛ 2r ⎞⎤ exp ⎢− j ( r − rc ) ⎜τ − c ⎟⎥ ⋅ exp ⎢ j 2 ( r − rc )2 ⎥ c c ⎠⎦ ⎝ ⎣ c ⎦ ⎣
where rect(⋅) represents a unit rectangular window, τ is the range time, t is the azimuth time centered at tc = xc / v , c is
the speed of light ， r = r ( t ) (or rc = rc ( t ) ) is the instantaneous distance between the antenna phase center (APC) and the target (or the scene center). Then it is a very important point in this paper that a pair of m and k corresponds to a certain t uniquely by: t = tc −
Tr aj ec to ry
z
P
θref = π 2
¦×ref
O
fw
¦× θ
y
r
Ground Target
(3)
wavenumber f w should be a function of (τ , k ) : f w = f w (τ , k ) =
4π c
⎡ 2rc ⎛ ⎢ f c (k ) + γ ⎜τ − c ⎝ ⎣
⎞⎤ ⎟⎥ , ⎠⎦
(4)
The input samples for successive transmitted pulses can be placed in an annular shape at a radial position according to their corresponding range wavenumber via the polar format approach [5]. Herein the data can be considered as stored in data collection plane OPx in Fig.1. Taking N =3 as an example, Fig.2 (a) depicts the samples of 6 pulses on the polar radii. The polar radius at the reference squint angle is denoted as U axis, and the overall center range wavenumber 4π / c ⋅ f c is denoted as f cw .
x
U
U P
Fig.1 Spotlight SAR data acquisition geometry
Fig.1 illustrates the spotlight SAR imaging geometry, where the flying trajectory is parallel to Ox and OP is the closest approach. The SAR sensor traveling at a fixed velocity v transmits a series of the bursts of the narrowbandwidth chirps defined above to illuminate a ground target area. The instantaneous squint angle and grazing angle are denoted as θ and ψ respectively. For the use of PCS, the reference angle ψ ref is defined in the yOz plane, and then θ ref is π / 2 , and the stabilized scene polar formatting (SSPF) [8] should be implemented. The azimuth coordinate of aperture center is xc . Assuming the burst number is L and aperture time is Ta , for the kth transmitted sub-chirp in the mth (m ≤ L) burst, the dechirped echo signal of a point target is
Ta + ( m ⋅ N + k ) ⋅ PRI 2
The input to range resampling would be achieved after the removal of Residual Video Phase [8], which would be still in (τ , t ) domain. Then the input sample’s corresponding range
Each sub-chirp is supposed to have a bandwidth of Bn no narrower than f step . Besides, the chirp rate is still γ , pulse duration T pn = Bn / γ .
(2)
P
trajectory(t)
trajectory(t)
4π ⋅ fc ( 2 )
T2
c 4π ⋅ f c (1)
fcw
c
f cw
sampled point
Δf w
T1
4π ⋅ f c ( 0 )
T0
c
O
x
(a) Before range resampling
O
x
(b) After range resampling
Fig.2 Data geometry in the data collection plane
Range resampling manages to produce new samples constantly spaced along U axis as in Fig.2 (b), with all virtual samples of the kth sub-pulses distributed in trapezoid Tk . The range resampling could be accomplished by range scaling [12]. Each polar pulse intersects the horizontal line for U = f cw at a point, which is depicted with the diamond mark of in Fig.2 (b) for the convenience of the following processing, and the distance between each two neighboring is a constant denoted as Δf w here:
⎛ v ⋅ PRI ⎞ 4π Δf w = ⎜ ⋅ fc ⎟⋅ ⎝ OP ⎠ c III.
A
another point of view, yy (τ , t ) represents the value for a sampled point where polar radius at time t intersects the horizontal line for U = f w _ ref in Fig.2 (b), while
⎡ 2 ⋅ OP ⎞ ⎤ ⎛ ⎢ f c (k ) + γ ⎜τ − c ⎟ ⎥ (6) ⎝ ⎠⎦ ⎣
Azimuth scaling aims to resample to convert the data into the desired rectangular data array, and the azimuth-time-domain relationship between the input and output has to be dug out. First of all, it is noteworthy that all samples are supposed to be divided into N blocks, while the kth one contains all samples belonging to the kth sub-pulse in all bursts, i.e. regarding a trapezoid as a block in Fig.2 (b). Thus, the wavenumber samples belonging to a block are horizontally aligned that are easily accessible to azimuth processing. Azimuth scaling ought to be implemented block by block. Denote the time points for pulses in the kth block as t ( k ) , which means regarding k as a constant in (3): t
T = tc − a + (m ⋅ N + k ) ⋅ PRI m=0……L-1 2
(7)
in which m is the only variable. Then the data of the kth
(
)
block should be formulated as yy τ , t ( k ) . Observing Fig.2
(b), azimuth scaling for this block is executed on the data in Tk row by row, while any row of input is consisted of L samples. For the convenience of formulating the resampled azimuth signal, azimuth scaling is divided into two steps of resampling, which will be demonstrated one by one as follows: A.
⎫ ⎪ ⎬ ⎪ ⎭
The First Step of Resampling
Just focus on the kth block in Fig.2 (b), it contains L sub-pulses and L diamond marks. PCS can be applied only if the input and output are both uniform and contain the same amount of samples, so the output grid lines for this block are constructed as the vertical lines traversing these L diamond marks, which means choosing different output grid lines for different blocks.
trajectory(t)
P
k=2
f cw
f cw
The range resampled signal of Fig.2 (b) is the input to azimuth scaling. If we denote it as yy (τ , t ) , then from
(k )
trajectory(t)
P
B
δ a (τ 1 , 2)
AZIMUTH SCALING FOR STEPPED CHIRPS
4π f w _ ref = f w _ ref (τ , k ) = c
U
U
(5)
input output
x
O
⎫ ⎪ ⎬ ⎪ ⎭
k=2
⎫ ⎪ ⎬ ⎪ ⎭
k=1
⎫ ⎪ ⎬ ⎪ ⎭
k=0
x
O
(a) for the block of k=2
(b) for all blocks
Fig.3 First resampling step of azimuth scaling
In order to formulate the output azimuth signal, Fig.3 (a) takes the block of k=2 as an example, sign the output grid lines for this block with the grayish thick line, and selects an arbitrary row of samples with the input denoted as yy τ1 , t (2) . Especially sign the leftmost input sample’s corresponding time as A, and the leftmost output sample’s corresponding time as B. It could be found that f cw fc BP (8) = = 2 ⋅ OP ⎞ AP f w _ ref (τ1 ,2 ) ⎛ f c ( 2 ) + γ ⎜ τ1 − c ⎟⎠ ⎝ This quotient features the resampling for this row as an azimuth scaling factor, according to which the input sample spacing for the azimuth time t ought to be resized.
(
)
yy (τ , t ( k ) )
h1 ( t ( k ) )
FFT
Φ scl ( f t )
h2 (t
(k )
)
Φ ins ( f t ) the first step of resampling to achieve
(
yy τ , δ at(k)
IFFT
FFT
Φ ' ( ft )
a
)
the second step of resampling to achieve
IFFT
(
yy τ , δ at (k) − Δ t
)
Fig.4 The two resampling steps of azimuth scaling for the kth block
Following the idea above, the general formulation of the expected output for the k th block could be deduced similarly. The general formulation of the expected output to
(
)
this step for the k th block is supposed to be yy τ , δa t ( k ) , with the azimuth scaling factor δa defined as δa = δ a (τ , k ) =
fc 2 ⋅ OP ⎞ ⎛ f c ( k ) + γ ⎜τ − c ⎟⎠ ⎝
(9)
Thus, the scaling flow of this step could be designed as Fig.4 depicts (in the left part), where
( )
(
h1 t ( k ) = exp[ jπγ a' t ( k ) − tck
)] 2
(10)
⎛ 1 − δa 2 ⎞ ft ⎟ Φscl ( ft ) = exp ⎜ − jπ ⎜ ⎟ δaγ a' ⎝ ⎠
( )
h2 t
(k )
= exp[− jπ δa γ a'
(t
(k )
− tck
(11)
)] 2
(12)
⎡ 1 − δa 2 ⎤ ⎛ δ −1 ⎞ f ⎥ ⋅ exp ⎜ j 2πft a tck ⎟ , (13) Φins ( ft ) = exp ⎢ jπ 2 ' t δa ⎢⎣ δa γ a ⎥⎦ ⎝ ⎠ where tck is the center time for t ( k ) , ft is the frequency
corresponding f w _ ref . Here denote the appropriate azimuth time shift as Δt , the formulation of which is educed as
Δt = Δt (τ , k ) =
γ a' is defined with the purpose of recovering the chirp rate in azimuth [14]. If γ a is the Doppler rate at aperture center, Bs is the scene bandwidth after azimuth dechirp, then γ a' is defined as γ a' =
γ aTa γa added to prevent the , with ξ = PRF /N − Bs ξ
rechirped signal from aliasing.
(
U
Δt C
δ a (τ 1 , 2)
P
trajectory(t)
B
2Δfw
f cw
Δfw
input output
⎫ ⎪ ⎬ ⎪ ⎭
k=2
⎫ ⎪ ⎬ ⎪ ⎭
k=1
⎫ ⎪ ⎬ ⎪ ⎭
k=0
O
x
Fig.5 Second resampling step of azimuth scaling
This step aims to produce new samples distributed uniformly in two dimensions to exploit the efficiency of the 2-D IFFT, which is depicted in Fig.5. The signal produced last step is the input here, and the output grid lines are selected as the vertical lines traversing the diamond marks of the first block, which is explained in the previous part. Observing Fig.5, for any arbitrary row in the kth block, it is obvious that the expected output would be achieved merely by shifting the input to the left by k Δf w along the azimuth dimension in the wavenumber domain. However, in order to formulate the expected resampled azimuth signal with the
(
known signal yy τ , δa t ( k )
)
, the azimuth-time-domain
relationship between the input and output has to be dug out. Actually, for a definite row of input and output samples, the constant shift between them in wavenumber domain implies a constant shift in azimuth time domain, and this shift is dependent on this row’s coordinate on U axis, i.e. its
)
be expressed as yy τ , δa t ( k ) − Δt . With this objective, we add a complex multiplier before the IFFT of Fig.4
⎡ Δt ⎤ Φ ' ( ft ) = exp ⎢ − j 2π ft ⎥. δ a⎦ ⎣
(16)
Observing Fig.5, the output samples are distributed uniformly in two dimensions in a rectangular, accomplishing the azimuth scaling. Obviously the bandwidth synthesis is merged into the scaling and the output is ready for the following 2-D Fourier transform, which is implemented through independent range and azimuth FFT's. It is quite useful to notice that the azimuth FFT will be canceled by the last IFFT in Fig.4, which means we can obtain the azimuth focused image for the kth block at point a in Fig.4. IV.
B. The Second Step of Resampling
(15)
Thus the expected output signal for this block is supposed to
variable corresponding to t ( k ) , sampled on PRF / N l ⋅ PRF / N ， l = 0 ,1, 2 , ......, L - 1 . (14) + ft = − L 2
k Δ f w ⋅ OP 2 ⋅ OP ⎞ ⎛ [ fc ( k ) + γ ⎜τ − ]⋅ v c ⎟⎠ ⎝
SIMULATION RESULTS
In this section, point target simulation is employed to validate the presented approach. A spotlight SAR system adopting stepped chirps in the step of 4 is assumed. Table I PARAMETERS IN THE SIMULATION Parameter Description Value step number 4 Total bandwidth 1.6GHz (range resolution 0.09375m) Sub bandwidth 480MHz Frequency step size 400MHz Center frequency for the total 10GHz bandwidth Center frequency for the sub 9.4、9.8、10.2、10.6 GHz bandwidth Forward velocity 200m/s Squint angle at aperture center 90° Scene center range at aperture 10000m center Altitude 3000m Scene radius 90m Coherent integration angle 9.16°(azimuth resolution 0.09375m) Range sampling rate 200MHz for dechirp Pulse repetition frequency 2000Hz
The classic interpolation based approach (with 8-point sinc interpolation) and our PCS based one are both used to process the simulated echo signals on the premise that the SSPF is applied. The resultant images obtained from the PCS based approach are shown in Fig. 7, which clearly indicate the arrangement of the 9 simulated targets, which are selected in an allowable scene radius to limit the quadratic phase error induced by range curvature within π 2 [8], [9].
[1]
A 88m
[2]
range(-Y)
O
[3]
azimuth(X) 88m Fig.6 Processing result of the simulated scene, using the PCS based PFA. 0
0
interpolation PCS
interpolation PCS
magnitude (dB)
magnitude (dB)
[5]
-10
-10
-20
-30
[4]
-20
[6]
-30
-40
-40
0
2
4
6
8
10
0
2
4
6
8
10
normalized azimuth(0.1442m/sample)
normalized range(0.1154m/sample)
[7]
Fig.7 Impulse response function of the point A, using the dechirped
The comparative results suggest that in terms of precision, both approaches are almost equivalent. It proves that the application of the PCS in the image formation of stepped chirps in spotlight SAR is feasible. Table II. Execution Time(sec) Approach based on interpolation Approach based on PCS 101.828000 57.704000
It is obvious that the implementation based on PCS is much more efficient than that based on interpolation, and it could save 43.3% of the processing time. Changing the parameters to simulate several times, we get the conclusion that with respect to the conventional interpolation based approach, the presented methodology could save at least 30% of the processing time. Since the FFT operation is more suitable for modern parallel DSP hardware than the interpolation, the presented methodology has good popularization using value in some engineering tasks in the presented high resolution SAR systems adopting stepped chirps. V.
CONCLUSIONS
The innovative contribution of this manuscript with respect to [12] is the extension of application extent of the PCS in processing stepped chirps; meanwhile, it could be regarded that the algorithm in [12] corresponds to a particular situation when the step number N=1. Clearly, the merit of the new algorithm over the classic interpolation based technique is the improvement of processing efficiency, since no interpolation but only FFT’s and vector multiplications are required during the processing. Hence for the sake of efficiency improvement, it has good popularization using value in some engineering tasks in the presented high resolution SAR systems adopting stepped chirps. REFERENCES
[8]
[9]
[10]
[11] [12]
[13]
[14]
Richard T. Lord, and Michael R. Inggs, “High Resolution SAR Processing Using stepped-Frequencies,” Geoscience and Remote Sensing, IGARSS '97. Remote Sensing-A Scientific Vision for Sustainable Development, IEEE International, pp.490～492, 1997. Andrew J. Wilkinson, Richard T. Lord, and Michael R. Inggs, “Stepped-Frequency Processing Reconstruction of Target Reflectivity Spectrum,” Communications and Signal Processing, 1998, COMSIG '98, pp.101-104, Sep. 1998. Alfred Wahlen, Helmut Essen, Thorsten Brehm, “High Resolution Millimeterwave SAR,” European Radar Conference, Amsterdam, pp.217- 220, 2004. H. Schimpf, A. Wahlen, H. Essen, “High range resolution by means of synthetic bandwidth generated by frequency-stepped chirps,” Electronics Letters, Vol.39, no.18, pp.1346-1348, Sept. 2003. Armin W. Doerry, SAR Processing with Stepped Chirps and Phased Array Antennas. Sandia National Laboratories, Technical Report, September 2006. J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Transactions on Aerospace and Electronic systems, vol. 16, no. 1, pp. 23-52, January 1980. D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Development in radar imaging,” IEEE Transactions on Aerospace and Electronic Systems, vol. 20, no.4, pp.363-400, July 1984. W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing algorithms. Norwood, MA: Artech House, 1995. C. V. Jokowartz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach. Kluwer Academic Publishers, 1996. Daiyin Zhu, Zhaoda Zhu, “Range resampling in the polar format algorithm for spotlight SAR image formation using the chirp ztransform,” IEEE Transactions on Signal Processing, vol. 55, no. 3, pp.1011-1023, March 2007. A. Papoulis, Systems and Transforms with Application in Optics, New York: McGraw-Hill, 1968, pp. 203-204. Daiyin Zhu, “A novel approach to residual video phase removal in spotlight SAR image formation,” The IET International Conference on Radar Systems, Edinburgh, Oct. 2007. R. Lanari and G. Fornaro, “A short discussion on the exact compensation of the SAR range-dependent range cell migration effect”,IEEE Transactions on Geoscience and Remote Sensing, vol.35, pp.1446-1452, Nov.1997. Daiyin Zhu, Yong Li, Zhaoda Zhu, “A keystone transform without interpolation for SAR ground moving target imaging,” IEEE Geoscience and Remote Sensing Letters, vol. 4, no. 1, pp.18-22, Jan. 2007.
