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Scale transform algorithm used in FMCW SAR data processing
Journal of Systems Engineering and Electronics Vol. 18, No. 4, 2007, pp. 745-750
Scale transform algorithm used in FMCW SAR data processing Jiang Zhihong, Kan Huangfu & Wan Jianwei School of Electronic Science and Engineering, National Univ. of Defense Technology, Changsha 410073, P. R. China (Received December 21, 2006)
Abstract: The frequency-modulated continuous-wave (FMCW) synthetic aperture radar (SAR) is a light-weight, cost-effective, high-resolution imaging radar, which is suitable for a small flight platform. The signal model is derived for FMCW SAR used in unmanned aerial vehicles (UAV) reconnaissance and remote sensing. An appropriate algorithm is proposed. The algorithm performs the range cell migration correction (RCMC) for continuous nonchirped raw data using the energy invariance of the scaling of a signal in the scale domain. The azimuth processing is based on step transform without geometric resampling operation. The complete derivation of the algorithm is presented. The algorithm performance is shown by simulation results.
Keywords: FMCW SAR, radar imaging, scale transform, step transform.
1. Introduction Frequency-modulated continuous-wave synthetic aperture radars (FMCW SAR) are a kind of small, light-weight, cost-effective, high-resolution imaging radar[1]. They are suitable for small-scale application, such as observation of enemy lines with unmanned aerial vehicles (UAV). The FMCW SAR will be widely used in the future. In the initial application, the slant range of FMCW SAR is very short[2,3] . The image formation algorithm cannot be stressed. The basic signal model and signal processing flow for UAV-borne FMCW SAR have been discussed in Ref. [4], but the range cell migration (RCM) which is the key of an imaging algorithm has not been considered. Dechirp-on-receive is often used in FMCW radar to obtain good range resolution[4] . On the one hand, the method can reduce the range sampling frequency; on the other hand, it can obtain high range resolution using fast Fourier transformation (FFT). Some processing algorithms suited for non-chirped signal in SAR processor are polar format algorithm (PFA), range migration algorithm (RMA) and frequency scaling algorithm (FSA). The PFA[6] is attractive
since it requires only two FFT’s, but two interpolations are needed in range and azimuth. Three FFT’s and the Stolt interpolation are required for the RMA[7] . The major advantage is the complete correction of the RCM, and the disadvantage is the need of two-dimensional interpolation in the wavenumber domain. The FSA[8] makes the RCM trajectories for targets at all ranges to have the same shape as the trajectory at a reference range. The algorithm performs the RCMC for non-chirped raw data without interpolation using a new frequency scaling operation in which three chirp functions are applied. It requires only FFT’s and multiplications. The problem is the azimuth data increasing extensively in high squint mode. Especially, all the above-mentioned three algorithms are demonstrated on pulsed SAR signal model. FMCW SAR image formation algorithm using RMA is discussed in Ref. [9]. The modified range-Doppler algorithm has been used in FMCW SAR data processing[10] . This method is efficient, and in principle, solves the problems of azimuth focusing and RCMC. A disadvantage is the need of interpolation in performing RCMC. Another disadvantage is that the secondary range compression (SRC) cannot easily incorporate azimuth frequency
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Jiang Zhihong, Kan Huangfu & Wan Jianwei
dependence. The raw signal is modeled using an appropriate approximation. A new algorithm is proposed. The algorithm works with continuous wave non-chirped raw data in range. By means of scale transform and the scale function, the RCM is corrected precisely. The azimuth compression is performed by step transform [11,12] . The method processes azimuth processing without a very long reference function and geometric resampling operation.
time domain. The term may cause distortion and space-variant defocus. It turns out to be important for dechirp-on-receive pulsed SAR[6] , but it is can be negligible for FMCW SAR[13] . So Eq. (1) can be simplified as 4π · rt · ss (ta , tr ; r0 ) = C · exp −j λ (3) 4πkr 2rc exp −j · tr − c0 · (rt − rc ) c0 The following parameter K is defined as
2. The FMCW SAR signal model Having been handled with dechirp-on-receive, downconversion and A/D-conversion, the FMCW SAR signal of a point target can be expressed as ss (ta , tr ; r0 ) = C · exp −j 4π · rt · λ 2rc r · (rt − rc ) · exp −j 4πk c 0 · tr − c 0 2 r exp j 4πk · (r − r ) 2 t c c
(1)
0
where, λ is the wavelength, ta and tr are the azimuth and range time, respectively, which is also known as the slow time and fast time, kr is the chirp modulation rate, c0 is the velocity of light. The envelope of the transmitted signal and the azimuth antenna diagram are included in the constant C with constant phase but varying amplitude. This article is concerned with the phase functions required for the algorithm derivation, the amplitude is not explicitly considered. rc is the distance between the antenna and the swath center. The distance between the antenna and a point target at closest approach is r0 , while rt is the azimuth and range dependent distance to the point target 2 (2) rt (ta ; r0 ) = r02 + V 2 · (ta + tr ) where, V is the velocity of the platform. In Eq. (1), the first exponential term denotes the azimuth phase history, and the second term represents the range signal, which is a sinusoidal signal with a constant frequency value corresponding to the azimuth and range-dependent distance to the point target rt . The last exponential term represents the residual video phase (RVP) term. The RVP term is a consequence of dechirping the SAR signal in range by means of mixing with a reference function in the
K=
sin θsq m · sin θa
(4)
as the ratio between the Doppler frequency and the frequency resolution. In Eq. (4), θsq is the squint angle, θa is the azimuth beam width and m( 1) is the sampling coefficient. The Doppler frequency introduced by the continuous antenna motion is neglected when K << 1. At this position, the stop and go approximation is also valid for FMCW SAR[9] . Then, Eq. (2) can be rewritten by rt (ta , r0 ) =
r02 + v 2 · t2a
(5)
By substituting Eq. (5) into Eq. (3), the raw data of FMCW SAR can be simplified as 4π 2 2 2 ss (ta , tr ; r0 ) = C · exp −j · r0 + V · ta · λ
4πkr 2rc 2 2 2 exp −j · tr − c 0 · r0 + V · ta − rc c0 (6)
3. The scale transform for FMCW SAR processing The principle of RCMC used in the scale transform algorithm is similar to that in Ref. [8]. Having been handled using an azimuth Fourier transformation on FMCW SAR raw data, the RCM trajectories of all targets with the same range but different azimuth are equalized. But the targets with different range also have different RCM trajectories. After applying the time-scale operation to the range-Doppler signal, all targets in swath have the same RCM trajectory, which is independent on a target range, but instead on the reference range. The RCMC is completed by a bulk shift.
Scale transform algorithm used in FMCW SAR data processing Compared with other algorithms, the scale transform algorithm uses an appropriate method called scale transform to process the time-scale operation. The negligibility of the RVP term is the most important characteristic of FMCW SAR and the negligibility can be used for simplifying data processing. The algorithm starts with continuous wave non-chirped raw data. The RCM is corrected using scale transform which can be calculated by FFT. The azimuth compression is performed using step transform without geometric re-sampling. Figure 1 shows the block diagram of the scale transform algorithm for FMCW SAR.
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where, fsa is the sweep frequency of FMCW SAR and fDC is the Doppler frequency centroid. The secondary range compression term src(fa , tr ; r0 ) in Eq. (7) is defined by
2π · r0 ·kr2 · λ β 2 − 1 · t2r · · src (fa , tr ; r0 ) = exp −j· c20 β3
2π · r0 · kr3 · λ3 β 2 − 1 exp +j · · t3r · c03 β5 (9) The dependency on the factor β expresses the RCM. The factor β is dependent on the azimuth frequency and is defined by f 2 λ2 (10) β (fa ) = 1 − a · 2 4 V After making a range scale transform, Eq. (7) becomes e−j·sr ·ln tr (11) √ dtr SS(fa , sr ; r0 ) = Ss(fa , tr ; r0 ) · tr where, sr represents the range scale. The definition and properties of the scale transform are discussed in detail in Ref. [14]. By introducing the substitution a Ssk (fa , tr ; r0 ) = Ss(fa , etr ; r0 ) · etr /2
(12)
into Eq. (11), the equation can be expressed as SS(fa , sr ; r0 ) = Ssk (fa , tr ; r0 ) · e−j·sr ·tr dtr Fig. 1
The block diagram of scale transform algorithm for FMCW SAR
3.1
Processing in range
After applying the principle of stationary phase for the azimuth Fourier transformation, the raw data in range-Doppler domain can be expressed by Ss(fa , tr ; r0 ) = C · exp [−j4πr0 · βλ] · exp [−j4π · kr c0 (r0 β − rc ) · (tr − 2rc c)] ·
(7)
src (fa , tr − 2rc c0 ; r0 ) where, fa represents the azimuth frequency, which varies within the following range −
fsa fsa + fDC fa + + fDC 2 2
(8)
(13) According to Eq. (13), the scale transform may also be viewed as the Fourier transformation. A complete implementation of a discrete scale transform using FFT has been presented Ref. [15]. From the properties of the scale transform, a signal and its time-scaled signal have coefficients of scale identical within a phase factor. Thus, after multiplying the phase factor and inverse scale transform, Eq. (13) can be expressed by Ss(fa , β tr ; r0 ), which is the time-scaled signal of the raw signal in range-Doppler domain described by Eq. (7). At this time, all targets in swath have the same RCM trajectories. The RCM correction is completed by a bulk shift. The phase factor HST is defined by HST (sr ; r0 ) = exp [j · sr · ln β]
(14)
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Jiang Zhihong, Kan Huangfu & Wan Jianwei
After multiplying HST and inverse scale transform, Eq. (11) becomes
Ss (fa , tr ; r0 ) = ej·sr ·ln tr dsr = SS(fa , sr ; r0 ) · ej·sr ·ln β · √ tr −j·sr ·ln tr
fr SS(fa , ; r0 ) = β 4πr0 · β C · exp −j · λ 2kr T · (r0 − rc ) si π · · fr + β c0
e √ · ej·sr ·ln β dtr . Ss(fa , tr ; r0 ) · tr ej·sr ·ln tr e−j·sr ·ln tr √ dsr = Ss(fa , β ·tr ; r0 )· √ ·dtr · tr tr ej·sr ·ln tr √ dsr = Ss (fa , β · tr ; r0 ) tr
(15) Inverse scale transform can also be calculated using FFT[15] . The above-mentioned equation proves that the time-scaled signal of Eq.(7) is obtained after correcting HST in sr domain. The time-scaled signal in range-Doppler can be rewritten as Ss(fa , β · tr ; r0 ) = 4πr0 · β C · exp −j · λ 2rc 4π · kr r0 − rc · β · tr − · exp −j c0 β c0 2rc ; r0 src fa , β · tr − c0
range processing is completed. At this time, the signal can be written as
3.2 Compression in azimuth The main application background of FMCW SAR is small flight platforms such as UAV. On the one hand, motion compensation is more difficult. On the other hand, the real time image formation is often necessary. The step transform[11,12] algorithm is suitable for processing FMCW SAR azimuth data, because it offers advantages, such as platform motion compensation, multilook processing, minimal energy loss, memory economy, and easy to implement real time processing. After making an inverse azimuth Fourier transformation, the signal Eq.(19) becomes sS (ta , fr ; r0 ) = C · exp j · π · ka · t2a · T 2kr si π · · fr + · (r0 − rc ) β c0
(16)
At this position, the secondary range compression can be performed together with the bulk shift since the RCM trajectories of all targets are equalized. The swath of FMCW SAR is often much narrower than pulsed SAR. So the approximation ro ≈ rc is appropriate. The phase multiplication for secondary range compression is given by HSRC (fa , tr ; rc ) =
2 2π · rc · kr2 · λ β 2 − 1 2 · rc exp +j · · · tr · β − · c02 β3 c0
3 2π · rc · kr3 · λ3 β 2 − 1 2 · rc exp −j · · · tr · β − c03 β5 c0
(19)
(20)
where,
2V 2 (21) λ · r0 is the chirp modulation rate of the azimuth signal. The azimuth processing starts in ta − fr domain with subaperture formation. The data after making the range processing is divided into separate blocks with small azimuth extension. The deramp reference in the frequency versus time plane is an overlapped saw tooth function, as shown in Fig. 2. ka = −
(17) while the bulk range shift phase function HBV is given by HBV (fa , tr ) = 4π · kr 1 2rc exp j − 1 · β · tr − · rc · c0 β c0
(18)
After making the secondary range compression, the bulk RCM shift and the last FFT operation in range,
Fig. 2
signal and overlapped swooth subaperture reference
A segment of the saw tooth reference, centered at ta = nM ∆, is used to deramp the input signal. The
Scale transform algorithm used in FMCW SAR data processing reference is given by 2 HDER (ta ) = exp −jπka · (ta − nM ∆) nM ∆ − T /2 ta nM ∆ + T /2
(22)
where, T is the length of the subaperture, ∆ is the azimuth sampling interval, M denotes the samples between two adjacent subapertures center, and n represents the sequence number of a subaperture. Mixing an incoming linear FM ramp with the saw tooth reference gives rise to a step-like waveform in the frequency versus space plane, as shown in Fig. 3.
749
by means of a fine-resolution FFT (FRFFT) along a diagonal line. The final resolution of the step transform algorithm is the same as conventional algorithms such as matched filtering via fast convolution[11] .
4. Simulation results Point target simulation is carried out to verify the validity of the proposed algorithm. The parameters used in the raw data generation are listed in Table 1. The parameters are similar to the FMCW SAR demonstrator system developed by DELFT university of Technology of Netherlands described in Ref. [4]. Table 1
Fig. 3
System parameters
Wavelength
0.03 m
Length of Synthetic aperture
149.9 m
Sweep frequency
1 kHz
Reference range
3 000 m
Platform velocity
60 m/s
Range chirp bandwidth
500 MHz
Sampling frequency
5 MHz
Signal after deramping by overlapped saw tooth reference function
The coarse-resolution FFT (CRFFT) is performed on the constant frequency data contained in each subaperture after completion of the mixing operation. The spectrum of each deramped subaperture represents a low resolution image with duration of one subaperture period. This is shown in Fig. 4 for the case of an input signal comprising of a signal point target.
The raw data of a target is simulated using the system parameters listed in Table 1. The raw data is processed using the algorithm proposed in the article. Hanning window is applied to the raw data before processing. Figure 5 shows the two-dimensional contour plot of a single processed point target in far range.
Fig. 5
Two-dimensional contour plot of impulse response function of the target in far range.
Fig. 4
DFT coefficients versus subapertures after CRFFT
During the procedure of subaperture combination, a two-dimensional processing scheme indexed by spectral frequency and subaperture number is required, as shown in Fig. 4. To form the final image, all the spectral energy associated with a single target can then be recaptured by coherently summing the subapertures
The point target analysis results are summarized in Table 2. The measured resolutions in azimuth and range show a deviation smaller than 5 % compared with the values calculated from the processed bandwidth. The peak sidelobe ratios (PSLR) and the integrated sidelobe ratios (ISLR) show almost ideal performance.
750
Jiang Zhihong, Kan Huangfu & Wan Jianwei Table 2
Results of point analysis for the
target in far range
1992, 139(5): 343−350. [7] Carrara G, Goodman R S, Majewski R M. Spotlight synthetic aperture radar. Norwood, MA: Artech House, 1995.
azimuth
range
Resolution calculated
0.513 1 m
0.522 8 m
[8] Cafforio C, Prati C, Rocca R. SAR data focusing using seis-
Resolution measured
0.500 0 m
0.500 0 m
mic migration techniques. IEEE Trans. Aerosp. Electron.
deviation
−2.621 4 %
−4.56 %
PSLR[dB]
−30.471 0
−30.475 4
[9] Mittermayer J, Moreira A, Loffeld O. Spotlight SAR data
ISLR[dB]
−27.888 6
−27.897 1
processing using the frequency scaling algorithm. IEEE
5. Conclusions and future activities The FMCW SAR received signal model has been derived. Compared with the pulsed SAR signal, the RVP term in FMCW SAR can be neglected. According to the characteristic, a new algorithm called scale transform algorithm is proposed. The algorithm is based upon equalizing the RCM trajectories of all targets. Then the complete RCMC can be performed by bulk range shift. The azimuth processing is based on a spectral analysis approach without geometric resampling operation. The computer simulation has been developed to perform the validation of the algorithm. Future activities include the processing of practical raw data. The phase errors in the different steps of the processing will be calculated to explore the limitations of the algorithm. The computational complexity will be evaluated.
References
Trans. Geosc. Remote Sensing, 1999, 37(5): 2198−2214. [10] Meta A, Hoogeboom P. Development of signal processing algorithms for high resolution airborne millimeter wave FMCW SAR. Proceeding of IEEE Radar Conference, 2005: 326−331. [11] de Wit J J M, Meta A, Hoogeboom P. Modified range Doppler processing for FM-CW synthetic aperture radar. IEEE Geosc. Remote Sensing Letters, 2006, 3(1): 83−87. [12] Sack M, Ito M R, Cumming I G. Application of efficient linear FM matched filtering algorithms to SAR processing. IEE Proc. F, 1985, 132(2): 45−57. [13] Wu K H, Vant M R. Extensions to the step transform SAR processing technique. IEEE Trans. Aerosp. Navig. Electron., 1985, 21(2): 338−344. [14] Meta A, Hoogeboom P. Signal processing for high resolution FMCW SAR and moving target. Proceeding of IRS, 2005: 263−268. [15] Cohen L.Time-frequency analysis. Englewood Cliff, NJ: Prentice hall, 1995.
[1] Meta A, Wit J de J M, Hoogeboom P. Development of a high resolution airborne millimeter wave FM-CW SAR.
[16] Zalubas E J, Williams W J. Discrete scale transform for signal analysis. Proceeding of ICASSP, 1995: 1557−1560.
Jiang Zhihong was born in 1977. He received his
Proceeding of EURAD, 2004:209−212. [2] Yamaguchi Y, Mitsumoto M, Sengoku M. Synthetic aperture FM-CW radar applied to the detection of objects buried in snowpack. IEEE Trans. Geosci. Remote Sensing, 1994, 32(1): 11−18. [3] Conna G, Griffiths H D, Brennan P V. FMCW SAR development for internal wave imaging. Proceeding of IEEE OCEANS Conference, 1997: 73−78. [4] Meta A, de Wit J J M, Hoogeboom P. Time analysis and processing of FM-CW SAR signal.
Syst., 1991,27(2): 194−206.
Proceeding of IRS,
M. S. from National University of Defense Technology in 2003. He is a Ph. D. candidate at Shanghai University now. His research interests include communication signal processing and multiple antenna systems. E-mail: [email protected] Kan Huangfu was born in 1939. He has published several scientific papers in various periodicals. His research interests include modern signal processing and radar system design and signal processing.
2003: 263−268. [5] Hwang K C, Lee K B . Efficient weight vector representation for closed-loop transmit diversity.
IEEE Trans.
Commu., 2004, 52 (1): 9–16. [6] Stove A G. Linear FMCW radar techniques. IEE Proc. F,
Wan Jianwei was born in 1964.
He received his
Ph. D. from National university of Defense Technology in 1998. His research interests include modern signal processing and radar signal processing.
Scale transform algorithm used in FMCW SAR data processing Jiang Zhihong, Kan Huangfu & Wan Jianwei School of Electronic Science and Engineering, National Univ. of Defense Technology, Changsha 410073, P. R. China (Received December 21, 2006)
Abstract: The frequency-modulated continuous-wave (FMCW) synthetic aperture radar (SAR) is a light-weight, cost-effective, high-resolution imaging radar, which is suitable for a small flight platform. The signal model is derived for FMCW SAR used in unmanned aerial vehicles (UAV) reconnaissance and remote sensing. An appropriate algorithm is proposed. The algorithm performs the range cell migration correction (RCMC) for continuous nonchirped raw data using the energy invariance of the scaling of a signal in the scale domain. The azimuth processing is based on step transform without geometric resampling operation. The complete derivation of the algorithm is presented. The algorithm performance is shown by simulation results.
Keywords: FMCW SAR, radar imaging, scale transform, step transform.
1. Introduction Frequency-modulated continuous-wave synthetic aperture radars (FMCW SAR) are a kind of small, light-weight, cost-effective, high-resolution imaging radar[1]. They are suitable for small-scale application, such as observation of enemy lines with unmanned aerial vehicles (UAV). The FMCW SAR will be widely used in the future. In the initial application, the slant range of FMCW SAR is very short[2,3] . The image formation algorithm cannot be stressed. The basic signal model and signal processing flow for UAV-borne FMCW SAR have been discussed in Ref. [4], but the range cell migration (RCM) which is the key of an imaging algorithm has not been considered. Dechirp-on-receive is often used in FMCW radar to obtain good range resolution[4] . On the one hand, the method can reduce the range sampling frequency; on the other hand, it can obtain high range resolution using fast Fourier transformation (FFT). Some processing algorithms suited for non-chirped signal in SAR processor are polar format algorithm (PFA), range migration algorithm (RMA) and frequency scaling algorithm (FSA). The PFA[6] is attractive
since it requires only two FFT’s, but two interpolations are needed in range and azimuth. Three FFT’s and the Stolt interpolation are required for the RMA[7] . The major advantage is the complete correction of the RCM, and the disadvantage is the need of two-dimensional interpolation in the wavenumber domain. The FSA[8] makes the RCM trajectories for targets at all ranges to have the same shape as the trajectory at a reference range. The algorithm performs the RCMC for non-chirped raw data without interpolation using a new frequency scaling operation in which three chirp functions are applied. It requires only FFT’s and multiplications. The problem is the azimuth data increasing extensively in high squint mode. Especially, all the above-mentioned three algorithms are demonstrated on pulsed SAR signal model. FMCW SAR image formation algorithm using RMA is discussed in Ref. [9]. The modified range-Doppler algorithm has been used in FMCW SAR data processing[10] . This method is efficient, and in principle, solves the problems of azimuth focusing and RCMC. A disadvantage is the need of interpolation in performing RCMC. Another disadvantage is that the secondary range compression (SRC) cannot easily incorporate azimuth frequency
746
Jiang Zhihong, Kan Huangfu & Wan Jianwei
dependence. The raw signal is modeled using an appropriate approximation. A new algorithm is proposed. The algorithm works with continuous wave non-chirped raw data in range. By means of scale transform and the scale function, the RCM is corrected precisely. The azimuth compression is performed by step transform [11,12] . The method processes azimuth processing without a very long reference function and geometric resampling operation.
time domain. The term may cause distortion and space-variant defocus. It turns out to be important for dechirp-on-receive pulsed SAR[6] , but it is can be negligible for FMCW SAR[13] . So Eq. (1) can be simplified as 4π · rt · ss (ta , tr ; r0 ) = C · exp −j λ (3) 4πkr 2rc exp −j · tr − c0 · (rt − rc ) c0 The following parameter K is defined as
2. The FMCW SAR signal model Having been handled with dechirp-on-receive, downconversion and A/D-conversion, the FMCW SAR signal of a point target can be expressed as ss (ta , tr ; r0 ) = C · exp −j 4π · rt · λ 2rc r · (rt − rc ) · exp −j 4πk c 0 · tr − c 0 2 r exp j 4πk · (r − r ) 2 t c c
(1)
0
where, λ is the wavelength, ta and tr are the azimuth and range time, respectively, which is also known as the slow time and fast time, kr is the chirp modulation rate, c0 is the velocity of light. The envelope of the transmitted signal and the azimuth antenna diagram are included in the constant C with constant phase but varying amplitude. This article is concerned with the phase functions required for the algorithm derivation, the amplitude is not explicitly considered. rc is the distance between the antenna and the swath center. The distance between the antenna and a point target at closest approach is r0 , while rt is the azimuth and range dependent distance to the point target 2 (2) rt (ta ; r0 ) = r02 + V 2 · (ta + tr ) where, V is the velocity of the platform. In Eq. (1), the first exponential term denotes the azimuth phase history, and the second term represents the range signal, which is a sinusoidal signal with a constant frequency value corresponding to the azimuth and range-dependent distance to the point target rt . The last exponential term represents the residual video phase (RVP) term. The RVP term is a consequence of dechirping the SAR signal in range by means of mixing with a reference function in the
K=
sin θsq m · sin θa
(4)
as the ratio between the Doppler frequency and the frequency resolution. In Eq. (4), θsq is the squint angle, θa is the azimuth beam width and m( 1) is the sampling coefficient. The Doppler frequency introduced by the continuous antenna motion is neglected when K << 1. At this position, the stop and go approximation is also valid for FMCW SAR[9] . Then, Eq. (2) can be rewritten by rt (ta , r0 ) =
r02 + v 2 · t2a
(5)
By substituting Eq. (5) into Eq. (3), the raw data of FMCW SAR can be simplified as 4π 2 2 2 ss (ta , tr ; r0 ) = C · exp −j · r0 + V · ta · λ
4πkr 2rc 2 2 2 exp −j · tr − c 0 · r0 + V · ta − rc c0 (6)
3. The scale transform for FMCW SAR processing The principle of RCMC used in the scale transform algorithm is similar to that in Ref. [8]. Having been handled using an azimuth Fourier transformation on FMCW SAR raw data, the RCM trajectories of all targets with the same range but different azimuth are equalized. But the targets with different range also have different RCM trajectories. After applying the time-scale operation to the range-Doppler signal, all targets in swath have the same RCM trajectory, which is independent on a target range, but instead on the reference range. The RCMC is completed by a bulk shift.
Scale transform algorithm used in FMCW SAR data processing Compared with other algorithms, the scale transform algorithm uses an appropriate method called scale transform to process the time-scale operation. The negligibility of the RVP term is the most important characteristic of FMCW SAR and the negligibility can be used for simplifying data processing. The algorithm starts with continuous wave non-chirped raw data. The RCM is corrected using scale transform which can be calculated by FFT. The azimuth compression is performed using step transform without geometric re-sampling. Figure 1 shows the block diagram of the scale transform algorithm for FMCW SAR.
747
where, fsa is the sweep frequency of FMCW SAR and fDC is the Doppler frequency centroid. The secondary range compression term src(fa , tr ; r0 ) in Eq. (7) is defined by
2π · r0 ·kr2 · λ β 2 − 1 · t2r · · src (fa , tr ; r0 ) = exp −j· c20 β3
2π · r0 · kr3 · λ3 β 2 − 1 exp +j · · t3r · c03 β5 (9) The dependency on the factor β expresses the RCM. The factor β is dependent on the azimuth frequency and is defined by f 2 λ2 (10) β (fa ) = 1 − a · 2 4 V After making a range scale transform, Eq. (7) becomes e−j·sr ·ln tr (11) √ dtr SS(fa , sr ; r0 ) = Ss(fa , tr ; r0 ) · tr where, sr represents the range scale. The definition and properties of the scale transform are discussed in detail in Ref. [14]. By introducing the substitution a Ssk (fa , tr ; r0 ) = Ss(fa , etr ; r0 ) · etr /2
(12)
into Eq. (11), the equation can be expressed as SS(fa , sr ; r0 ) = Ssk (fa , tr ; r0 ) · e−j·sr ·tr dtr Fig. 1
The block diagram of scale transform algorithm for FMCW SAR
3.1
Processing in range
After applying the principle of stationary phase for the azimuth Fourier transformation, the raw data in range-Doppler domain can be expressed by Ss(fa , tr ; r0 ) = C · exp [−j4πr0 · βλ] · exp [−j4π · kr c0 (r0 β − rc ) · (tr − 2rc c)] ·
(7)
src (fa , tr − 2rc c0 ; r0 ) where, fa represents the azimuth frequency, which varies within the following range −
fsa fsa + fDC fa + + fDC 2 2
(8)
(13) According to Eq. (13), the scale transform may also be viewed as the Fourier transformation. A complete implementation of a discrete scale transform using FFT has been presented Ref. [15]. From the properties of the scale transform, a signal and its time-scaled signal have coefficients of scale identical within a phase factor. Thus, after multiplying the phase factor and inverse scale transform, Eq. (13) can be expressed by Ss(fa , β tr ; r0 ), which is the time-scaled signal of the raw signal in range-Doppler domain described by Eq. (7). At this time, all targets in swath have the same RCM trajectories. The RCM correction is completed by a bulk shift. The phase factor HST is defined by HST (sr ; r0 ) = exp [j · sr · ln β]
(14)
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Jiang Zhihong, Kan Huangfu & Wan Jianwei
After multiplying HST and inverse scale transform, Eq. (11) becomes
Ss (fa , tr ; r0 ) = ej·sr ·ln tr dsr = SS(fa , sr ; r0 ) · ej·sr ·ln β · √ tr −j·sr ·ln tr
fr SS(fa , ; r0 ) = β 4πr0 · β C · exp −j · λ 2kr T · (r0 − rc ) si π · · fr + β c0
e √ · ej·sr ·ln β dtr . Ss(fa , tr ; r0 ) · tr ej·sr ·ln tr e−j·sr ·ln tr √ dsr = Ss(fa , β ·tr ; r0 )· √ ·dtr · tr tr ej·sr ·ln tr √ dsr = Ss (fa , β · tr ; r0 ) tr
(15) Inverse scale transform can also be calculated using FFT[15] . The above-mentioned equation proves that the time-scaled signal of Eq.(7) is obtained after correcting HST in sr domain. The time-scaled signal in range-Doppler can be rewritten as Ss(fa , β · tr ; r0 ) = 4πr0 · β C · exp −j · λ 2rc 4π · kr r0 − rc · β · tr − · exp −j c0 β c0 2rc ; r0 src fa , β · tr − c0
range processing is completed. At this time, the signal can be written as
3.2 Compression in azimuth The main application background of FMCW SAR is small flight platforms such as UAV. On the one hand, motion compensation is more difficult. On the other hand, the real time image formation is often necessary. The step transform[11,12] algorithm is suitable for processing FMCW SAR azimuth data, because it offers advantages, such as platform motion compensation, multilook processing, minimal energy loss, memory economy, and easy to implement real time processing. After making an inverse azimuth Fourier transformation, the signal Eq.(19) becomes sS (ta , fr ; r0 ) = C · exp j · π · ka · t2a · T 2kr si π · · fr + · (r0 − rc ) β c0
(16)
At this position, the secondary range compression can be performed together with the bulk shift since the RCM trajectories of all targets are equalized. The swath of FMCW SAR is often much narrower than pulsed SAR. So the approximation ro ≈ rc is appropriate. The phase multiplication for secondary range compression is given by HSRC (fa , tr ; rc ) =
2 2π · rc · kr2 · λ β 2 − 1 2 · rc exp +j · · · tr · β − · c02 β3 c0
3 2π · rc · kr3 · λ3 β 2 − 1 2 · rc exp −j · · · tr · β − c03 β5 c0
(19)
(20)
where,
2V 2 (21) λ · r0 is the chirp modulation rate of the azimuth signal. The azimuth processing starts in ta − fr domain with subaperture formation. The data after making the range processing is divided into separate blocks with small azimuth extension. The deramp reference in the frequency versus time plane is an overlapped saw tooth function, as shown in Fig. 2. ka = −
(17) while the bulk range shift phase function HBV is given by HBV (fa , tr ) = 4π · kr 1 2rc exp j − 1 · β · tr − · rc · c0 β c0
(18)
After making the secondary range compression, the bulk RCM shift and the last FFT operation in range,
Fig. 2
signal and overlapped swooth subaperture reference
A segment of the saw tooth reference, centered at ta = nM ∆, is used to deramp the input signal. The
Scale transform algorithm used in FMCW SAR data processing reference is given by 2 HDER (ta ) = exp −jπka · (ta − nM ∆) nM ∆ − T /2 ta nM ∆ + T /2
(22)
where, T is the length of the subaperture, ∆ is the azimuth sampling interval, M denotes the samples between two adjacent subapertures center, and n represents the sequence number of a subaperture. Mixing an incoming linear FM ramp with the saw tooth reference gives rise to a step-like waveform in the frequency versus space plane, as shown in Fig. 3.
749
by means of a fine-resolution FFT (FRFFT) along a diagonal line. The final resolution of the step transform algorithm is the same as conventional algorithms such as matched filtering via fast convolution[11] .
4. Simulation results Point target simulation is carried out to verify the validity of the proposed algorithm. The parameters used in the raw data generation are listed in Table 1. The parameters are similar to the FMCW SAR demonstrator system developed by DELFT university of Technology of Netherlands described in Ref. [4]. Table 1
Fig. 3
System parameters
Wavelength
0.03 m
Length of Synthetic aperture
149.9 m
Sweep frequency
1 kHz
Reference range
3 000 m
Platform velocity
60 m/s
Range chirp bandwidth
500 MHz
Sampling frequency
5 MHz
Signal after deramping by overlapped saw tooth reference function
The coarse-resolution FFT (CRFFT) is performed on the constant frequency data contained in each subaperture after completion of the mixing operation. The spectrum of each deramped subaperture represents a low resolution image with duration of one subaperture period. This is shown in Fig. 4 for the case of an input signal comprising of a signal point target.
The raw data of a target is simulated using the system parameters listed in Table 1. The raw data is processed using the algorithm proposed in the article. Hanning window is applied to the raw data before processing. Figure 5 shows the two-dimensional contour plot of a single processed point target in far range.
Fig. 5
Two-dimensional contour plot of impulse response function of the target in far range.
Fig. 4
DFT coefficients versus subapertures after CRFFT
During the procedure of subaperture combination, a two-dimensional processing scheme indexed by spectral frequency and subaperture number is required, as shown in Fig. 4. To form the final image, all the spectral energy associated with a single target can then be recaptured by coherently summing the subapertures
The point target analysis results are summarized in Table 2. The measured resolutions in azimuth and range show a deviation smaller than 5 % compared with the values calculated from the processed bandwidth. The peak sidelobe ratios (PSLR) and the integrated sidelobe ratios (ISLR) show almost ideal performance.
750
Jiang Zhihong, Kan Huangfu & Wan Jianwei Table 2
Results of point analysis for the
target in far range
1992, 139(5): 343−350. [7] Carrara G, Goodman R S, Majewski R M. Spotlight synthetic aperture radar. Norwood, MA: Artech House, 1995.
azimuth
range
Resolution calculated
0.513 1 m
0.522 8 m
[8] Cafforio C, Prati C, Rocca R. SAR data focusing using seis-
Resolution measured
0.500 0 m
0.500 0 m
mic migration techniques. IEEE Trans. Aerosp. Electron.
deviation
−2.621 4 %
−4.56 %
PSLR[dB]
−30.471 0
−30.475 4
[9] Mittermayer J, Moreira A, Loffeld O. Spotlight SAR data
ISLR[dB]
−27.888 6
−27.897 1
processing using the frequency scaling algorithm. IEEE
5. Conclusions and future activities The FMCW SAR received signal model has been derived. Compared with the pulsed SAR signal, the RVP term in FMCW SAR can be neglected. According to the characteristic, a new algorithm called scale transform algorithm is proposed. The algorithm is based upon equalizing the RCM trajectories of all targets. Then the complete RCMC can be performed by bulk range shift. The azimuth processing is based on a spectral analysis approach without geometric resampling operation. The computer simulation has been developed to perform the validation of the algorithm. Future activities include the processing of practical raw data. The phase errors in the different steps of the processing will be calculated to explore the limitations of the algorithm. The computational complexity will be evaluated.
References
Trans. Geosc. Remote Sensing, 1999, 37(5): 2198−2214. [10] Meta A, Hoogeboom P. Development of signal processing algorithms for high resolution airborne millimeter wave FMCW SAR. Proceeding of IEEE Radar Conference, 2005: 326−331. [11] de Wit J J M, Meta A, Hoogeboom P. Modified range Doppler processing for FM-CW synthetic aperture radar. IEEE Geosc. Remote Sensing Letters, 2006, 3(1): 83−87. [12] Sack M, Ito M R, Cumming I G. Application of efficient linear FM matched filtering algorithms to SAR processing. IEE Proc. F, 1985, 132(2): 45−57. [13] Wu K H, Vant M R. Extensions to the step transform SAR processing technique. IEEE Trans. Aerosp. Navig. Electron., 1985, 21(2): 338−344. [14] Meta A, Hoogeboom P. Signal processing for high resolution FMCW SAR and moving target. Proceeding of IRS, 2005: 263−268. [15] Cohen L.Time-frequency analysis. Englewood Cliff, NJ: Prentice hall, 1995.
[1] Meta A, Wit J de J M, Hoogeboom P. Development of a high resolution airborne millimeter wave FM-CW SAR.
[16] Zalubas E J, Williams W J. Discrete scale transform for signal analysis. Proceeding of ICASSP, 1995: 1557−1560.
Jiang Zhihong was born in 1977. He received his
Proceeding of EURAD, 2004:209−212. [2] Yamaguchi Y, Mitsumoto M, Sengoku M. Synthetic aperture FM-CW radar applied to the detection of objects buried in snowpack. IEEE Trans. Geosci. Remote Sensing, 1994, 32(1): 11−18. [3] Conna G, Griffiths H D, Brennan P V. FMCW SAR development for internal wave imaging. Proceeding of IEEE OCEANS Conference, 1997: 73−78. [4] Meta A, de Wit J J M, Hoogeboom P. Time analysis and processing of FM-CW SAR signal.
Syst., 1991,27(2): 194−206.
Proceeding of IRS,
M. S. from National University of Defense Technology in 2003. He is a Ph. D. candidate at Shanghai University now. His research interests include communication signal processing and multiple antenna systems. E-mail: [email protected] Kan Huangfu was born in 1939. He has published several scientific papers in various periodicals. His research interests include modern signal processing and radar system design and signal processing.
2003: 263−268. [5] Hwang K C, Lee K B . Efficient weight vector representation for closed-loop transmit diversity.
IEEE Trans.
Commu., 2004, 52 (1): 9–16. [6] Stove A G. Linear FMCW radar techniques. IEE Proc. F,
Wan Jianwei was born in 1964.
He received his
Ph. D. from National university of Defense Technology in 1998. His research interests include modern signal processing and radar signal processing.