使用CS算法机载SAR高精度成像处理的运动补偿

【实例简介】
Airborne SAR processing of highly squinted data using a chirp scaling approach with integrated motion compensation
MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION the fight direction, and ts is the time at the center of the where C is a complex constant and fa is the azimuth fre- azimuth illumination path, which is related to the Doppler quency, which varies within the following range centroid fac by the following equation PRF PRE 入·ro'fas 2+f≤f≤f+ The last term in(4)corresponds to the azimuth modula tion in the frequency domain. The first exponential term in(4)shows that the frequency modulation of the range where A is thc radar wavelength. After demodulation, the chirp is now dependent on the azimuth frequency and the two-dimensional received SAR signal s(T, L; To)of a point range distance. If this variation is not considered,the target can be written as range IRF for high squint data processing will be deto cused. The modified range-frequency modulation basi s(7,r;r)=a-(了 2·R(t;ro) cally consists of two terms 2·入·(2-1) 2.R(;m)21k;o5=k xp 1"J R(G; ro) (3)wh The azimuth antenna pattern aa and the envelope a of the 入·fn transmitted pulse are slowly varying functions relative to the signal variations in the azimuth time t and range time (range delay)T. In(3), the first exponential term accounts The correction of the term ksre in the range processing is for the range chirp with frequency modulation rate k, and called secondary range compression(SRC). The tradi- for the range migration The last term in(3)is the azimuth tional chirp scaling algorithm assumes one reference range for the SrC and updates it with the azimuth frequency (Doppler)modulation, In a special case, where the squint The missing update of the SrC with range causes a phase angle and azimuth illumination time are small, the range migration in the first exponential term of (3) can be ne error in the range compression for ranges different from glected and the received signal is approximated to two the reference range. This phase error is insignificant for a quint angle up to approximately 20 in the case of one-dimensional functions. In practical cases, however, L-band spaceborne SAR systems[71 the range migration leads to a coupling between the range function The range migration in the range-Doppler domain can Due to the large time-bandwidth product of the re- be expressed ceived SAR signal, the principle of the stationary phase R(f;0)=ro·[1+a(f (8) [16] can be used to obtain a signal formulation in the wavenumber domain. In this domain the form of the azi- where a(fa) is the linear chirp scaling factor given by muth antenna pattern and the envelope of the transmitted pulse remains the same. By performing a series expansion a(fa) (9) in the range frequency and an inverse Fourer transfor mation in range, the signal formulation in the range Doppler domain is given by [ 16] In the case of airborne SAR, the scaling factor a(a)is not S(r,f;0o)=C·a 2·R(fa;ro) independent on range, so that a linear scaling in range C leads to a perfect equalization of the rCmc, if the sro term can be neglected The traditional chirp scaling ro·λ·f method performs this equalization by means of a qua dratic phase term in the range-Doppler domain. actually 2·t2√1-[(λ·f)/(2·U) the scaling consists of changing the position of the phase ·R(fa;ro minimum of each chirp signal. No explicit interpolation k(fa C The quadratic phase term of the chirp scaling intro duces a frequency offset in the range chirp signal, which can assume values as high as several megahertz for high P uint angle If the f offset is high enough so that the signal bandwidth is aliased or shifted outside of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL32.NO. 5. SEPTEMBER 1994 processed bandwidth, then the range IRF will deteriorate. C. Range Compression and Bulk rCmc In order to minimize this effect, the offset scaling factor One basic characteristic of the extended chirp scaling due to the squint angle should be removed from the orig- algorithm is that no azimuth compression is performed in inal chirp scaling term a( fa), so that the new scaling fac- the wavenumber domain. Only the range compression and tor for the processing is defined by the bulk range cell migration(for the reference range)are a'(f)=a(f)-c() performed in this domain. The removal of the azimuth ompression from this step is ncccssary, since second-or The scaling factor leads to an equalization of the range der motion compensation must be performed after range migration. However, the range positioning of the targets compression(due to the range update), but before the azi in the final image will be according to the slant range at muth compression the time te(at the center of the azimuth illumination path). The phase-correction term of the ECS algorithm in the he range positioning can be changed to that of the broad- wavenumber domain consists of the range compression side geometry (range distances according to closest ap- with the src for the reference range and an additional proach) during the slant to ground range transformation. linear phase term with respect to the range frequency The scaling factor a'(a)is now used for the range mi- which corresponds to a linear time shift in the range time gration representation in the frcqucncy domain according Using the wavenumber formulation of the SAR signal Lo( 8), L.e., R'(fa: ro)=. 11+ a'(f). The quadratic this phase correction can be calculated as phase function for RCMC can be written as HI(T, fa, rref)= exp k(fa;rsf)·a'(Jfa H2(f,;rn)=exp|-j·丌 r.[1+a'(fa) 2·R(f后 2·入·(62-1) (11) [l+a'(f)·β After this phase correction, all targets have their range 4·丌·rf·a'(fa) migration trajectories equalized to that of the reference exp range rref. Since the scaling factor for airborne SAr is exactly lincar, the only approximation made in(11)is (14) k(Ja;r0)≈k(后;rer) (12) When transforming the SAR signal back to the range- The errors caused by this approximation are analyzed in Doppler domain, a residual phase correction is applied the Appendix. Basically, if a phase error of 25 is allowed that compensates a slowly varying range-dependent azi in the range compression, high-quality images can be ob- muth phase and is given by tained for up to a 20 squint angle in the c-band mode of the E-SAR system [21 However, for higher squint an H2(r,;70)=cxpj·丌·k(;re)·[1a( gles and for the L-band mode, this error will affect the image quality. Besides the phase error, the approximation in(11) leads to an inaccurate equalization of the rCMc a'(a) (r0 The analysis in the Al that cubic term can be added to the quadratic chirp scaling 8·入·(82- exp-j phase function, so that the new chirp scaling phase cor rection is changed to (r-r) (l5) H1(r,f;r)=exp-j·丌·k(f,re)·a'() The second term in the above equation corrects a residual 2·R'( phase error that was introduced by the cubic phase term in the scaling function of(13). After the above correction the sar signal can be transformed to th azimuth and range time)to perforin accurate motion p compensation 2·R(f;rer) (13) D. Motion Compensation and Azimuth compression With the additional phase term, up to a 30 squint angle The motion errors induced by the atmospheric turbu can be processed without deterioriation of the image qual- lence are a crucial problem in most airborne SAr sys ity. At the end of this section, a quantitative analysis of tems. If not corrected, the image quality will considerably the irf for different squint angles will be presented ade [14], [23]. The main effects observed are a loss MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION of geometric resolution and radiometric accuracy, reduc- tion of image contrast, azimuth ambiguities, and geomet ric and phase distortions. Once the velocity variations of FIRST ORDER MOTION COMPENSATION KHm(t, Tre the aircraft are compensated by means of an on-line I Azimuth FFTs variation of PRF, which leads to a constant pixel spacing"CHIRP SCALING WITH QUADRATIC in azimuth, the aircraft trajectory is corrected to a straight AND CUBIC PHASE TERMS 8+H(τ,后;r) line by applying a time-domain phase-correction func I Range FFT ion. If the deviations of the aircraft trajectory are greater . RANGE COMPRESSION than one range bin then the range del ECONDARY RANGE COMPRESSION B-H2lU fa; rrep) LINEAR RANGE SHIFT FOR BULK RCMC compensated. The first-order motion compensation is de- Range IFFT fined as being the phase correction for a reference range te uncom-*PHASE CORRECTION DUE TO and it can be directly carried out with the range uncom- 8h2([,A;r CHIRP SCALING pressed data(.e, before the processing starts). The sec ond-order motion compensation includes the update of the Azimuth IFFTs phase correction as a function of the range distance SECOND OADER MOTON ERROR 6-Hn(,) Let m(r; ro)be the total phase for motion compensa- CORRECTION tion and puc(t; rrer) be the first-order motion compensa Azimuth FFr's tion, then the second-order motion compensation, which "AZIMUTH COMPRESSION 6l(,m) is applied after transforming the range-compressed SAR data to the signal domain is formulated as Azimuth IFFTs Hme(t, ro) 句·[φmc(G;7o)-φm(,ref)]}.(16) Fig. I. Block diagram of the extended chirp scaling(ECS)algorithm for high-precision airborne Sar processing. In this diagram, a constant After performing the second-order motion compensatoin Doppler centroid value is assumed for processing only a one-dimensional azimuth compression must be per formed. In the case where the squint angle is constant, the SAR Signal is transformed again to the range - Doppler do Radar PrF: 1100 Hz main. and the azimuth modulation is corrected in this do- Swath width: 3000 m main by a hyperbolic phase modulation Flight altitude: 3000 m Azimuth and range resolution: 0.3 X 2. 5 m(I look) Ha(T, fa; ro)=exp j The impulse response functions obtained by processing with traditional chirp scaling and the ecs algorithms for squint angle of 30 are shown in Fig. 2(a)and(b) (17) spectively. No weighting function was used in the pro cessing and a rectangular pulse envelope and azimuth an Fig. 1 shows the block diagram of the ECS algorithm with tenna pattern were adopted so that a two-dimensional the corresponding phase-correction functions. The extra sin (r)/x function is expected for an error-free compres computation consists of the additional phase corrections sion. In both cases, the target is located at the edge of the and of the transformation to the signal domain before azi- swath, and the reference range for processing is at the muth compression (additional azimuth IFFT's and center. This situation corresponds to the worst case for FFTS). If the azimuth time-bandwidth product is low, processing as far as the accuracy of the algorithm is con then the azimuth compression is carried out more efi- cerned. In the chirp scaling algorithm [Fig 2(a), the geo ciently by time-domain correlation approaches (e. g. sub- metric resolution of the range irF is deteriorated by ap aperture processing [13) proximately 12 percent and the peak sidelobe ratio(PSlr) is increased to 7. 3 dB(for an error free sin (r)/x function it should be 13.2 dB) E. Analysis of the Impulse response Function For the ECS algorithm, the simulation results up to 30 The following parameters were used for the simulation squint angle are almost perfect. The deterioration of the of the IrF according to the specifications of the E-sAr azimuth and range resolution are al ways lower than 1.7 system of dir in the c-band mode [10] and 1. 4 percent, respectively The PSlr is not worse than 12. 6 dB and the measured phase accuracy is better than Radar wavelength: x=0.0566 m Sensor velocity:U=75 m/s For higher squint angles, a. more accurate algorithm Antenna depression angle 0;=37 proposed in [7] should be used, which requires an addi Squint angle: variable, from -30 to 30 in 5 steps. tional transformation to the wavenumber domain for ap- Chirp frequency modulation rate: k =2.10H/ plying a cubic phase correction before the chirp scaling operation is carried out in the range-Doppler domain 1034 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL 32, NO 5, SEPTEMBER 1994 0000010CC000000 range timc[μs]…> ange(samples expanded by 4)---> ange time Iμs】-> E 主时(级以 0000cc0c000⊙0000 是影经 百 range frequency MHz ]-- range(samples expanded by 4)---> Fig. 2. Contour plot of the impulse response function (IRF) for 30 squint angle (a)iRF obta ned with the chirp scaling algorithm using a quadratic hase for equa!ization of the range cell migration. (b) IRF obtained with he extended chirp scaling algorithm using a quadratic and cubic phase term for equalization of the range cell Migration. The sensor and processing range m I paramctcrs of the E-SAR system are used for the simulation(2.5 X0.3 m (d) resolution in range and azimuth with 1-look processing II. ACCOMMODATION OF DOPPLER CENTROID VARIATIONS WITH RANGE Due to the large variation of the look angle in the air bome Sar geometry, the Doppler centroid can vary sc 2500 m eral hundred hertz from the near to the far range. The update of the Doppler centroid avoids a loss of signal-to noise ratio, an azimuth defocusing, and azimuth amh Fig. 3. Cuntuur plot of the signal of four puint targets in the different do guities. In the CS algorithm, this variation could be ac- Range-Doppler domain.(c)Wavenumber domain. (d)Range-Doppler do- commodated in an inaccurate and inefficient way by a main. (e) Signal domain (final image). The sensor and processing param eters of the E-SAR system were used for the simulation. A squint angle of block processing in the range direction with an overlap go was used, so that the PRF ambiguity number is equal to onc. For reading berween blocks equal to the length of the range reference the unambiguous azimuth frequency values in(b), (c), and(d), an offset of function. A simple update of the Doppler centroid value PRF/4(275 Hz)must in the chirp scaling phase would not work well, since the uncompressed range chirp signals from different range po- Due to the variation of the look angle 0, from the near sitions are overlapped in the range-doppler domain to the far range, the Doppler centroid will vary according In order to clarify this effect, an illustrative represen- to tation of the Sar signal of four point targets in the dif fcrent domains of the Cs algorithm are represented in Fig B(a)-(e). The pa of the E-sar fac(ru) Isin,·si8a+cos6n·sin6l the previous section were used with imuth pre suming factor of four, so that the effective PRF is reduced (18) to 275 Hz where ad is the drift angle(which is not dependent on the MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION PRF of 275 Hz, the ambiguous Doppler centroid values in Fig. 3(b)are 30 Hz for the first target in the near range and 150 Hz for the last target in the far range. The azi- e chirp scallng pnase ap plied in the range-Doppler domain was selected to be cen rangc time[μsl tered around the Doppler centroid fac(ri) of the first target in near range(i.e,fdc(r1)+ PRF/2). Sincc the bandwidth of the first target in near range is within this variation, the RCMC will be performed accurately and the focusing quality will not deteriorate [see Fig 3(c)and(d). How- ever, for the second, third and fourth target from the near to the far range the azimuth frequency variation will not be matched to that of the signal itself, leading to an in correct RCMC [see Fig 3(d)], defocused IRF's, and azi muth ambiguities [see Fig 3(e) The solution to this problem is very simple and accu range time[μsl--> rate. After transforming the sar raw data to th Doppler domain, the azimuth frequency variation is arti ficially increased by means of an azimuth spectral-length 们 extension which accommodates the variation of the Doppler centroid with range. The azimuth spectral-length extension should be at least as great as the variation of the doppler centroid from the near to thc far range. In thc example of Fig. 4, the azimuth frequency variation wa increased by PrF, i.e., from fdel- Prf/2 to fdcl+ 3 PRF/2 [see Fig. 4(b) -(d). In the general case, the azi ange frequency MHz 1--> muth spcctrum must bc extended according to the follow Ing limits PRE PRF min [f(o) <fa max lfac(ro]+ (19 8 where min [d] and max [de] are thc minimum and max imum Doppler centroid values used in the processing. Due to the azimuth spectral-length extension, all of the phase functions (HI new, H21, and H22) applied in the range- Doppler and wavenumber domain are unambiguous, so nge m 1 that the measured quality of the IrF's in Fig 4(e)are the same as in the case presented in the previous section with constant Doppler centroid. The only limitation of this proach is that the Doppler centroid variation within range chirp length must be less than the PRf. Otherwis the azimuth frequencies cannot be represented in an ambigous way after the azimuth spectral extension IV, ACCOMMODATION OF DOPPLER CENTROID Fig. 4. Contour plot of the signal of four point targets in the different do- VARIATIONS IN AZIMUTH WITH INTEGRATED MOTION mains of the extended chirp scaling algorthm. (a) Signal domain(raw data COMPENSATION (I Range.doppier domain, (c) Wavenumber domain. (d) Range-Doppler A. Doppler Centroid variation in Azimuth tension, the variation of the Doppler centroid can be exactly accommodated Due to the motion errors of the aircraft the antenna y the ECS algorithm. The sensor and processing parameters are the same for Fig. 3 must be steered in order to keep the squint angle constant In the case of the E-sar system the antenna is fixed on he fuselage of the aircraft and a wide beam is used in range distance) and 6, is the pitch angle. In the example azimuth, so that a constant squint angle can be adopted in Fig.3(8 squint angle in middle range, 0 pitch an- for the processing. The variations of the squint angle lead gle), the Doppler centroid varies from approximately 305 to a small-signal loss, which is radiometrically corrected to 426Hz (near to far range, respectively). Due to the after the processing. Additionally, a deterioration of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 32. NO. 5. SEPTEMRER 1994 IRF is observed, since the RCMC is a function of the compensation equation (16), the data is compressed in squint angle azimuth using the reference value of the Doppler centroid Then, for accurate processing, the squint-angle value (ay avg). The azimuth phase correction H3 is then applied for the processing must be updated with azimuth. The according to (I7)in the range-Doppler domain basic assumption in the following approach is that the Doppler centroid for the processing is conslant within the B. Motion Error Compensation synthetic aperture length but it can vary several hundred The subaperture structure in azimuth allows the incor- hertz along the entire data acquisition interval(data take). poration of the so-called reflectivity displacement method which is several minutes long. Since the processed azi- (RDM. [14] )in the processing for determination of the muth bandwidth for the E-SAR system is less than one- second-order motion compensation. This is based on the fourth of the total azimuth signal bandwidth, this assump- fact that the rdm uses the results of the cross correlation tion leads to very accurate results. With the update of the between adjacent azimuth spectra in order to determine Doppler centroid with azimuth the signal-to-noise ratio the doppler shift afa that occurred in this time interval the ambiguity level and the image resolution are opti- In the case where there are no motion errors, the Doppler mized for the entire data take shift between the subapertures is constant and equal to the The proposed approach for accommodating the Dopp nominal value Af a nominal. When there are motion errors ler centroid variations with azimuth consists of dividing it can be easily shown that the doppler deviations from the entire data take in small azimuth subapertures (e. g the nominal value, 1. e, Afa(t; ro)-Afa nominal (ro)are 128 points), which are much smaller than one synthetic proportional to the acceleration in line-of-sight(LOS)di aperture length. A small overlap between the subapertures rection. The double integration of the Doppler deviations is used in order to guarantee a phase continuity between from the nominal value multiplied by a constant term leads the subapertures. Defining Tsub as the duration of each azi. to the displacement in Los direction muth subaperture, Tovt as the overlap between the azimuth subapertures, and Tdata take as the duration of the whole Ad(t; ro) A·AJf(;r)- fa nominal(列 data take we obtain S;(,t;r0)=s(7,t+i·(-To); where At is the time interval between adjacent data scts data take t T 20)(spectra). The function Amc for second-order motion con pensation is calculated according to(16), whereby the For each subaperture e azimuth frequency variation is phase m(L: 1o) is obtained by multiplying the displace- centered around the actual Doppler centroid value for that m 入 subaperture, I.e The second-order motion compensation not only cor- rects the phase errors for accuratc azimuth comprcssion PRE Jas;≤J1<f:+ PRE ( 21) but ensures the correct azimuth positioning of the com- pressed signal [23] where i is the subaperture number. The abuve variation of the azimuth frequency is used for all the phase correc- V. RESULTS OF IMAGE PROCESSING tions in the range-Doppler and wavenumber domain (i. e, The complete software for SAr processing was devel- H, new, H21, and H22). However, the chirp scaling oper- oped using an interactive data language (IDL), which runs ation of each subaperture must be performed with respect on a UNIX operating system. The sensor and processing to one Doppler centroid reference value in order to guar- parameters for the processing are defined in an input file antee the same range scaling for the complete image. The software includes processing with the 1)rangc-Dopp- Then, the Doppler centroid value fac is the last term of ler, 2)hybrid, 3)chirp scaling, and 4) extended chirp (10)is kept constant for the entire image. The reference scaling algorithms. In all cases, with exception of the f aci, so that the scaling amount is minimized. Let yd_ avg be be optionally introduced into the processiltrcction can Doppler centroid should be selected as the mean value of chirp scaling algorithm, the motion crror correction can the mean value of fac, then(10) is rewritten as a flight of the esar system over the airfield of a'(fa= a(fa)-a( fdc ave) Oberpfaffenhofen, Germany was used to test the proach. The selected data set for processing have a squin It is important to note that the azimuth frequency variation angle of 7.8 and very strong motion errors, which were f a, must be updated for each subaperture, although the ref intentionally induced by the pilot, The phase correction erence Doppler centroid fac avg is kept constant for motion compensation was obtained by the approach After multiplication with the functions Hi new, H21, and described in [2] H22, the subapertures are transformed to the signal do Fig. 5(a)and(b)show the processed image(FiRE re main. In this domain the time overlap is removed and the corder output)using the extended chirp scaling algorithm subaperture signals are joined in order to reconstruct the without and with motion compensation. The image sizes full data take. After applying the second-order motion are 2895 x 3766 m(range x azimuth). The evaluation MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION 1037 NE→HF SAR-SYSTEM 2D PROCESSOR Obe pfaffenhofen, Ger mo gtt11-0491 Piret Spacing in Range: 10 m +Prel Spacing in Az imuth: 0.7 m Altitude·2205m Gra and Sped.74 m/s De pression Ar eok siggloDe Roto 少岁 Fig. 5. Processed E-SAR image of the runway at Oberpfaffcnhofcn. Ger many, using the extended chirp scaling algorithm. (a) without correction of motion crrors.(b)with correction of the motion errors. Main sensor and pmcessing parameters are 2205 m flight altitude, 74 round spee 7.8 squint angle, C-band, 8 looks with 50 percent overlap, vv polariza tion, 2.5 4.0 m resolution(range X azimuth of the image quality(with motion compensation) and the The first step consists of the first-order motion com- comparison to the results obtained by processing with the pensation, which compensates the motion errors for a ref- range-Doppler and hybrid algorithms shows that no sig- erence range nificant differences can be measured as far as the geo- .'the data is transformed from the signal domain to metric and radiometric resolution of the images are con- the range-Doppler domain by means of azimuth FFTs cerned. The azimuth geometry from the different Multiplication with HI new(T fa; rrei )performs the algorithms was also compared and the diffcrences arc less chirp scaling for a reference range than 1 pixel spacing(measurement accuracy) Range FFT's are carried out to map the data from the range-Doppler domain to the two-dimensional frequency domain(wavenumber domain) ⅵI,. CONCLUSION Multiplication with H2(f, fa, rref )carries out an ac The extended chirp scaling algorithm has a dedicated curate range compression with SrC update. Additionally structure with accurate phase-correction functions in or- this function includes a linear phase term in range, which der to accommodate the motion error compensation re- removes all of the bulk range migration. No azimuth quired by airbome SaR as woll as the Doppler centroid compression is performed in the wavenumber domain variations in range and azimuth. The processing steps are Range IFFTs are performed to transform the data summarized in the followin into the range-Doppler domain IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL 32, NO. 5, SEPTEMBER 1994 DLR NE-HFE-SAR-SYSTEM 2D PROCESSOR Oberofaifenhofen, Cermony Coordinates >481130"N 1'0830E Image porometers Pirel Spacing n Ronge: 1 0 m Pirel Spacing in azimuth: 2.7 m RF Certer Freq Jency: 5.3 GHz Polorisotoon vv Number ot Locks: t with Motion Compensation / 2D Compression 2 Fig. 5.(Continued Multiplication with the function H2(T, fa; rref)com Doppler centroid update with az imuth is introduced pensates for a slowly varying azimuth phase, which was by means of azimuth subaperture proccssing troduced by the chirp scaling operation e Motion error extraction using the Reflectivity Dis By means of azimuth IFFT,'s, the data is transformed placement Method is incorporated by means of azimuth rom the range-Doppler domain into the range and azi- subaperture processing muth time domain(signal domain). At this step all tar- Data processing with squint angle up to 30 can b gets are range compressed and have their azimuth trajec- carried out in the C-band without deterioration of the IrF tories without any range migration by means of an additional cubic phase term in the chirp The second-order motion compensation accounts for scaling operation the accurate phase correction with range update and com Future work includes the image processing of satellite pensates the residual azimuth phase error. data with the extended chirp scaling algorithm(e. g, SIR The azimuth FFt's transform the motion-compen- C/X-SAR). A feasibility study of dedicated hardware for sated data into the range-Doppler domain a real-time Sar processing, which is suitable for airborne Multiplication with H3(T, fu; ro) performs the azi- and spaceborne SAR systems, will be carried out. In ad muth compression dition, the processing of spotlight Sar data and inverse The final image is obtained after azimuth IFFt SAR will be considered in connection with the flexibility The above description of the extended chirp scaling al- of the extended chirp scaling algorithm for Doppler cen gorithm allows the following steps to be included in the troid update and motion error correction processing Doppler centroid update with range is accommo APPENDIX dated by means of an azimuth spectral length extension in In the following text, an analysis of the phase errors the rangc-Doppler domain duced by the chirp scaling operation with quadratic 【实例截图】
【核心代码】