High-resolution imaging based on coherent processing for distributed multi-band radar data.
Nowadays, modern wideband radar systems have the capability of obtaining more features of the targets to carry out real-time discrimination and target identification. This capability is primarily the result of target resolution improvement. Ultra-wideband (UWB) imaging is a good way to improve target range resolution. However, due to specific engineering or technology constraints, it is usually difficult for a single radar system to operate over a very large bandwidth. As the next generation radar, distributed radar has great promise for various sensing applications. Distributed multi-band radar can be used to synthesize large bandwidth for achieving high-resolution radar imagery of many targets [1-5]. Actually, there are two challenges for this application. First, in real radar applications, the relative bandwidth of radar is limited by its center frequency and it is about of 10% of the center frequency, the subbands may have different bandwidth and large bandwidth gaps. Accurate coherent processing for these subbands must be performed at first. Second, different data gaps between subbands should be filled to form the final high-resolution images.
Among the existing coherent processing methods, coherence function  is used to acquire incoherent parameters (ICP), but large extrapolation errors may occur, and it is easily trapped in local optimization. Pole rotation method [3, 4] is used to obtain phase ICP; however, it is required that all the subbands have the same bandwidth. In , a data based coherent compensation method is used for coherent processing, but it is easily influenced by noise due to lack of de-noising process. For the data fusion methods, the available algorithms include nonparametric spectrum estimation [6-10], the parametric spectrum estimation method [1,12,13] and p-norm regularization method . In , Burg algorithm is used to find the linear prediction parameters and a iterative procedure is used to improve the estimation of the parameters and the extrapolation of the data, but the gaps between subbands can not be too large. In [7,8], minimum weighted norm (MWN) method is used for optimizing one-dimensional (1-D) aperture extrapolation, but it may not suitable for multi-band data with few data samples and large gaps. In [9-11], the method called gapped-data APES (GAPES) is used for data fusion, which is based on an interpolation of the gapped data under certain constraints. However, the computational burden is a problem for real-time processing. Reference  uses the all-pole model and the root-MUSIC for model parameter estimation. In [12, 13], an 1-D gapped-data state space approach (1-D GSSA) is used to estimate parameters of complex exponential (CE) model for gaps data estimation. In [14,15], the Lp-norm regularization method with 0 < p < 1 is applied for partial-aperture and sparse band imaging, where the choice of p remains an open problem. With a large number of missing samples, simulations have shown that the regularization method may be unable to obtain a satisfying outcome.
In this paper, a coherent processing technique is proposed to process multi-band data of different bandwidth and different gaps between subbands. We use de-noising cross-correlation (DNCC) algorithm and statistical method to obtain estimation of phase ICP along every azimuth. A coherence function is used to estimate amplitude ICP after phase ICP is compensated and statistical method is used to obtain optimal amplitude ICP. Finally, data fusion method via two-dimensional gapped-data state space approach (2-D GSSA) is applied to subbands and high-resolution imaging is achieved to enhance the capability of target discrimination and identification.
2. DISTRIBUTED MULTI-BAND RADAR
In this section, we will describe the general imaging geometry for distributed collocated multi-band radar.
2.1. Distributed Collocated Radar Imaging Geometry
Figure 1 illustrates a scenario in which three collocated radars are interrogating an unknown target in the aerospace, simultaneously. Each radar can produce a 'snapshot' image by using range-Doppler algorithm when target rotates a small angle . As is shown in Figure 1, the radars operate at the different frequency band and share the same target aperture. Here, different subband signals mean the signals transmitted by different collocated radars. The radars operate at different center frequency with different bandwidth. For instance, S-band centered at 3 GHz with 300 MHz bandwidth, C-band data centered at 6 GHz with 600 MHz bandwidth, X-band data centered at 10GHz with 1 GHz bandwidth. For application considered here, the target motion in the aerospace is determined by two basis components: a trajectory motion, generally characterized by the motion of the center of gravity of the target, and a localized motion about center of rotation of the body. In this paper, we assume that the trajectory motion is compensated and the radar pulses are range aligned such that body motion is centered about the body axis of rotation. As we known, subband data coherence is essential before coherent combination of the data, mutual-coherence errors occur because of range-estimation bias and radar amplitude gain bias. This paper studies the methods for mutually cohering the subband radar data. The goal is to obtain the best-possible image from multi-sensor data, so that the resultant image can be fed into a target characterization process for full characterization.
2.2. Signal Model
The received echo signal y(f, [??]) of a target at frequency f and aspect angle [section] can be expressed as follows :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where P is the number of scattering centers on the target, [[sigma].sub.i] the amplitude of the i-th scattering center, and ([x.sub.i]; [y.sub.i]) the position of the i-th scattering center in the spatial domain. v is the additive white Gaussian noise (AWGN) and c the speed of light.
Equation (1) illustrates that the phase of the received signal varies as a function of aspect angle. When the target rotates, the received radar data is collected in polar format. Although such data may be collected at uniformly spaced frequency steps and rotation angle positions, the samples are nonuniformly spaced in the spatial frequency domain. The results is that two-dimensional Fourier transform to target-space reflectivity results in unfocused images for the large rotation angles. An optional method for inverse synthetic aperture radar (ISAR) imaging is the natural double integral imaging method
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Although the natural integration method provides better resolved images in range and azimuth domains for large rotation angles, the main drawback is the considerable computation time in evaluating the ISAR integral.
To improve the computation efficiency, we should make full use of fast Fourier transform (FFT) for ISAR imaging. For 'snapshot' imaging, i.e., small rotational angle case, FFT can be used to form ISAR image of the target. For large rotation angle case, the received backscattering data uniformly sampled in the frequency-aspect domain does not correspond to the uniformly sampled data in the spatial frequency domain, so the polar formatting algorithm (PFA)  should be used that allows us to retain the computational advantage of the 2-D FFT exploited in the imaging algorithm.
3. SUBBAND DATA COHERENT PROCESSING AND DATA FUSION
Figure 2 shows the diagram of data processing flow. Coherent processing between subbands data must be carried on first. In this step, it estimates and compensates the phase and amplitude ICP between subband data. After that, subband data can be combined to form large bandwidth data. In the last step, 2-D FFT is used to form ISAR image of the target.
3.1. Coherent Processing of Subband Data
According to the configuration in 2.1, the target is illuminated by three collocated radars which operate at different subbands with different bandwidth, simultaneously. The subband with largest bandwidth, i.e., X-band data is used to be a reference. Coherent processing is applied to compensate for the lack of mutual coherence between the reference radar and others. The data mutual-coherence procedure between subband data is shown in Figure 3. After optimal phase and amplitude ICP are estimated and compensated, subband data can be combined to form a large bandwidth data.
Without loss of generality, we consider mutual incoherence between 5-band data and X-band data. The method can also be extended to arbitrary subbands data mutual-coherence. Here, we treat X-band radar as a reference one. The radar returns from S-band data and X-band data at a certain azimuth can be written as 
[y.sub.nS] (k) = Ay(k)[e.sup.j(k-1)[phi]] + [[upsilon].sub.1] (k) = 1,2, ..., [N.sub.1] (3)
[y.sub.nX] (k) = y(k) + [[upsilon].sub.2](k); k = N - [N.sub.2] + 1, ..., N (4)
where y(k) denotes fullband radar return without noise; [[upsilon].sub.1](k) and [[upsilon].sub.2](k) are AWGN; A and [phi] are amplitude and phase ICP, respectively; N denotes frequency samples number of fullband. S-band data contains [N.sub.1] frequency data samples, and reference one contains [N.sub.2] frequency data samples. Data mutual-coherence between them is as follows.
The radar returns of two subband data can be modeled by CE model [18,19]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [a.sub.i] and [[alpha].sub.i] are respectively the amplitude and dispersion factor of ith scattering center, and Ti the relative range and c the speed of light. The frequency vector [f.sub.k] is specified in terms of the carrier frequency [f.sub.c] as [f.sub.k] = [f.sub.c] + (k - 1)[DELTA]f, where [DELTA]f is frequency step. p denotes the number of the scattering centers. Order selection method is used to estimate p, here, we use two methods combined together to estimate it, one is singular value decomposition (SVD) method through selecting proper threshold, the other one is information criterion [20, 21]. SSA  is used to estimate those parameters in (5) except p, and subband signals after de-noising are obtained as [y.sub.S](k) and [y.sub.X](k). If we consider that the target characteristic at certain aspects is similar according to each radar.
[y.sub.X](k) = [1/A] [y.sub.S] (k)[e.sup.-j(k-1)[phi]] (6)
We compute [N.sub.c] point inverse fast Fourier transform (IFFT) of [y.sub.S](k) and [y.sub.X] (k) to obtain [y.sub.S](n) and [y.sub.X](n), so the correlation function can be obtained as R(n) = [y.sub.X](n) [cross product] [Y.sup.*.sub.S] (-n), where [cross product] denotes circular convolution. * denotes conjugate transpose. The function can also be rewritten as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
The formula after substituting (6) to (7) has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
Equation (8) indicates that the maximum of the function can be achieved when n = [N.sub.c][phi]/2[pi] + 1, so [phi] = 2[pi](n - 1)/[N.sub.c]. If the radar echo contains [N.sub.a] pair of aspects, the final estimation of phase ICP can be obtained as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where 'median' denotes computing median of the array.
A coherence function is defined, and the S-band data is compensated with the optimal phase ICP
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Optimization is performed to estimate amplitude ICP. The optimal estimation of the amplitude ICP can be obtained as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where 'mode' denotes computation of the most frequent values in array.
3.2. Subband Data Fusion
After all the ICP corresponding to S-band data and C-band data are estimated and compensated with X-band data, data fusion by using 2-D GSSA can be carried on mutually cohered subband data. Figure 4 shows flow chart of subband data fusion.
For linear imaging, formula (1) can be written as 2-D data samples y(m, n)  that comprise P scattering centers corrupted with white Gaussian noise [upsilon](m, n)
y (m, n) = [P.summation over (i=1)] [a.sub.i][s.sup.m.sub.i][q.sup.n.sub.i] + [upsilon] (m, n) (10)
where [a.sub.i] refers to the complex amplitude associated with the of ith scattering center with pole pair ([s.sub.i], [p.sub.i]). We assume that the length of S-band, C-band and X-band are [N.sub.1], [N.sub.2] and [N.sub.3], respectively. The length of data gaps between them are [N.sub.g1] and [N.sub.g2], respectively. The set of three subband data may be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
where M denotes the number of azimuth samples, [Y.sub.bs] the S-band data, [Y.sub.bC] the C-band data, and [Y.sub.bX] the X-band data.
The state matrices ([A.sub.cj], [B.sub.cj], [C.sub.cj]) and column enhanced matrices [H.sup.col.sub.bj] can be obtained from two dimensional data sets , where j = S, C, X. The extended state matrices are constructed as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], blkdiag denotes the block diagonal matrix. The Hankel matrices along every range cells can be expressed as
[H.sup.col.sub.n] = [CA.sup.n-1]B; n = 1, ..., N (15)
where N denotes frequency number of fullband samples. We decompose the Hankel matrices along every column to retrieve the estimation of range cell data, then, the data along every column can be deduced as
[Y.sub.eb] (:,n) = [[[H.sup.col.sub.n] (1 : [d.sub.r] - 1,1) [H.sup.col.sub.n] [([d.sub.r],:).sup.T]].sup.T]; n = 1, ..., N (16)
where [d.sub.r] denotes the dimension of row of [H.sup.col.sub.n].
The final ISAR image of radar echo with subband data fusion can be generated by using 2-D FFT method, and improved ISAR images can be obtained.
4. EXPERIMENTAL RESULTS
In this section, two examples are presented to demonstrate the effectiveness and superiority of the proposed method.
4.1. Numerical Simulation Using Electromagnetic Calculation Target
In the first example, the proposed method is applied to a target calculated from physical optics (PO) and equivalent edge current (EEC) . Figure 5 shows the shape and geometry of the target. The target model is a blunt nosed cone-cylinder-frustum (BNCCF) target. As we can see from Figure 5(a), the target is composed of three parts. The first part is a blunt nosed cone which has a sphere tip with a radius of 51 mm. The second part is a cylinder connecting with the first part and the third part. The third part is a frustum. The total length of the target is 1400 mm. The joint of cone and cylinder is located 692 mm from the target base. The joint of cylinder and frustum is located 92 mm from the target base. The other parameters such as diameter of the cylinder and target base are shown in Figure 5(b).
The primary aims of our experiment are to verify the proposed processing method for the distributed multi-band radar data to obtain high-resolution images of rigid targets in the aerospace. Calculations were taken from 2 to 20 GHz in 20 MHz increments, HH polarization. The target viewing angles, relative to nose-on, ranged from -180[degrees] to 180[degrees] in 0.25[degrees] increments.
4.1.1. Comparison of Pole Rotation and DNCC Method
In this section, we investigate the performance of pole rotation  and DNCC method for estimating phase ICP. We consider two cases: case 1, the subband 3 ~ 3.3 GHz and 8 ~ 9 GHz are mutually incoherent; case 2, the subband 13 ~ 14 GHz and 16 ~ 17 GHz are mutually incoherent. The target viewing angle is 5[degrees] and lower subband is modulated by phase bias 60[degrees] for two cases. AWGN is added to subband signals. The simulation number of Monte Carlo is 1000. Figure 6 shows the comparison of pole rotation and DNCC method. Note that the phase ICP estimation from DNCC method is more reliable than pole rotation method in all SNR circumstances.
4.1.2. The Statistical Method for ICP Estimation
To obtain optimal phase ICP, statistical method is used in dwell time. Subband data used for simulation are from 4 ~ 5 GHz and 7 ~ 8 GHz, respectively. The cumulative angle is from 0 ~ 10[degrees]. To simulate the amplitude and phase incoherence between subband data, the lower subband data is modulated by a gain of 5 and phase bias of 45[degrees]. AWGN is added to subband signals. The simulation number of Monte Carlo is 1000.
Figure 7 shows the root mean squared errors (RMSE) of phase and amplitude ICP estimation derived by statistical method and arithmetic mean method. It is seen that the proposed statistical method has better performance over arithmetic mean method.
4.1.3. Imaging Results of BNCCF Target
In order to simulate the real-world situation, we assume that there are three collocated radars which operate at S-band (3 ~ 3.3 GHz), C-band (5.7 ~ 6.3 GHz) and X-band (8 ~ 9GHz), respectively. The target is illuminated by these radars simultaneously and the target viewing angle ranging from -3[degrees] to 3[degrees]. The bandwidth of these radars are 300 MHz, 600 MHz, 1GHz, respectively and all these radars are mutually incoherent. To simulate the amplitude and phase incoherence between the subbands, X-band is treated as a reference subband, S-band signals are modulated by a gain of 5 and phase bias [pi]/3, C-band signals are modulated by a gain of 3 and phase bias [pi]/4. Signal noise ratio (SNR) is set to be 20 dB.
Figures 8(a)-(c) depict ISAR images of these subbands by using two-dimensional (2-D) FFT, respectively. The resolution of these images are too coarse, especially the S-band data. They do not explicitly resolve the subtle features of the target. The full band data image is shown in Figure 8(d), which shows the target features including target base, joint of cylinder & frustum, joint of cone & cylinder and sphere tip. Compared to Figure 8(d), Figure 8(e) shows the image results by using 2-D GSSA without mutual-coherence, which can not show the correct distribution of scattering centers. As is known, mutual-coherence must be done before data fusion, Figures 8(f)-(i) compare different fusion algorithms after mutual-coherence processing. Figure 8(f) shows the result of GAPES, as we can see, the method can not improve range resolution and show subtle feature of the target, whereas induces grating lobes. Figures 8(g)-(h) show the results of 1-D GSSA and 1-D SSA combined with MWN, respectively. They can also improve the range resolution of sphere tip and target base, but lose the information of middle joints which is because the data is too sparse. By using 2-D GSSA, we can obtain all the subtle information about the target as shown in Figure 8(i), which shows high fidelity comparable to Figure 8(d).
In real-time imaging applications, time cost is a key point for evaluating algorithm performance. After mutual-coherence processing, we compare the time cost of different data fusion algorithms. The numerical simulation experiments were conducted in the same personal computer and the simulation condition is as above. The time cost is as follows: 1-D GSSA is 0.8699 s, 1-D SSA+MWN is 1.9674 s, GAPES is 431.6062 s and 2-D SSA is 3.1126 s. From above analysis, 1-D SSA and 1-D SSA+MWN cost less time, but they may lose important scattering centers information and can not show subtle feature of the target. The time cost of GAPES is huge, so it's not suitable for real-time processing. 2-D GSSA supplies the best imaging outcome and the time cost is reasonable for the imaging applications.
4.2. Real Data Imaging Results
In this section, the proposed method is applied to a target measured from microwave chamber. Figure 9 shows the optical picture and geometry of the target. The target model is a blunt nosed cone-cylinder target. As we can see from Figure 9(a), the target is composed of three parts. The first part is a blunt nosed cone which has a sphere tip with a radius of 22 mm and a piece of foil. The second part is a frustum, which connected the first and third parts. The total length of the target is 1200 mm. The joint A is located 930 mm from the target base. The joint B is located 430 mm from the target base. The other parameters are shown in Figure 9(b).
In this experiment, radar echo data is collected from radar operating at X-band. The measurement is taken from 8 to 12 GHz in 20 MHz increments, HH polarization. The target viewing angles, relative to nose-on, ranged from 0[degrees] to 180[degrees] in 0.2[degrees] increments.
We assume that the target is illuminated by two collocated X-band wideband radars, simultaneously. Each radar operates at a different center frequency with 1 GHz bandwidth. The target viewing angle is 0 ~ 15[degrees]. To simulate the amplitude and phase incoherence between multi-band data, the lower subband data is modulated by amplitude and phase ICP. After ICP estimation, PFA should be applied to subbands before data fusion. Figures 10(a) and 10(b) show the lower and upper subband images, respectively. The resolution is insufficient to resolve subtle feature of important scattering centers, such as joint A and joint B, on the target along range direction. Compared to Figure 10(c), Figure 10(d) shows the image results by using 2-D GSSA without mutual-coherence, which cannot show the correct distribution of target scattering centers. As known, mutual-coherence must be done before data fusion. Figures 10(e)~(h) compare different fusion algorithms after mutual-coherence processing. Figure 10(e) shows the result of GAPES. As we can see, the method cannot improve range resolution, and target scattering centers are smeared. In the meanwhile, it needs huge time cost. Figures 10(f)~(g) show the results of 1-D GSSA and 1-D SSA combined with MWN, respectively. The target scattering centers are embedded in large amount of side lobes and fake scattering centers. As shown, 1-D method is not accurate enough to reconstruct target scattering centers. By using 2-D GSSA, we can obtain almost all the scattering centers information about the target, especially the joint A and joint B, edge of the base and high-order diffraction behind target base, as shown in Figure 10(h).
In this paper, we present a coherent processing method for distributed multi-band radar data to generate high-resolution images, in which DNCC algorithm and coherence function combined with statistical method are employed. Then, cohered subband signals are combined by 2-D GSSA method. The proposed method can be used to process multi-band data of different bandwidth and different gaps between them, which is verified by using a BNCCF target. The proposed data mutual-coherence method performs better than pole rotation method under different SNR levels. After data mutual-coherence process, 2-D GSSA exhibits better performance than GAPES, 1-D GSSA and 1-D SSA+MWN methods. The results validate the superiority of this method. Apart from electromagnetic calculation data, a real target measured in microwave chamber is used to demonstrate the effectiveness of the proposed method.
[1.] Cuomo, K. M., J. E. Piou, and J. T. Mayhan, "Ultrawide-band coherent processing," IEEE Trans. Antennas Propag., Vol. 47, No. 6, 1094-1107, 1999.
[2.] Xu, X. J. and L. Jia, "Ultrawide-band radar imagery from multiple incoherent frequency subband measurements," Journal of Systems Engineering and Electronics, Vol. 22, No. 3, 398-404, 2011.
[3.] Liang, F. L., X. T. Huang, and P. Z. Lei, "A novel coherent algorithm of multiband radar echo," Signal Processing, Vol. 26, No. 6, 863-868, 2010.
[4.] Tian, J. H., J. P. Sun, G. H. Wang, Y. P. Wang, and W. X. Tan, "Multiband radar signal coherent fusion processing with IAA and apFFT," IEEE Signal Process. Lett., Vol. 20, No. 5, 463-466, 2013.
[5.] Liu, C. L., F. He, and X. Z. Wei, "Research on multiple radar fusion imaging coherence compensation based on data correlation," Systems Engineering and Electronics, Vol. 32, No. 6, 1266-1271, 2010.
[6.] Li, H. J., N. H. Farhat, and Y. Shen, "A new iterative algorithm for extrapolation of data available in multiple restricted regions with application to radar imaging," IEEE Trans. Antennas Propag., Vol. 35, No. 5, 581-588, 1987.
[7.] Cabrera, S. D. and T. W. Parks, "Extrapolation and spectral estimation with iterative weighted norm modification," IEEE Trans. Signal Process., Vol. 39, No. 4, 842-851, 1991.
[8.] Wang, Q., R. B. Wu, M. D. Xing, and Z. Bao, "A new algorithm for sparse aperture interpolation," IEEE Goesci. Remote Sensing Letters, Vol. 4, No. 3, 480-484, 2007.
[9.] Larsson, E. G., P. Stoica, and J. Li, "Amplitude spectrum estimation for two-dimensional gapped data," IEEE Trans. Signal Process., Vol. 50, No. 6, 1343-1354, 2002.
[10.] Larsson, E. G., G. Q. Liu, P. Stoica, and J. Li, "High-resolution SAR imaging with angular diversity," IEEE Trans. Aerosp. Electron. Syst., Vol. 37, No. 4, 1359-1372, 2001.
[11.] Bai, X. R., F. Zhou, M. D. Xing, and Z. Bao, "High-resolution radar imaging of air targets from sparse azimuth data," IEEE Trans. Aerosp. Electron. Syst., Vol. 48, No. 2, 1643-1655, 2012.
[12.] Piou, J. E., "A state-space technique for ultrawide-bandwidth coherent processing," Technical Report TR 1054, MIT Lincoln Laboratory, 1999.
[13.] Piou, J. E., "A state identification method for 1-D measurements with gaps," AIAA Guidance Navigation and Control Conference, 1-11, San Francisco, California, 2005.
[14.] Cetin, M. and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process., Vol. 10, No. 4, 623-631, 2001.
[15.] Cetin, M. and R. L. Moses, "SAR imaging from partial-aperture data with frequency-band omissions," Proceedings of the SPIE Defense and Security Symposium, Orlando, FL, 2005.
[16.] Mayhan, J. T., M. L. Burrows, K. M. Cuomo, and J. E. Piou, "High resolution 3D snapshot ISAR imaging and feature extraction," IEEE Trans. Aerosp. Electron. Syst., Vol. 37, No. 2, 630-650, 2001.
[17.] Park, J.-I. and K.-T. Kim, "A comparative study on ISAR imaging algorithms for radar target identification," Progress In Electromagnetics Research, Vol. 108, 155-175, 2010.
[18.] Naishadham, K. and J. E. Piou, "A robust state space model for the characterization of extended returns in radar target signatures," IEEE Trans. Antennas Propag., Vol. 56, No. 6, 17421751, 2008.
[19.] He, F. Y. and X. J. Xu, "A comparative study of two scattering center models," IEEE 11th International Conference on Signal Processing, 1931-1935, Beijing, 2012.
[20.] Akaike, H., "A new look at the statistical model identification," IEEE Trans. Autom. Control., Vol. 19, No. 6, 716-723, 1974.
[21.] Wax, M. and T. Kailath, "Detection of signals by information theoretic criteria," IEEE Trans. Acoust., Speech, Signal Process., Vol. 33, No. 2, 387-392, 1985.
[22.] Piou, J. E., "System realization using 2-D output measurements," Proceedings of the 2004 American Control Conference, 2840-2845, Boston, MA, USA, Jun. 2004.
[23.] Michaeli, A., "Equivalent edge currents for arbitrary aspects of observation," IEEE Trans. Antennas Propag., Vol. 32, No. 3, 252-258, 1984.
[24.] Ausherman, D. A., A. Kozma, J. L. Walker, et al, "Developments in radar imaging," IEEE Trans. Aerosp. Electron. Syst., Vol. 20, No. 4, 363-400, 1984.
Feiyang He * and Xiaojian Xu
School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
* Corresponding author: Feiyang He (email@example.com).
Received 25 June 2013, Accepted 20 July 2013, Scheduled 25 July 2013
|Printer friendly Cite/link Email Feedback|
|Author:||He, Feiyang; Xu, Xiaojian|
|Publication:||Progress In Electromagnetics Research|
|Date:||Sep 1, 2013|
|Previous Article:||A fast simulation method of silicon nanophotonic echelle gratings and its applications in the design of on-chip spectrometers.|
|Next Article:||Evolution of cos-Gaussian beams in a strongly nonlocal nonlinear medium.|