Model-Based Stripmap Synthetic Aperture Radar Processing

Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2011 Model-Based Stripmap Synthetic Aperture Radar Processing Roger D West Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Electrical and Computer Engineering Commons Recommended Citation West, Roger D., "Model-Based Stripmap Synthetic Aperture Radar Processing" (2011) All Graduate Theses and Dissertations 962 https://digitalcommons.usu.edu/etd/962 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU For more information, please contact digitalcommons@usu.edu MODEL-BASED STRIPMAP SYNTHETIC APERTURE RADAR PROCESSING by Roger D West A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Electrical Engineering Approved: Dr Jacob H Gunther Major Professor Dr Todd K Moon Committee Member Dr Randy J Jost Committee Member Dr Wei Ren Committee Member Dr Mark E Fels Committee Member Dr Byron R Burnham Dean of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2011 ii Copyright c Roger D West 2011 All Rights Reserved iii Abstract Model-Based Stripmap Synthetic Aperture Radar Processing by Roger D West, Doctor of Philosophy Utah State University, 2011 Major Professor: Dr Jacob H Gunther Department: Electrical and Computer Engineering Synthetic aperture radar (SAR) is a type of remote sensor that provides its own illumination and is capable of forming high resolution images of the reflectivity of a scene The reflectivity of the scene that is measured is dependent on the choice of carrier frequency; different carrier frequencies will yield different images of the same scene There are different modes for SAR sensors; two common modes are spotlight mode and stripmap mode Furthermore, SAR sensors can either be continuously transmitting a signal, or they can transmit a pulse at some pulse repetition frequency (PRF) The work in this dissertation is for pulsed stripmap SAR sensors The resolvable limit of closely spaced reflectors in range is determined by the bandwidth of the transmitted signal and the resolvable limit in azimuth is determined by the bandwidth of the induced azimuth signal, which is strongly dependent on the length of the physical antenna on the SAR sensor The point-spread function (PSF) of a SAR system is determined by these resolvable limits and is limited by the physical attributes of the SAR sensor The PSF of a SAR system can be defined in different ways For example, it can be defined in terms of the SAR system including the image processing algorithm By using this definition, the PSF is an algorithm-specific sinc-like function and produces the bright, star-like artifacts that are noticeable around strong reflectors in the focused image The iv PSF can also be defined in terms of just the SAR system before any image processing algorithm is applied This second definition of the PSF will be used in this dissertation Using this definition, the bright, algorithm-specific, star-like artifacts will be denoted as the inter-pixel interference (IPI) of the algorithm To be specific, the combined effect of the second definition of PSF and the algorithm-dependent IPI is a decomposition of the first definition of PSF A new comprehensive forward model for stripmap SAR is derived in this dissertation New image formation methods are derived in this dissertation that invert this forward model and it is shown that the IPI that corrupts traditionally processed stripmap SAR images can be removed The removal of the IPI can increase the resolvability to the resolution limit, thus making image analysis much easier SAR data is inherently corrupted by uncompensated phase errors These phase errors lower the contrast of the image and corrupt the azimuth processing which inhibits proper focusing (to the point of the reconstructed image being unusable) If these phase errors are not compensated for, the images formed by system inversion are useless, as well A model-based autofocus method is also derived in this dissertation that complements the forward model and corrects these phase errors before system inversion (167 pages) v To Jenny, Kaitlynd, Ethan, and Dylan vi Acknowledgments I would first like to thank my advisor, Dr Jake Gunther, for all of his ideas, suggestions, and patience over the years; this dissertation would never have materialized without his continual guidance I am very thankful to Dr Todd Moon for all of the help and ideas that he has provided over the years I would also like to thank the rest of my committee for the discussions that we have had and for their willingness to be on my committee I am very grateful for the patience and understanding that my wife and kids have shown throughout my PhD program; I could not have finished without their constant support and daily encouragement I am also very thankful for the encouragement from my parents and my in-laws I am deeply grateful for all the divine inspiration that I have received; it is the only explanation for some of the roadblocks that I have been able to overcome I am very grateful to Scott A Anderson and Chad Knight for all the SAR discussions we have had and the suggestions they have offered I would also like to thank the countless others that have offered suggestions, advice, and ideas I am very grateful to the powers-that-be at the Space Dynamics Laboratory for granting the Tomorrow Fellowship to me; the financial assistance is another very important component that has made this dissertation possible Finally, I would like to thank Mary Lee Anderson for her guidance through all the required paperwork and for reviewing my dissertation for publication Roger D West vii Contents Page Abstract iii Acknowledgments vi List of Figures x Notation xiv Acronyms xv Introduction 1.1 Introduction to SAR 1.2 Advantages of Model-Based SAR Processing 1.3 Contributions of this Dissertation 1.4 Outline of Dissertation Pulsed Synthetic Aperture Radar Preliminaries 3.1 SAR Antenna 3.2 SAR Coordinate Frames 3.3 Induced Azimuth Signal 3.4 Bandwidth of the Induced Azimuth Signal 3.5 Pulse Transmission and Range Sampling 3.6 Model for the Range Sampled Data 3.7 SAR Image Formation 3.7.1 SAR Point-Spread Function 3.7.2 SAR Resolution Radar Preliminaries 2.1 Ideal Point Reflectors 2.2 Range Resolution 2.3 Pulsed LFM Transmitted Signals 2.4 Received Pulsed LFM Signal 2.5 Matched Filtering and Pulse Compression Pulsed LFM Stripmap Synthetic Aperture Radar 4.1 Stripmap Geometry 4.2 Synthetic Aperture for Stripmap SAR 4.3 Induced Azimuth Signal for Stripmap SAR 4.4 Bandwidth of the Induced Azimuth Signal 4.5 Induced Discrete Doppler Signal for Pulsed LFM Stripmap SAR 4.6 Resolution Limits 1 4 7 10 13 19 19 21 25 28 29 29 33 34 36 37 37 39 41 42 50 52 viii 4.7 4.8 Basic Steps for Image Formation Image Formation Algorithms 54 56 Forward Model for Pulsed LFM Stripmap SAR 5.1 Forward Model for Stripmap SAR 5.1.1 Circularly Symmetric Additive White Gaussian Noise 5.1.2 Signal-to-Noise Ratio 5.2 Region of Interest 5.3 Summary 60 60 63 65 67 68 Maximum Likelihood Image Formation 6.1 Likelihood Function 6.2 Simulated Results 6.3 Cram´er-Rao Lower Bound Analysis 6.4 Insights into the ML Estimation Method 6.5 Summary 69 69 74 78 79 82 Maximum A Posteriori Image Formation 7.1 A Posteriori Distribution 7.2 A More General Cost Function 7.3 Stripmap SAR Data Matrix 7.4 Block Recursive Least-Squares Algorithm for Stripmap SAR 7.4.1 Block RLS Algorithm Derivation 7.4.2 Block RLS Algorithm Results 7.5 Block Fast Array RLS Algorithm for Stripmap SAR 7.5.1 Block FARLS Algorithm Derivation 7.5.2 Block FARLS Algorithm Results 7.6 Comparison Between the BRLS and FARLS Methods 7.7 Summary 83 83 84 88 91 91 95 98 99 104 105 109 Model-Based Stripmap Autofocus 8.1 Background 8.2 Phase Error Model Development 8.2.1 Phase Error Free Model 8.2.2 Phase Error Model 8.3 Subspace Fitting Autofocus 8.3.1 Subspace Fitting Autofocus Derivation 8.3.2 Subspace Fitting Autofocus Optimization Strategy 8.4 Results of Proposed Autofocus Methods 8.4.1 Results Using Gradient Ascent 8.4.2 Results Using Regularized Newton’s Method 8.4.3 Results Using Convex Optimization 8.5 Comparison of Optimization Strategies 8.5.1 Optimization Results 8.5.2 Comparison of Computational Complexity 8.6 Summary 110 110 112 112 113 114 115 117 122 122 122 123 123 123 132 133 ix Summary and Future Work 134 9.1 Summary 134 9.2 Future Work 137 References 139 Appendix Appendix Circular and Hyperbolic Transformations A.1 Traditional Circular and Hyperbolic Householder Transformations A.2 Block Circular and Hyperbolic Householder Transformations 142 143 143 145 Vita 151 138 The Cram´er-Rao lower bound (CRLB) on the variance of the ML estimates was briefly presented in Chapter Future work in this area would be to a full development of the CRLB on an actual stripmap SAR sensor and compare the variance of the ground reflectivity estimates from actual images to the CRLB In Chapter it was shown that MAP ground reflectivity estimation has a close connection to regularized algorithms Future work in the area of MAP image estimation would be to explore the dependence of the regularization parameter on the SAR system parameters so the optimal value of the regularization parameter could be computed directly Also, future work in this area would be to explore other algorithms that would form MAP estimated images if the priors were not Gaussian, but still in the family of exponential distributions It was stated in Chapter that the convex solver that was used for the linearized convex optimization was sensitive to the conditioning of M It was also suggested that the linearized convex optimization method could be made more efficient if a solver were tailored to the problem that used the structure of the objective function and the constraints Future work in the area of model-based autofocus would be create a solver tailored to the linearized convex optimization method If the residual phase error in Chapter is known to be band-limited “enough” (with respect to the PRF), then future work may be to see if it would be possible to estimate a decimated phase error vector, then interpolate to obtain the full phase error vector If this were possible, then the size of the problem could be controlled There are several avenues for future work that were not mentioned in this dissertation One area would be to explore the idea of extending the forward model to coherent change detection (CCD) to see if it would be possible to estimate the phase difference between scenes without explicitly forming either image Another area would be to explore the possibility of altering the forward model to allow for multiple antennas, then use it to model-based interferometric SAR and ground moving target indication (GMTI) 139 References [1] C V Jakowatz and D E Wahl, “Three-dimensional tomographic imaging for foliage penetration using multiple-pass spotlight-mode SAR,” in Signals, Systems and Computers, Conference Record of the Thirty-Fifth Asilomar Conference, 2001 [2] A Elsherbini and K Sarabandi, “Mapping of sand layer thickness in deserts using SAR interferometry,” Geoscience and Remote Sensing, IEEE Transactions, vol 48, no 9, pp 3550–3559, 2010 [3] V Chen and H Ling, Time-Frequency Transformations for Radar Imaging and Signal Analysis Norwood, MA: Artech House, 2002 [4] C V Jakowatz, D E Wahl, P H Eichel, D C Ghiglia, and P A Thompson, Spotlightmode Synthetic Aperture Radar: A Signal Processing Approach New York, NY: Springer, 1996 [5] H Maitre, Processing of Synthetic Aperture Radar Images NJ: ISTE; John Wiley and Sons, 2008 London, UK; Hoboken, [6] M Richards, J Scheer, and W Holm, Principles of Modern Radar: Basic Principles Raleigh, NC: Scitech, 2010 [7] R Sullivan, Radar Foundations for Imaging and Advanced Concepts Scitech, 2000 [8] S Kingsley and S Quegan, Understanding Radar Systems 1999 [9] C Balanis, Antenna Theory: Analysis and Design Sons, 2005 Raleigh, NC: Raleigh, NC: Scitech, Hoboken, NJ: John Wiley and [10] M C ¸ etin, Feature-Enhanced Synthetic Aperture Radar Imaging Ph.D dissertation, Boston University, 2001 [11] M C ¸ etin and W Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” Image Processing, IEEE Transactions, vol 10, no 4, pp 623–631, Apr 2001 [12] J Skinner, B Kent, R Wittmann, D Mensa, and D Andersh, “Normalization and interpretation of radar images,” Antennas and Propagation, IEEE Transactions, vol 46, no 4, pp 502–506, Apr 1998 [13] B Wang, Digital Signal Processing Techniques and Applications in Radar Image Processing Hoboken, NJ: John Wiley and Sons, 2008 [14] M Richards, Fundamentals of Radar Signal Processing New York, NY: McGraw-Hill, 2005 140 [15] I G Cumming and F H Wong, Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation Boston, MA: Artech House, 2005 [16] M Bertero and P Boccacci, Introduction To Inverse Problems in Imaging UK: Institute of Physics Publishing (IoP), 1998 Bristol, [17] S Luttrell, “A Bayesian derivation of an iterative autofocus/super resolution algorithm,” Inverse Problems, vol 6, Dec 1990 [18] C Oliver and S Quegan, Understanding Synthetic Aperture Radar Images NC: SciTech, 2004 Raleigh, [19] J Horrell, A Knight, and M Inggs, “Motion compensation for airborne SAR,” in Communications and Signal Processing, Proceedings of the 1994 IEEE South African Symposium, pp 128–131, 1994 [20] A Potsis, A Reigber, J Mittermayer, A Moreira, and N Uzunoglou, “Sub-aperture algorithm for motion compensation improvement in wide-beam SAR data processing,” Electronics Letters, vol 37, no 23, pp 1405–1407, 2001 [21] R K Raney, H Runge, R Bamler, I G Cumming, and F H Wong, “Precision SAR processing using chirp scaling,” Geoscience and Remote Sensing, IEEE Transactions, vol 23, no 4, July 1994 [22] A Moriera, J Mittermayer, and R Scheiber, “Extended chirp scaling algorithm for air- and spaceborne SAR data processing in stripmap and scansar imaging modes,” Geoscience and Remote Sensing, IEEE Transactions, vol 34, no 5, Sept 1996 [23] M Soumekh, Synthetic Aperture Radar Signal Processing: With Matlab Algorithms Hoboken, NJ: John Wiley and Sons, 1999 [24] O Frey, E H Meier, and D R Nuesch, “Processing SAR data of rugged terrain by time-domain back-projection,” Proceedings SPIE, vol 5980, 2005 [25] O Frey, E H Meier, and D R Nuesch, “A study on integrated SAR processing and geocoding by means of time-domain backprojection,” in Proceedings of the International Radar Symposium, Berlin, 2005 [26] S Xiao, J Munson, D.C., S Basu, and Y Bresler, “An n2 log(n) back-projection algorithm for SAR image formation,” in Signals, Systems and Computers, Conference Record of the Thirty-Fourth Asilomar Conference, 2000 [27] L Ulander, H Hellsten, and G Stenstrom, “Synthetic-aperture radar processing using fast factorized back-projection,” Aerospace and Electronic Systems, IEEE Transactions, vol 39, no 3, pp 760–776, 2003 [28] S Bin and J Haibo, “Feature enhanced synthetic aperture radar image formation,” in Electronic Measurement and Instruments, 8th International Conference, 2007 [29] T K Moon and W C Stirling, Mathematical Methods and Algorithms for Signal Processing Upper Saddle River, NJ: Prentice Hall, 2000 141 [30] A H Sayed, Adaptive Filters Hoboken, NJ: John Wiley and Sons, 2008 [31] A H Sayed, Fundamentals of Adaptive Filtering Hoboken, NJ: John Wiley and Sons, 2003 [32] R Morrison, M Do, and D Munson, “MCA: A multichannel approach to SAR autofocus,” Image Processing, IEEE Transactions, vol 18, no 4, pp 840–853, 2009 [33] P Samczynski and K Kulpa, “Coherent mapdrift technique,” Geoscience and Remote Sensing, IEEE Transactions, vol 48, no 3, pp 1505–1517, 2010 [34] T Calloway and G Donohoe, “Subaperture autofocus for synthetic aperture radar,” Aerospace and Electronic Systems, IEEE Transactions, vol 30, no 2, pp 617–621, Apr 1994 [35] J Wang and X Liu, “SAR minimum-entropy autofocus using an adaptive-order polynomial model,” Geoscience and Remote Sensing Letters, IEEE, vol 3, no 4, pp 512– 516, 2006 [36] R L Morrison and M N Do, “A multichannel approach to metric-based SAR autofocus,” in Image Processing, IEEE International Conference, vol 2, 2005 [37] C V Jakowatz and D E.Wahl, “Eigenvector method for maximum-likelihood estimation of phase errors in synthetic-aperture-radar imagery,” Journal of the Optical Society of America A, vol 10, no 12, 1993 [38] F Berizzi, G Corsini, M Diani, and M Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimisation,” in Geoscience and Remote Sensing Symposium, ‘Remote Sensing for a Sustainable Future,’ International, vol 2, pp 1230–1232, May 1996 [39] R L Morrison, M N Do, and D C Munson, “SAR image autofocus by sharpness optimization: a theoretical study,” Image Processing, IEEE Transactions, vol 16, no 9, pp 2309–2321, 2007 [40] M Viberg and B Ottersten, “Sensor array processing based on subspace fitting,” Signal Processing, IEEE Transactions, vol 39, no 5, pp 1110–1121, May 1991 [41] S Boyd and L Vandenberghe, Convex Optimization 2004 Cambridge, UK: Cambridge, [42] M Grant and S Boyd, “CVX: Matlab software for disciplined convex programming, version 1.21,” http://cvxr.com/cvx, Feb 2011 [43] G H Golub and C F V Loan, Matrix Computations Baltimore, MD: Johns Hopkins University Press, 1989 [44] R Schreiber and B Parlett, “Block reflectors: theory and computation,” SIAM Journal on Numerical Analysis, vol 25, no 1, 1988 [45] S Singer and S Singer, “Orthosymmetric block reflectors,” Linear Algebra and its Applications, vol 429, no 5-6, pp 1354–1385, 2008 142 Appendix 143 Appendix Circular and Hyperbolic Transformations This appendix derives the block transformations that are needed in the BFARLS algorithm Both the scalar FARLS and the block FARLS algorithms need a J-unitary matrix Θ in order to transform the pre-array into the post-array In both cases Θ can be factored into the product of a unitary circular transformation and a J-unitary hyperbolic transformation Θ = Θc Θh , (A.1) where Θc are the circular transformations and Θh are the hyperbolic transformations In the scalar FARLS, the circular transformations can be accomplished using either Givens or Householder transformations and the hyperbolic transformations can be accomplished by using the Givens or Householder hyperbolic transformations In this appendix, we will be working with Householder transformations A.1 Traditional Circular and Hyperbolic Householder Transformations The circular Householder transformation is well known, [29, 43] Most often the trans- formation is employed to “compress” the energy of a vector or a column of a matrix onto a coordinate axis, thereby introducing zeros into the transformed vector or column Consider the complex vector a ∈ Cn To compress the energy onto the ith coordinate, the Householder vector is formed as v = a ± ejφ(ai ) ||a||2 ei , (A.2) 144 where φ(ai ) is the angle of the ith element of a, [30] The Householder transformation is then computed as Q=I −2 vvH vH v (A.3) It is straightforward to verify that Q is a unitary matrix (length preserving) and involutary (applied twice to a vector returns the original vector) Applied to a generic vector x, Qx is the resulting vector that has been reflected about v⊥ However, applied to the vector a (from which Q was constructed), Qa = ||a||2 ei As an illustration, consider the following matrix (written in terms of its columns) A= a1 a2 · · · an (A.4) Applying a circular Householder transformation with the intention of “compressing” the energy of the first column into the first element gives QA = ||a1 ||2 e1 Qa2 · · · Qan ˜ = A (A.5) Notice that A˜H A˜ = AH QH QA = AH A; thus the Householder transformation applied to a matrix preserves the Frobenius norm and the matrix 2-norm The hyperbolic Householder transformation is a slight modification to the circular transformation Replacing the Euclidean norm with the J-norm, the construction of the hyperbolic Householder vector that will “compress” the energy of a (with respect to the J-norm) onto the ith coordinate axis is v = a ± ejφ(ai ) ||a||J ei , (A.6) where ||v||J = |vH Jv| (A.7) 145 The hyperbolic Householder transformation is also a slight modification of the circular transformation QJ = I − vvH J vH Jv (A.8) H It is straightforward to verify that QJ is J-unitary (i.e QH J JQJ = QJ JQJ = J) and is also H H H −1 = JQ J involutary The inverse of QJ is Q−1 J J = JQJ J and the inverse of QJ is (QJ ) A.2 Block Circular and Hyperbolic Householder Transformations The generalization of Householder transformations are called block reflectors Block reflector transformations are common in the literature and can be either symmetric or nonsymmetric [44, 45]; however, most produce a resulting matrix that has special properties such as being upper-triangular or block upper-triangular and zeros elsewhere Sometimes all that is needed is a transformation that compresses the energy of a matrix into a square submatrix, regardless of the symmetry of the transformation and with no other requirement other than being matrix norm preserving (this is analogous to compressing all the energy of a vector into a single element) To illustrate this result, consider the matrix A ∈ Cm×n If A does not have full row rank (or m > n), then we can zero the rows of A until A has the following form   ΘA =  A˜n×n 0(m−n)×n    (A.9) If A˜n×n were upper-triangular, then the traditional Householder matrix could be used or if A˜n×n were to be block upper-triangular many of the block reflector algorithms in the literature could be used [45] Likewise, if A˜n×n were to be symmetric We now develop a method for generating a generic A˜n×n 146 Without loss of generality, let A be a tall matrix (m > n) and have full column rank The range of A is the subspace spanned by the columns of A R(A) = span a1 a2 · · · , an (A.10) where is the ith column of A The range of A can be extended to form a basis for the m-dimensional vector space S S = span a1 a2 · · · an q q · · · qm−n (A.11) H The vectors qi can be found such that qH i qj = δij and A qi = 0, ∀i (i.e the vectors qi are orthonormal (with respect to each other) and span the null-space of AH ) Thus, S is the direct sum of the range of A and the null-space of AH S = R(A) ⊕ N (AH ) (A.12) Let U denote the matrix whose columns are composed of the qi U= q1 q2 · · · qm−n (A.13) The conjugate transpose of the Householder matrix Q such that   QH U =  0n×(m−n) Λ(m−n)×(m−n)   , (A.14) where Λ(m−n)×(m−n) is an (m − n) × (m − n) reverse-diagonal matrix, such that ΛH Λ = I, is exactly the matrix that produces the result we desire to have in equation (A.9) To be explicit,   QH A =  A˜n×n 0(m−n)×n    (A.15) 147 Due to the qi being orthonormal, the resulting matrix in equation (A.14) can be produced by simple Householder transformations on U where the energy in each column is being compressed down the columns (instead of up the columns as is done in QR decompositions) To show this result, note that since the matrix Q is constructed from Householder transformations, it has the factorization Q = Q1 Q2 · · · Qm−n , (A.16) ˜ = q1 ± ejφ(qm ) em (where ||qi ||2 = has been used) To be where Q1 is constructed from q explicit, Q1 = I − ˜1q ˜H q ˜H q ˜ q (A.17) ˜1 = Let v be a vector that is orthogonal to q1 Then applying Q1 constructed from q q1 + ejφ(qm ) em to v gives QH v = v−2 = v−2 ˜H q v ˜1 q H ˜1 q ˜1 q (A.18) (q1 + ejφ(qm ) em )H v (q1 + ejφ(qm ) em ) (q1 + ejφ(qm ) em )H (q1 + ejφ(qm ) em ) (A.19) e−jφ(qm ) vm (q1 + ejφ(qm ) em ) + |q1,m |   −jφ(q ) m q1,1:m−1  vm  e = v−   + |q1,m | |q1,m | +     e−jφ(qm ) vm  v1:m−1   1+|q1,m | q1,1:m−1  =   − vm vm   e−jφ(qm ) vm  v1:m−1 − 1+|q1,m | q1,1:m−1  =   = v− (A.20) (A.21) (A.22) (A.23) 148 ˜ = q1 − ejφ(qm ) em and applying to v gives Similarly, constructing Q1 from q   v1:m−1 + QH v = e−jφ(qm ) vm 1−|q1,m | q1,1:m−1    (A.24) ˜ = q1 ± ejφ(qm ) em , then These two cases can be combined; let q   v1:m−1 ∓ QH v = e−jφ(qm ) vm 1±|q1,m | q1,1:m−1   , (A.25) where v1:m−1 is the vector formed from the first m − elements of v and vm is the mth element of v Applying Q1 to U gives QH U = ∓ejφ(qm ) em QH q2 · · · QH qm−n (A.26) Using that the columns of U are orthonormal, substituting qi (2 ≤ i ≤ m − n) for v in equation (A.25) gives   qi,1:m−1 ∓ QH qi =  qi,m e−jφ(q1,m ) q1,1:m−1 1±|q1,m |   , ≤ i ≤ m − n, (A.27) H where ql,m is the mth element of vector ql Forming Q2 from QH q2 and applying to Q1 U gives H QH Q1 U = Hq ∓ejφ(qm ) em ∓ejφ([Q1 ]m−1 ) H em−1 QH Q1 q · · · H QH Q1 qm−n (A.28) 149 Continuing on in this manner shows that   0n×(m−n)  H H QH U = QH m−n · · · Q2 Q1 U =   , Λ(m−n)×(m−n) (A.29) as claimed Applying Q1 to the matrix A gives QH A= H QH a1 Q1 a2 · · · QH am−n (A.30) Using that the columns of A are orthogonal to the qi (by construction) and substituting for v in equation (A.25) gives   ai,1:m−1 ∓ QH =  e−jφ(qm ) ai,m 1±|q1,m | q1,1:m−1    (A.31) Since equation (A.31) holds for any column of A, it is clear that QH A has compressed all of the energy in the bottom row of A into the rows above Continuing in this manner with Q2 through Qm−n shows that   H H QH A = Q H m−n · · · Q2 Q1 A =  A˜n×n 0(m−n)×n    (A.32) The geometric interpretation of this is that the orthonormal vectors that span the null space of AH are being aligned with the coordinate axes en+1 through em and thus the reflected range cannot have any components on those axes The block FARLS needs the J-unitary equivalent to equation (A.15) Under a strict condition (which will be stated below), a similar result holds for J-unitary block reflectors Assume that the condition holds, let U be the same as above (i.e the columns of U are orthonormal and span the null-space of AH ), then the inverse of the complex transpose of 150 the J-unitary block reflection matrix QJ such that   QJ U =  0n×(m−n) C(m−n)×(m−n)   , (A.33) where C(m−n)×(m−n) is an (m − n) × (m − n) reverse-lower triangular matrix, is the matrix that produces the desired result   −1 (QH J ) A = JQJ JA =  A˜n×n 0(m−n)×n    (A.34) The matrix QJ can be factored into a product of J-unitary Householder transformations QJ = QJ,(m−n) · · · QJ,1 , (A.35) where each QJ,i has the same construction as in equation (A.8) The condition that must hold to produce the results in equation (A.34) is ˆH ˆ i = sign(Ji,i ), q i Jq (A.36) ˆ i is the vector that QJ,i is constructed from where q The result in equation (A.34) is exactly what we need for the block FARLS algorithm If the stated condition fails, it is an indication that numerical stability has set in and the block FARLS starts to diverge 151 Vita Roger D West Education • Ph.D Electrical and Computer Engineering, Utah State University, Logan, UT, 2011 • M.S Electrical and Computer Engineering, Utah State University, Logan, UT, 2008 • B.S Electrical and Computer Engineering (Summa Cum Laude, Minor in Mathematics), Utah State University, Logan, UT, 2008 • A.S Electrical and Computer Engineering, Salt Lake Community College, Salt Lake City, UT, 2005 Awards • Space Dynamics Laboratory Tomorrow Fellowship recipient, 2008-2011 • Utah State University Outstanding Graduate Teaching Assistant, 2007-2008 • Utah State University Presidential Transfer Scholarship recipient, 2005-2006 Published Conference Papers • Maximum Likelihood Estimation of Ground Reflectivity from Synthetic Aperture Radar Data, Roger West, Jake Gunther, and Todd Moon, in 56th Annual Meeting of the MSS Tri-Services Radar Symposium, 2010 152 • A Novel Block Fast Array RLS Algorithm Applied to Linear Flight Strip-Map SAR Images, Roger West, Todd Moon, and Jake Gunther, in Signals, Systems and Computers (ASILOMAR), Conference Record of the Forty Fourth Asilomar Conference, 2010 • Forming Regularized Maximum Likelihood Strip-Map Synthetic Aperture Radar Images Using the Block RLS Algorithm, Roger West, Jake Gunther, and Todd Moon, in Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), 2011 • Maximum Likelihood Synthetic Aperture Radar Image Formation for Highly Nonlinear Flight Tracks, Jake Gunther, Roger West, Nate Crookston, and Todd Moon, in Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), 2011 • Model-Based Autofocus for Strip-Map SAR Images Formed via Convolution BackProjection, Roger West, Jake Gunther, and Todd Moon, in 57th Annual Meeting of the MSS Tri-Services Radar Symposium, 2011 (Accepted) • STARTLE: An Open-Source Software Tool for SAR Signal Processing and Imaging, Jake Gunther, Nate Crookston, Roger West, and Todd Moon, in 57th Annual Meeting of the MSS Tri-Services Radar Symposium, 2011 (Accepted) ... Pulsed LFM Stripmap Synthetic Aperture Radar 4.1 Stripmap Geometry 4.2 Synthetic Aperture for Stripmap SAR 4.3 Induced Azimuth Signal for Stripmap. .. Utah 2011 ii Copyright c Roger D West 2011 All Rights Reserved iii Abstract Model-Based Stripmap Synthetic Aperture Radar Processing by Roger D West, Doctor of Philosophy Utah State University, 2011.. .MODEL-BASED STRIPMAP SYNTHETIC APERTURE RADAR PROCESSING by Roger D West A dissertation submitted in partial fulfillment