• Sonuç bulunamadı

4.5 Numerical Results

4.5.3 Summary of the Comparison Study

sector. The results are presented in Fig. 4.11. The FDOA outperforms all other techniques which can achieve RMSE = 3.5o while the second best technique, the A-MMD, can achieve RMSE = 6o. Compared to the previous scenario, since L(i)r

is increased, the general performance of the FDOA, the A-MMD, and the EA-MMD degrades. For the A-EA-MMD and the EA-EA-MMD, the reason is the increasing uncertainty in the prior information. For the FDOA, focusing on larger sectors is expected to be more difficult which explains the performance degradation.

0 10 20 30 40

SNR (dB) 0

5 10 15 20 25 30

RMSE (o )

FDOA IRG A-MMD EA-MMD RGD

Figure 4.11: DOA estimation performance comparison among the FDOA, the IRG, the A-MMD, the EA-MMD and the RGD for K = 4 where the first two targets emerge at the 1st sector and the remaining two targets emerge at the 3rd sector with L = 120 and L(i)r = 40

achieves the lowest CRLBs in all scenarios. After the CRLB comparison, the DOA estimation performance of different algorithms is investigated. The FDOA can achieve RMSE ≈ 0o in various scenarios and outperforms the alternatives including the A-MMD and the EA-MMD that were proposed in Chapter 3. Fur-thermore, even at the low SNR regime, the FDOA achieves high performance even in the most challenging scenarios despite the highly noisy environment.

Chapter 5

Conclusions and Future Research Directions

In this thesis, two essential essential questions are addressed: why and how the CS based techniques should be used in DOA estimation. In Chapter 2, it is shown that the CS based DOA estimation techniques outperform the classical methods under various scenarios. Furthermore, unlike the well-known classical techniques the MUSIC and the Capon’s beamformer, the CS based techniques can typically work with a single snapshot and they are not affected by the multipath effects.

That is esentially “why” the CS based DOA estimation should be preferred to the classical methods.

To improve the performance of a CS based DOA estimation system, the natu-ral attempt would be to improve the encoding and the decoding strategies. The encoding of data in a CS based system is determined by measurement matri-ces; therefore, measurement matrix design is among the most important aspects of the CS based sensor processing applications. Various measurement matrix de-sign algorithms are proposed in this thesis by using the mutual coherence and the CRLB criteria. The mutual coherence based design, namely the A-MMD adapts to the prior information on the target scene. Hence, a computationally feasible,

closed-form expression for the measurement matrix is derived enabling online up-dates of the measurement matrix at a high compression rate for each snapshot of the sensor data. Further improvements on the software and the hardware complexity of the proposed measurement matrix methodology are done which allows for more efficient hardware and software implementation of the proposed system, namely, the more software-efficient EA-MMD and the more hardware-efficient A-MMD-BD algorithms are presented. The CRLB based measurement matrix design methodology requires solving a more complicated non-convex op-timization compared to the A-MMD algorithm and does not have a closed-form solution. However, the CRLB based design does not need a prior information on the target scene, while it is able to adapt to the prior information depending on the application. To sum up, the measurement matrix design is extensively studied in this thesis. For the decoding part, an iterative reconstruction algo-rithm that can work with the proposed CRLB based measurement matrix design methodology is proposed leading to the FDOA algorithm. The FDOA finds the DOAs of the sources by focusing on particular sectors and iteratively canceling out the interferences with the received signal leading to the successive estimations on the varying residual signal. The proposed methodologies in the thesis provide alternative answers to the question of “how” the CS based techniques should be used in DOA estimation.

The proposed techniques are eligible for the joint optimizization over both the measurement matrix and the signal dictionary. While a detailed investigation of the dictionary design is not performed in the presented thesis, a simple adaptive dictionary design algorithm is proposed and shown to improve the performance under some scenarios when it is used with the A-MMD algorithm. As a future study, dictionary design in the CS based DOA estimation can be studied in more detail and the resulting designs can be integrated with the techniques proposed in the thesis. Furthermore, while the hardware complexity constraints are taken into account, the other restrictions caused by the hardware implementation of the presented system such as mutual coupling effects, quantization limits and the calibration errors in the antenna displacement are not investigated. Another research direction can be the wideband extensions of the proposed results since

a narrowband DOA estimation signal model is assumed throughout the thesis.

Finally, although the DOA estimation has been the main focus, in the future, the proposed methodologies can be applied to other areas of sensor processing including detection of sparse signals in video and audio streams.

Bibliography

[1] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.

[2] D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory, vol. 52, no. 4, pp. 1289–1306, April 2006.

[3] E. J. Cand`es, and T. Tao, “Near-optimal signal recovery from random pro-jections: Universal encoding strategies?,” IEEE Trans. Inform. Theory, vol.

52, no. 12, pp. 5406–5425, Dec. 2006.

[4] E. J. Candes, and T. Tao, “Decoding by linear programming,” IEEE Trans.

Inform. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005.

[5] R. G. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag., vol. 24, no. 4, pp. 118–120, Jul. 2017.

[6] M. Elad, Sparse and Reduntant Representations: From Theory to Applica-tions in Signal and Image Processing. New York: Springer, 2010.

[7] M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An algorithm for design-ing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4311–4322, Nov. 2006.

[8] J. Mairal, F. Bach, J. Ponce, and G. Sapiro, “Online Learning for matrix Factorization and sparse coding,” J. Machine Learning Research, vol. 11, pp.

19-60, 2010.

[9] J. M. Duarte-Carvajalino, and G. Sapiro, “Learning to sense sparse signals:

simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process., vol. 18, no. 7, pp. 1395–1408, 2009.

[10] D. L. Donoho, and M. Elad, “Optimally sparse representation in general (non-orthogonal) dictionaries via `1 minimization,” Proc. Nat. Acad. Sci., vol. 100, pp. 2197–2202, 2002.

[11] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Ap-proximation, vol. 28, no. 3, pp. 253–263, 2008.

[12] M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Sig-nal Process., vol. 55, no. 12, pp. 5695–5701, Dec. 2007.

[13] B. ¨Ozer, A. Lavrenko, S. Gezici, G. Del Galdo, and O. Arıkan, “Adaptive measurement matrix design for compressed doa estimation with sensor ar-rays,” in Proc. Asilomar Conf. Signals, Syst. Comput., 2015, pp. 1769–1773.

[14] Y. Gu, Y. D. Zhang, and N. A. Goodman, “Optimized compressive sensing based direction-of-arrival estimation in massive MIMO,” in Proc. IEEE Int.

Conf. Acoust., Speech, Signal Process., New Orleans, LA, USA, Mar. 2017, pp. 3181–3185.

[15] J. Tropp, and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inform. Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007.

[16] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society, vol. 58, no. 1, pp. 267–288, 1996.

[17] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1998.

[18] H. Krim, and M. Viberg, “Two decades of array signal processing research:

The parametric approach,” IEEE Signal Process. Mag., vol. 13, July 1996.

[19] B. D. Van Veen, and K. M. Buckley, “Beamforming: a versatile approach to spatial filtering,” IEEE ASSP Mag. pp. 4–24, Apr. 1988.

[20] J. Capon, “High resolution frequency-wavenumber spectrum analysis,” Proc.

IEEE, vol. 57, pp. 1408–1418, Aug. 1969.

[21] R. Schmidt, “Multiple emitter location and signal parameter estimation,”

IEEE Trans. Antennas Propagat., vol. 34, no. 3, pp. 276–280, Mar. 1986.

[22] R. G. Lorenz, and S. P. Boyd, “Robust minimum variance beamforming,”

IEEE Trans. on Signal Process., vol. 53, no. 5, pp. 1684–1696, May 2005.

[23] M. Orton, and W. Fitzgerald, “A Bayesian approach to tracking multiple targets using sensor arrays and particle filters,” IEEE Trans. on Signal Pro-cess., vol. 50, no.2, pp. 216–223, Feb. 2002.

[24] N. Yuen, and B. Friedlander, “Doa estimation in multipath: An approach using fourth-order cumulants,” IEEE Trans. on Signal Process., vol. 45, no.

5, pp. 1253–1263, May 1997.

[25] P. Palanisamy, N. Kalyanasundaram, and P. M. Swetha, “Two-dimensional DOA estimation of coherent signals using acoustic vector sensor array,” Sig-nal Process.,, vol. 92, no. 1, pp. 19–28, Jan. 2012.

[26] D. Malioutov, M. Cetin, and A. Willsky, “A sparse signal reconstruction per-spective for source localization with sensor arrays,” IEEE Trans. on Signal Process., vol. 53, no. 8, pp. 3010–3022, Aug. 2005.

[27] Y. Wang, G. Leus, and A. Pandharipande, “Direction estimation using com-pressive sampling array processing,” in Proc. IEEE/SP Workshop on Statis-tical Signal Process., Aug. 2009, pp. 626–629.

[28] M. Ibrahim, F. Roemer, and G. D. Galdo, “On the design of the measurement matrix for compressed sensing based DOA estimation,” in Proc. Int. Conf.

Acoust., Speech, Signal Process., pp. 3631–3635, Apr. 2015.

[29] M. Ibrahim, F. Roemer, and G. D. Galdo, “An adaptively focusing measure-ment design for compressed sensing based doa estimation,” in Proc. Eur.

Signal Process. Conf., 2015, pp. 859–863.

[30] M. Ibrahim, V. Ramireddy, A. Lavrenko, J. K¨onig, F. R¨omer, M. Landmann, M. Grossmann, G. D. Galdo, and R. S. Thoma, “Design and analysis of com-pressive antenna arrays for direction of arrival estimation,” Signal Process., vol. 138, pp.35–47, Sep. 2017.

[31] M. Guo, Y. D. Zhang, and T. Chen, “Doa estimation using compressed sparse array,” IEEE Trans. on Signal Process., vol. 66, no. 15, pp. 4133–4146, Aug.

2018.

[32] B. Kılı¸c, A. G¨ung¨or, M. Kalfa, and O. Arıkan, “Adaptive measurement mat-rix design in compressed sensing based direction of arrival estimation,” in Proc. Eur. Signal Process. Conf., 2020, pp. 1881–1885.

[33] M. A. Davenport, A. K. Massimino, D. Needell, and T. Woolf, “Constrained adaptive sensing,” IEEE Trans. on Signal Process., vol. 64, no. 20, pp.

5437–5449, Oct. 2016.

[34] E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted `1 minimization,” J. Fourier Anal. Appl., vol. 14, pp. 877–906, 2008.

[35] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans.

Signal Process., vol. 56, no. 6, pp. 2346–2356, June 2008.

[36] B. Kılı¸c, M. Kalfa, and O. Arıkan, “Prior based grid selection algorithm for compressed sensing based direction of arrival estimation methods,” in 10th International Symposium on Phased Array Systems and Technology, Waltham, MA USA, Oct. 2019.

[37] S. Boyd, and L. Vandenberghe, Convex Optimization, Cambridge, U.K.:

Cambridge Univ. Press, 2004.

[38] A. M. Tillmann, and M. E. Pfetsch, “The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing,” IEEE Trans. Inf. Theory, vol. 60, no. 2, pp. 1248–1259, Feb. 2014.

[39] O. Teke, “Robust compressive sensing techniques,” M.S. thesis, Electrical and Electronic Engineering Dept., Bilkent Univ., Ankara, Turkey, 2014.

[40] R. Roy, and T. Kailath, “ESPRIT - Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. on Acoustics, Speech and Signal Process., vol. 37, no. 7, July 1989.

[41] D. Feng, M. Bao, Z. Ye, L. Guan, and X. Li, “A novel wideband DOA estimator based on Khatri–Rao subspace approach,” Signal Process., vol.

91, no. 10, pp. 2415-2419, Oct. 2011.

[42] J. Selva, “Efficient wideband doa estimation through function evaluation techniques,” IEEE Trans. on Signal Process., vol. 66, no.12, pp. 3112–3123, June 2018.

[43] M. Wax, T. J. Shan, and T. Kailath, “Spatio temporal spectral analysis by eigenstructure methods,” IEEE Trans. on Acoustics, Speech and Signal Process., vol. 32, no. 4, pp. 817-827, Aug 1984.

[44] H. Wang, and M. Kaveh, “Coherent signal-subspace processing for the de-tection and estimation of angles of arrival of multiple wide-band sources,”

IEEE Trans. on Acoustics, Speech and Signal Process., vol. 33, no. 4, pp.

823-831, Aug 1985.

[45] Y. S. Yoon, L. M. Kaplan, and J. H. McClellan, “TOPS: New DOA estima-tor for wideband signals,” IEEE Trans. Signal Process., vol. 54, no. 6, pp.

1977–1989, Jun. 2006.

[46] Q. Shen, W. Liu, W. Cui, and S. Wu, “Underdetermined DOA estimation under the compressive sensing framework: A review,” IEEE Access, vol. 4, pp. 8865–8878, 2016.

[47] P. Stoica, and A. Nehorai, “Music, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. on Acoustics, Speech and Signal Process., vol. 37, no.

5, May 1989.

[48] S. Haykin, J. Litva, and T. J. Shepherd, Radar Array Processing, 1st ed.

New York: Springer-Verlag, 1993.

[49] P. Stoica, and K. Sharman, “Maximum likelihood methods for direction-of-arrival estimation,” IEEE Trans. ASSP, vol. 38, no. 7, pp. 1132–1143, July 1990.

[50] A. E. Gonnouni, M. M. Ramon, J. L. Rojo-Alvarez, G. Camps-Valls, A.

R. Figueiras-Vidal, C. G. Christodoulou, “A support vector machine music algorithm,” IEEE Trans. Antennas and Propagation, vol. 60, no. 10, Oct.

2012.

[51] A. J. Barabell, J. Capon, D. F. Delong, J. R. Johnson, and K. Senne, “Per-formance comparison of superresolution array processing algorithms,” Tech.

Rep. TST-72, Lincoln Lab., M.I.T., 1998.

[52] Y. C. Eldar, A. Nehorai, and P. S. La Rosa, “A competitive mean-squared error approach to beamforming,” IEEE Trans. on Signal Process., vol. 55, no. 11, pp. 5143–5154, Nov. 2007.

[53] K. Lange, MM Optimization Algorithms, Philadelphia, PA, USA: SIAM, 2016.

[54] S. Foucart, and H. Rauhut, A Mathemathical Introduction to Compressive Sensing, Basel, Switzerland: Birkh¨auser, 2013.

[55] M. Grant, and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.0 beta [Online]. Available: http://cvxr.com/cvx, September 2013.

[56] M. Grant, and S. Boyd. “Graph implementations for nonsmooth convex pro-grams,” in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, Eds. New York: Springer, 2008, Lecture Notes in Control and Information Sciences, pp. 95–110.

[57] E. van den Berg, and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” [Online]. Available: https://friedlander.io/spgl1 [58] E. van den Berg, and M. P. Friedlander, “Probing the Pareto frontier for

basis pursuit solutions,” SIAM J. on Scientific Computing, vol. 31, no. 2, pp. 890–912, 2008.

[59] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed op-timization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning, vol. 3, no. 1, pp.

1–122, 2011.

[60] G. Li, Z. Zhu, X. Wu, and B. Hou, “On joint optimization of sensing matrix and sparsifying dictionary for robust compressed sensing systems,” Digital Signal Processing, vol. 73, pp. 62–71, Feb. 2018.

[61] Z. Yang, J. Li, P. Stoica, and L. Xie, “Sparse methods for direction-of-arrival estimation,” 2017, arXiv:1609.09596v2.

[62] M. Wagner, Y. Park, and P. Gerstoft, “Gridless DOA estimation and root-MUSIC for non-uniform arrays,” IEEE Trans. on Signal Process., vol. 69, pp. 2144–2157, 2021.

[63] Y. Park, and P. Gerstoft, “Alternating projections gridless covariance-based estimation for DOA,” in Proc. Int. Conf. Acoust., Speech, Signal Process., 2021, pp. 4385–4389.

[64] M. Kalfa, and H. E. G¨uven, “Fast 2-d direction of arrival estimation using two-stage gridless compressive sensing,” in Proc. Int. Conference on Radar, 2018.

[65] B. N. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with ap-plications to line spectral estimation,” IEEE Trans. Signal Process., vol. 61, no. 23, pp. 5987-5999, 2013.

[66] Y. Chen, and Y. Chi, “Robust spectral compressed sensing via structured matrix completion,” IEEE Trans. Inform. Theory, vol. 60, no. 10, pp.

6576–6601, 2014.

[67] Z. Yang, L. Xie, and C. Zhang, “A discretization-free sparse and parametric approach for linear array signal processing,” IEEE Trans. Signal Process., vol. 62 no. 19, pp. 4959–4973, 2014.

[68] Z. Yang, and L. Xie, “On gridless sparse methods for line spectral estimation from complete and incomplete data,” IEEE Trans. Signal Process., vol. 63, no. 12, pp. 3139–3153, Jun. 2015.

[69] V. F. Pisarenko, “The retrieval of harmonics from a covariance function,”

Geophysical Journal International, vol. 33, no. 3, pp. 347–366, 1973.

[70] J. H. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for gen-eralized linear models via coordinate descent,” Journal of Statistical Soft-ware, vol. 33, no. 1, pp. 1–22, Jan. 2010.

[71] M. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process., vol. 20, no. 3, pp. 681–695, March 2011.

[72] M. Mahadi, T. Ballal, M. Moinuddin, T. Y. Al-Naffouri, and U. Al-Saggaf,

“Low-complexity robust beamforming for a moving source,” in Proc. Eur.

Signal Process. Conf., 2020, pp. 1846–1850.

[73] A. J. Weiss, and B. Friedlander, “Preprocessing for direction finding with minimal variance degradation,” IEEE Trans. on Signal Process., vol. 42, no.

6, June 1994.

[74] J. Sheinvald, and M. Wax, “Direction finding with fewer receivers via time-varying preprocessing,” IEEE Trans. on Signal Process., vol. 47, no. 1, pp.

2–9, Jan. 1999.

[75] V. Abolghasemi, S. Ferdowsi, B. Makkiabadi, and S. Sanei, “On optimization of the measurement matrix for compressive sensing,” in Proc. Eur. Signal Process. Conf., Aalborg, Denmark, Aug. 2010, pp. 427–431.

[76] L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimiza-tion for block-sparse decoding,” IEEE Trans. on Signal Process., vol. 59, no.

6, pp. 4300–4312, Sep. 2011.

[77] G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix opti-mization for compressive sensing systems,” IEEE Trans. on Signal Process., vol. 61, no. 11, pp. 2887–2898, June 2013.

[78] N. Cleju, “Optimized projections for compressed sensing via rank-constrained nearest correlation matrix,” Appl. Computat. Harmon. Anal., vol. 36, no. 3, pp. 495–507, May 2014.

[79] M. Sustik, J. A. Tropp, I. S. Dhillon, and R. W. Heath, Jr., “On the existence of equiangular tight frames,” Linear Algebra and Its Appl., vol. 426, no. 2–

3, pp. 619–635, 2007.

[80] J. A. Tropp, I. S. Dhillon, R. W. Heath, Jr., and T. Strohmer, “Designing structured tight frames via an alternating projection method,” IEEE Trans.

Inf. Theory, vol. 51, no. 1, pp. 188–209, 2005.

[81] J. Xu, Y. Pi, and Z. Cao, “Optimized projection matrix for compressive sensing,” in EURASIP J. on Adv. in Signal Process., pp. 1–8, 2010.

[82] V. Abolghasemi, S. Ferdowsi, and S. Sanei, “A gradient-based alternating minimization approach for optimization of the measurement matrix in com-pressive sensing,” Signal Process., vol. 92, pp. 999–1009, 2012.

[83] L. R. Welch, “Lower bounds on the maximum cross correlation of signals,”

IEEE Trans. Inf. Theory, vol. 20, no. 3, pp. 397–399, May 1974.

[84] B. Li, L. Zhang, T. Kirubarajan, and S. Rajan, “Projection matrix design using prior information in compressive sensing,” Signal Processing, vol. 135, pp. 36–47, 2017.

[85] T. Huang, Y. Liu, H. Meng, and X. Wang, “Adaptive compressed sensing via minimizing Cramer-Rao bound,” IEEE Signal Process. Letters, vol. 21, no. 3, pp. 270–274, Mar. 2014.

[86] T. M. Cover, and J. A. Thomas, Elements of Information Theory, 2nd ed.

New York, NY, USA: Wiley, 2006.

[87] R. Obermeier, and J. A. Martinez-Lorenzo, “Sensing matrix design via capacity maximization for block compressive sensing applications,” IEEE Trans. on Computational Imaging, vol. 5, no. 1, pp. 27–36, Mar. 2019.

[88] X. Duan, J. Li, Q. Wang, and X. Zhang, “Low rank approximation of the symmetric positive semidefinite matrix,” J. Comput. Appl. Math., vol. 260, pp. 236–243, 2014.

[89] A. Dax, “Low-rank positive approximants of symmetrix matrices,” Advances in Linear Algebra & Matrix Tehory, vol. 4, no. 3, pp. 172–185, 2014.

[90] B. Kılı¸c, M. Kalfa, and O. Arıkan, “Adaptive sensing matrix design in com-pressive sensing based direction of arrival estimation with hardware con-straints,” in Proc. IEEE International Symposium on Antennas and Propa-gation and North American Radio Science Meeting, 2020, pp. 149–150.

[91] M. A. Richards, Fundamentals of Radar Signal Processing, New York, NY, USA: McGraw Hill, 2005.

[92] A. Eftekhari, H. L. Yap, C. J. Rozell, and M. B. Wakin “The restricted isom-etry property for random block diagonal matrices,” 2014, arXiv:1210.3395v2.

[93] M. Shaghaghi, and S. A. Vorobyov, “Cramer-Rao bound for sparse signals fitting the low-rank model with small number of parameters,” IEEE Signal Process. Letters, vol. 22, no. 9, Sep. 2015.

[94] P. Pakrooh, A. Pezeshki, L. L. Scharf, D. Cochran, and S. D. Howard, “Anal-ysis of Fisher information and the Cramer-Rao bound for nonlinear parame-ter estimation afparame-ter random compression,” IEEE Trans. on Signal Process., vol. 63, no. 23, pp. 6423–6428, Dec. 2015.

[95] D. Ramasamy, S. Venkateswaran, and U. Madhow, “Compressive parameter estimation in awgn,” IEEE Trans. on Signal Process., vol. 62, no. 8, pp.

2012–2027, Apr. 2014.

[96] J. Zhu, L. Han, R. S. Blum, and Z. Xu, “On the analysis of the Fisher information of a perturbed linear model after random compression,” IEEE Signal Process. Letters, vol. 25, no. 1, pp. 100–105, Jan. 2018.

[97] Z. Ben-Haim and Y. Eldar, “The Cramer–Rao bound for estimating a sparse parameter vector,” IEEE Trans. Signal Process., vol. 58, no. 6, pp.

3384–3389, Jun. 2010.

[98] A. Poudel and D. R. Fuhrmann, “Adaptive sensing and target tracking of a simple point target with online measurement selection,” in Proc. Asilomar Conf. Signals, Syst. Comput., 2010, pp. 2017–2020.

[99] S. S. Dragomir, “Some trace inequalities for operators in Hilbert spaces,”

Kragujevac J. Math, vol. 41, no. 1, pp. 33–55, 2017.

[100] P. Stoica, J. Li, and X. Zhu, “Waveform synthesis for diversity-based trans-mit beampattern design,” IEEE Trans. on Signal Process., vol. 56, no. 6, pp.

2593–2598, June 2008.

[101] B. F. Green, “The orthogonal approximation of an oblique structure in factor analysis,” Psychometrika, vol. 17, no. 4, pp. 429–440, Dec. 1952.

[102] P. H. Schonemann, “A generalized solution of the orthogonal procrustes problem,” Psychometrika, vol. 31, no. 1, pp. 1–10, Mar. 1966.

[103] T. Viklands, “Algorithms for the weighted orthogonal Procrustes problem and other least squares problems,” Ph.D. dissertation, Comput. Sci. Dept., Umea Univ., Umea, Sweden, 2006.

[104] K. Kume, and I. Yamada, “A Nesterov-type acceleration with adaptive localized Cayley parametrization for optimization over the Stiefel manifold,”

in Proc. Eur. Signal Process. Conf., 2020, pp. 2105–2109.

[105] F. Nie, R. Zhang, and X. Li, “A generalized power iteration method for solv-ing quadratic problem on the Stiefel manifold,” 2017, arXiv:1701.00381v1.

[106] D. Higham, and N. Higham, MATLAB Guide. Philadelphia: SIAM, 2005.

[107] C. Eckart, and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika, vol. 1, pp. 211-218, 1936.

[108] L. Mirsky, “Symmetric gauge functions and unitarily invariant norms,”

Quart. J. Math., vol. 11, no. 1, pp. 50–59, 1960.

[109] A. A. Mohsenipour, “On the distribution of quadratic expressions in variaous types of random vectors,” Ph.D. dissertation, Univ. Western On-tario, London, ON, Canada, 2012.

Appendix A

To achieve the result given in (3.23), we first write the eigendecomposition of Z = QΛQH for QQH = QHQ = IM and Λ ≡ diag(λ). Since the Frobenius norm is a unitarily invariant norm, we can write

kGΨ− Zk2F = kQHGΨQ − Λk2F, (A.1) and optimize over GΨQ ≡ QHGΨQ since GΨQ satisfies the constraints if and only if GΨ satisfies them. Let

Λ ≡¯

λ1 . . . 0 ... . .. ...

0 . . . λm

0

0 0

 , ˜Λ ≡

λ1 . . . 0 ... . .. ...

0 . . . λz

0

0 0

. (A.2)

The proof consists of two parts for m < z and m ≥ z.

I) For m < z, let GΨQ be any feasible matrix. Our aim is to show that k ¯Λ − Λk2F ≤ kGΨQ − Λk2F for any GΨQ to prove that ¯Λ is the solution. We write GΨQ and Λ as:

GΨQ =

"

GzzΨQzzΨQrrΨQ GrrΨQ

# , Λ =

"

Λzz 0 0 Λrr

#

, (A.3)

where GzzΨQ and Λzz consist of the first z rows and the first z columns of the matrices GΨQ and Λ while GrrΨQ and Λrr include rest of the entries that are on the block diagonal of GΨQ and Λ respectively. G˘zzΨQ and ˘GrrΨQ denote the