{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T14:45:59Z","timestamp":1775745959628,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T00:00:00Z","timestamp":1698710400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61871384"],"award-info":[{"award-number":["61871384"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61901487"],"award-info":[{"award-number":["61901487"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61901498"],"award-info":[{"award-number":["61901498"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61921001"],"award-info":[{"award-number":["61921001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022RC1092"],"award-info":[{"award-number":["2022RC1092"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["61871384"],"award-info":[{"award-number":["61871384"]}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["61901487"],"award-info":[{"award-number":["61901487"]}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["61901498"],"award-info":[{"award-number":["61901498"]}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["61921001"],"award-info":[{"award-number":["61921001"]}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["2022RC1092"],"award-info":[{"award-number":["2022RC1092"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The water-filling (WF) algorithm is a widely used design strategy in the radar waveform design field to maximize the signal-to-interference-plus-noise ratio (SINR). To address the problem of the poor resolution performance of the waveform caused by the inability to effectively control the bandwidth, a novel waveform-related optimization model is established in this paper. Specifically, a corrected SINR expression is first derived to construct the objective function in our optimization model. Then, equivalent bandwidth and energy constraints are imposed on the waveform to formulate the waveform-related non-convex optimization model. Next, the optimal frequency spectrum is obtained using the Karush\u2013Kuhn\u2013Tucker condition of our non-convex model. Finally, the transmit waveform in the time domain is synthesized under the constant modulus constraint. Different experiments based on simulated and real-measured data are constructed to demonstrate the superior performance of the designed waveform on the SINR and equivalent bandwidth compared to the linear frequency modulated signal and waveform designed by the WF algorithm. In addition, to further evaluate the effectiveness of the proposed algorithm in the application of cognitive radar (CR), a closed-loop radar system design strategy is introduced based on our waveform design method. The experiments under real-measured data confirm the advantages of CR compared to the traditional open-loop radar structure.<\/jats:p>","DOI":"10.3390\/rs15215187","type":"journal-article","created":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T12:48:31Z","timestamp":1698756511000},"page":"5187","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Cognitive Radar Waveform Design Method under the Joint Constraints of Transmit Energy and Spectrum Bandwidth"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3294-7181","authenticated-orcid":false,"given":"Chen","family":"Yang","sequence":"first","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Wei","family":"Yang","sequence":"additional","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Xiangfeng","family":"Qiu","sequence":"additional","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Wenpeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Zhejun","family":"Lu","sequence":"additional","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Weidong","family":"Jiang","sequence":"additional","affiliation":[{"name":"The College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MAES.2019.2953762","article-title":"An Overview of Cognitive Radar: Past, Present, and Future","volume":"34","author":"Gurbuz","year":"2019","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"He, H., Li, J., and Stocia, P. (2012). Waveform Design for Active Sensing Systems: A Computational Approach, Cambridge University Press.","DOI":"10.1017\/CBO9781139095174"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5603","DOI":"10.1109\/TSP.2013.2273885","article-title":"Ambiguity function shaping for cognitive radar via complex quartic optimization","volume":"61","author":"Aubry","year":"2013","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"108145","DOI":"10.1016\/j.sigpro.2021.108145","article-title":"A new approach for design of constant modulus discrete phase radar waveform with low WISL","volume":"187","author":"Bu","year":"2021","journal-title":"Sig. Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1109\/TSP.2015.2510982","article-title":"Sequence design to minimize the weighted integrated and peak sidelobe levels","volume":"64","author":"Song","year":"2015","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1109\/TSP.2016.2621723","article-title":"Cognitive radar-based sequence design via SINR maximization","volume":"65","author":"Wu","year":"2016","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zhang, H., Zhang, W., Liu, Y., Yang, W., and Yong, S. (2023). Scatterer-Level Time-Frequency-Frequency Rate Representation for Micro-Motion Identification. Remote Sens., 6615.","DOI":"10.3390\/rs15204917"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhang, H., Tang, M., Zhang, W., Yang, W., and Jiang, W. (2023, January 8\u201310). Micro-motion Signal Enhancement via Convolutional Autoencoder Equipped with Multi-scale Feature Pyramid. Proceedings of the 2023 8th International Conference on Signal and Image Processing (ICSIP), Wuxi, China.","DOI":"10.1109\/ICSIP57908.2023.10271027"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1109\/TSP.2017.2787115","article-title":"Transmit waveform\/receive filter design for MIMO radar with multiple waveform constraints","volume":"66","author":"Wu","year":"2017","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1109\/TSP.2016.2620113","article-title":"A unified framework for low autocorrelation sequence design via majorization\u2013minimization","volume":"65","author":"Zhao","year":"2016","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3783","DOI":"10.1109\/TSP.2015.2423257","article-title":"Relative Entropy-Based Waveform Design for MIMO Radar Detection in the Presence of Clutter and Interference","volume":"63","author":"Tang","year":"2015","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1109\/TSP.2016.2601299","article-title":"Majorization-minimization algorithms in signal processing, communications, and machine learning","volume":"65","author":"Sun","year":"2016","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_13","first-page":"1","article-title":"Waveform and Filter Joint Design Method for Pulse Compression Sidelobe Reduction","volume":"60","author":"Zhou","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1109\/TSP.2017.2787104","article-title":"Fast Algorithms for Designing Unimodular Waveform(s) with Good Correlation Properties","volume":"66","author":"Li","year":"2018","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5367","DOI":"10.1109\/TSP.2016.2597123","article-title":"Unimodular sequence design based on alternating direction method of multipliers","volume":"64","author":"Liang","year":"2016","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4912","DOI":"10.1109\/TSP.2017.2718976","article-title":"Constant Modulus Waveform Design for MIMO Radar Transmit Beampattern","volume":"65","author":"Cheng","year":"2017","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3487","DOI":"10.1109\/TSP.2019.2916732","article-title":"Wideband MIMO Radar Waveform Design","volume":"67","author":"Yu","year":"2019","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5942","DOI":"10.1109\/TSP.2017.2723354","article-title":"A Coordinate-Descent Framework to Design Low PSL\/ISL Sequences","volume":"65","author":"Kerahroodi","year":"2017","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3347","DOI":"10.1109\/TSP.2019.2914878","article-title":"Designing Sets of Binary Sequences for MIMO Radar Systems","volume":"67","author":"Naghsh","year":"2019","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.1109\/TSP.2020.2983642","article-title":"On the Design of Multi-Spectrally Constrained Constant Modulus Radar Signals","volume":"68","author":"Aubry","year":"2020","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_21","first-page":"4707","article-title":"Joint Design of the Receive Filter and Transmit Sequence for Active Sensing","volume":"20","author":"Liang","year":"2013","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sigpro.2019.02.006","article-title":"A Max-min Fractional Quadratic Programming Framework with Applications in Signal and Information Processing","volume":"160","author":"Liang","year":"2019","journal-title":"Sig. Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Qiu, X., Zhang, X., and Huo, K. (2023). Quartic Riemannian Adaptive Regularization with Cubics for Radar Waveform Design. IEEE Trans. Aerosp. Electron. Syst., 1\u20135.","DOI":"10.1109\/TAES.2023.3289779"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"5602","DOI":"10.1109\/TSP.2020.3022821","article-title":"Geometric Optimization Methods for Joint Design of Transmit Sequence and Receive Filter on MIMO Radar","volume":"68","author":"Li","year":"2020","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3313","DOI":"10.1109\/TAES.2022.3223887","article-title":"Joint Design of Horizontal and Vertical Polarization Waveforms for Polarimetric Radar via SINR Maximization","volume":"59","author":"Xu","year":"2023","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_26","first-page":"386","article-title":"Joint Optimization Design Method for Cognitive MIMO Radar Transmit Waveform and Receive Filter","volume":"45","author":"Qiu","year":"2023","journal-title":"Syst. Eng. Electron."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/MAES.2016.150216","article-title":"Optimization Theory-based Radar Waveform Design for Spectrally Dense Environments","volume":"31","author":"Aubry","year":"2016","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1109\/TAES.2014.120731","article-title":"Radar Waveform Design in a Spectrally Crowded Environment Via Nonconvex Quadratic Optimization","volume":"50","author":"Aubry","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1109\/TSP.2016.2569431","article-title":"Joint Design of Transmit Waveforms and Receive Filters for MIMO Radar Space Time Adaptive Processing","volume":"64","author":"Tang","year":"2016","journal-title":"IEEE Trans. Sig. Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4022005","DOI":"10.1109\/LGRS.2022.3151679","article-title":"Quartic Riemannian Trust Region Algorithm for Cognitive Radar Ambiguity Function Shaping","volume":"19","author":"Qiu","year":"2022","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1109\/TAES.2007.4383615","article-title":"Waveform Design for MIMO Radars","volume":"43","author":"Friedlander","year":"2007","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"147250","DOI":"10.1109\/ACCESS.2021.3122837","article-title":"Phase-Only Space-Time Adaptive Processing","volume":"9","author":"Pallotta","year":"2021","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1109\/TAES.2011.5751234","article-title":"Theory and application of SNR and mutual information matched illumination waveforms","volume":"47","author":"Romero","year":"2011","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"48","DOI":"10.21629\/JSEE.2018.01.05","article-title":"Waveform design for radar and extended target in the environment of electronic warfare","volume":"29","author":"Wang","year":"2018","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/JSTSP.2007.897046","article-title":"Optimal signal design for detection of gaussian point targets in stationary gaussian clutter\/reverberation","volume":"1","author":"Kay","year":"2007","journal-title":"IEEE J. Sel. Top. Sig. Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1578","DOI":"10.1109\/18.259642","article-title":"Information theory and radar waveform design","volume":"39","author":"Bell","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_37","first-page":"354","article-title":"Spectrum Optimization Via FFT-based Conjugate Gradient Method for Unimodular Sequence Design","volume":"142","author":"Zhao","year":"2018","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Farina, A., Maio, A.D., and Haykin, S.S. (2017). The Impact of Cognition on Radar Technology, IET.","DOI":"10.1049\/SBRA520E"},{"key":"ref_39","first-page":"726","article-title":"A Review on Cognitive Waveform Optimization for Different Radar Missions","volume":"50","author":"Yu","year":"2022","journal-title":"Acta Electron. Sin."},{"key":"ref_40","unstructured":"Oppenheim, A.V. (1997). Waveform Design for Active Sensing Systems: A Computational Approach, Prentice Hall International Inc."},{"key":"ref_41","unstructured":"Woodward, P.M. (1964). Probability and Information Theory, with Applications to Radar: International Series of Monographs on Electronics and Instrumentation, Elsevier."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1109\/TAES.2010.5461666","article-title":"Iterative Method for Nonlinear FM Synthesis of Radar Signals","volume":"46","author":"Jackson","year":"2010","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1109\/TSP.2009.2012562","article-title":"New Algorithms for Designing Unimodular Sequences with Good Correlation Properties","volume":"57","author":"Stoica","year":"2009","journal-title":"IEEE Trans. Sig. Process."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/21\/5187\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:14:46Z","timestamp":1760130886000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/21\/5187"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,31]]},"references-count":43,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["rs15215187"],"URL":"https:\/\/doi.org\/10.3390\/rs15215187","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,31]]}}}