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Optimal power allocation (OPA) is one of the general methods to achieve EET. But when OPA is only considered in EET, it maybe cannot analyze EET problems accurately. This paper aims at analyzing energy-efficient bidirectional direct and relay transmission (RT) through joint optimization of transmit power (TP) and transmit time (TT) allocation. In RT, direct links (DLs) are existed. The EET problems are given with three optimization cases: (i) maximizing sum throughput (ST) to maximize energy efficiency (EE), in such case, optimal TT (OTT), optimal relay position (ORP), and outage probability analysis are given; (ii) minimizing total energy consumption (TEC) to maximize EE, in such case, optimal TP (OTP), OTT with gradient-descent algorithm, and influences of asymmetry and asymmetry transmission tasks are given; and (iii) maximizing ST and minimizing TEC simultaneously to maximize EE, in such case, Dinkelbach\u2019s algorithm and one-by-one optimal algorithm are given. Simulation results are presented to validate theoretical analysis. Results reveal that relay technique and DLs in RT can improve system\u2019s EE.<\/jats:p>","DOI":"10.1186\/s13638-020-01772-8","type":"journal-article","created":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T10:04:11Z","timestamp":1596189851000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Energy-efficiency maximization bidirectional direct and relay transmission"],"prefix":"10.1186","volume":"2020","author":[{"given":"Caixia","family":"Cai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6528-8514","authenticated-orcid":false,"given":"Runhe","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xue-Qin","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuyang","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,31]]},"reference":[{"key":"1772_CR1","volume-title":"Green Communications: Theoretical Fundamentals, Algorithms, and Applications","author":"J. 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