I am from an engineering field (EE, specifically signal processing), and I have been working on a paper since March 2014.

Back then, I spent a significant amount of time reviewing the literature and didn't find any existing work that discussed my topic. I actually finished the bulk of the paper several months ago (Jan 2015), but since then I have been editing and reviewing it with my coauthors before we planned to submit to journals.

I recently, however, found a similar paper that was "published" or posted on arXiv in April 2015. I believe that it is extremely similar (~80% of the ideas). I am not sure how much of an issue this is, as I have heard that arXiv is not a "legitimate" domain (in my field, almost everyone just uses the IEEE searching engine).

My original plan was to submit my work to journals soon. Do you think I can still submit my paper to a journal? How should I acknowledge of otherwise handle the existence of this similar arXiv publication?

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    But I need to wait 5 more months until my reviewers finish the reviewing process before submiting to public journal. — Who are these reviewers you mentioned? Your co-authors? – Mad Jack May 27 '15 at 18:15
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    This, right here, is a primary example of why using arXiv is highly encouraged, regardless of its "domain" – CGCampbell May 27 '15 at 18:43
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    in my field, almost everyone justs the IEEE searching engine — Trust me, everyone in your field uses Google. – JeffE May 27 '15 at 21:11
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    Oooh, yeah, I'm going to use search tool X that requires a login and restricts my results to those from one publisher. Way better than Google Scholar. – David Ketcheson May 28 '15 at 5:02
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    @Aragon: I think they are co-authors – You think they are? Shouldn’t you know? – Wrzlprmft May 28 '15 at 5:57

Sure, you can go ahead and submit your paper. Of course you are obligated to discuss the arXiv paper, and its relation to your work -- but it sounds like you can honestly describe it as independent work. It would be a good idea to also make your paper public at this point (perhaps by posting a preprint on arXiv).

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