Should I sign this letter or force them to fire me? What are the differences between my prospects with the two options? I was a first year PhD student and have mentioned before I was forced to quit although hardwork and grants. Now the head of the school admit that the supervisor was wrong and he had nothing to do with the supervisor which is awkward, he also confused about another student who hadnot passed his viva after years of work and publishing in top tier conference.

In to the retrospect, they asked me whether to sign a resignation letter or to be fired and asked me to sign in the moment, but I refused and asked them I will take the letter and do it later during this week.

They asked you have to do it during this week ( obligatory), could any one recommend me whether this is good direction to sign this letter as it is first time for me and I am not aware of many LEGAL stuff.

PS. For more details why this happened:

I am a first year PhD student, I was working in a new topic, I found a problem in research and then I proposed a methodology and verified it. My PI said from few months ago that it was good what I did verbatim. Then I got two grants from top conferences although I waited a lot of time to do experiment in other lab, and this lab was not happy because I proposed different approach than they use, but I explained what I proposed and how it could solve. Honestly, I worked so much hard as you can imagine, but in the same time I noticed that my PI is underestimating from my ideas sometimes and have sharp looking at me without any reason, at the beginning he didnot want to publish anything or attend conference without any reason, sometimes he didnot listen to me and I made correction for him which was critical for research, but I did in very polite way

He began to catch small mistakes in report for instance which was very trivial and inflate it and never acknowledge the two grants I got and I did other stuff for free like videos for the group and sketching which he was astonished by it. 

Until I present my work in the committee which every one told me that they do it to see every thing is okay and every one pass, this committee is my PI, another professor who have never worked in my research topic and asked questions  was very sketchy and the head of the school also not related to my domain they see presentation and report no exams no other things, so just presentation and report. Of course this PI made badmouthing behind me and that what I know later and it was mentioned that I didnot do a good bibliography and he doubt the methodology although I got the same results as the real experiment based on physics approach, but he doubts. I was shocked and I confronted this PI, why you didnot tell me from few months ago when you said it is good, he lied and I told this PI, if you have doubts could you please tell me what are these doubts like 1,2,3 and what you suggested, he didnot say anyword and go. In conclusion, he wasnot specialized in this new topic, even when I am doing any proposed formula he accuse me that I stole it or from where I had come with this equation, I did that on my own, it is quite hurting because he accused me many times of any schematics drawings, sketches formula that may be I stole it, and I tell him that I did that on my own, and BTW I have never ever stole something so that can interpret his actions. In conclusion,  this PI told me your are independent  and he didnot like that and also know that I worked hard but he dont want to continue in this research this is after making badmouthing about me as I am not eligible to continue and even the committee yesterday which the head of the school told me I know you are very good and you did a good work, but no one can speak, the head of school recommended me to find another lab in different country. Sorry for English grammar mistakes and typos.
If you can afford to do so, consult a lawyer. That way you will know all of your options, not just the most apparent ones. If you have no option other than to leave, then your decision should rest on whether you want to fight/contest your removal or not. If you don't contest it, then it is probably (subject to legal advice) better to sign the letter. But if you sign it, you are probably giving up options. 

But advice like this should come from a lawyer.