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Basically my question is the same as the title: Are the 4 types of linear phase FIR filters the only linear phase FIR filters one can come up with? And if so, then why?

I'm pretty sure that they are the only ones but I really can't explain why. Can someone articulate it clearly?

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    $\begingroup$ You haven't accepted or upvoted a single answer to your previous questions. People might think twice before answering any more of your questions. $\endgroup$ – Matt L. Mar 1 '16 at 19:33
  • $\begingroup$ I upvoted before twice but I it doesn't publicly appear because I need 15 rep or at least that's what the error message says $\endgroup$ – zer0square Mar 1 '16 at 20:35
  • $\begingroup$ OK. If you accept answers (by clicking the green check mark) you also earn points, which allows you to do more on this site, including upvoting. $\endgroup$ – Matt L. Mar 1 '16 at 20:39
  • $\begingroup$ Some complements on their choices are given in FIR filter with linear phase, 4 types $\endgroup$ – Laurent Duval Mar 1 '16 at 21:07
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It can be shown that in order for an FIR filter to have linear phase, its impulse response must be symmetric or anti-symmetric. The impulse response length $L$ can be either odd or even. Those two variables lead to 4 combinations, hence the 4 types of linear-phase filters.

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The answer by @Jason R is short and perfect. Just adding a pictorial version, from Openstax: Four types of linear-phase FIR filters: enter image description here

For more text, I suggest the short text Linear-Phase FIR Filters, by Ivan Selesnick.

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