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This might be a strange question - but most of the Type 3 LPF I'm seeing are having this in common for $h[n]$, assuming $h[n]$ is real:

  1. We cannot have outermost element of h[n] lower than middle elements
  2. (or) Have middle element lower than outermost elements.

Eg: if filter coefficients are $[3,2,1,0,-1,-2,-3]$, it there any restriction for a type $3$ filter to not have $[1,2,3,0,-3,-2,-1]$ as a possible LPF $h[n]$?

(Please excuse my ignorance if this is a basic question - I'm quite new FIR Filter design). And I've gone through quite a few design docs before asking this question - none of them mention this, so I'm guessing this is not a req? Yet the diagrams seem to indicate this, hence the question...

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There is no such restriction for a type III linear phase filter. Neither is there such a restriction for any other type of linear phase filter. Type III linear phase filters have an odd filter length $N$ and odd symmetry. I.e., as long as the filter's impulse response $h[n]$ satisfies

$$h\left[\frac{N-1}{2}+n\right]=-h\left[\frac{N-1}{2}-n\right]$$

you have a proper type III linear phase filter.

If you look at the impulse responses of type III differentiators or Hilbert transformers then you'll notice that in both cases the filter taps are decreasing in magnitude as you move away from the center tap. So these common applications of type III filters are counterexamples to your assumption.

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