symmetric impulse response vs. antisymmetric impulse response

[robismyname]

impulse response h = {-4,1,-1,-2,5,6,5,-2,-1,1,-4}
↑
The solution says that M=11, which is odd, and since h is symmetric about about α = (11−1)/2 =
5, this is a Type-1 linear phase FIR filter.

Can someone tell me how it was derived that this filter is symmetric?

impulse response h = {−4, 1,−1,−2, 5, 6,−6,−5, 2, 1,−1, 4}.
↑
The solution says that M=12, and since h is anti-symmetric with respect to α = (12−1)/2 =
5.5, this is a Type-4 linear phase FIR filter.

Can someone tell me how it was derived that this filter is anti-symmetric?

Is symmetry and antisymmetry based on the principal of even and odd signal?

 

[o_0]

Yes, symmetry and antisymmetry are related to even and odd.

Notice the symmetric filter reads the same forward and backward. Also if you graphed it with the middle number graphed on the y-axis, then you would see symmetry about the y-axis (that is, even) I thought it would be symmetric about 6 though, based on what the graph would look like but I didn't calculate.

A graph of the anti symmetric filter, on the other hand, would be symmetric about y=x probably, that is odd symmetry.