How do we identify the four different types of FIR Filter (Type I,II,III and IV) based on the location of Zero(s) in the Pole-Zero plot?
FIR filters can be assigned to four different types regarding symmetry and length of the impulse response:
Type 1: symmetric, odd
Type 2: symmetric, even
Type 3: antisymmetric, odd
Type 4: antisymmetric, even
The positions of zeros in the complex plane are as follows:
Type 1: Either an even number or no zeros at z = 1 and z=-1
Type 2: Either an even number or no zeros at z = 1, and an odd number of zeros at z=-1
Type 3: An odd number of zeros at z = 1 and z=-1
Type 4: An odd number of zeros at z = 1, and either an even number or no zeros at z=-1
When deriving the 4 linear phase filter FIR filter types, didn't you also derive any rules regarding the position of the zeros?
For example: type II always has a zero at z=-1?
Maybe take a look at this website: http://cnx.org/content/m10700/latest/
FIR filters can be assigned to four different types regarding symmetry and length of the impulse response:
Type 1: symmetric, odd
Type 2: symmetric, even
Type 3: antisymmetric, odd
Type 4: antisymmetric, even
The positions of zeros in the complex plane are as follows:
Type 1: Either an even number or no zeros at z = 1 and z=-1
Type 2: Either an even number or no zeros at z = 1, and an odd number of zeros at z=-1
Type 3: An odd number of zeros at z = 1 and z=-1
Type 4: An odd number of zeros at z = 1, and either an even number or no zeros at z=-1
I found this which may be useful for you. It discusses the locations of the zeros.
http://dsp.stackexchange.com/questions/9408/fir-filter-with-linear-phase-4-types
And, I note that these "types" are strictly linear phase filters. There are others which are not linear phase. e.g. minimum phase, maximum phase with linear phase being somewhere in between so to speak.
Then between {y(n+1)-y(n-1)}--Type III and {y(n)-y(n-1)} --Type IV which difference operation is better?
Did you read it? From the link I gave above:
"Type III filters cannot be used for standard frequency selective filters because in these cases the 90 degrees phase shift is usually undesirable. For Hilbert transformers, type III filters have a relatively bad magnitude approximation at very low and very high frequencies due to the zeros at z=1 z=1 and z=−1 z=−1 . On the other hand, a type III Hilbert transformer can be implemented more efficiently than a type IV Hilbert transformer because in this case every other tap is zero.
Type IV filters cannot be used for standard frequency selective filters, for the same reasons as type III filters. They are well suited for differentiators and Hilbert transformers, and their magnitude approximation is usually better because, unlike type III filters, they have no zero at z=−1 z=−1 ."
Based on the symmetry(for type I,II) and anti-symmetry(type-III,IV) conditions, presence of zero at z=1 and z=-1 can be found. For type-I there is no pole at z=1 and z=-1. For type-II there is one pole at z=-1. For type-III there are two poles at z=1 and z=-1. For type-IV there is one pole at z=1. I this way from the pole location the type filter can be found
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