# OpenStax_CNX

You are here: Home » Content » Four Types of Linear-Phase FIR Filters

### Recently Viewed

This feature requires Javascript to be enabled.

# Four Types of Linear-Phase FIR Filters

Module by: Ivan Selesnick. E-mail the author

Summary: (Blank Abstract)

Note: You are viewing an old version of this document. The latest version is available here.

## FOUR TYPES OF LINEAR-PHASE FIR FILTERS

Linear-phase FIR filter can be divided into four basic types.

Table 1
Type impulse response
I symmetric length is odd
II symmetric length is even
III anti-symmetric length is odd
IV anti-symmetric length is even

When hn h n is nonzero for 0nN1 0 n N 1 (the length of the impulse response hn h n is NN), then the symmetry of the impulse response can be written as

hn=hN1n h n h N 1 n
(1)
and the anti-symmetry can be written as
hn=hN1n h n h N 1 n
(2)

## TYPE I: ODD-LENGTH SYMMETRIC

The frequency response of a length N=5 N 5 FIR Type I filter can be written as follows.

H f ω= h 0 + h 1 e(i)ω+ h 2 e-2iω+ h 1 e-3iω+ h 0 e-4ω H f ω h 0 h 1 ω h 2 -2 ω h 1 -3 ω h 0 -4 ω
(3)
H f ω=e-2iω( h 0 e2iω+ h 1 eiω+ h 2 + h 1 e(i)ω+ h 0 e-2iω) H f ω -2 ω h 0 2 ω h 1 ω h 2 h 1 ω h 0 -2 ω
(4)
H f ω=e-2iω( h 0 (e2iω+e-2iω)+ h 1 (eiω+e(i)ω)+ h 2 ) H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω h 2
(5)
H f ω=e-2iω(2 h 0 cos2ω+2 h 1 cosω+ h 2 ) H f ω -2 ω 2 h 0 2 ω 2 h 1 ω h 2
(6)
H f ω=Aωeiθω H f ω A ω θ ω
(7)
where θω=-2ω θ ω -2 ω Aω=2 h 0 cos2ω+2 h 1 cosω+ h 2 A ω 2 h 0 2 ω 2 h 1 ω h 2 Note that Aω A ω is real-valued and can be both positive and negative. In general, for a Type I FIR filters of length NN: H f ω=Aωeiθω H f ω A ω θ ω
Aω=hM+2n=0M1hncos(Mn)ω A ω h M 2 n 0 M 1 h n M n ω
(8)
θω=(M)ω θ ω M ω M=N12 M N 1 2

## TYPE II: EVEN-LENGTH SYMMETRIC

The frequency response of a length N=4 N 4 FIR Type II filter can be written as follows.

H f ω= h 0 + h 1 e(i)ω+ h 1 e-2iω+ h 0 e-3iω H f ω h 0 h 1 ω h 1 -2 ω h 0 -3 ω
(9)
H f ω=e-32iω( h 0 e32iω+ h 1 e12iω+ h 1 e-12iω+ h 0 e-32iω) H f ω -3 2 ω h 0 3 2 ω h 1 1 2 ω h 1 -1 2 ω h 0 -3 2 ω
(10)
H f ω=e-32iω( h 0 (e32iω+e-32iω)+ h 1 (e12iω+e-12iω)) H f ω -3 2 ω h 0 3 2 ω -3 2 ω h 1 1 2 ω -1 2 ω
(11)
H f ω=e-32iω(2 h 0 cos32ω+2 h 1 cos12ω) H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω
(12)
H f ω=Aωeiθω H f ω A ω θ ω
(13)
where θω=-32ω θ ω -3 2 ω Aω=2 h 0 cos32ω+2 h 1 cos12ω A ω 2 h 0 3 2 ω 2 h 1 1 2 ω In general, for a Type II FIR filters of length NN: H f ω=Aωeiθω H f ω A ω θ ω
Aω=2n=0N21hncos(Mn)ω A ω 2 n 0 N 2 1 h n M n ω
(14)
θω=(M)ω θ ω M ω M=N12 M N 1 2

## TYPE III: ODD-LENGTH ANTI-SYMMETRIC

The frequency response of a length N=5 N 5 FIR Type III filter can be written as follows.

H f ω= h 0 + h 1 e(i)ω( h 1 e-3iω h 0 e-4ω) H f ω h 0 h 1 ω h 1 -3 ω h 0 -4 ω
(15)
H f ω=e-2iω( h 0 e2iω+ h 1 eiω( h 1 e(i)ω h 0 e-2iω)) H f ω -2 ω h 0 2 ω h 1 ω h 1 ω h 0 -2 ω
(16)
H f ω=e-2iω( h 0 (e2iωe-2iω)+ h 1 (eiωe(i)ω)) H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω
(17)
H f ω=e-2iω(2i h 0 sin2ω+2i h 1 sinω) H f ω -2 ω 2 h 0 2 ω 2 h 1 ω
(18)
H f ω=e-2iωi(2 h 0 sin2ω+2 h 1 sinω) H f ω -2 ω 2 h 0 2 ω 2 h 1 ω
(19)
H f ω=e-2iωeiπ2(2 h 0 sin2ω+2 h 1 sinω) H f ω -2 ω 2 2 h 0 2 ω 2 h 1 ω
(20)
H f ω=Aωeiθω H f ω A ω θ ω
(21)
where θω=2ω+π2 θ ω -2 ω 2 Aω=2 h 0 sin2ω+2 h 1 sinω A ω 2 h 0 2 ω 2 h 1 ω In general, for a Type III FIR filters of length NN: H f ω=Aωeiθω H f ω A ω θ ω
Aω=2n=0M1hnsin(Mn)ω A ω 2 n 0 M 1 h n M n ω
(22)
θω=Mω+π2 θ ω M ω 2 M=N12 M N 1 2

## TYPE IV: EVEN-LENGTH ANTI-SYMMETRIC

The frequency response of a length N=4 N 4 FIR Type IV filter can be written as follows.

H f ω= h 0 + h 1 e(i)ω( h 1 e-2iω h 0 e-3iω) H f ω h 0 h 1 ω h 1 -2 ω h 0 -3 ω
(23)
H f ω=e-32iω( h 0 e32iω+ h 1 e12iω( h 1 e-12iω h 0 e-32iω)) H f ω -3 2 ω h 0 3 2 ω h 1 1 2 ω h 1 -1 2 ω h 0 -3 2 ω
(24)
H f ω=e-32iω( h 0 (e32iωe-32iω)+ h 1 (e12iωe-12iω)) H f ω -3 2 ω h 0 3 2 ω -3 2 ω h 1 1 2 ω -1 2 ω
(25)
H f ω=e-32iω(2i h 0 sin32ω+2i h 1 sin12ω) H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω
(26)
H f ω=e-32iωi(2 h 0 sin32ω+2 h 1 sin12ω) H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω
(27)
H f ω=e-32iωeiπ2(2 h 0 sin32ω+2 h 1 sin12ω) H f ω -3 2 ω 2 2 h 0 3 2 ω 2 h 1 1 2 ω
(28)
H f ω=Aωeiθω H f ω A ω θ ω
(29)
where θω=-32ω+π2 θ ω -3 2 ω 2 Aω=2 h 0 sin32ω+2 h 1 sin12ω A ω 2 h 0 3 2 ω 2 h 1 1 2 ω In general, for a Type IV FIR filters of length NN: H f ω=Aωeiθω H f ω A ω θ ω
Aω=2n=0N21hnsin(Mn)ω A ω 2 n 0 N 2 1 h n M n ω
(30)
θω=Mω+π2 θ ω M ω 2 M=N12 M N 1 2

## Content actions

### Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks