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Four Types of Linear-Phase FIR Filters

Module by: Ivan Selesnick

Summary: (Blank Abstract)

FOUR TYPES OF LINEAR-PHASE FIR FILTERS

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

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
Figure 1
Figure 1 (fourstem.png)

When hn h n is nonzero for 0≤n≤N-1 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=hN-1-n h n h N 1 n (1)
and the anti-symmetry can be written as
hn=-hN-1-n 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-ⅈω+ 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-ⅈω+ 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-ⅈω+ h 2 H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω h 2 (5)
H f ω=ⅇ-2ⅈω2 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+2∑n=0M-1hncosM-nω A ω h M 2 n 0 M 1 h n M n ω (8)
θω=-Mω θ ω M ω M=N-12 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-ⅈω+ 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-32ⅈω2 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=2∑n=0N2-1hncosM-nω A ω 2 n 0 N 2 1 h n M n ω (14)
θω=-Mω θ ω M ω M=N-12 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-ⅈω- 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-ⅈω- 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 ⅇ2ⅈω-ⅇ-2ⅈω+ h 1 ⅇⅈω-ⅇ-ⅈω H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω (17)
H f ω=ⅇ-2ⅈω2ⅈ 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-2ⅈωⅈ2 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-2ⅈωⅇⅈπ22 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=2∑n=0M-1hnsinM-nω A ω 2 n 0 M 1 h n M n ω (22)
θω=-Mω+π2 θ ω M ω 2 M=N-12 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-ⅈω- 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 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 (25)
H f ω=ⅇ-32ⅈω2ⅈ 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-32ⅈωⅈ2 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-32ⅈωⅇⅈπ22 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=2∑n=0N2-1hnsinM-nω A ω 2 n 0 N 2 1 h n M n ω (30)
θω=-Mω+π2 θ ω M ω 2 M=N-12 M N 1 2

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