Four Types of Linear-Phase FIR Filters 2.1 2002/07/01 2002/07/03 Ivan Selesnick selesi@taco.poly.edu Charlet Reedstrom charlet@rice.edu Ivan Selesnick selesi@taco.poly.edu linear-phase FIR filters (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

When h n is nonzero for 0 n N 1 (the length of the impulse response h n is N), then the symmetry of the impulse response can be written as h n h N 1 n and the anti-symmetry can be written as h n h N 1 n

TYPE I: ODD-LENGTH SYMMETRIC The frequency response of a length N 5 FIR Type I filter can be written as follows. H f ω h 0 h 1 ω h 2 -2 ω h 1 -3 ω h 0 -4 ω H f ω -2 ω h 0 2 ω h 1 ω h 2 h 1 ω h 0 -2 ω H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω h 2 H f ω -2 ω 2 h 0 2 ω 2 h 1 ω h 2 H f ω A ω θ ω where θ ω -2 ω A ω 2 h 0 2 ω 2 h 1 ω h 2 Note that A ω is real-valued and can be both positive and negative. In general, for a Type I FIR filters of length N: H f ω A ω θ ω A ω h M 2 n 0 M 1 h n M n ω θ ω M ω M N 1 2

TYPE II: EVEN-LENGTH SYMMETRIC The frequency response of a length N 4 FIR Type II filter can be written as follows. H f ω h 0 h 1 ω h 1 -2 ω h 0 -3 ω 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 H f ω -3 2 ω h 0 3 2 ω -3 2 ω h 1 1 2 ω -1 2 ω H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω H f ω A ω θ ω where θ ω -3 2 ω A ω 2 h 0 3 2 ω 2 h 1 1 2 ω In general, for a Type II FIR filters of length N: H f ω A ω θ ω A ω 2 n 0 N 2 1 h n M n ω θ ω M ω M N 1 2

TYPE III: ODD-LENGTH ANTI-SYMMETRIC The frequency response of a length N 5 FIR Type III filter can be written as follows. H f ω h 0 h 1 ω h 1 -3 ω h 0 -4 ω H f ω -2 ω h 0 2 ω h 1 ω h 1 ω h 0 -2 ω H f ω -2 ω h 0 2 ω -2 ω h 1 ω ω H f ω -2 ω 2 h 0 2 ω 2 h 1 ω H f ω -2 ω 2 h 0 2 ω 2 h 1 ω H f ω -2 ω 2 2 h 0 2 ω 2 h 1 ω H f ω A ω θ ω where θ ω -2 ω 2 A ω 2 h 0 2 ω 2 h 1 ω In general, for a Type III FIR filters of length N: H f ω A ω θ ω A ω 2 n 0 M 1 h n M n ω θ ω M ω 2 M N 1 2

TYPE IV: EVEN-LENGTH ANTI-SYMMETRIC The frequency response of a length N 4 FIR Type IV filter can be written as follows. H f ω h 0 h 1 ω h 1 -2 ω h 0 -3 ω 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 H f ω -3 2 ω h 0 3 2 ω -3 2 ω h 1 1 2 ω -1 2 ω H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω H f ω -3 2 ω 2 h 0 3 2 ω 2 h 1 1 2 ω H f ω -3 2 ω 2 2 h 0 3 2 ω 2 h 1 1 2 ω H f ω A ω θ ω where θ ω -3 2 ω 2 A ω 2 h 0 3 2 ω 2 h 1 1 2 ω In general, for a Type IV FIR filters of length N: H f ω A ω θ ω A ω 2 n 0 N 2 1 h n M n ω θ ω M ω 2 M N 1 2