Understanding Digital Signal Processing, 3rd Edition

Appendix G. Frequency Sampling Filter Derivations

While much of the algebra related to frequency sampling filters is justifiably omitted in the literature, several derivations are included here for two reasons: first, to validate the equations used in Section 7.5; and second, to show the various algebraic acrobatics that may be useful in your future digital signal processing analysis efforts.

G.1 Frequency Response of a Comb Filter

The frequency response of a comb filter is H_comb(z) evaluated on the unit circle. We start by substituting e^jω for z in H_comb(z) from Eq. (7-37), because z = e^jω defines the unit circle, giving

(G-1)

Factoring out ...