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Understanding Digital Signal Processing, Third Edition by Richard G. Lyons

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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 Hcomb(z) evaluated on the unit circle. We start by substituting ejω for z in Hcomb(z) from Eq. (7-37), because z = ejω defines the unit circle, giving

(G-1)

image

Factoring out ...

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