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This chapter analyzes how core multirate operations—sampling (with and without zero removal), modulation, and time reversal—map into the z-domain. The z-transform generalizes the discrete-time Fourier transform (DTFT) by extending analysis off the unit circle, exposing pole–zero geometry and enabling compact, algebraic descriptions of aliasing (via rotations), frequency-axis scaling (via exponent changes), spectral shifts (via rotations), and time reversal (via \(z \mapsto z^{-1}\) ). These identities are central to perfect reconstruction filter banks and wavelet systems.

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Effects in the z-Domain

  • 章节

  • 发表日期：

    2026年

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  • Gerald Schuller

摘要

This chapter analyzes how core multirate operations—sampling (with and without zero removal), modulation, and time reversal—map into the z-domain. The z-transform generalizes the discrete-time Fourier transform (DTFT) by extending analysis off the unit circle, exposing pole–zero geometry and enabling compact, algebraic descriptions of aliasing (via rotations), frequency-axis scaling (via exponent changes), spectral shifts (via rotations), and time reversal (via \(z \mapsto z^{-1}\) ). These identities are central to perfect reconstruction filter banks and wavelet systems.