### Abstract

The present thesis addresses mixed phase deconvolution of speech by z-transform zeros. This includes investigations into stability, accuracy, and time complexity of a numerical bijection between time domain and the domain of z-transform zeros.

Z-transform factorization is by no means esoteric, but employing zeros of the z-transform (ZZT) as a signal representation, analysis, and processing domain per se, is only scarcely researched. A notable property of this domain is the translation of time domain convolution into union of sets; thus, the ZZT domain is appropriate for convolving and deconvolving signal sequences.

By suitable windowing in pitch synchronous speech processing, the ZZT representation pertaining to the glottal flow opening phase lies outside the open unit disc and the ZZT representation pertaining to the vocal tract filter and the glottal flow closing phase lies inside. Unit disc discrimination achieves mixed phase deconvolution and equivalates complex cepstrum based deconvolution by causality, which has lower time and space complexities as demonstrated. However, deconvolution by ZZT prevents phase wrapping.

Existence and persistence of ZZT domain immiscibility of the opening and closing phases of the glottal flow derivative is investigated analytically with regard to model parameters and sequence lengths. It is shown that time domain concatenation and convolution of the phases are tightly related. Therefore, immiscibility remains regardless of considering entire glottal flow derivative sequences or individual opening and closing sequences.

To counter the computational burden associated with z-transform factorization, an adaptive ZZT estimation method is proposed for time-varying z-transforms. The method bounds estimation drifting as accuracy is the cost of lowered time complexity. The accuracy degradation correlates positively with time complexity reduction and temporal magnitude variations in coefficient vectors.

Even in well-conditioned cases, numerical stability and accuracy is of primary concern when estimating time domain signals from ZZT representations. Leja ordering coefficient vectors prior to polynomial expansion achieve persistent estimation accuracies near machine epsilon. The associated time complexity cost is effectively countered by proposing a refined Leja ordering obtained via `1 maximization.

Z-transform factorization is by no means esoteric, but employing zeros of the z-transform (ZZT) as a signal representation, analysis, and processing domain per se, is only scarcely researched. A notable property of this domain is the translation of time domain convolution into union of sets; thus, the ZZT domain is appropriate for convolving and deconvolving signal sequences.

By suitable windowing in pitch synchronous speech processing, the ZZT representation pertaining to the glottal flow opening phase lies outside the open unit disc and the ZZT representation pertaining to the vocal tract filter and the glottal flow closing phase lies inside. Unit disc discrimination achieves mixed phase deconvolution and equivalates complex cepstrum based deconvolution by causality, which has lower time and space complexities as demonstrated. However, deconvolution by ZZT prevents phase wrapping.

Existence and persistence of ZZT domain immiscibility of the opening and closing phases of the glottal flow derivative is investigated analytically with regard to model parameters and sequence lengths. It is shown that time domain concatenation and convolution of the phases are tightly related. Therefore, immiscibility remains regardless of considering entire glottal flow derivative sequences or individual opening and closing sequences.

To counter the computational burden associated with z-transform factorization, an adaptive ZZT estimation method is proposed for time-varying z-transforms. The method bounds estimation drifting as accuracy is the cost of lowered time complexity. The accuracy degradation correlates positively with time complexity reduction and temporal magnitude variations in coefficient vectors.

Even in well-conditioned cases, numerical stability and accuracy is of primary concern when estimating time domain signals from ZZT representations. Leja ordering coefficient vectors prior to polynomial expansion achieve persistent estimation accuracies near machine epsilon. The associated time complexity cost is effectively countered by proposing a refined Leja ordering obtained via `1 maximization.

Original language | Danish |
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Number of pages | 143 |
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Publication status | Published - 2013 |

## Cite this

Pedersen, C. F. (2013).

*Z-transform Zeros in Mixed Phase Deconvolution of Speech*.