Modeling for Zeros in Complex Time and Frequency Plane

Abstract

AbstractThe poles, in principle, correspond to the resonant frequencies, whereas the zeros are created by local interaction between the responses of two adjacent poles. The occurrences of the zeros can be formulated using the residues of the adjacent poles and the remainder function that approximates the sum of responses from other poles. Clustered line-spectral modeling (CLSM) synthesizes the pole/zero responses in the frequency domain with some modeling error. An example of piano-vibration analysis by CLSM uncovers a significant zero in the modeling error from the source signature, which determines an initial portion of the time sequence for the response record, but mostly determines the entire spectral envelope of the response. Complementarity under complex conjugation holds between the complex-time and frequency planes. Like CLSM in the frequency domain, clustered time sequence modeling (CTSM) can be formulated on the complex-time domain in accordance with complementarity. CTSM of piano-string vibration analysis extracts the source signature from the initial portion of the vibration corresponds to the residual error by CLSM. Adjacent pairing time-pulse modeling (APTM) is formulated for spectral trough selection as well as the spectral peak selection on the frequency plane. The significant zeros generated by the piano-string vibration hence can be identified using APTM.KeywordsResiduesRemainder functionModeling errorPoles and zerosTransfer functionSource separationSinusoidal modelingSpectral peak selectionSpectral trough selectionSpectral envelopePiano-string vibrationFactorization of z-transformLeast-squares-error solutionComplementarity between time and frequency planesComplex-time regionAnalytic signalInstantaneous frequencyEnvelopeMinimum and non-minimum phases