Analysis of the asymptotic impulse and frequency responses of polynomial predictors

Authors:
Konsta Koppinen
Affiliations:
Tampere University of Technology, Institute of Signal Processing, FIN-33101 Tampere, Finland
Venue:
Signal Processing
Year:
2004

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Abstract

Explicit formulas for the asymptotic impulse and frequency responses of polynomial predictive FIR filters with minimal noise gain are derived. It is shown that the impulse response is asymptotically a polynomial with coefficients given by the first column of the inverse of the Hilbert matrix, that the magnitude response of predictors behaves as O(1/ω) and that the asymptotic frequency response is independent of the prediction step. The values of the magnitude-response peak are calculated numerically for low-order predictors and the group delay is shown to achieve all real values at arbitrarily small frequencies for long enough predictors. A conjecture on the exact formula for the asymptotic noise gain is proved and another conjecture on the inverse relationship between the prediction bandwidth and passband magnitude peak is refuted for long predictors.