The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences

Demeure, C.J.; Mullis, C.T.

doi:10.1109/29.17535

Cited by 51 publications

(22 citation statements)

References 12 publications

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“…The integral in the above expression has a well known closed-form expression in terms of the reflection coefficients (See for example [10], [22]). The following closed-form expression for the coding gain can be therefore obtained…”

Section: The Special Case Of First-order Filters With Equal Coeffimentioning

confidence: 99%

Statistically optimum pre- and postfiltering in quantization

Tuqan

Vaidyanathan²

1997

IEEE Trans. Circuits Syst. II

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Abstract-We consider the optimization of pre-and postfilters surrounding a quantization system. The goal is to optimize the filters such that the mean square error is minimized under the key constraint that the quantization noise variance is directly proportional to the variance of the quantization system input. Unlike some previous work, the postfilter is not restricted to be the inverse of the prefilter. With no order constraint on the filters, we present closed-form solutions for the optimum pre-and postfilters when the quantization system is a uniform quantizer. Using these optimum solutions, we obtain a coding gain expression for the system under study. The coding gain expression clearly indicates that, at high bit rates, there is no loss in generality in restricting the postfilter to be the inverse of the prefilter. We then repeat the same analysis with first-order preand postfilters in the form 1+z 01 and 1=(1+ z 01 ). In specific, we study two cases: 1) FIR prefilter, IIR postfilter and 2) IIR prefilter, FIR postfilter. For each case, we obtain a mean square error expression, optimize the coefficients and and provide some examples where we compare the coding gain performance with the case of = . In the last section, we assume that the quantization system is an orthonormal perfect reconstruction filter bank. To apply the optimum pre-and postfilters derived earlier, the output of the filter bank must be wide-sense stationary WSS which, in general, is not true. We provide two theorems, each under a different set of assumptions, that guarantee the wide sense stationarity of the filter bank output. We then propose a suboptimum procedure to increase the coding gain of the orthonormal filter bank.Index Terms-Half-whitening scheme, noise shaping, optimum pre-and postfiltering, subband coding.

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Section: The Special Case Of First-order Filters With Equal Coeffimentioning

confidence: 99%

Statistically optimum pre- and postfiltering in quantization

Tuqan

Vaidyanathan²

1997

IEEE Trans. Circuits Syst. II

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“…This method was generalized by Demeure and Mullis [16] to include computation of cross-correlation sequences. In addition, Demeure and Mullis derived fast algorithms for performing the required computations.…”

Section: Algorithm For Generating the Asymptotic Fisher Matrixmentioning

confidence: 99%

“…The coefficients of Y(z) can be obtained by solving the linear set of equations in (3.22) using any standard technique. A fast algorithm for solving (3.22) can be found in [16]. Let {Oil} denote the impulse response of the system Y(z) / G(z) where Y(z) is the polynomial formed from y of (3.22).…”

Section: Algorithm For Generating the Asymptotic Fisher Matrixmentioning

confidence: 99%

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Cramer-Rao bounds for deterministic modal analysis

McWhorter

Scharf

1993

IEEE Trans. Signal Process.

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Abstract-How accurately can deterministic modes be identified from a finite record of noisy data? In this paper we answer this question by computing the Cramer-Rae bound on the error covariance matrix of any unbiased estimator of mode parameters. The bound is computed for many of the standard parametric descriptions of a mode, including autoregressive and moving average parameters, poles and residues, and poles and zeros. Asymptotic, frequency domain versions of the CramerRao bound bring insight into the role played by poles and zeros. Application of the bound to second-and fourth-order systems illustrates the coupling between estimator errors and illuminates the influence of mode locations on our ability to identify them. Application of the bound to the estimation of an energy spectrum illuminates the accuracy of estimators that presume to resolve spectral peaks.

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“…Several authors, like Maravall (1994) and McElroy (2008), have mentioned a link with the autocovariance function of some auxiliary processes based on the innovations of the corrected series, and on the polynomials of the ARIMA models of the corrected series and of the components. These autocovariances can be computed by a straightforward algorithm (McLeod 1975) of solving a linear system of equations or by fast algorithms due to Tunnicliffe Wilson (1979) or Demeure and Mullis (1989). These simple approaches do not appear often in the literature (e.g.…”

Section: Principles Behind Seatsmentioning

confidence: 99%

On some remarks about SEATS signal extraction

Mélard

2015

SERIEs

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In seasonal adjustment a time series is considered as a juxtaposition of several components, the trend-cycle, and the seasonal and irregular components. The Bureau of the Census X-11 method, based on moving averages, correction of large errors and trading day adjustments, has long dominated. With the success of ARIMA modelling at the end of the 20th century, methods with better outlier detection and trading day corrections by regression with ARIMA errors have appeared, with the regARIMA module of Census X-12-ARIMA or Bank of Spain TRAMO-SEATS. SEATS consists of extracting the components by an ARIMA-model-based unobserved components approach. This means that models are used for each component such that the sum of the components is compatible with the ARIMA model for the corrected time series. The underlying theory of the SEATS program is studied in many papers but there is no complete and systematic description of its output. Our purpose is to examine SEATS text output and to explain the results in simple words and formulas. This is done on a simple example, a time series with a non-seasonal model so that the computations can be verified step by step. The principles behind SEATS are first described, including the admissible decompositions and the canonical decomposition, and the derivation of the Wiener-Kolmogorov filter. Then the example is introduced: the interest rates of US certificates of deposits. The text output from SEATS is presented in edited form in several tables. Finally, the main results are checked on the example by means of a Microsoft Excel workbook and direct computations. In particular, the forecasts and backcasts are obtained; the admissible and canonical decompositions with two components are discussed; the filters are first derived using autocorrelations of two auxiliary ARMA processes, then applied on the prolonged time series; and the characteristics of the estimates, the revisions and the growth rates are analyzed.

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The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences

Cited by 51 publications

References 12 publications

Statistically optimum pre- and postfiltering in quantization

Statistically optimum pre- and postfiltering in quantization

Cramer-Rao bounds for deterministic modal analysis

On some remarks about SEATS signal extraction

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