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## Publikācija: A Band Pass Filter for Real-Time Signal Extraction

Publikācijas veids Raksts konferenču tēžu krājumā Nav zināms , English (en) A Band Pass Filter for Real-Time Signal Extraction 1. Dabaszinātnes 1.1. Matemātika Ginters Bušs real time estimation; Christiano-Fitzgerald filter; Monte Carlo simulation; band pass filter The paper proposes an extension of the symmetric Baxter-King band pass filter to an asymmetric Baxter-King filter. In order to optimally approximate an ideal symmetric linear filter in a Baxter-King sense, the problem is to minimize \begin{align} Q&=\int_{-\pi}^{\pi} \delta(\omega)\delta(-\omega) d\omega \label{6.0} \\ & \text{s.t.}
onumber \\ &\hat{\beta}(0)=\sum_{h=-p}^{f}\hat{b}_h=0, \label{6.1} \end{align} where $\delta(\omega)=\beta(\omega)-\hat{\beta}(\omega)$ is the discrepancy between frequency response functions of the exact and the approximate filters at frequency $\omega$, and the constraint \eqref{6.1} is to ensure zero weight on the trend frequency. The FOCs are $-4 \pi [b_h - \hat{b}_h] - \lambda =0$, where $\lambda$ is Lagrangian multiplier. If there is no constraint \eqref{6.1}, the optimal approximate (in Baxter-King sense) filter is simply derived by truncation of the ideal filter's weights. If there is a constraint \eqref{6.1}, then rewrite FOCs as $\hat{b}_h = b_h + \theta$, where $\theta=\lambda/(4 \pi)$. In order to have $\hat{\beta}(0) = \sum_{h=-p}^f \hat{b}_h = 0$, the required adjustment is $\theta = - \sum_{h=-p}^f b_h /(p+f+1)$, which yields the same optimal weight adjustment scheme as in the symmetric Baxter-King filter case, i.e, cut the ideal filter at the appropriate length and add a constant to all filter weights to ensure zero weight on zero frequency. Since the symmetric Baxter-King filter is unable to extract the desired signal at the very ends of the series, the extension to an asymmetric filter is useful whenever the real time estimation is needed. The paper uses Monte Carlo simulation to compare the proposed filter's properties in extracting business cycle frequencies to the ones of the original Baxter-King filter and Christiano-Fitzgerald filter. Simulation results show that the asymmetric Baxter-King filter is superior to the asymmetric default specification of Christiano-Fitzgerald filter in real time signal extraction exercises. Bušs, G. A Band Pass Filter for Real-Time Signal Extraction. No: Abstracts of 16th International Conference on Mathematical Modelling and Analysis, Latvija, Sigulda, 25.-28. maijs, 2011. Riga: University of Latvia, 2011, 22.-22.lpp. ISBN 978-9984-45-351-4. 10623