Search

Summary

The objective of this chapter is to extend the ad hoc least squares method of somewhat arbitrarily selected base functions to a more generic method applicable to a broad range of functions – the Fourier series, which is an expansion of a relatively arbitrary function (with certain smoothness requirement and finite jumps at worst) with a series of sinusoidal functions. An important mathematical reason for using Fourier series is its “completeness” and almost guaranteed convergence. Here “completeness” means that the error goes to zero when the whole Fourier series with infinite base function is used. In other words, the Fourier series formed by the selected sinusoidal functions is sufficient to linearly combine into a function that converges to an arbitrary continuous function. This chapter on Fourier series will lay out a foundation that will lead to Fourier Transform and spectrum analysis. In this sense, this chapter is important as it provides background information and theoretical preparation.

The purpose of this note is to analyze the capability of bandpass filters to extract a known periodicity. The specific bandpass filters considered are a conventional discrete Fourier transform (DFT) filter and the filter recently proposed by Iacobucci and Noullez. We employ simulation methods to investigate cycle extraction properties. We also examine the implications arising from the Gibbs effect in practical settings that typically confront applied macroeconomists

Search Results

Refine search

Refine search

Actions for selected content:

2 results

12 - Fourier Series

Summary

DISCRETE FOURIER TRANSFORM FILTERS: CYCLE EXTRACTION AND GIBBS EFFECT CONSIDERATIONS

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

12 - Fourier Series

Summary

DISCRETE FOURIER TRANSFORM FILTERS: CYCLE EXTRACTION AND GIBBS EFFECT CONSIDERATIONS