On the backwards difference filter

TL;DR

This paper analyzes the backwards difference filter used in helioseismology, deriving correction methods for its effects on amplitude and phase in the Fourier domain to improve signal analysis accuracy.

Contribution

It provides a mathematical derivation of the filter's transfer function and introduces correction techniques for amplitude and phase distortions caused by the filter.

Findings

Derived the transfer function of the backwards difference filter.

Proposed correction methods for amplitude bias in the power spectrum.

Provided phase correction techniques for the amplitude spectrum.

Abstract

It is usual in helioseismology to remove unwanted instrumental low-frequency trends by applying high-pass filters to the time series. However, the choice of the filter is very important because it can keep the periodic signals throughout the spectrum. At the same time, these filters should not introduce any spurious effects on the remaining signal, which would modify the periodic signatures in the Fourier domain. One of the most used filters is the so-called backwards difference filter that can be applied when the time series are regularly sampled. The problem of this filter is that the amplitudes and the phases of the periodic signals in the Fourier domain are modified and, therefore, their extracted amplitudes are biased when the frequency of the periodic signals decreases. The objective of this research note is to give a correction that could be applied to the power spectrum to over come this problem. We properly derive the transfer function of the backwards difference filter and we show how this can be applied to the power spectrum to correct the amplitudes. For the amplitude spectrum and the time series, we also derive a correction for the phase. The amplitudes of the periodic signals in the resultant power spectrum density are corrected from the cut-off frequency down to zero.