# Alias filters

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 9.5a

The standard alias filter such as that shown in Figure 9.5a has a 3-dB point at about half the Nyquist frequency ${\displaystyle f_{N}}$ and a very steep slope, so that noise above ${\displaystyle f_{N}}$ is highly attenuated relative to the passband of the system. Assuming an original flat spectrum, alias filtering with a 125-Hz, 72-dB/octave filter, and subsequent resampling from 2 to 4 ms (without additional alias filtering), graph the resulting alias noise versus frequency.

### Background

The passband of a system is the range of frequencies that are unattenuated by passage through the system. The limits are usually taken as the frequencies for which the attenuation is 3 dB.

Figure 9.5a.  Seismic filter responses.

### Solution

With the 3-dB point on the high-cut filter at 125 Hz and a 72 dB/octave slope, the attenuation at ${\displaystyle f_{N}=250}$ Hz is 75 dB. Frequencies ${\displaystyle \Delta {f}}$ higher than the Nyquist frequency, ${\displaystyle f_{N}+\Delta f}$, alias to appear as the frequencies ${\displaystyle f_{N}-\Delta {f}}$, i.e., they fold back about ${\displaystyle f_{N}}$, as shown in Figure 9.5a.

A 65-Hz 72-dB/octave filter is usually applied before resampling to prevent aliasing. If resampling to 4 ms is done without this additional filtering, ${\displaystyle f_{N}}$ for the resampled data is ${\displaystyle 1/(2\times 0.004)=125}$ Hz; the aliased noise is shown by the foldback curves in Figure 9.5a.

## Problem 9.5b

Some believe that standard alias filters may be unnecessarily restrictive. The standard alias filter for 4-ms sampling is about 65 Hz, 72-dB/octave. Graph the alias noise versus frequency for a 90-Hz, 72-dB/octave filter for 4-ms sampling and draw conclusions.

### Solution

The 90-Hz filter is shown in Figure 9.5a. Because the sampling interval is 4 ms, ${\displaystyle f_{N}}$ is 125 Hz and the alias noise is given by the foldback curves shown. The 90-Hz filter gives a broader passband than the standard 65-Hz alias filter if signal frequencies above about 80 Hz and signals attenuated more than 60 dB are not important.

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