# Alias filters

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 $f_{N}$ and a very steep slope, so that noise above $f_{N}$ 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.

### Solution

With the 3-dB point on the high-cut filter at 125 Hz and a 72 dB/octave slope, the attenuation at $f_{N}=250$ Hz is 75 dB. Frequencies $\Delta {f}$ higher than the Nyquist frequency, $f_{N}+\Delta f$ , alias to appear as the frequencies $f_{N}-\Delta {f}$ , i.e., they fold back about $f_{N}$ , 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, $f_{N}$ for the resampled data is $1/(2\times 0.004)=125$ 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, $f_{N}$ 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.