Alias filters

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{N} =250 Hz is 75 dB. Frequencies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta{f} higher than the Nyquist frequency, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{N} +\Delta f , alias to appear as the frequencies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{N} -\Delta{f} , i.e., they fold back about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{N} for the resampled data is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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.

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