Filter Slopes

Not all filters have the same initial slopes, even if they are rated as having the same ultimate slope rates.

The cut-off frequency of a high or low pass filter is defined as the -3 dB point of the initial slope.

Ultimate filter slope rates are determined by the number of "poles" in the filter design, with each pole contributing 6 dB to the ultimate slope rate of a high or low pass filter. The ultimate slope rate may not be reached until far away from the cut-off frequency.

For example, the "12 dB per octave" slope rating of a two pole filter refers to its ultimate slope rate, not to its initial slope rate. The amount of slope in the first or initial octave varies according to the filter type or characteristic.

For a high pass filter the first or initial octave of the slope will end at 1/2 the cut-off frequency while for a low pass filter it will end at twice the cut-off frequency.

Butterworth filters have the steepest initial slope that can be obtained without getting ripple in the passband (the part of the response we want flat).

For example below you will see three different types of two pole high pass filters all of which have the same 12 dB per octave ultimate slope rates and are set to 1 kHz. First is the Butterworth characteristic. It is followed by the Bessel and Linkwitz-Riley filter types. All three filter types are shown twice. First with the cursor set to the -3 dB or cutoff frequency of 1 kHz, and then with the cursor moved down in frequency one octave so you can see the attenuation gained in the first octave of the slope.

12 dB per octave Butterworth filter with the cursor set to 1 kHz.

12 dB per octave Butterworth filter with the cursor set to 500 Hz.

Note that the total attenuation is -12.5 dB. 9.5 dB of attenuation was added in the first octave of the slope.

12 dB per octave Bessel filter with the cursor set to 1 kHz.

12 dB per octave Bessel filter with the cursor set to 500 Hz.

Note that the total attenuation is -9.9 dB. 6.9 dB of attenuation was added in the first octave of the slope.

12 dB per octave Linkwitz-Riley filter with the cursor set to 1 kHz.

12 dB per octave Linkwitz-Riley filter with the cursor set to 500 Hz.

Note that the total attenuation is -8.6 dB. 5.6 dB of attenuation was added in the first octave of the slope.

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Last edited on 4/7/2013
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