Filter slope, bandwidth and Q

Filter specifications mix several terms that sound alike: slope, order, bandwidth and Q. They describe different things, and confusing them leads to wrong conclusions.

Slope

The slope (or steepness) describes how quickly a filter attenuates beyond its cutoff. It is given in dB per octave or dB per decade.

• 6 dB per octave and 20 dB per decade are the same slope. A decade is 3.32 octaves, and 6.02 dB × 3.32 = 20 dB.
• An octave is a doubling of frequency, a decade a tenfold increase.

Order

A filter's order is the number of reactive elements (capacitors and inductors) that form it. Each order contributes 6 dB per octave:

A simple RC low-pass is first order, see the RC filter tool. A Sallen-Key stage is second order, see the active filter designer.

Cutoff frequency

For low-pass and high-pass filters, the cutoff frequency is usually the point where the level has dropped by 3 dB. The slope only applies well beyond it. At the cutoff of a first-order filter the attenuation is 3 dB, one octave above it 7 dB, and two octaves above it about 12 dB. The step from one octave to the next approaches 6 dB only well above the cutoff.

Bandwidth and Q

Bandwidth and Q describe the width of a band-pass or peaking filter: the range around its centre frequency where the response is within 3 dB of the maximum.

Q = centre frequency / bandwidth in Hz

A high Q means a narrow filter. One octave of bandwidth is a Q of 1.41; one third of an octave, a Q of 4.3. Convert with the Q and bandwidth tool.

They are not the same thing

• Slope says how steep the flank is.
• Bandwidth and Q say how wide the pass band is.

A band-pass filter with a very high Q can have gentle slopes, and a wide filter can have very steep slopes. Quoting "24 dB/octave" for a band-pass does not tell you whether it is narrow or wide, and quoting "Q = 4" does not tell you how fast the response falls beyond the pass band.

Peaking equalisers

In a parametric equaliser, the slopes at the sides of a peak are tied to the Q and the gain: a narrow, strong boost has steep sides, a wide, gentle boost has gentle sides. See Equalisation basics.

Why steep filters need care

Steep slopes bring phase shift near the cutoff and, for analogue filters, ringing. A 24 dB/octave high-pass filter has a noticeable phase shift well above its cutoff, which can influence the sound of the bass in combination with other signals.

Draft: written by an AI assistant, awaiting technical review.