|Formula for the lower cutoff frequency:|
|Formula for the upper cutoff frequency:|
|Formula for the Q factor:|
|Formula for the bandwidth:|
| A high filter quality means narrow-band filtering (notch), with a large Q factor.
This results in steep filter flanks with a small bandwidth.
A low filter quality means broad-band filtering, with a small Q factor.
This results in flat filter flanks with a large bandwidth.
A low Q factor gives a broad band (wide) bandwidth or a high Q factor gives a narrow band (small) bandwidth.
Q factor as a function of the bandwidth in octaves N (octave bandwidth)
| Bandwidth in
|3.0 wide||0.404 low|
|1/12 small||17.310 high|
Conversion: 'bandwidth in octaves' N to quality factor Q
Interrelationship of 'octave bandwidth' N and the quality factor Q
Formulas for conversion of bandwidth in octaves to quality factor
Questions on "Parametric filter adjustment"
Conversion table Q to N and N to Q for parametric filters
Filter slope or steepness (dB/oct) is not bandwidth
Excel conversion - quality factor Q to bandwidth in octaves N
Calculating the center frequency from a given bandwidth
Finding the filter center frequency - geometric mean
Conversion RC-pad − R × C to Corner frequency fc and Cutoff frequency to R × C − Time constant t (tau) = R × C
|Why is the bandwidth and the cutoff frequency found at the level of "−3 dB"?
Why we always take 3 dB down gain of a filter?
Full width at half maximum (FWHM).
Answer: That is the point where the energy (power) is fallen to the value ½ or 0.5 = 50 percent of the initial power as energy quantity, that is equivalent to (−)3 dB = 10×log(0.5). A (−)3 dB power drop is a decrease of 50 % to the value of 50 %.
There the voltage is fallen to the value of √(½) or 0.7071 = 70.71 percent of the initial voltage as field quantity equivalent to (−)3 dB = 20×log(0.7071). A (−)3 dB voltage drop is a decrease of 29.29 % to the value of 70.71 %.
|(−)3 dB implies ½ the electric power and since the power is proportional to the
square of voltage, the value will be 0.7071 or 70.71 % of the passband voltage.
√½ = 1/√2 = √0.5 = 0,7071. P ~ V2, that is 0,5 ~ 0,70712.
|Sound engineers and sound designers ("ear people") mostly use the usual (sound) field quantity. That'swhy they say:
The cutoff frequency of a device (microphone, amplifier, loudspeaker) is the frequency at which the output voltage level is decreased to a value of (−)3 dB below the input voltage level (0 dB).
● (−)3 dB corresponds to a factor of √½ = 1/√2 = 0.7071, which is 70.71% of the input voltage.
Acousticians and sound protectors ("noise fighters") seem to like more the (sound) energy quantity. They tell us:
The cutoff frequency of a device (microphone, amplifier, loudspeaker) is the frequency at which the output power level is decreased to a value of (−)3 dB below the input power level (0 dB).
● (−)3 dB corresponds to a factor of ½ = 0.5, which is 50% of the input power (half the value).
|Note: Power gain (power amplification) is not common in audio engineering.
Even power amplifiers for loudspeakers don't amplify the power.
They amplify the audio voltage that moves the voice coil.
|Sound field quantities
Sound pressure, sound or particle velocity, particle displacement or displacement amplitude, (voltage, current, electric resistance).
Inverse Distance Law 1/r
|Sound energy quantities
Sound intensity, sound energy density, sound energy, acoustic power.
Inverse Square Law 1/r²
|Note: A sound field quantity (sound pressure p, electric voltage V) is not a sound energy
quantity (sound intensity I, sound power Pak). I ~ p2 or P ~ V2. Sometimes you can hear
the statement: The cutoff frequency is there where the level L is decreased by (−)3 dB.
Whatever the user wants to tell us so accurately: Level is level or dB is dB.
Bandwidth for Yamaha Parametric Equalizer
|For a Yamaha parametric equalizer EQ there is the filter bandwidth of an
octave divided in 60/60 (12 semitones).
One half tone step (semitone) is then 5/60 − 01V Digital Mixing Console.
N = "bandwidth in octaves" (semi tone or half tone distance). Q = Q factor
|Filter EQ||N||Q||Interval||Filter EQ||N||Q||Interval|
|30/60||0.5||2.871||1/2 octave||120/60||2||0.667|| 2 octaves
|60/60||1||1.,414||1 octave||150/60||2.5||0.511||2.5 octaves|
|90/60||1.5||0.92||1.5 octaves||180/60||3||0.404||3 octaves|
|The "BW/60" control replicates the effect of the Behringer Pro DSP1124P - Feedback Destroyer bandwidth setting.
This control sets the bandwidth of the filter between the half-gain points with:
Note that the Behringer DSP1100 - 24 band parametric equalizer software package does NOT correctly reproduce the way the bandwidth control actually operates, its bandwidths are too small by a factor of √2.
Defining filter bandwidth in this way is not uncommon (the TMREQ filters use a similar definition).
The relationship between Q and BW for the DSP1124P is:
Quality Factor Q = f0 / BW
BW = f0/Q Q = f0/BW f0= BW × Q
Please enter two values, the third value will be calculated.