-3dB bandwidth calculating the -3 dB cutoff frequencies - quality factor Q factor filter center frequency and q factor formula bandwidth in octaves 3 db bandwidth frequency EQ equalizer bandpass filter octave 3 dB bandwidth conversion calculator corner frequency half-power frequency

● Calculation − Equalization − Bandpass − Filter ●

Calculating the −3 dB cut-off frequencies f₁ and f₂
when center frequency f₀ and Q factor is given.

3 dB Bandwidth BW = f₂ − f₁ = f₀/Q and quality factor is Q factor
EQ filter conversion Q factor to bandwidth in octaves N

Parametric peak equalizer and notch (dip) equalizer
People use 'Q' and 'bandwidth' interchangeably, though they're not.
Defining the bandwidth for a bandpass as the −3 dB points cannot be correct for a boost gain of 3 dB or less.

Corner frequency = cut-off frequency = crossover frequency
= half-power frequency = 3 dB frequency = break frequency is all the same.
The center frequency f₀ is the geometric mean of f₁ and f₂
BW = f₀/Q Q = f₀/BW f₀= BW × Q

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Formula for the lower cutoff frequency:

Formula for the upper cutoff frequency:

Formulas mailed by Daniel Fournier

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.

Notice:
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 octaves N	Q factor
3.0 wide	0.404 low
2.5	0.511
2.0	0.667
1.5	0.920
1.0	1.414
2/3	2.145
1/2	2.871
1/3	4.318
1/6	8.651
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

3 dB implies 1/2 the power and since the power is proportional to the square of voltage, the voltage will be 0.707 or 70.7 % of the passband voltage.
√ (0.5) = 0.707

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.
Conversion:
N = "bandwidth in octaves" (semi tone or half tone distance). Q = Q factor

Filter EQ	N	Q	Interval	Filter EQ	N	Q	Interval
5/60	0.083	17.31	Semitone step	95/60	1.583	0.867
10/60	0.167	8.651	Whole tone	100/60	1.667	0.819
15/60	0.25	5.764		105/60	1.75	0.776
20/60	0.333	4.318	1/3 octave	110/60	1.833	0.736
25/60	0.417	3.45		115/60	1.917	0.7
30/60	0.5	2.871	1/2 octave	120/60	2	0.667	2 octaves
35/60	0.583	2.456	Fifth	125/60	2.083	0.636
40/60	0.667	2.145		130/60	2.167	0.607
45/60	0.75	1.902		135/60	2.25	0.581
50/60	0.833	1.707		140/60	2.333	0.556
55/60	0.917	1.548		145/60	2.417	0.532
60/60	1	1.,414	1 octave	150/60	2.5	0.511	2.5 octaves
65/60	1.083	1.301		155/60	2.583	0.49
70/60	1.67	1.204		160/60	2.667	0.471
75/60	1.25	1.119		165/60	2.,75	0.453
80/60	1.333	1.044		170/60	2.883	0.436
85/60	1.417	0.979		175/60	2.917	0.419
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:
BW (Hz) = f₀ × (BW / 60) × √2 For example, at a bandwidth setting of 60/60 a filter centred on 1 kHz with a gain of −6 dB will have a bandwidth of 1,414 Hz between the points where its response crosses −3 dB. This bandwidth remains constant as the filter's gain is adjusted.
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:
Q = 60 / [(BW / 60) × √2] So the bandwidth range of 1/60 to 120/60 gives a range from Q = 42.4 to 0.35.

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 1/2 value or 0.5 = 50 percent of the initial energy quantity.
There the voltage is fallen to the value of √(1/2) = 1/√2 or 0.7071 = 70.71 percent of the initial voltage as field quantity. A 3 dB voltage drop is a decrease of 29.29 % to 70,71 %.

At the cut-off frequency f_c of a drop the voltage V is always fallen to the value
1/√2 = 0.7071 ≡ 70.71 % and the voltage level is damped by
20 × log (1/√2) = (−)3.0103 dB.

At the cut-off frequency (half-power frequency) the less interesting power P
is always fallen to
1/2 = 0.5 ≡ 50 % and the power level is damped by
10 × log (1/2) = (−)3.0103 dB − that is the same dB value.

This is often confusing. 0.7071 × 0.7071 is 0.5 and P = V²/R; P ~ V².
What do you mean by 3 dB cutoff frequency? Why is it 3 dB, not 1 dB?
Answer: The power P is always fallen there to 1/2 = 0.5 = 50 %.

Quality Factor Q = f₀/ BW
BW = f₀/Q Q = f₀/BW f₀= BW × Q

Please enter two values, the third value will be calculated.

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