Audio Filter Design Formulas & Phase Relationships
Audio Filter Design Formulas
Bessel Filters
12dB/octave High-Pass 2nd Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = 0.7071 / (2 * pi * Fc * C)
- Rb = 1.4142 / (2 * pi * Fc * C)
Units: R [Ohm], C [Farads], Fc [Hz]
12dB/octave Low-Pass 2nd Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 0.9076 / (2 * pi * Fc * R)
- Cb = 0.6809 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
18dB/octave High-Pass 3rd Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = 1.0474 / (2 * pi * Fc * C)
- Rb = 2.008 / (2 * pi * Fc * C)
- Rc = 1.3228 / (2 * pi * Fc * C)
Units: R [Ohm], C [Farads], Fc [Hz]
18dB/octave Low-Pass 3rd Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 0.9548 / (2 * pi * Fc * R)
- Cb = 0.4998 / (2 * pi * Fc * R)
- Cc = 0.7560 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
24dB/octave High-Pass 4th Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = 1.3701 / (2 * pi * Fc * R)
- Rb = 1.4729 / (2 * pi * Fc * R)
- Rc = 0.9952 / (2 * pi * Fc * R)
- Rd = 2.5830 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
24dB/octave Low-Pass 4th Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 0.7298 / (2 * pi * Fc * C)
- Cb = 0.6699 / (2 * pi * Fc * C)
- Cc = 1.0046 / (2 * pi * Fc * C)
- Cd = 0.3872 / (2 * pi * Fc * C)
Units: R [Ohm], C [Farads], Fc [Hz]
Butterworth Filters
6dB/octave High-Pass
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- R = 1.000 / (2 * pi * Fc * C)
Units: R [Ohm], C [Farads], Fc [Hz]
6dB/octave Low-Pass
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- C = 1.000 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
12dB/octave High-Pass 2nd Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- C = 1.000 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
12dB/octave Low-Pass 2nd Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 1.4142 / (2 * pi * Fc * R)
- Cb = 0.7071 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
18dB/octave High-Pass 3rd Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = 0.500 / (2 * pi * Fc * R)
- Rb = 2.000 / (2 * pi * Fc * R)
- Rc = 1.000 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
18dB/octave Low-Pass 3rd Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 2.000 / (2 * pi * Fc * R)
- Cb = 0.500 / (2 * pi * Fc * R)
- Cc = 1.000 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
24dB/octave High-Pass 4th Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = 0.9239 / (2 * pi * Fc * C)
- Rb = 1.0824 / (2 * pi * Fc * C)
- Rc = 0.3827 / (2 * pi * Fc * C)
- Rd = 2.6130 / (2 * pi * Fc * C)
Units: R [Ohm], C [Farads], Fc [Hz]
24dB/octave Low-Pass 4th Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = 1.0824 / (2 * pi * Fc * R)
- Cb = 0.9239 / (2 * pi * Fc * R)
- Cc = 2.6130 / (2 * pi * Fc * R)
- Cd = 0.3872 / (2 * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
Linkwitz-Riley Filters
24dB/octave High-Pass 4th Order
Cutoff frequency (Fc):
- C = 4.7nF – 10nF
- Ra = Rc = 1 / (2 * sqr(2) * pi * Fc * C)
- Rb = Rd = 2Ra
Units: R [Ohm], C [Farads], Fc [Hz]
24dB/octave Low-Pass 4th Order
Cutoff frequency (Fc):
- R = 4.7K – 10KΩ
- Ca = Cc = 2 * Cb
- Cb = Cd = 1 / (2 * sqr(2) * pi * Fc * R)
Units: R [Ohm], C [Farads], Fc [Hz]
Phase Relationships in Filters
Filters are classified by their slope (decibels per octave) and affect the phase relationship of an audio signal. The slope translates into a delay measured in degrees. The types of filters discussed above (Butterworth, Bessel, Linkwitz-Riley) exhibit different phase characteristics.