Index DigitalParametric EQ Biquad

Parametric EQ Biquad

Design a filter band, read the response and copy the coefficients.

Frequency1kHz

Bandwidth1.01oct

Gain at f06.0dB

Normalised ω00.131rad

Magnitude response

Coefficients, a0 normalised to 1

b0 =  1.03179626113
b1 = -1.91954111760
b2 =  0.904308501105
a1 = -1.91954111760
a2 =  0.936104762233

Input

Filter type

Centre or corner

f_0

GaindB

Quality factor

Q

Sample rate (kHz)

How it works

The biquad

Nearly every digital equaliser is a chain of second-order sections. Each one computes an output sample from the current input, two past inputs and two past outputs:

y[n] = b_0x[n] + b_1x[n-1] + b_2x[n-2] - a_1y[n-1] - a_2y[n-2]

which corresponds to the transfer function

H(z) = \frac{b_0 + b_1z^{-1} + b_2z^{-2}}{1 + a_1z^{-1} + a_2z^{-2}}

From knobs to coefficients

The coefficients follow Robert Bristow-Johnson's Audio EQ Cookbook. Three intermediate values are shared by every filter type:

A = 10^{G/40},

\omega_0 = \frac{2\pi f_0}{f_s},

\alpha = \frac{\sin\omega_0}{2Q}

For the peaking band, for example:

\begin{aligned} b_0 &= 1 + \alpha A & b_1 &= -2\cos\omega_0 & b_2 &= 1 - \alpha A \\ a_0 &= 1 + \alpha / A & a_1 &= -2\cos\omega_0 & a_2 &= 1 - \alpha / A \end{aligned}

All five are then divided by $a_0$ . The plotted curve is $|H(e^{j\omega})|$ evaluated from those same numbers, so what you see is what the filter does.

Q and bandwidth

\mathrm{BW}_{oct} = \frac{2}{\ln 2}\,\sinh^{-1}\!\left(\frac{1}{2Q}\right)

Check the sign convention of your DSP before pasting. Some platforms store $-a_1$ and $-a_2$ instead.

References

R. Bristow-Johnson, “Cookbook Formulae for Audio EQ Biquad Filter Coefficients,” Audio EQ Cookbook. W3C