# Derive Time-Domain Biquad Coefficients from Z-Domain

I'm trying to derive the time-domain coefficients of a biquad filter based on its z-domain coefficients. The purpose is to import externally-generated EQ filters (in z-domain) into Max/MSP (in time-domain).

Could someone kindly explain how to do so?

Z-domain: $$H(z) = \frac{b_0+b_1 z^{-1}+b_2 z^{-2}}{a_0+a_1 z^{-1}+a_2 z^{-2}}$$

Time-domain: $$y[n] = a_0 x[n] + a_1 x[n-1] + a_2 x[n-2] - b_1 y[n-1] - b_2 y[n-2]$$

• Two things. 1. you gotta learn to use $\LaTeX$ markup here. it's easy. 2. the common convention now puts $a_0, a_1, a_2$ in the denominator and the $b_m$ coefficients into the numerator. (i just figured out that you swapped it. Z-domain okay but Time-domain you swapped the $a_m$ and $b_m$) 3. these coefficients are coefficients. they are both time domain and frequency domain, as your "Z-domain" equations depicts. 4. there is the Audio EQ Cookbook. try googling it. and 5. i can't count to two. Jan 1, 2021 at 1:49
• Thanks for updating with the correct formatting. I wasn't aware that Latex could be used here. Thanks for catching my mistake with the coefficients. From the answer below, I realize now that the value of, eg. a0, is not identical between H(z) and y(n). Thanks again!
– Joel
Jan 2, 2021 at 2:25

I'm not entirely sure what you are asking but if your Z-domain representation is $$H(z) = \frac{b_0+b_1 z^{-1}+b_2 z^{-2}}{a_0+a_1 z^{-1}+a_2 z^{-2}}$$
Then your difference equation is $$a_0 \cdot y[n] + a_1 \cdot y[n-1] + a_2 \cdot y[n-2] = b_0 \cdot x[n] + b_1 \cdot x[n-1] + b_2 \cdot x[n-2]$$
And solving for $$y[n]$$ we get $$y[n] = \frac{1}{a_0}(b_0 \cdot x[n] + b_1 \cdot x[n-1] + b_2 \cdot x[n-2] - a_1 \cdot y[n-1] - a_2 \cdot y[n-2])$$