The biquad transformation is the substitution

$$s = \frac{1}{K}\frac{z-1}{z+1}, \, K=tan(\frac{\omega_c}{2})$$

used to convert the representation of a continuous (or "analog", Laplace transform) system into the representation of a discrete (or "digital", Z transform) system.

The biquad transformation is similar to the bilinear transformation, but does not exhibit the same warping of the frequency domain as the bilinear transformation.

An example of applying the biquad transformation on the transfer function of an infinite impulse response filter can be found in the topic Shelving filter.

## Biquad transformation and normalized transfer functions

The biquad transformation works when applied to the normalized transfer function of the filter. It cannot, for example, be applied to the transfer function of the phaser all pass filter below, as that transfer function is not normalized.

$$H(s)=\frac{s-\omega_0}{s+\omega_0}$$

It can be applied to the following transfer function.

$$H(s)=\frac{s-1}{s+1}$$ $$H(z)= \frac{\frac{1}{K} \frac{z-1}{z+1}-1}{\frac{1}{K} \frac{z-1}{z+1}+1}= \frac{\frac{1-K}{1+K}-z^{-1}}{1-\frac{1-K}{1+K} z^{-1}}$$

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