# What is absolute ripple specification and how does it arrive in low pass prototype for Butterworth?

Digital Signal Processing by John G. Proakis, Dimitris G. Manolakis(4th Edition) gives the prototype function for Butterworth low pass filter in an analog system which is fine till the point where the cutoff radian frequency is present in the equation. But then an epsilon and pass band radian frequency come into the equation. Digital Signal Processing by Li Tan mentions this epsilon as 'Absolute ripple specification'. So, my question is, we know about the stop band attenuation, pass band ripple and the pass band, stop band frequencies, what is this epsilon and how does the equation containing it (in the image) come to be?

In the first formulation, $$\Omega_c$$ is the $$3$$ $$\textrm{dB}$$ cut-off frequency, because for $$\Omega=\Omega_c$$ you obtain $$|H(\Omega_c)|^2=\frac12$$. In the other formulation, the attenuation at the pass band edge $$\Omega_p$$ is determined by the constant $$\epsilon$$. For $$\Omega=\Omega_p$$ you have $$|H(\Omega_p)|=\frac{1}{1+\epsilon^2}$$. I.e., by choosing the constant $$\epsilon$$ appropriately, any desired attenuation at the passband edge can be achieved.