I have a basic question.

The "well known" lowpass to bandpass transformation is $$ s \longmapsto \frac{\bar{s}^2 + \omega_1\omega_2}{\bar{s}(\omega_1 - \omega_2)}, $$ which gives a bandpass transfer function of $$ \frac{1}{s + 1} \longmapsto \frac{\bar{s}(\omega_1 - \omega_2)}{\bar{s}^2 + \bar{s}(\omega_1 - \omega_2) + \omega_1 \omega_2}. $$

My intuition is that a bandpass should be the product of a lowpass and a highpass. However, this product gives a different transfer function: $$ \frac{\omega_1}{s + \omega_1} \frac{s}{s + \omega_2} = \frac{\omega_1 s}{s^2 + (\omega_1 + \omega_2) s + \omega_1 \omega_2}, $$ which indicates that the bandpass transformation does not give this cascade of lowpass and highpass.

• My question is, how is the bandpass transformation designed, in terms of either combining lowpass filters or by pole placement?

• Related question, but using a different derivation technique, and reference is made to the lowpass/highpass derivation, but it is not shown: How is the lowpass to bandpass transformation derived?