minimum phase channel

notes from "Minimum-Phase Impulse Response Channels":

• $h\left(t\right) \leftrightarrow H\left(w\right)$ 
• if the inverse system of a causal system is also causal
• zeros and poles of $H\left(w\right)$ and $H^{-1}\left(w\right)$ being interchanged
• both zeros and poles of $H\left(w\right)$ have negative imaginary parts
• then the system is "minimum phase"
• for a multi-path channel: $h\left(t\right) = a_0 \delta\left(t\right) + \sum\limits_{l=1}^{L_p}a_l \delta\left(t-\tau_l\right)$
• Fourier transform is given by $H\left(w\right) = a_0+\sum\limits_{l=1}^{L_p}a_l \text{exp}\left(j w \tau_l\right)$   
• if channel is minimum phase then $\text{ln}\left[\frac{\left|H\left(w\right)\right|}{a_0}\right]$ and $\phi\left(w\right)-\phi_0$ are Hilbert transform and inverse transform of one another, where $H\left(w\right)=\left|H\left(w\right)\right| \text{exp}\left[j\phi\left(w\right)\right]$ and $a_0 = \left|a_0\right| \text{exp}\left(j\phi_0\right)$;
• in other words, if channel is minimum phase, channel phase information can be extracted from its amplitude response;
• Hilbert transform of  a function $G\left(w\right) = \int_{-\infty}^{+\infty}\frac{G\left(w'\right)}{\pi \left(w-w'\right)}dw'$; the inverse Hilbert transform is given by the same expression with a minus sign at the right hand side;
• the paper give a sufficient condition for a wireless channel to be minimum phase:
• the energy of the first path to be larger than the power spectral density of sum of all the subsequent paths;
• $\left|a_0\right|^2 > \left| \sum\limits_{l=1}^{L_p}a_l \text{exp}\left(j w \tau_l\right) \right|^2$ or $\left|a_0\right|^2 > \left| H\left(w\right)-a_0\right|^2$;
•  this condition is more likely to be met if transmitter and receiver are in LOS than when they are in NLOS