Tcoil Topics

here is an introduction to tcoil and its applications. it also covers asymmetric Tcoil.

for beginners: "Broadband ESD Protection Circuits in CMOS Technology" and "The Bridged T-Coil" are helpful.

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

comb filter

notes from "A Comb Filter Design Using Fractional-Sample Delay" [1]:

• transfer function of the filter: $H_C\left(z\right)=\frac{1-z^{-D}}{1-\rho^D z^{-D}}$, where $D$ is the period of harmonic interference (see block diagram below);
• if $D$ is integer, filter realization is straight forward;
• if $D$ is a fraction, we can use the the techniques in "Splitting the unit delay" [2] to realize $z^{-D}$; FIR frac delay filter or All-Pass frac delay filter;
•  an extra optimization that [1] proposes is to make sure that the filter response has an actual zero at  $w_0=\frac{2\pi}{D}$ frequency;

comb filter to remove periodic interference, where $D$ is the period of the interference harmonic