Multipath fading in wireless channel - parameter needed

Question

I’m trying to implement in Matlab the magnitude of the channel taps of the wireless communication system that is described at page 23 (Section 2.1.4) in the classic book of D. Tse, Fundamentals of Wireless Communications.

The $l^{\rm th}$ channel taps at time $m$ is given by eq. (2.51) p. 44:

$$ h_l[m] = \sum_i a_i\left(\frac mW\right)e^{-j2\pi f_c \tau_i\left(\frac mW\right)} \mathrm{sinc}\left[l-\tau_i\left(\frac mW\right)\right] $$

In this particular case we have $i = 2$ paths and attetuation $a_i(t)$ for these paths are given from eq. (2.15):

$$ a_1(t) = \frac{\lvert\alpha\rvert}{r_0 + vt}, \quad a_2(t) = \frac{\lvert\alpha\rvert}{2d - r_0 - vt} $$

And propagation delays by eq.(2.16):

$$ \tau_1(t) = \frac{r_0 + vt}{c} - \frac{\phi_1}{2\pi f }, \quad \tau_2(t) = \frac{ 2d -r_0 - vt}{c} - \frac{\phi_2}{2\pi f} $$

I have calculated everything apart from $\lvert a\rvert $. I don’t really know how to calculate it.

Any suggestions?

Answer 1

You don't need to compute the parameter $|\alpha|$, because it is the antenna gain, which is part of the antenna's specification. This is also mentioned on p.21 of the book you quoted (above Eq. $(6)$).