I have a simple question that confuses me.
Assume that we have the following results of a measurement campaign, distance (km) and the corresponding path loss $L_{e}$ (dB). I want to estimate the path loss exponent of the simplified path loss model
$$L_{m,\text{dB}} = K_\text{dB} - 10 n \log_{10}\left(\frac{d}{d_0}\right)\text ,$$
where $K$ is a constant that depends on antenna characteristics and average channel attenuation at reference distance $d_0$ and is given by $K_\text{dB} = 20 \log_{10}\left(\frac{\lambda}{4 \pi d_0}\right)$ or $K_\text{dB} = 20 \log_{10}\left(\frac{\lambda \sqrt{G_t G_r}}{4 \pi d_0}\right)$, where $G_t$ and $G_r$ are the transmit and receive antenna gains, respectively, and $\lambda$ is the wavelength.
To estimate the path loss exponent $n$, I am finding $n$ that minimizes the MMSE error for the dB power measurement
$$\begin{align} \text{MMSE}(n) &= \sum_i^N \left[L_e(d_i) - L_m(d_i)\right]^2\\ &= \sum_i^N{ \left[L_e(d_i) - K + 10 n \log_{10}\left(\frac{d_i}{d_0}\right)\right]^2}\text , \end{align}$$
where $N$ is the total number of measurements.
- It seems the exact value of $n$ depends on the reference distance $d_0$ and the antenna gains $G_t$ and $G_r$, right?
- In other words, for different $d_0, G_t, G_r$, I will have different value for $n$. Is this in conflict with the fact that on a log-log scale, the path loss is represented by a straight line with a slope of $10 n$ per decade?