Adaptive Cooperation for Millimeter Wave Communications

Abstract

In this paper, we suggest two adaptive cooperation protocols for millimeter wave communications. The first protocol chooses, between non cooperative and cooperative communications, the protocol that offers the highest instantaneous throughput. The second protocol chooses, between non cooperative and cooperative communications, the protocol maximizing the average throughput. Both protocols use adaptive modulation and coding to select the best modulation and coding scheme maximizing the average or instantaneous throughput. Our results are valid for interference limited millimeter wave communications in the presence of Nakagami fading channels.

Appendix A

Let Y and Z be two independent Gamma r.v. with joint PDF expressed as [26]

$$\begin{aligned} f_{Y,Z}(y,z)=\frac{m_1^{m_1}m_2^{m_2}y^{m_1-1}z^{m_2-1}e^{-\frac{ym_1}{\beta _1}}e^{-\frac{zm_2}{\beta _2}}}{\Gamma (m_1)\Gamma (m_2)\beta _1^{m_1}\beta _2^{m_2}} \end{aligned}$$

(42)

Let $U=\frac{Y}{Z}$ and $V=Y+Z$, the determinant of Jacobian matrix is equal to

$$\begin{aligned} |J|=\begin{vmatrix} \frac{\partial U}{\partial Y}&\frac{\partial U}{\partial Z} \\ \frac{\partial V}{\partial Y}&\frac{\partial V}{\partial Z} \\ \end{vmatrix}=\begin{vmatrix} \frac{1}{Z}&\frac{-Y}{Z^2} \\ 1&1 \\ \end{vmatrix}=\frac{Y+Z}{Z^2}=\frac{(1+U)^2}{V} \end{aligned}$$

(43)

We have $Z=\frac{V}{1+U}$ and $Y=\frac{UV}{1+U}$. We deduce the PDF of (U, V) as

$$\begin{aligned} f_{U,V}(u,v)= & {} \frac{f_{Y,Z}(y,z)}{|J|}\nonumber \\= & {} \frac{v}{(1+u)^2}\left( \frac{v}{1+u}\right) ^{m_2-1}\left( \frac{vu}{1+u}\right) ^{m_1-1}\nonumber \\&\displaystyle \frac{m_1^{m_1}m_2^{m_2}e^{-\frac{vum_1}{(1+u)\beta _1}}e^{-\frac{vm_2}{(1+u)\beta _2}}}{\Gamma (m_1)\Gamma (m_2)\beta _1^{m_1}\beta _2^{m_2}} \end{aligned}$$

(44)

The PDF of U is computed as

$$\begin{aligned} f_U(u)=\int _0^{+\infty }f_{U,V}(u,v)\hbox {d}v. \end{aligned}$$

(45)

We have [25]

$$\begin{aligned} \int _0^{+\infty }e^{-av}v^b\hbox {d}v=\frac{\Gamma (b+1)}{a^{b+1}} \end{aligned}$$

(46)

Using (45) and (46), we obtain

$$\begin{aligned} f_U(u)=\frac{m_1^{m_1}u^{m_1-1}\Gamma (m_1+m_2)}{\beta _1^{m_1}\Gamma (m_1)\Gamma (m_2)}(1+\frac{u\beta _2m_1}{\beta _1m_2})^{-m_1-m_2}\nonumber \\ \end{aligned}$$

(47)

The PDF of SNR at the destination given in (6) can be deduced from (47) by setting : $m_1=m_{SD}$, $m_2=P_Da_{D}$, $\frac{\beta _1}{m_1}=\beta _{SD}$ and $\frac{\beta _2}{m_2}=b_{D}$.