This section studies the channel average rate in the case where the scheduler is provided with quantized CSI about the fading channels. In this way, considering

*N* quantization regions, the quantization encoder function

$C\left({g}_{k}\right)=i\phantom{\rule{0.3em}{0ex}}\text{if}{g}_{k}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{S}_{i}=\left[{\stackrel{\u0303}{g}}_{i-1},{\stackrel{\u0303}{g}}_{i}\right),\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{g}}_{0}=0,\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{g}}_{N}=\infty $

(6)

is implemented by each receiver. Here, ${\stackrel{\u0303}{g}}_{i}$'s denote the quantization boundaries and *S*_{
i
} is the *i*-th quantization region. The quantization indices are sent back to the scheduler which selects the user with the higher quantization index (max-rate scheduler). Also, if the channel gains are in the same quantization regions, one of them is selected randomly.

*Remark 1:* The optimal max-rate scheduler should select the users with the highest SNR. However, as stated in the following, the water-filling properties imply that higher powers are allocated to the higher quantization regions (see, e.g., (14), (19) and [34–38]). Therefore, the SNR increases with the quantization index and scheduling based on the quantization indices works the same as scheduling based on the SNRs.

Provided that the scheduled user channel gain is in the region

*S*_{
i
} , a fixed gain

$\widehat{{g}_{i}}\in \left[{\stackrel{\u0303}{g}}_{i-1},{\stackrel{\u0303}{g}}_{i}\right)$is considered by the transmitter and the data is sent with power

*T*_{
i
} and rate

*R*_{
i
} = log(1 +

*ĝ*_{
i
}*T*_{
i
} ). The data is successfully decoded at the corresponding receiver if

*G*_{
k
} ≥

*ĝ*_{
i
} where

*k* represents the selected user index. Therefore, considering all quantization regions, the channel average rate is found as

$\stackrel{\u0304}{R}=2\sum _{i=2}^{N}{P}_{i}{R}_{i}+\sum _{i=1}^{N}{Q}_{i}{R}_{i}.$

(7)

Here,

$\begin{array}{ll}\hfill {P}_{i}& =\text{Pr}\left\{{G}_{1}\in \left[{\u011d}_{i},{\stackrel{\u0303}{g}}_{i}\right)\&{G}_{2}\in \left[0,{\stackrel{\u0303}{g}}_{i-1}\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{{\u011d}_{i}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}\underset{0}{\overset{{\stackrel{\u0303}{g}}_{i-1}}{\int}}{f}_{{G}_{1},{G}_{2}}\left(x,y\right)\text{d}x\text{d}y\phantom{\rule{2em}{0ex}}\end{array}$

(8)

is the probability that (1) for instance, channel

*G*_{1} is in the

*i*-th quantization region, (2) its corresponding channel gain is higher than the considered value

*ĝ*_{
i
} , that is,

${G}_{1}\in \left[\widehat{{g}_{i}},\stackrel{\u0303}{{g}_{i}}\right)$ and (3) the second user channel gain is in one of the quantization regions

*S*_{
j
} ,

*j < i*, such that the first user is selected by the scheduler. Then, the first summation term is multiplied by two, as the same thing can happen for the other user. Furthermore,

*Q*_{
i
} is found as

$\begin{array}{ll}\hfill {Q}_{i}& =\text{Pr}\left\{{G}_{1}\in \left[{\u011d}_{i},{\stackrel{\u0303}{g}}_{i}\right)\&{G}_{2}\in \left[{\stackrel{\u0303}{g}}_{i-1},{\stackrel{\u0303}{g}}_{i}\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{{\u011d}_{i}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}\underset{{\stackrel{\u0303}{g}}_{i-1}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}{f}_{{G}_{1},{G}_{2}}\left(x,y\right)\text{d}x\text{d}y\phantom{\rule{2em}{0ex}}\end{array}$

(9)

which is the probability that (1) both users are in the

*i*-th quantization region and (2) the channel gain of the selected user supports the rate, e.g.,

${G}_{1}\in \left[\widehat{{g}_{i}},{\stackrel{\u0303}{g}}_{i}\right)$ if the first user is selected by the scheduler. Note that in this case one of the users is scheduled randomly with probability

$\frac{1}{2}$. Therefore, the second summation term in (7) is not multiplied by two. Correspondingly, the average transmission power is obtained by

$\stackrel{\u0304}{T}=2\sum _{i=2}^{N}{{P}^{\prime}}_{i}\phantom{\rule{0.3em}{0ex}}{T}_{i}+\sum _{i=1}^{N}{{Q}^{\prime}}_{i}\phantom{\rule{0.3em}{0ex}}{T}_{i}$

(10)

where

$\begin{array}{c}{{P}^{\prime}}_{i}=\text{Pr}\left\{{G}_{1}\in {S}_{i}\&{G}_{2}\in {S}_{j},\phantom{\rule{0.3em}{0ex}}j<i\right\}\\ =\underset{{\stackrel{\u0303}{g}}_{i-1}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}\underset{0}{\overset{{\stackrel{\u0303}{g}}_{i-1}}{\int}}{f}_{{G}_{1},{G}_{2}}\left(x,y\right)\text{d}x\text{d}y\end{array}$

(11)

denotes the probability that, for instance, channel

*G*_{1} is in the

*i*-th quantization region while the second user channel gain is in one of the lower regions. Also,

$\begin{array}{ll}\hfill {{Q}^{\prime}}_{i}& =\text{Pr}\left\{{G}_{1}\in {S}_{i}\&{G}_{2}\in {S}_{i}\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{{\stackrel{\u0303}{g}}_{i-1}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}\underset{{\stackrel{\u0303}{g}}_{i-1}}{\overset{{\stackrel{\u0303}{g}}_{i}}{\int}}{f}_{{G}_{1},{G}_{2}}\left(x,y\right)\text{d}x\text{d}y\phantom{\rule{2em}{0ex}}\end{array}$

(12)

is the probability that both channels are in the *i*-th quantization region where one of them is selected randomly.

Using (7), (10) and the power constraint

$\stackrel{\u0304}{T}\le T,$the power-limited average rate maximization problem can be stated as

$\begin{array}{c}\underset{{\stackrel{\u0303}{g}}_{i},{\u011d}_{i},{T}_{i}}{\text{max}}\left\{2\sum _{i=2}^{N}{P}_{i}{R}_{i}+\sum _{i=1}^{N}{Q}_{i}{R}_{i}\right\}\phantom{\rule{0.3em}{0ex}}\\ \text{subject}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}\left\{2\sum _{i=2}^{N}{{P}^{\prime}}_{i}{T}_{i}+\sum _{i=1}^{N}{{Q}^{\prime}}_{i}{T}_{i}\le T\right\}\end{array}$

(13)

which, as discussed in [

34–

38], [[

41], Section 9.4], is a convex problem in terms of transmission powers

*T*_{
i
} . Therefore, the optimal transmission powers can be determined based on the Lagrange multiplier function

$\Upsilon =\stackrel{\u0304}{R}-\lambda \stackrel{\u0304}{T}$ which leads to the water-filling equations

$\begin{array}{c}\frac{\partial \Upsilon}{\partial {T}_{i}}=0\Rightarrow \left\{\begin{array}{c}{T}_{1}={\u2308\frac{{Q}_{1}}{\lambda {{Q}^{\prime}}_{1}}-\frac{1}{{\u011d}_{1}}\u2309}^{+},\phantom{\rule{0.3em}{0ex}}i=1\\ {T}_{i}={\u2308\frac{2{P}_{i}+{Q}_{i}}{\lambda \left(2{{P}^{\prime}}_{i}+{{Q}^{\prime}}_{i}\right)}-\frac{1}{{\u011d}_{1}}\u2309}^{+},\phantom{\rule{0.3em}{0ex}}i>1\end{array}\right.\phantom{\rule{0.3em}{0ex}}.\end{array}$

(14)

Here, *λ* is the Lagrange multiplier satisfying $\stackrel{\u0304}{T}\le T\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{\u2308x\u2309}^{+}\doteq \text{max}\left(0,x\right).$ Intuitively, using optimal power allocation the power is not wasted on *weak* channel realization and the saved power is spent on *strong* gain realizations. Therefore, there will be a quantization index $\stackrel{\u2322}{i}$ where *T*_{
i
} = 0 if $i<\stackrel{\u2322}{i}$and ${T}_{i}>0$ if $i\ge \stackrel{\u2322}{i}$. This point is helpful for simplifying the water-filling power allocation algorithm.

Considering (13), the main problem is to find the probability terms in (7) and (10) which can be found according to the following procedure

$\begin{array}{l}{\displaystyle {\int}_{u}^{v}{\displaystyle {\int}_{w}^{z}{f}_{{G}_{1},{G}_{2}}}}(x,\phantom{\rule{0.25em}{0ex}}y)\text{d}x\text{d}y\\ {\underset{\xaf}{\underset{\xaf}{(a)}}}^{}{\displaystyle {\int}_{u}^{v}{\scriptscriptstyle \frac{1}{\mu}}}{e}^{-\frac{x}{r}}\left({\int}_{\sqrt{\frac{2w}{r}}}^{\sqrt{\frac{2z}{r}}}\theta {e}^{-\frac{{\theta}^{2}}{2}}{I}_{0}(s\sqrt{x}\theta )\text{d}\theta \right)\text{d}x\\ {\underset{\xaf}{\underset{\xaf}{(b)}}}^{}{\displaystyle {\int}_{u}^{v}{\scriptscriptstyle \frac{1}{\mu}}}{e}^{-\frac{x}{\mu}}\left\{\xi \left(s\sqrt{x},\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2w}{r}}}\right)-\xi \left(s\sqrt{x},\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2z}{r}}}\right)\right\}\text{d}x\\ {\underset{\xaf}{\underset{\xaf}{(c)}}}^{}(1-{\beta}^{2}){e}^{-\frac{w}{\mu}}\left\{\xi \phantom{\rule{0.25em}{0ex}}\left(\sqrt{{\scriptscriptstyle \frac{2w}{r}}}\beta ,\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2u}{r}}}\right)-\xi \left(\sqrt{{\scriptscriptstyle \frac{2w}{r}}}\beta ,\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2v}{r}}}\right)\right\}\\ -(1-{\beta}^{2}){e}^{-\frac{z}{\mu}}\left\{\xi \left(\sqrt{{\scriptscriptstyle \frac{2z}{r}}}\beta ,\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2u}{r}}}\right)-\xi \left(\sqrt{{\scriptscriptstyle \frac{2z}{r}}}\beta ,\phantom{\rule{0.25em}{0ex}}\sqrt{{\scriptscriptstyle \frac{2v}{r}}}\right)\right\}\\ +{\scriptscriptstyle \frac{1}{\mu}}{\displaystyle {\int}_{u}^{v}{e}^{-\frac{x}{\mu}}}\left\{\xi \phantom{\rule{0.25em}{0ex}}\left(\sqrt{{\scriptscriptstyle \frac{2z}{r}}},\phantom{\rule{0.25em}{0ex}}s\sqrt{x}\right)-\xi \left(\sqrt{{\scriptscriptstyle \frac{2w}{r}}},\phantom{\rule{0.25em}{0ex}}s\sqrt{x}\right)\right\}\text{d}x\\ {\underset{\xaf}{\underset{\xaf}{(d)}}}^{}{e}^{-\frac{w}{\mu}}\{\varphi (w{\beta}^{2},\phantom{\rule{0.25em}{0ex}}u)-\varphi (w{\beta}^{2},\phantom{\rule{0.25em}{0ex}}v)\}-{e}^{-\frac{z}{\mu}}\{\varphi (z{\beta}^{2},\phantom{\rule{0.25em}{0ex}}u)-\varphi (z{\beta}^{2},\phantom{\rule{0.25em}{0ex}}v)\}\\ +{e}^{-\frac{v}{\mu}}\varphi (w,\phantom{\rule{0.25em}{0ex}}v{\beta}^{2})-{e}^{-\frac{u}{\mu}}\varphi (w,\phantom{\rule{0.25em}{0ex}}u{\beta}^{2})-{e}^{-\frac{v}{\mu}}\varphi (z,\phantom{\rule{0.25em}{0ex}}v{\beta}^{2})+{e}^{-\frac{u}{\mu}}\varphi (z,\phantom{\rule{0.25em}{0ex}}u{\beta}^{2}).\end{array}$

(15)

Here, (

*a*) is obtained by defining

$r\doteq \left(1-{\beta}^{2}\right)\mu ,\phantom{\rule{0.3em}{0ex}}s\doteq \sqrt{2/r}\beta $ and using variable transform

$\theta =\sqrt{2y/r}$. Then, (

*b*) is directly obtained from the definition of the Marcum

*Q*-function

$\xi \left(x,y\right)=\underset{y}{\overset{\infty}{\int}}t{{e}^{-}}^{\frac{{t}^{2}+{x}^{2}}{2}}{I}_{0}\left(xt\right)\text{d}t.$

(16)

Also, (

*c*) is based on the fact that

$\xi \left(x,y\right)=1+{e}^{-\left({x}^{2}+{y}^{2}\right)/2}{I}_{0}\left(xy\right)-\xi \left(y,x\right)$

(17)

and finally, (*d*) is derived by using variable transform $t=\sqrt{x}$, partial integration, defining $\varphi \left(x,y\right)\doteq \xi \left(\sqrt{\frac{2x}{r}},\sqrt{\frac{2y}{r}}\right)$and some calculations.

*A simple average rate optimization algorithm:* In contrast to transmission power parameters, the power-limited average rate optimization problem of quantized CSI-based systems, e.g., (13), is not a convex optimization problem in terms of quantization parameters ${\u011d}_{i},\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{g}}_{i}$ ∀_{
i
}[34–38]. Therefore, although implementable, gradient-based algorithms are not efficient in determining the optimal quantization parameters. In order to tackle this problem, we propose an iterative algorithm, illustrated in Algorithm 1.

*Remark 2:* Similar to other techniques for solving non-convex optimization problems, it can not be guaranteed that the algorithm leads to the globally optimal solution for all channel conditions. However, by extensive testing, it is observed that for many different initial parameter settings and vector generation procedures, the algorithm leads to unique solutions. Furthermore, our experiments show that the algorithm is much more efficient than using greedy search scheme which requires a large number of initial random seeds due to the non-convexity of (13). Finally, although it may be time-consuming when the number of optimization parameters increases, the proposed algorithm has been shown to be efficient in many complex optimization problems dealing with local minima issues [42].

In the following, the channel average rate in the presence of perfect CSI available at the transmitter is studied and then the simulation results are presented in Section 5.