Simple precoding algorithms using Gram–Schmidt orthonormalization process for multiuser relay communications with optimized power allocation

Abstract

This paper proposes a new simple precoding solution based on the Gram–Schmidt orthonormalization to be used at the relay station of a multirelay wireless networks where the different mobile stations belong to the same network, in order to mitigate the multiuser interference at each mobile station. The strength of this method is that it only requires the knowledge of all channel impulse responses from a given relay to all the mobile stations. In other words, to compute its precoding vectors, each relay does not need to know the channel impulse responses of the channels of other relays. Unlike the centralized reference method where each mobile station benefits from the same diversity gain, using this algorithm, some mobile stations will improve their diversity gain at the cost of a loss in the diversity gain of other users. This constitutes a simple solution to supply different qualities of service in the case of a multiservices network. Furthermore, this work proposes an optimized power allocation between the relays. Analytical and accurate performance analyses for the different studied contexts are provided.

Appendices

Appendix 1: Proof of the characteristics of $\lambda^2_{k,i}$

Using Eq. 33, we can obtain the characterization of $\lambda _{k,i}^{2}$ by mathematical induction

• For i = 1, we have $\mathbf e_{k,1}=\mathbf{h}_{k1} /\left\| \mathbf{h}_{k1}\right\|$ thus

  $$ \label{eq:4-GS-lambda-i1} \lambda_{k,1}^{2} =\left|\left\langle \mathbf{h}_{k1},\mathbf e_{k,1} \right\rangle \right|^{2} =\left|\left\langle \mathbf{h}_{k1},\frac{\mathbf{h}_{k1}}{\left\| \mathbf{h}_{k,1}\right\| } \right\rangle \right|^{2} =\left\| \mathbf{h}_{k1} \right\| ^{2}. $$

  (92)

  Since h _k1 is a vector of R complex Gaussian random components each of which with zero mean and a variance equal to 0.5, the random variable $\lambda _{k,1}^{2}$ may be written as

  $$ \label{eq:4-GS-lambda-i1-2} \lambda _{k,1}^{2} =\sum\limits_{i=1}^{R}\left|h_{k1}(i)\right|^{2} =\sum\limits_{i=1}^{R}\left[\left(h_{k1}^{R} (i)\right)^{2} +\left(h_{k1}^{I} (i)\right)^{2} \right]. $$

  (93)

  It is straightforward to conclude that $\lambda _{k,1}^{2}$ is a chi-square variable with 2R degrees of freedom.

• For i = 2, we have

  $$ \label{eq:4-GS-E-i2} \mathbf e_{k,2}=\frac{\mathbf{h}_{k2} -\left\langle \mathbf e_{k,1} ,\mathbf{h}_{k2} \right\rangle\mathbf e_{k,1}}{\left\| \mathbf{h}_{k2} -\left\langle \mathbf e_{k,1} ,\mathbf{h}_{k2} \right\rangle \mathbf e_{k,1} \right\|}. $$

  (94)

  Hence, we obtain

  $$\begin{array}{lll} \label{eq:4-GS-lambda-i2-1} \lambda _{k,2}^{2} &=&\left|\left\langle \mathbf{h}_{k2} ,\mathbf e_{k,2} \right\rangle \right|^{2}\\ & =&\left|\left\langle \mathbf{h}_{k2},\frac{\mathbf{h}_{k2} -\left\langle \mathbf e_{k,1} ,\mathbf{h}_{k2} \right\rangle \mathbf e_{k,1}}{\left\| \mathbf{h}_{k2} -\left\langle \mathbf e_{k,1} ,\mathbf{h}_{k2} \right\rangle \mathbf e_{k,1} \right\|} \right\rangle \right|^{2} \\ &=&\mu _{k,2}^{2}\left(\left\langle \mathbf{h}_{k2} ,\mathbf{h}_{k2} \right\rangle -\left|\left\langle \frac{\mathbf{h}_{k1}}{\left\| \mathbf{h}_{k1} \right\|} ,\mathbf{h}_{k2}\right\rangle \right|^{2} \right)\\ &=&\mu _{k,2}^{2}\left(\left\langle \mathbf{h}_{k2} ,\mathbf{h}_{k2} \right\rangle -\frac{\left|\left\langle \mathbf{h}_{k1} ,\mathbf{h}_{k2} \right\rangle \right|^{2} }{\left\| \mathbf{h}_{k1} \right\| ^{2} }\right). \end{array} $$

  (95)

  The term $\left|\left\langle \mathbf{h}_{k1} ,\mathbf{h}_{k2} \right\rangle \right|^{2}$ can be calculated as

  $$ \label{eq:4-GS-lambda-i2-2} \begin{array}{lll} \left\langle \mathbf{h}_{k1} ,\mathbf{h}_{k2}\right\rangle& =\sum\limits_{i=1}^{R}\left[h_{k1}^{R} (i)+j h_{k1}^{I} (i)\right]\left[h_{k2}^{R} (i)+jh_{k2}^{I} (i)\right] \\ &=\sum\limits_{i=1}^{R}\left[h_{k1}^{R} (i)h_{k2}^{R} (i)- h_{k1}^{I} (i)h_{k2}^{I} (i)\right]\\ &{\kern10pt}+j\sum\limits_{i=1}^{R}\left[h_{k,1}^{R} (i).h_{k,2}^{I} (i)+h_{k,1}^{I} (i).h_{k,2}^{R} (i)\right]. \end{array} $$

  (96)

  And this yields

  $$ \label{eq:4-GS-lambda-i2-3} \begin{array}{lll}\frac{\left|\left\langle \mathbf{h}_{k1},\mathbf{h}_{k2}\right\rangle \right|^{2} }{\left\| \mathbf{h}_{k1} \right\| ^{2} } &=\frac{\sum_{i=1}^{R}\left[h_{k1}^{R} (i)h_{k,2}^{R} (i)- h_{k1}^{I} (i)h_{k2}^{I} (i)\right]^{2}}{\sum_{i=1}^{R}\left[\left|h_{k1}^{R} (i)\right|^{2} +\left|h_{k1}^{I} (i)\right|^{2} \right] } \\ &{\kern10pt}+\frac{\sum _{i=1}^{R}\left[h_{k1}^{R} (i)h_{k2}^{I} (i)+h_{k1}^{I} (i)h_{k2}^{R} (i)\right]^{2} }{\sum _{i=1}^{R}\left[\left|h_{k1}^{R} (i)\right|^{2} +\left|h_{k1}^{I} (i)\right|^{2} \right] }. \end{array} $$

  (97)

  Taking the mean of this expression and using the same approximation as in [8],

  $$ \label{eq:4-GS-lambda-i2-approx} \mathbb{E}\left\{{\frac{\left|\left\langle \mathbf{h}_{k1}^{} ,\mathbf{h}_{k2}^{} \right\rangle \right|^{2} }{\left\| \mathbf{h}_{k1}^{} \right\| ^{2} } }\right\}\approx \frac{\mathbb{E}\left\{{\left|\left\langle \mathbf{h}_{k1}^{} ,\mathbf{h}_{k2}^{} \right\rangle \right|^{2}}\right\}}{\mathbb{E}\left\{{\left\| \mathbf{h}_{k1}^{} \right\| ^{2}}\right\}}. $$

  (98)

  Using the approximation in Eq. 98 after some mathematical developments, we get $\mathbb{E}\left\{{\frac{\left|\left\langle \mathbf{h}_{k1}^{} ,\mathbf{h}_{k2}^{} \right\rangle \right|^{2} }{\left\| \mathbf{h}_{k1}^{} \right\| ^{2} } }\right\}=1$. This entails that the random variable $\lambda _{k,2}^{2}$ is a chi-square variable with 2(R − 1) degrees of freedom.

• As the last step of the induction, let us assume that the random variable $\lambda _{k,i-1}^{2}$ is a chi-square random variable with $2\left[R-(i-2)\right]$ degrees of freedom. We search to obtain the distribution of $\lambda _{k,i}^{2}$. We have

  $$ \label{eq:4-GS-lambda-i3-1} \begin{array}{lll} \lambda _{k,i}^{2} &=\,\mu _{k,i}^{2} \left[\,\left\langle \mathbf{h}_{ki}^{} ,\mathbf{h}_{ki}^{} \right\rangle -\sum\limits_{j=1}^{i-1}\left|\left\langle \mathbf e_{k,j}^{} ,\mathbf{h}_{ki}^{} \right\rangle \right| ^{2} \,\right]\\ &=\,\mu _{k,i}^{2}\left[\,\left\langle \mathbf{h}_{ki}^{} ,\mathbf{h}_{ki}^{} \right\rangle -\sum\limits_{j=1}^{i-2}\left|\left\langle \mathbf e_{k,j}^{} ,\mathbf{h}_{ki}^{} \right\rangle \right| ^{2} -\left|\left\langle \mathbf e_{k,i-1}^{} ,\mathbf{h}_{ki}^{} \right\rangle \right|^{2} \,\right]\,. \end{array} $$

  (99)

  Considering the hypothesis on $\lambda _{k,i-1}^{2}$, we know that $\left\langle \mathbf{h}_{ki}^{} ,\mathbf{h}_{ki}^{} \right\rangle -\sum _{j=1}^{i-2}\left|\left\langle \mathbf e_{k,j}^{} ,\mathbf{h}_{ki}^{} \right\rangle \right| ^{2}$ is a chi-square variable with $2\left[R-(\textit{i}-2)\right]$ degrees of freedom. Subtracting the quantity $\left|\left\langle \mathbf e_{k,i-1}^{} ,\mathbf{h}_{ki}^{} \right\rangle \right|^{2}$ and using the same method as the case i = 2, we can prove that $\lambda _{k,i}^{2}$ is a chi-square random variable with $2\left[R-(i-1)\right]$ degrees of freedom.

We proved that if the approximation in Eq. 98 is verified, the random variable $\lambda _{k,i}^{2}$ is a chi-square random variable with $2\left[R-(i-1)\right]$ degrees of freedom.

Appendix 2: Expectation–maximization algorithm

We want to approximate the pdf of random variables $\lambda_{k,i}^2$. Based of the demonstration given under Appendix 1, we know that $\lambda_{k,i}^2$ is a chi-square distributed random variable. Knowing that the chi-square distribution is a special case of gamma distribution

$$ \label{eq:A-Gamma} \lambda _{k,i}^{2}(x)\sim g(\alpha,\beta,x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, $$

(100)

we will use the EM algorithm to calculate the values of α and β. EM is an iterative algorithm which alternates between performing an expectation step and a maximization step. Given a likelihood function $L(\boldsymbol{\theta}; \mathbf{x})$, where $\boldsymbol{\theta}$ is the parameter vector (α and β for a gamma distribution) and x is the n samples of observed data, the EM algorithm seeks to find the Maximum likelihood estimate (MLE) of the parameters by iteratively applying the following two steps:

• Expectation step: Calculate the expected value of the log-likelihood function given x under the current estimate of the parameters $\boldsymbol{\theta}^{(t)}=\left\{\alpha,\beta\right\}$. The likelihood function $L(\boldsymbol{\theta};\mathbf{x},\mathbf{z})$ is related to the probability density function by

  $$\label{eq:3-Likelihood1} L(\boldsymbol{\theta};\mathbf{x})=p(\mathbf{x}|\boldsymbol{\theta})=\prod\limits_{i=1}^n\frac{\beta^{\alpha}}{\Gamma(\alpha)}x_i^{\alpha-1}e^{-\beta x_i}. $$

  (101)

  The log-likelihood function is then determined by

  $$\begin{array}{lll}\label{eq:3-logLikelihood1} \log L(\boldsymbol{\theta};\mathbf{x})= \sum\limits_{i=1}^n && \left[ \alpha \log\beta-\log\Gamma(\alpha)\right.\\ &&{\kern3pt}\left.+\,(\alpha-1)\log x_i-\beta x_i\right]. \end{array} $$

  (102)

  Since we are considering a single gamma distribution (and not a mixture of gamma distributions), there is no hidden variable, thus $\mathbb{E}\left\{{\log L(\boldsymbol{\theta};\mathbf{x})}\right\}=\log L(\boldsymbol{\theta};\mathbf{x})$. Finally the expectation step then yields

  $$\begin{array}{lll}\label{eq:3-E_step} &&Q(\theta|\theta^{(t)}) \triangleq \mathbb{E}\left\{{\log L(\theta;\mathbf{x})}\right\} \\ &&\sum\limits_{i=1}^n \left[\alpha \log\beta-\log\Gamma(\alpha)+(\alpha-1)\log x_i-\beta x_i\right].\\ \end{array} $$

  (103)

• Maximization step: Find the parameters which maximize the quantity Q(θ|θ ^(t)). In order to maximize Q(θ|θ ^(t)), the partial derivatives of this function must be calculated:

  $$\begin{array}{lll}\label{eq:3-Beta_max} \frac{\partial Q(\theta|\theta^{(t)})}{\partial\beta}&=&0 \Longrightarrow \sum\limits_{i=1}^n\left(\alpha/\beta-x_i\right)\\&=&0\Longrightarrow \alpha=\beta\frac{\displaystyle\sum\limits_{i=1}^nx_i}{n}=\beta\bar{x_i} \end{array} $$

  (104)

  and

  $$\label{eq:3-alpha_max} \begin{array}{lll} \frac{\partial Q(\theta|\theta^{(t)})}{\partial\alpha}&\,=\,0 \Longrightarrow\,\sum\limits_{i=1}^n\,\left[\,\log \beta - \frac{d\left(\log\Gamma(\alpha)\right)}{d\alpha}+\log x_i\,\right]\\ &=0\Longrightarrow\,\log\beta=\Psi(\alpha)-\frac{\displaystyle\sum\limits_{i=1}^n\log x_i}{n} \end{array} $$

  (105)

  where Ψ(x) denotes the digamma function (i.e., the polygamma function of order 0): $\Psi(x)\equiv\psi_0(x)=\frac{\Gamma'(x)}{\Gamma(x)}$.

  Substituting the value of β from Eq. 105 into Eq. 104, we obtain

  $$\label{eq:x_fx} \alpha=\beta\bar{x_i}=\exp\left[\Psi(\alpha)-\frac{\displaystyle\sum\limits_{i=1}^n\log x_i}{n}\right]\bar{x_i} $$

  (106)

  which is clearly of form α = f(α) and thus can be calculated using the well-known Newton–Raphson method. The simulation results confirm that the gamma distribution obtained from this method fits the Monte Carlo results.