New Design Method: Information Outage Minimization

For a fixed rotation $S$ and $n_c$ fixed MIMO channel matrices $H_i$, $i=1 \ldots n_c$, defined by the $n_c$ fading blocks, let $\mathcal{I}_{SH}=I(z;y)$ denote the average mutual information of the equivalent channel with QAM input $z$ and complex output $y$ as in (1). The expression of $\mathcal{I}_{SH}$ is

$$\mathcal{I}_{SH} = s.m.n_t - \frac{1}{n_c} \sum_{i=1}^{n_c} E_{z,... ...ft( \frac{\sum_{z^{'}} p(y\vert z^{'},SH_i)} {p(y\vert z,SH_i)} \right) \right]$$ (6)

where $E_{z,y\vert SH_i}$ is the conditional mathematical expectation over $z$ and $y$. The channel likelihood is written in its classical form

$$p(y\vert z,SH) \propto \exp \left( -\frac{\Vert y-zSH\Vert^2}{2\sigma^2} \right)$$ (7)

Expression (6) assumes that the precoder $S$ does space-time spreading within the same fading block $H_i$. Its main role is to collect transmit diversity. Time diversity $n_c$ is collected by the convolutional code whereas receive diversity is naturally collected by the detector. The information rate transmitted by the space-time BICM is $R=s.m.n_t.R_c$ bits per $s$ time periods. An outage occurs if the instantaneous capacity, i.e. $\mathcal{I}_{SH}$ in our case, is less than $R$. The outage probability associated to the rotation $S$ at a given signal-to-noise ratio is

Figure 2: Outage limits for $n_t=n_c=s=2$, $n_r=1$, and $R_c=1/2$.

$$P_{out}(S) = P \left( \mathcal{I}_{SH} < s.m.n_t.R_c\right)$$ (8)

The new design, called IOM (Information Outage Minimization), selects a matrix $S_{IOM}$ within the ensemble $\aleph$ of random unitary matrices such that

$$S_{IOM}= \arg ~\min_{S \in \aleph} ~ P_{out}(S)$$ (9)

As an example, choosing the best rotation within an ensemble $\aleph$ limited to 2000 matrices yields the matrix written below, for QPSK alphabet with $n_t=s=2$ and coding rate $R_c=1/2$

$$S_{IOM} = \left[\begin{array}{cccc} 0.57e^{+j1.71} & 0.64e^{+j1.5... ...0.84} & 0.57e^{+j1.74} & 0.53e^{+j3.05} & 0.43e^{-j2.66}\\ \end{array}\right]$$

A smaller set $\aleph_G$ of random unitary matrices is obtained by adding to $\aleph$ the first genie constraint, i.e. orthogonal sub-rows in $S$. This second design, called G-IOM, selects a matrix $S_{G-IOM}$ satisfying

$$S_{G-IOM}= \arg ~\min_{S \in \aleph_G} ~ P_{out}(S)$$ (10)

As an example, choosing the best rotation within an ensemble $\aleph_G$ limited to 2000 matrices yields the matrix written below, for QPSK alphabet with $n_t=s=2$ and coding rate $R_c=1/2$

$$S_{G-IOM} = \left[\begin{array}{cccc} 0.88e^{-j0.30} & 0 & 0 & 0.... ...^{+j2.85} & 0\\ 0 & 0.88e^{+j2.96} & 0.47e^{-j1.49} & 0\\ \end{array}\right]$$

Figure 3: Outage limits for $n_t=n_c=s=2$, $n_r=1$, and $R_c=3/4$.

In a similar fashion, a DNA-IOM precoder minimizes the information outage and satisfies DNA constraints [10]. The matrix $S_{DNA}$ given below corresponds to $n_t=4$ and $s=2$. The DNA-IOM precoder is obtained by combining $S_{DNA}$ with $\xi_{DNA-IOM}$. Also, the DNA cyclotomic precoder is constructed by combining $S_{DNA}$ to $\xi_{DNA-Cyclo}=S_{Cyclo}$ given previously in (3).

$$S_{DNA} = \left[\begin{array}{cccccccc} \xi_{11} & \xi_{12} & 0 &... ... 0 & 0 & \xi_{41} & \xi_{42} & 0 & 0 & \xi_{43} & \xi_{44} \end{array}\right]$$

$$\xi_{DNA-IOM} = \left[\begin{array}{cccc} 0.73e^{-j0.81} & 0.22e^... ...+j1.01} & 0.78e^{+j3.49} & 0.51e^{+j2.27} & 0.20e^{+j0.91} \end{array}\right]$$

Figures 2 and 3 show the outage limit for different type of precoders in terms of Word Error Rate versus signal-to-noise ratio. The outage probability has been also evaluated for other system parameters. Surprisingly, algebraic precoders satisfying the first genie condition perform as good as G-IOM precoding. Cyclotomic rotation and Golden code are superimposed with G-IOM in Figure 2. All outage evaluations have been made by (6) and (8), without gaussian and analytical approximations when the channel input is a gaussian alphabet as in [18][20].