3 An information theoretic perspective.

In this chapter, we will see that two main features are important, to ensure the good performance of a coding scheme from an information theoretic point of view: (i) reaching the diversity-multiplexing gain trade-off and (ii) using information lossless codes. We will see that both these properties will correspond to other properties already required. Namely, the non-vanishing determinant property will be shown to be a sufficient condition to reach the diversity-multiplexing gain, and the cubic shaping will give information lossless codes. (1)

Historically, the first 2 x 2 Space Time code to achieve the diversity-multiplexing gain trade-off has been found by Yao and Wornell in [63], where they show that for a 2 x 2 code, having a minimum determinant bounded away from zero when the constellation size increases with SNR is a sufficient condition to reach the trade-off. This notion later on appeared to be similar to the non-vanishing determinant property introduced independently by Belfere and Rekaya [6]. In [20], the non-vanishing determinant property is shown to be in general a sufficient condition for division algebra codes to reach the trade-off.

3.1 Mutual Information of a Gaussian MIMO Channel

Let us start by considering a MIMO Gaussian channel characterized by a fired [n.sub.r] x [n.sub.t] complex matrix H = [h2j]. Recall that each term [h.sub.ij] is the complex attenuation factor between receive antenna i and transmit antenna j. At each symbol time, the received signal is the [n.sub.r]-dimensional vector

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

where x is the transmitted vector of dimension [n.sub.t] and z, which represents the noise, is a Gaussian vector with [n.sub.r] i.i.d. components.

Note that this is a particular realization of the original channel (2.1), where the coherence time is T = 1, and the channel matrix is fixed.

Theorem 3.1 [58]. Assume that the vector x has circularly complex Gaussian distributed components and H is deterministic. Then, the expression of the mutual information is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the identity matrix with dimension [n.sub.r], 9 2 is the variance of each real component of the noise z and Q is the covariance matrix of x,

Q = IE[xx[dagger]]. (3.3)

When the transmitter knows perfectly H, then it can optimize mutual information with water-filing [58, 59] which achieves

max I (x,y), Q, Tr(Q) [less than or equal to] [P.sub.x] (3.4)

where [P.sub.x] is the maximum power available at the transmitter.

We now consider the case where the receiver knows the channel, but this channel is random. Here, we follow Telatar in [58, p. 22] who conjectures that when the channel matrix is random, non-ergodic, then, in the high SNR region, the optimal covariance matrix for the source is [Q.sub.opt] = ([P.sub.x]/[n.sub.t]) * [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. By using this value of [Q.sub.opt], we deduce the value of mutual information

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

which corresponds to a strategy where the source transmits the signal isotropically (that is, its probability density function is invariant to multiplication by a unitary matrix).

3.2 The Outage Probability

The outage probability is a key concept in wireless communications. If we assume that the channel is not Eked but can be represented by a random matrix H, then the mutual information given by (3.5) becomes a random variable which is denoted by C(H). When we consider a quasistatic channel, we assume that the channel matrix remains constant during the transmission of a codeword, say of length T, as in (2.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Whenever the data rate R is lower than C(H), then it is possible to End a code which achieves an arbitrarily low error probability. But when C(H) < R, then we say that the channel is in outage. We define the outage probability as

Definition 3.1. The outage probability of a MIMO channel is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

The optimal covariance matrix depends on the SNR and on the rate R. The choice Q = ([P.sub.x]/[n.sub.t]) - In, is often used as it is a good approximation of the optimal covariance matrix. Since we are interested in the SNR exponent of the outage which is the same in both cases, we will use a definition of the outage probability when using an i.i.d. source,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7)

Definition 3.2. The diversity order of a channel is defined as the negative of the slope of the outage probability when plotted in a log-log scale versus SNR.

Similarly to this definition of the channel diversity, we can define the diversity order of a coding scheme,

Definition 3.3. The diversity order of a coding scheme is defined as the negative of the slope of the word error probability when plotted in a log-log scale versus SNR.

We note that the diversity order of the channel gives the maximum diversity gain achievable by any coding scheme operating over such channel.

3.2.1 The SISO Case

We consider the case where we have only one transmit and one receive antenna (Single Input Single Output), that is

y = hx + z.

Let h be a zero-mean Gaussian random variable with variance 1. The fading of the channel is thus assumed to be Rayleigh distributed, which means that |h|[sup.2] is exponentially distributed. The outage probability is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which gives at high SNR,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

We remark that the outage probability asymptotically decays as 1/SNR for a fixed rate. This channel has diversity order one.

3.2.2 Receive Diversity: The SIMO Case

In the Single Input Multiple Output case (SIMO), the receiver is assumed to be equipped with an antenna array in order to increase the spatial diversity order of the channel. Here, the transmitted vector is in fact a scalar and the channel is a column vector h with [n.sub.t] components,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The outage probability for the SIMO case is

[P.sup.SIMO.sub.out] (R) = Pr {[log.sub.2] (1 + SNR ||h||[sup.2] < R} (3.9)

which yields

[P.sup.SIMO.sub.out] (R) = Pr {||h||[sup.2] < [2.sup.R] - 1/SNR}. (3.10)

If we suppose that the channel coefficients are Gaussian zero-mean and spatially uncorrelated, then ||h||[sup.2] is a X-square distributed random variable with [2n.sub.t] degrees of freedom. Its probability density function (pdf) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11)

where [1.sub.S] is is the indicator function of the set S. Let E be an arbitrarily small positive real number. Then, by approximating ex by I for x small, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12)

By applying (3.12) to the expression of the outage probability at high SNR, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13)

Now, the outage probability asymptotically decays as [1/SNR.sup.n.sub.r], hence [n.sub.r] is the diversity order of this channel.

3.2.3 Transmit Diversity: The MISO Case

In the Multiple Input Single Output (MISO) case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the channel is a row vector h with [n.sub.t] components, which are assumed to be i. i. d. zero-mean Gaussian. The outage probability for the MISO case is

[P.sup.SIMO.sub.out] (R) = Pr {[log.sub.2] (1 + SNR/[n.sub.t] ||h||[sup.2] < R}. (3.14)

which yields

[P.sup.SIMO.sub.out] (R) = Pr {||h||[sup.2] < [n.sub.t]([2.sup.R] - 1)/SNR}. (3.15)

The same calculation as for the SIMO case yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.16)

enlightening a transmit diversity order equal to [n.sub.t].

3.2.4 The MIMO Case

The calculation of the outage probability for the MIMO case is more difficult than for the previous above cases, but we can start by intuitively explaining the behavior of the SNR exponent. We follow the method developed in [591. In the MIMO case, the channel matrix H is a [n.sub.r] x [n.sub.t] matrix with zero-mean Gaussian i.i.d. components. Let q = min {[n.sub.t], [n.sub.r]}. Then the outage probability is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.17)

where [[lambda].sub.i]s are the singular values of the matrix H. The MIMO channel exhibits q modes of transmission, each corresponding to an instantaneous SNR equal to (SNR?) /[n.sub.t]. How effective each mode is depends on how large the instantaneous SNR is. For large values of SNR, we say that mode i is effective if (SNR[[lambda].sup.2.sub.i]) /[n.sub.t] is of order SNR and not effective if (SNR[[lambda].sup.2.sub.i]) /[n.sub.t] is of order I or less. Consider (3.17), there is an outage event when none of the modes are effective. That means that all [[lambda].sup.2.sub.i]are of order I/SNR or less. Remark that

[q.summation over (i=1)] [[lambda].sup.2.sub.i] = Tr (HH[dagger]) = [summation over (i,j)] |[h.sub.ij]|[sup.2].

So there is an outage event when each |[h.sub.ij]|[sup.2] is of order I/SNR or less. Since all |[h.sub.ij]|[sup.2] are independent and Pr{|h.sub.ij]|[sup.2] < I/SNR} [approximately equal to] 1/SNR, the outage probability is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.18)