2 The MIMO system model.

2.1 Introduction

Multiple transmit and multiple receive antennas have emerged as a promising technique for improving the performance of wireless digital transmission systems [25, 58]. The limited resources of a wireless communication system, such as spectrum and power, can be efficiently used with multiple antennas to provide good quality and large capacity to a wide range of applications requiring high data rate.

Multiple antenna systems are described by a multiple-input multiple output (MIMO) system model, where the propagation environment is a quasi-static and frequency-flat Rayleigh fading channel [8]. This assumption is necessary to establish simple code design criteria. Nevertheless, the codes designed under this simplifying assumption yield good performance in a wide variety of real world scenarios.

Consider a system with [n.sub.t] transmit antennas and [n.sub.r] receive antennas. The complex baseband channel, within a single fading block of T symbol durations, can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

The subscripts indicate the corresponding matrix dimensions and will be omitted for simplicity in the following. The [h.sub.ij] element of the channel matrix H corresponds to the channel coefficient between the jth transmit and the ith receive antenna and it is modeled as a complex Gaussian random variable with zero mean and unit variance [N.sub.c],1).

The matrix Z corresponds to the spatially and temporally additive white noise, whose independent entries are complex Gaussian random variables [N.sub.c],(0, [N.sub.0]), where No is the noise power spectral density. The [X.sub.ik] entry of X corresponds to the signal transmitted from the ith antenna during the kth symbol interval for 1 [less than or equal to] k [less than or equal to] T. We let the time T coincide with the coherence time, i.e., the time during which the channel coefficients remain constant. It is also assumed that H is independent of both X and Z.

Let [E.sub.s] = E[|[[x.sub.ik]|[sup.2]] denote the signal energy transmitted from each antenna. We define the signal-to-noise ratio (SNR) at the receiver as

E [||HX||[sup.2]]/E[||Z||[sup.2]] = [n.sub.t][E.sub.s]/[N.sub.0] (2.2)

where ||*|| denotes the Frobenius norm of the matrix argument. For this kind of channels, the capacity at high SNR scales with min([n.sub.t],[n.sub.r]), [25]

C([n.sub.t], [n.sub.r],SNR) ~ min([n.sub.t],[n.sub.r])log(SNR). (2.3)

This means that by using appropriate processing, the additional spatial degrees of freedom (with respect to single transmit single receive antenna) allow the transmission of independent data flows through the channel and the separation of these flows at the receiver side. Equation (2.3) indicates how MIMO techniques enable the data rate of wireless systems to increase. In fact, we can observe that MIMO offers approximately min([n.sub.t], [n.sub.r]) parallel spatial channels between the transmitter and the receiver. Several schemes have been proposed to effectively exploit this spatial multiplexing [24, 61].

In the case of sufficiently spaced transmit and receive antenna arrays, the [n.sub.t][n.sub.r] channels between all pairs of transmit and receive antennas are independent. This suggests that MIMO can be also used to combat fading using diversity techniques, i.e., different independently faded replicas of the information symbols are sent over the independent channels and are available at the receiver side. In other words, a signal is lost only when all its copies are lost, resulting in higher immunity against channel fades.

A maximum diversity advantage of [n.sub.t][n.sub.r], corresponding to the number of channels between the transmitter and the receiver, can be achieved. In order to exploit the transmit diversity, several SpaceTime Block Codes (STBC) have been proposed in the literature [1, 12, 14, 26, 27, 33, 38, 40, 55]. Space-Time Trellis Codes (STTC) have also been extensively studied in the literature developing from [56]. In this work, we will focus on algebraic constructions of STBCs. Note that the terminology Space-Time Codes for multiple antennas codes comes from the fact that we are indeed coding over "pace" (since we have several antennas) and "time." The same codes can be applied over a multipath channel by swapping time with frequency and using a multicarrier modulation technique such as OFDM. In this setting these codes are known as Space-Frequency codes [9].

2.2 Design Criteria for Space-Time Codes

Design criteria for Space Time codes depend on the type of receiver that is considered. Two major classes of receivers have been considered in the literature: coherent and non-coherent. In the fist case, considered throughout this work, it is assumed that the receiver has recovered the exact information about the state of the channel (this is also known as perfect Channel State Information (CSI)). In practice this can be obtained by introducing some pilot symbols that enable accurate channel estimation, so that we can assume that the channel matrix H is known at the receiver. For the non-coherent case, many different solutions are available and we address the reader to [5, 29].

Definition 2.1. An STBC is a finite set C of [n.sub.t] x T complex matrices X and we denote by |C| the cardinality of the codebook.

Under the assumption of perfect CSI, maximum likelihood (ML) decoding corresponds to choosing the codeword X that minimizes the squared Frobenius norm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

An estimate of the error probability P(e) can be obtained using the union bound

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

where P(X [right arrow] X) is the pairwise error probability, i.e., the probability that, when a codeword X is transmitted, the ML receiver decides erroneously in favor of another codeword X, assuming only X and X are in the codebook. It can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

where Q is the Gaussian tail function, X - X is the codeword difference matrix and the average is over all realizations of H.

In the case of independent Rayleigh fading (h2j -N,(0,1)), we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

Let r denote the rank of the codeword difference matrix. If r = [n.sub.t] for all pairs (X, X), we say that the code is full rank. If we denote by [[lambda].sub.j], j = 1, ... , r the non-zero eigenvalues of the codeword distance matrix

A = (X - X) (X - X) [dagger] (2.7)

we can rewrite (2.6) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

For high signal-to-noise ratios (small No), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

where [DELTA]=[[product].sup.r.sub.j=1][[lambda].sub.j]. Then we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

In the union bound (2.4) the asymptotically dominant terms in the sum have the lowest exponent [rn.sub.r].

Definition 2.2. We call min{[rn.sub.r]} the diversity gain of the code, which represents the asymptotic negative slope of the error probability in a log-log scale plot versus SNR.

In the case of full rank codes (r = [n.sub.t]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and we say that the code has full diversity. This means that we can exploit all the [n.sub.t][n.sub.r] independent channels available in the MIMO system.

It is well known that the truncated union bound, taking into account only some of the terms in the sum (2.4), is not very accurate with fading channels. Nevertheless it provides a reasonably simple code design criterion if only the dominant term in the sum is considered. In the case of full diversity codes, the dominant term in the union bound (2.4) is given by the so called minimum determinant of the code,

[[DELTA].sub.min] = min det(A) X [not equal to] X (2.11)

The term ([[DELTA].sub.min])[sup.1/[n.sub.t]] is also known as the coding gain [56].

Definition 2.3. We define a linear STBC as an STBC C such that [for all]X,X' [member of] C X [+ or -] X' [member of] C.

This linearity property can only be true for an infinite code C[infinity] i.e., a code with an infinite number of codewords.

This definition parallels the one of lattice constellations carved from infinite lattices: a finite STBC can be carved from a linear STBC. We will see below that the performance analysis of these codes can be greatly simplified.

In the case of linear codes the sum or difference of any pair of codewords is a codeword, hence the union bound reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.12)

and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.13)

A finite STB code C [subset] C[infinity] has a minimum determinant [[DELTA].sub.min](C) [greater than or equal to] [[DELTA].sub.min](C[infinity]). In order to simplify the design problem, we will only consider linear infinite codes. Moreover, linearity implies a lattice structure and enables the application of the Sphere Decoder (see Section 2.4).

Remark 2.1. The 'pseudo-distance" [[DELTA].sub.min] is similar to the minimum Euclidean distance in the case of fnite constellations carved from infinite lattices.

In order to increase reliability, we will focus on full diversity linear codes with large minimum determinant [[DELTA].sub.min]