The DMT classification of real and quaternionic lattice codes

01/09/2018 ∙ by Laura Luzzi, et al. ∙ aalto ENSEA 0

In this paper we consider space-time codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also point out that lattice codes based on Q-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central division algebra codes that are based on standard embeddings.

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I Introduction

In [1] the authors proved that for every number of transmit antennas there exist a DMT optimal code in the space . These codes are derived from division algebras where the center of the division algebra is a complex quadratic field. However, this result is actually more general, and their proof revealed that as long as a -dimensional lattice code in has the non-vanishing determinant property (NVD), it is DMT optimal. Yet, this result does not tell us anything about space-time lattice codes that are not full dimensional in . Such codes naturally appear in the scenario where we have less receive than transmit antennas and try to keep the decoding complexity limited.

One natural class of such space-time codes are the codes derived from -central division algebras. In this paper we will measure their DMT. Unlike the case of complex quadratic center, -central division algebras are divided into two categories with respect to their DMT performance. This division is based on the ramification of the infinite Hasse-invariant of the division algebra, which decides if the lattice code corresponding to the division algebra can be embedded into real or quaternionic space.

Our DMT classification holds for any multiplexing gain, extending previous partial results in [2, 3] which were based on the theory of Lie algebras. We note that the approach used in this paper is quite different and more general. In the spirit of [1] we are not just considering division algebra codes, but all space-time codes where the code matrices are restricted to (resp. ), and provide two different upper bounds for the DMT of such codes. We then prove that if we have a degree -dimensional NVD lattice inside (resp. ) then this code achieves the respective upper bound. As the -central division algebra codes are of this type, we get their DMT as a corollary.

Ii Notation and preliminaries

Notation

Given a matrix , we denote its complex conjugate by , its transpose by and its conjugate transpose by .
We use the the dotted inequality to mean and similarly for equality.

Ii-a Subspaces and lattices

In this paper we will consider space-time codes that are subsets of certain subspaces of the

-dimensional real vector space

. The first such subspace consists of all the real matrices inside and we denote it with . The other subspace of interest consists of quaternionic matrices.

Let us assume that . We denote with the set of quaternionic matrices

where refers to complex conjugation and and are complex matrices in . Note that quaternionic matrices form a -dimensional subspace in .

The space-time codes we consider in this work are based on additive groups in .

Definition 1

A matrix lattice has the form

where the matrices are linearly independent over , i.e., form a lattice basis, and is called the dimension of the lattice.

We immediately see that if we have a lattice inside the space or the maximal dimension it can have is .

Definition 2

If the minimum determinant of the lattice is non-zero, i.e. satisfies

we say that the lattice satisfies the non-vanishing determinant (NVD) property.

Building high dimensional NVD lattices is a highly non-trivial task. A natural source of such lattices are division algebras. Let be a degree -central division algebra. We say that the algebra is ramified at the infinite place if . If it is not, then

Let be an order in .

Lemma 1

[2, Lemma 9.10] If the infinite prime is ramified in the algebra , then there exists an embedding

such that is a dimensional NVD lattice. If is not ramified at the infinite place, then there exists an embedding

such that is a dimensional NVD lattice.

Ii-B Channel model

We consider a MIMO system with transmit and receive antennas, and minimal delay . The received signal is

(1)

where is the transmitted codeword, and are the channel and noise matrices with i.i.d. circularly symmetric complex Gaussian entries , and is the signal-to-noise ratio (SNR). The set of transmitted codewords satisfies the average power constraint

(2)

We suppose that perfect channel state information is available at the receiver but not at the transmitter, and that maximum likelihood decoding is performed.

In the DMT setting [4], we consider codes whose size grows with the SNR, and define the multiplexing gain as

and the diversity gain as

where

is the average error probability.

Spherically shaped lattice codes

Let now be a lattice in . Given , consider the subset of elements whose Frobenius norm is bounded by :

Let be the dimension of as a -module. As in [2], we choose and consider codes of the form

which satisfy the power constraint (2). The multiplexing gain of this code is .

Iii Real lattice codes

In this section, we focus on the special case where , i.e. the code is a set of real matrices.

Iii-a Equivalent real channel

First, we show that the channel model (1) is equivalent to a real channel with transmit and receive antennas.
We can write , , where

have i.i.d. real Gaussian entries with variance

. If , with , we can write an equivalent real system with receive antennas:

(3)

where , have real i.i.d. Gaussian entries with variance .

Iii-B General DMT upper bound for real codes

Using the equivalent real channel in the previous section, we can now establish a general upper bound for the DMT of real codes.

Theorem 1

Suppose that , . Then the DMT of the code is upper bounded by the function connecting the points where .

Proof:

This part of the proof closely follows [4]. Given a rate , consider the outage probability [5]

(4)

where is the maximum mutual information per channel use of the real MIMO channel (3) with fixed and real input with fixed covariance matrix .111Unlike [5] and [4], we don’t use a strict inequality in the definition (4), but our definition is equivalent since the set of such that has measure zero. Following a similar reasoning as in [5, Section 3.2], it is not hard to see that

As in [4, Section III.B], since is increasing on the cone of positive definite symmetric matrices, for all such that we have and

Note that . Let , and . Let

be the nonzero eigenvalues of

. The joint probability distribution of

is given by [6]222We have slightly modified the expression to be consistent with our notation. In [6], the author considers a matrix where each element of is .:

(5)

for some constant . Consider the change of variables . The corresponding distribution for in the set is

(6)

Then we have

To simplify notation, we take . Note that , therefore

where

(7)

In fact, given , let be such that . Then ,
Consider . Then

where . The previous inequality follows from the fact that for , and if . (Note that for a fixed , there are possible values for such that .)

Lemma 2

Let . Then

where , , .

The proof of Lemma 2 can be found in Appendix -D.
Using Lemma 2 with , , we find that is equal to

This is the piecewise function connecting the points where .
Using the Laplace principle, we have

Note that , the point such that is in and when , . By continuity of ,

Iii-C DMT of real lattice codes with NVD

In this section, we show that real spherically shaped lattice codes with the NVD property achieve the DMT upper bound of Theorem 1. This result extends Proposition 4.2 in [3].

Theorem 2

Let be an -dimensional lattice in , and consider the spherically shaped code .
If has the NVD property, then the DMT of the code is the function connecting the points where .

Proof:

Since the upper bound has already been established in Theorem 1, we only need to prove that the DMT is lower bounded by . The following section follows very closely the proof in [1], and thus some details are omitted. To simplify notation, we assume that .
We consider the sphere bound for the error probability for the equivalent real channel (3): for a fixed channel realization ,

where is the squared minimum distance in the received constellation:

We denote . Let , and . Let be the non-zero eigenvalues of , and the eigenvalues of . Using the mismatched eigenvalue bound and the arithmetic-geometric inequality as in [1], for all

For all , , and

due to the NVD property. Consequently, for all

With the change of variables , we can write

where we have set and

(8)

To simplify the notation, we will take .
Since is a random variable, we have

Let be the distribution of in (6). Note that for , and for a fixed , there are possible values for . Consequently

(9)

where . By averaging over the channel, the error probability is bounded by

Finally, we get ,

(10)

where , and

(11)

The following Lemma is proven in Appendix -E:

Lemma 3

where is defined in (III-B).

The proof of the Theorem is concluded using Lemma 2 with , .

Fig. 1: DMT upper bounds for real (solid) and quaternion (dashed) codes for and . The dotted lines correspond to the optimal DMT.

Iv Quaternion lattice codes

Suppose that is even. We consider again the channel

(12)

and we suppose that the codewords are of the form

where .

Iv-a Equivalent quaternion channel

First, we derive an equivalent model where the channel has quaternionic form. We can write

where . Then

and we have the equivalent “quaternionic channel”:

Iv-B General DMT upper bound for quaternion codes

Theorem 3

Suppose that , . Then the DMT of the code is upper bounded by the function connecting the points for .

Proof:

The quaternionic channel can be written in the complex MIMO channel form

(13)

If is the multiplexing gain of the original system (12), then the multiplexing gain of this channel is , since the same number of symbols is transmitted using half the frame length.
Consider the eigenvalues of . Let the number of pairs of nonzero eigenvalues, and . For fixed , the capacity of this channel is [5]

The joint eigenvalue density of a quaternion Wishart matrix is [7]333The quaternion case corresponds to taking in [7, equation (4.5)]. Note that we modify the distribution to take into account the fact that each entry of has variance per real dimension.

for some constant . Considering the change of variables , the distribution of is

The output probability for rate is given by

where . Given , define . Then

where . Let . Using the Laplace principle, Using Lemma 2 with , , we find that is the piecewise linear function connecting the points for . Note that , the point such that is in and when , . By continuity of , .

Iv-C DMT of quaternionic lattice codes with NVD

We now show that quaternionic lattice codes with NVD achieve the upper bound of Theorem 3. This result extends Proposition 4.3 in [3].

Theorem 4

Let be an -dimensional lattice in , and consider the spherically shaped code . If has the NVD property, then the DMT of the code is the piecewise linear function connecting the points for .

Proof:

To simplify notation, assume . For a fixed realization , , where

Let . We denote by the eigenvalues of , and by the eigenvalues of . Both sets of eigenvalues have multiplicity since and are quaternion matrices. Again we set and .
Using the mismatched eigenvalue bound and the arithmetic-geometric inequality as in [1], we find that for all ,

As before, for all , , and using the NVD property of the code. Consequently, for all

With the change of variables , we have

where and