Nonasymptotic Probability Bounds for Fading Channels Exploiting Dedekind Zeta Functions

In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper’s correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algebraic number theory before, they are now brought into novel practice in the context of fading channel communications. Hence, the interest here is in small-dimensional lattices and finite constellations rather than in the asymptotic behavior. Index Terms Bounded height, Dedekind zeta function, lattices, norm forms, number fields, PEP, theta series, union bound, unit group, wiretap channel. I. BACKGROUND AND RELATED WORK It has been well known for many years that number field lattice codes provide an efficient and robust means for many applications in wireless communications. We refer to [2] for a thorough introduction to the topic. More recently, wiretap channels and number field based codes have been under study. Gaussian and fading wiretap channels have been considered for example in [3], [4], [5], [6], [7]. In [8] the authors propose to use number field lattice codes, which will also form the basis for our study and constructions. This paper can be seen, on one hand, C. Hollanti and D. Karpuk are with the Department of Mathematics and System Analysis, P.O. Box 11100, FI-00076 Aalto University, Finland (e-mails: [email protected], [email protected]). E. Viterbo is with the Monash University, Australia (e-mail: [email protected]). The research of D. Karpuk and C. Hollanti was partly supported by the Emil Aaltonen Foundation’s Young Researcher’s Project, and by the Academy of Finland grant #131745. This research was partly carried while C. Hollanti was visiting E. Viterbo at the Monash University in 2011. Part of this work was performed at the Monash Software Defined Telecommunications Lab and was supported by the Monash Professional Fellowship and the Australian Research Council under Discovery grants ARC DP 130100103. Part of the results in Section IV were presented at ICUMT 2011 [1]. AMS Classifications 14G50, 14G25. 2 as a continuation of [9], [1], where analysis on lattice codes in fast and block fading channels was carried out based on various explicit code constructions and, on the other hand, of [10], [11], where Vehkalahti and Lu showed how the unit group and diversity-multiplexing gain trade-off (DMT) of division algebra-based space-time codes are linked to each other through inverse determinant sums, and also demonstrated the connection to zeta functions and point counting. As continuation to [10], [11], the authors later showed that the density of unit group completely determines the growth of the inverse determinant sum, see [12], [13]. Our work differs from [13] in that it is targeted towards practical SNR region and small delay rather than to asymptotic behavior. In addition to the pairwise error probability case, we apply the same methods to the wiretap channel, where an eavesdropper is trying to intercept the data. While in [13] the authors concentrate on the number of units in a finite spherical subset of a lattice (as this is known to be the dominating factor), here we bound each individual term in the norm sum. This will enable us to estimate the number of points with each norm within a hyper-cubic constellation resulting in finer probability bounds. In summary, the contributions of this paper lie within • deriving (nonasymptotic) bounds for the probability expressions related to Rayleigh fading channels and to (Rayleigh fading) wiretap channels by using Dedekind zeta functions, • finding more accurate bounds through geometric analysis based on the unit lattice, Dedekind zeta functions, and bounded height norm sums, • deriving an estimate to the number of elements with certain norm as a byproduct, • demonstrating the accuracy of the estimate through numerical examples and showing that the estimation error is very small for small dimensions. While bounded height (cf. Def. 5) estimates of the same type as the ones derived in this paper are known in algebraic number theory [14], [15], they are far from being standard or well-known. Hence, we hope that our derivation and then the practical use of such estimates will boost further research leading to yet tighter bounds. One should keep in mind that in general the estimation error is not negligible when the lattice dimension grows. However, as we apply these estimates in the situation where the lattice dimension corresponds to the decoding delay, the error is less severe. We will show that the estimate is very good when the dimension is relatively low and hence the delay short. Notice that the lattice dimension is not limiting the data rate as we can always increase the constellation size by choosing a bigger hyper-cube. Actually, the bigger the cube, the better our estimate will be, since the edge error effect becomes more negligible. Another limitation to the lattice dimension is forced upon by decoding – it is known that the complexity of any ML decoder such as a sphere decoder grows exponentially in the lattice dimension. We refer the reader to [14], [15] for more details regarding norm forms and the treatment of related error terms. We also point out the similarity of the constant term obtained here to the one in [15], which well demonstrates the accuracy of our derivation. The rest of the paper is organized as follows. In Section II, we provide some algebraic preliminaries related to number field lattice codes. Section III shortly introduces lattice coset coding employed in wiretap communications 3 and revisits the probability expression for the fast fading wiretap channel as well as for the typical Rayleigh fading channel, finally unifying the treatment of both expressions as one to simplify the computations in the rest of the paper. In Section IV, first bounds are derived using Dedekind zeta functions. We then refine the bounds in Section V, where geometric analysis is carried out in order to estimate the number of constellation points with certain algebraic norm. The accuracy of the estimate in low dimensions is demonstrated in Section VI. Conclusions are provided in Section VII. II. ALGEBRAIC PRELIMINARIES Lattices will play a key role throughout the paper, so let us first recall the notion of a lattice. For our purposes, a lattice Λ is a discrete abelian subgroup of a real vector space, Λ = Zβ1 ⊕ Zβ2 · · · ⊕ Zβt ⊂ R, where the elements β1, . . . , βt ∈ Rn are linearly independent, i.e., form a lattice basis, and t ≤ n is called the rank of the lattice. Here, we consider full (t = n) totally real lattices arising from algebraic number fields (see Def. 3 below). For more details on lattices and lattice codes, refer to [2]. Definition 1: The minimum product distance of a lattice Λ is dp,min(Λ) = min 06=x∈Λ n ∏ i=1 |xi|, where x = (x1, . . . , xn) ∈ Λ. Definition 2: Let K be a number field. A real embedding is a field homomorphism σ : K →֒ R. A complex embedding is a field homomorphism σ : K →֒ C such that σ(K) ( R. Definition 3: Let K/Q be a totally real number field extension of degree n and σ1, . . . , σk its embeddings to R. Let OK denote the ring of integers in K. The canonical embedding ψ : K →֒ Rn defines a lattice Λ = ψ(OK) in Rn: ψ(x) = (σ1(x), . . . , σk(x)) ∈ ψ(OK) ⊂ R, where x ∈ OK . Definition 4: The algebraic norm of x ∈ K is defined as NK/Q(x) = n ∏ i=1 σi(x). We abbreviate N(x) = NK/Q(x) whenever there is no danger of confusion. If x ∈ OK , then NK/Q(x) ∈ Z. Hence, we have that dp,min(ψ(OK)) = min 06=x∈OK |NK/Q(x)| = 1. In what follows, a cubic constellation will be used, bounding the size of the vector components in the canonical embedding. To this end, we define the height of an algebraic integer as follows. 4 Definition 5: The height of x ∈ OK is H(x) = max 1≤i≤n |σi(x)|. We extend this notation to the height of a principal ideal I = (x), x ∈ OK , in a natural way by identifying an ideal I with its minimum-height generating element and simply defining H(I) = min{H(y) | I = (y)}. Let us denote by (r1, r2) the signature of K, i.e., [K : Q] = n = r1 + 2r2, where r1 is the number of real embeddings K →֒ R and r2 is the number of conjugate pairs of imaginary embeddings K →֒ C. The group O K of units of OK is described by the following well-known theorem, repeated here for the ease of reading. Theorem 1: ([16, Dirichlet Unit Theorem 1.9]) Let K be a number field and let (r1, r2) be the signature of K. There are units ǫ1, . . . , ǫr1+r2+1 ∈ O K such that O K ∼= WK × 〈ǫ1〉 × · · · × 〈ǫr1+r2−1〉 ∼= WK × Z12, where WK is the group of roots of units in K. The ǫj are called a fundamental system of units for K. The fundamental units are used for defining the regulator of K. Let {ǫ1, . . . , ǫr} be a fundamental system of units for K, where r = r1 + r2 − 1. Consider a matrix A = (log |σj(ǫi)|j) for 1 ≤ i ≤ r and 1 ≤ j ≤ r1 + r2, and where we have used the notation |x|j = 