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Calderbank, Robert Reconstruction of Multi-User Binary Subspace Chirps

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Calderbank, Robert Reconstruction of Multi-User Binary Subspace Chirps This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Pllaha, Tefjol; Tirkkonen, Olav; Calderbank, Robert Reconstruction of Multi-user Binary Subspace Chirps Published in: 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings DOI: 10.1109/ISIT44484.2020.9174412 Published: 01/06/2020 Document Version Peer reviewed version Please cite the original version: Pllaha, T., Tirkkonen, O., & Calderbank, R. (2020). Reconstruction of Multi-user Binary Subspace Chirps. In 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings (pp. 531-536). [9174412] (IEEE International Symposium on Information Theory). IEEE. https://doi.org/10.1109/ISIT44484.2020.9174412 This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) Reconstruction of Multi-user Binary Subspace Chirps Tefjol Pllaha∗, Olav Tirkkonen∗, Robert Calderbanky ∗Aalto University, Finland, e-mails: {tefjol.pllaha, olav.tirkkonen}@aalto.fi yDuke University, NC, USA, e-mail: [email protected] Abstract—We consider codebooks of Complex Grassmannian In this paper, we expand on [13]. Based on the underlying Lines consisting of Binary Subspace Chirps (BSSCs) in N = 2m binary symplectic geometry, we provide a systematic way of dimensions. BSSCs are generalizations of Binary Chirps (BCs), looking at the reconstruction algorithm by making use of stabi- their entries are either fourth-roots of unity, or zero. BSSCs consist of a BC in a non-zero subspace, described by an on-off lizer states [14] and related notions in quantum computation. pattern. Exploring the underlying binary symplectic geometry, we This combines the binary subspace reconstruction discussed provide a unified framework for BSSC reconstruction—both on- in [13] and the BC reconstruction algorithm of [5] under off pattern and BC identification are related to stabilizer states of the same algebraic framework. Furthermore, we investigate the underlying Heisenberg-Weyl algebra. In a multi-user random BSSC decoding in true random access scenarios, where there access scenario we show feasibility of reliable reconstruction of multiple simultaneously transmitted BSSCs with low complexity. are multiple randomly selected users simultaneously accessing the channel. We provide a compressive sensing multi-user detection algorithm for L simultaneously accessing randomly I. INTRODUCTION selected users with complexity O(N log2+L N). We find nu- Codebooks of complex projective (Grassmann) lines, or merically that in a scenario where the channels of the randomly tight frames, have applications in multiple problems of inter- accessing users come from a continuous complex valued est for communications and information processing, such as fading distribution, this multi-BSSC reconstruction algorithm code division multiple access sequence design [1], precoding is capable of reliable multi-user detection. for multi-antenna transmissions [2] and network coding [3]. Contemporary interest in such codes arise, e.g., from deter- II. PRELIMINARIES ministic compressed sensing [4]–[8], virtual full-duplex com- munication [9], mmWave communication [10], and random A. The Binary Grassmannian G(m; r; 2) access [11]. In this paper, the main motivation will come A binary subspace H 2 G(m; r; 2) is the column space of from a random access scenario, in particular from a Massive some matrix HI in column reduced echelon form, where I ⊂ Machine Type Communication (MTC) scenario [12], where f1; : : : ; mg records the leading positions. The dual subspace the number of potentially accessing users may be extremely of H in G(m; m − r; 2) is the column space of HgI , with high, while a majority of devices may be stationary. In such T (HI ) HgI = 0. By II we will denote the m × r consisting scenarios, encoding and decoding complexity is of particular of the r columns of the identity matrix indexed by I. Put interest. To limit complexity and power consumption for MTC Ie := f1; : : : ; mg n I. Then, devices, it is important that a limited alphabet with small power variation is applied for transmission. Low decoding (I )TH = I ; (I )TI = 0; H I = I ; (1) I I r I Ie gI Ie m−r complexity is important for receiver implementation; complex- and H can be completed to an invertible matrix ity should not grow as a function of the number of codewords. I Codebooks of Binary Chirps (BCs) [5] provide an al- P := H I 2 GL(m; 2): (2) I I Ie gebraically determined set of Grassmannian line codebooks in N = 2m dimensions, with desirable properties; all en- The transposed inverse is given by tries are fourth root of unity and the minimum distance −T h i p P = II HgI : (3) is 1= 2. The number of codewords is reasonably large, I growing as 2m(m+3)=2, while single-user decoding complexity B. Bruhat Decomposition of the Symplectic Group is O(N log2 N). Recently in [13], we expanded the set of F2m Binary Chirps to Binary Subspace Chirps (BSSCs). Taking We first briefly describe the symplectic structure of 2 via the underlying binary symplectic geometry fully into account, the symplectic bilinear form T T complex Grassmannian line codebooks are created with entries h a; b j c; d is := b c + a d: (4) being either scaled fourth-roots of unity, or zero. Comparing p T to BCs, the minimum distance remains 1= 2, the number of A 2m×2m matrix F preserves h • j • is iff FΩF = Ω where codewords is ≈ 2:38 times larger, and a single-user decoder 0m Im 3 Ω = : (5) with complexity O(N log N) is provided. Im 0m T We will denote the group of all such symplectic matrices F where E(a; b) := ia bD(a; b): Here we view the binary with Sp(2m; 2). To proceed, we use the Bruhat decomposition vectors as integer vectors and the exponent is taken modulo of Sp(2m; 2) [15]. For P 2 GL(m; 2) and S 2 Sym(m; 2) 4. It follows that we distinguish two types of elements in Sp(2m; 2): aTb X vTb T E(a; b) = i (−1) ev+aev : (14) Fm P 0m Im S v2 2 FD(P) = −T and FU (S) = : (6) 0m P 0m Im Let S = E(A; B) ⊂ HWN be a maximal stabilizer, that Then every F 2 Sp(2m; 2) can be written as is, a subgroup of N commuting Pauli matrices that does not contain −IN , and put F = FD(P1)FU (S1)FΩ(r)FU (S2)FD(P2); (7) V (S) := fv 2 CN j Ev = v; 8 E 2 Sg: (15) where Im|−r Imjr FΩ(r) = ; (8) It is well-known (see, e.g., [16]) that dim V (S) = 1. A unit Imjr Im|−r vector that generates it is called stabilizer state, and with a with Imjr being the block matrix with Ir in upper-left corner slight abuse of notation is also denoted by V (S). Because we and 0 else, and Im|−r = Im − Imjr. We are interested in the are disregarding scalars, it is beneficial to think of a stabilizer right cosets in the quotient group Sp(2m; 2)=P, where P is state as a Grassmannian line, that is, V (S) 2 G(CN ; 1). the subgroup generated by products FD(P)FU (S). It follows III. CLIFFORD GROUP that a coset representative will look like The Clifford group in N dimensions is defined to be the U U FD(P)FU (S)FΩ(r); (9) normalizer of HWN in the unitary group (N) modulo (1): y for some rank r, invertible P, and symmetric S. However, two CliffN = fG 2 U(N) j GHWN G = HWN g=U(1): different invertibles P may yield representatives of the same F2m Let fe1;:::; e2mg be the standard basis of 2 , and consider coset. We make this precise below. F2m G 2 CliffN . Let ci 2 2 be such that Lemma II.1 ([13]). A right coset in Sp(2m; 2)=P is uniquely y GE(ei)G = E(ci): (16) characterized by a rank r, a m × m symmetric matrix Ser that has Sr 2 Sym(r) in its upper-left corner and zero else, and Then the matrix FG whose ith row is ci is a symplectic matrix Fm an r-dimensional subspace H in 2 . such that y T GE(c)G = E(c FG) (17) We will use the coset representative F2m for all c 2 2 . We thus have a group homomorphism FO(PI ; Sr) := FD(PI )FU (Sfr)FΩ(r); (10) Φ : CliffN −! Sp(2m; 2); G 7−! FG; (18) where PI as in (2) describes H. with kernel ker Φ = HWN [17]. This map is also surjective; see Section III-A where specific preimages are given. C. The Heisenberg-Weyl Group Remark III.1. Since Φ is a homomorphism we have m 2 y −1 y Fix N = 2 , and let fe0; e1g be the standard basis of C . that Φ(G ) = FG and as a consequence G E(c)G = Fm Fm E(cTF−1). For v 2 2 set ev := ev1 ⊗ · · · ⊗ evm . Then fev j v 2 2 g G N is the standard basis of C . The Pauli matrices are A. Decomposition of the Clifford Group 0 1 1 0 In this section we will make use of the Bruhat decomposi- I2; σx = , σz = , σy = iσxσz: 1 0 0 −1 tion of Sp(2m; 2) to obtain a decomposition of CliffN .
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