Wireless Communications and Mobile Computing

Volume 2017 (2017), Article ID 9207108, 16 pages

https://doi.org/10.1155/2017/9207108

## Efficient Offline Waveform Design Using Quincunx/Hexagonal Time-Frequency Lattices

^{1}MEDIATRON Laboratory, Sup’Com, University of Carthage, 2083 Ariana, Tunisia^{2}LETI Laboratory, ENIS, 3038 Sfax, Tunisia

Correspondence should be addressed to Raouia Ayadi

Received 29 June 2017; Revised 16 October 2017; Accepted 5 November 2017; Published 26 November 2017

Academic Editor: Milos Tesanovic

Copyright © 2017 Raouia Ayadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Conventional orthogonal frequency division multiplexing (OFDM) may turn to be inappropriate for future wireless cellular systems services, because of extreme natural and artificial impairments they are expected to generate. Natural impairments result from higher Doppler and delay spreads, while artificial impairments result from multisource transmissions and synchronization relaxation for closed-loop signaling overhead reduction. These severe impairments induce a dramatic loss in orthogonality between subcarriers and OFDM symbols and lead to a strong increase in intercarrier interference (ICI) and intersymbol interference (ISI). To fight against these impairments, we propose here an optimization of the transmit/receive waveforms for filter-bank multicarrier (FBMC) systems, with hexagonal time-frequency (TF) lattices, operating over severe doubly dispersive channels. For this, we exploit the Ping-pong Optimized Pulse Shaping (POPS) paradigm, recently applied to rectangular TF lattices, to design waveforms maximizing the signal-to-interference-plus-noise ratio (SINR) for hexagonal TF lattices. We show that FBMC, with hexagonal lattices, offers a strong improvement in SINR with respect to conventional OFDM and an improvement of around 1 dB with respect to POPS-FBMC, with rectangular lattices. Furthermore, we show that hexagonal POPS-FBMC brings more robustness to frequency synchronization errors and offers a 10 dB reduction in out-of-band (OOB) emissions, with respect to rectangular POPS-FBMC.

#### 1. Introduction

OFDM is now a well-established technique that provides high-data-rate wireless communications through a transformation of the frequency-selective channel into several nonselective subchannels, thereby reducing the ISI [1]. For all standardized systems, OFDM uses a rectangular waveform, in order to ensure maximum spectral efficiency, while maintaining orthogonality between the different shifts of the used waveform in the TF plane. Unfortunately, in practice, the mobile radio channel is highly TF dispersive, causing an orthogonality loss between the TF shifted versions of this rectangular waveform. By adding a cyclic prefix (CP), the conventional OFDM system becomes less sensitive to delay spread and time synchronization errors, at the cost of a spectrum efficiency loss. However, in the presence of Doppler spread and frequency synchronization errors, the bad frequency localization of the rectangular waveform leads to a very important ICI, in both downlink and uplink channels. Moreover, at the uplink channel, guard bands between users accessing the spectrum resources are required because of the misalignment nature of multiple access and the strong out-of-band (OOB) power leakages between adjacent user bands, leading to an inefficient use of spectrum resources.

With respect to LTE-A, future wireless cellular systems are expected to support new applications and services, such as Tactile Internet, the Internet of things (IoT), and machine type communications (MTC) [2–4]. In order to reduce latency and efficiently use energy and radio spectrum resources for all these applications and services, it is necessary to alleviate the synchronization mechanism overhead for sporadic and small packet transmissions. Here, synchronization alleviation means the reduction of the signaling load by tolerating large synchronization errors. Unfortunately, decreasing signaling by synchronization alleviation or relaxation introduces artificial impairments, caused by timing and carrier frequency offset shifts. In addition, other extra artificial impairments, caused by delay spread and multisource transmissions, such as a CoMP, MBMS, and cloud-radio access networks (C-RAN), are expected to be prevalent. These delay and frequency spreads cannot be tolerated and endured by conventional OFDM, leading to either a dramatic loss in spectral efficiency or a strong decrease in SINR. To face all these impairments, a plethora of research activities have focused on waveform design. Waveform design is precisely the concern for this paper, which proposes a new technique to optimize the transmit/receive waveforms.

##### 1.1. Related Works on Waveform Design

Waveform design dates back to the work of Le Floch et al. in [5] and Haas and Belfiore [6] in 1995. In [5], a transformation of the Gaussian waveform into an orthogonal waveform with slightly worse localization was proposed as the so-called Isotropic Orthogonal Transform Algorithm (IOTA) approach. In [6], the most localized Hermite functions including the Gaussian waveform were combined in order to obtain a partially orthogonal and well TF-localized waveform. Since then, the problem of waveform optimization has been extensively investigated. The first series of research works focused on continuous-time waveform optimization. In [7], the authors proposed a method for orthogonalizing the Gaussian waveform and introduced an optimization of the TF lattices for TF dispersive channels, in order to minimize ICI/ISI. They concluded that the use of hexagonal lattices outperforms the use of rectangular ones. However, the continuous-time proposed waveforms are orthogonalized but not well TF-localized [8], leading to lower robustness to TF dispersive channels. This motivated some subsequent research works to abandon the strict orthogonality for the modulated waveforms and to focus on the good localization of these waveforms in TF plane. In [9], the authors have derived a TF well-localized continuous-time waveform for hexagonal lattices, by minimizing the mean power of ISI and ICI. However, they overlooked the mean power of the useful signal leading to a reduction in SIR. In the aforementioned works, the same waveform is considered at transmission and reception sides. To give an additional degree of freedom in waveform design, some research works proposed to consider different waveforms at transmission and reception. In [10], the authors presented two methods for optimal continuous-time design of the transmit and receive waveforms of BFDM systems over dispersive channels based on ISI/ICI-minimizing. Stojanovic et al. derived, in [11], TF well-localized waveforms for BFDM systems based on a quasi-Newton method with line search for the minimization of the interference power. In [12], based on SINR maximization, we proposed a novel approach to design continuous-time nonorthogonal well-localized transmit/receive waveforms for rectangular lattices. The proposed waveforms, which are expressed as linear combinations of the most TF-localized Hermite waveforms, demonstrated an extra robustness to doubly dispersive channels. The main drawbacks behind continuous-time optimization are significant optimization complexity and performance degradation following discretization, subsequently required in the system implementation stage. Accordingly, in more than one respect, a single-step discrete-time waveform design is more advantageous than a two-step continuous-time design. The discrete-time design is, therefore, an active area of research for either OFDM or filter-bank multicarrier (FBMC) systems. Although the theory of FBMC has a long history close to OFDM [13], it has recently been considered as a promising waveform for 5G systems, to enhance spectral efficiency with respect to the conventional CP-OFDM thanks to CP-less transmission and guard-band reduction [14]. In [15], Siala et al. optimized the discrete-time nonorthogonal waveforms, for rectangular lattices, operating over doubly dispersive channels. For this, they exploit the POPS paradigm based on SINR maximization. In [16], the authors proposed an optimization design of QAM-FBMC waveform that provides superior spectrum confinement and higher spectral efficiency compared to conventional OFDM. Furthermore, recent research projects, such as PHYDYAS [17] and 5GNOW [4], thoroughly studied FBMC in the 5G applications framework. They also considered some other waveforms suitable for 5G applications, such as generalized frequency division multiplexing (GFDM) [18] and universal filtered multicarrier (UFMC) [19]. These projects proposed off-the-shelf waveforms obtained empirically in order to reduce secondary lobes and therefore to decrease OOB emissions. These waveforms meet the requirements for which they have been designed, with no guarantee for the provision of good SINR over strongly dispersive channels.

##### 1.2. Work Motivation and Contributions of the Paper

This paper is concerned with the optimization of FBMC waveforms for hexagonal TF lattices, accounting for both natural and artificial impairments, expected in future wireless applications and services. The first objective of the paper is to specify an iterative method for waveform optimization for hexagonal lattices, using the POPS paradigm introduced in [15] for rectangular lattices and further investigated in [20]. The second objective is to assess whether the use of hexagonal lattices, instead of rectangular lattices, actually brings a noteworthy improvement in performance, as claimed in [7, 9], or not. Our main contributions can be outlined as follows:(i)We present a general mathematical framework for waveform optimization for hexagonal lattices and detail the approach for SINR computation. We stress the fact that the analytical derivation of the SINR is not straightforward and different from that for rectangular lattices [15].(ii)We evaluate the gain, in terms of SINR, brought by hexagonal lattices compared to rectangular ones. Moreover, we determine through numerical results the situations for which the hexagonal lattices outperform the rectangular lattices.(iii)We compare, through numerical results, the OOB emission of hexagonal and rectangular lattices, with optimized waveforms, and conventional OFDM. It is worth noting here that OOB is not included as a criterion in waveform optimization although it can be in practice, since OOB could be expressed as a quadratic form of the transmitted waveform.(iv)We characterize the robustness of POPS-FBMC with respect to frequency and time synchronization and channel spread factor estimation errors. We also compare the robustness of the optimized systems on hexagonal and rectangular lattices to that of conventional OFDM.

With respect to existing works, the proposed approach provides five main advantages:(i)It is based on SINR maximization instead of partial ICI/ISI minimization, as performed in [9].(ii)It considers the exact optimization in the discrete-time domain with a finite waveform support instead of continuous-time optimization with an infinite waveform support, followed by discretization and truncation in time [9] that can lead to a performance degradation. Thus, the resulting waveforms can be directly implementable in hardware.(iii)The strategy adopted in this paper for waveform optimization offers more simplicity and accuracy than the optimization method proposed in [12].(iv)Our work can be implemented for any frequency band and discrete-time channel that satisfies the Wide-Sense Stationary Uncorrelated Scattering (WSSUS) criterion.(v)Performed offline, the proposed iterative optimization strategy can be carried out for a finite bunch of representative propagation channel dispersion statistics, for all expected novel services (such as CoMP, MBMS, and C-RAN). The resulting optimized transmit/receive waveform pairs form a codebook and are then used online, in an adaptive fashion, to better adapt to the slowly varying channel propagation statistics. More specifically, as in adaptive modulation and coding (AMC), adaptive waveform communications (AWC) could become a reality in future wireless communications, whereby the most suitable pair of transmit-receive waveforms is selected from the codebook and used, as a function of an estimate of the current channel dispersion statistics.

##### 1.3. Organization of This Paper

The rest of this paper is organized as follows. In Section 2, we introduce the system model. We analyse the interference and noise statistical characteristics in Section 3. In Section 4, we present the optimization procedure of the transmit/receive waveforms, based on SINR maximization. Then, in Section 5, we detail the approach for signal and interference Kernels computation. Finally, we devote Section 6 to simulation results and Section 7 to concluding this paper.

##### 1.4. Notations

Throughout the paper, the norm of a vector is denoted by and the Hermitian scalar product of two vectors is represented by . The operators , , , and stand for transposition, complex conjugation, transconjugation, and expectation, respectively. We denote by the Kronecker matrix product and by the component-wise product (also known as the Hadamard matrix product). The notations and are used to refer to vector and components matrix entries generically indexed by and , respectively. The function produces a diagonal matrix of eigenvalues and a matrix whose columns are the corresponding eigenvectors of matrix , while the function returns the eigenvector associated with the maximum eigenvalue of matrix . Finally, represents the identity matrix and denotes the th-order Bessel function of the first kind.

#### 2. System Model

##### 2.1. Transmitter and Receiver Models

We consider a baseband model of a multicarrier system with subcarriers, regularly spaced by in frequency. The transmitted multicarrier signal is sampled at a sampling rate , where is the sampling period, is the OFDM symbol period, and is an integer accounting for the number of samples per OFDM symbol period. Due to the hexagonal nature of the TF lattices and the underlying half-symbol period shift between consecutive subcarriers, must be even, leading to a slight flexibility reduction with respect to the rectangular TF lattices, where could also be odd. The subcarrier spacing, , is related to the sampling period and to the number of subcarriers by . In this study, the TF lattice density, , is taken below unity, leading to undersampled TF lattices in the Weyl-Heisenberg frame theory jargon. The undersampled nature of the lattices, acquired by taking smaller than , offers flexibility and room to absorb and put up with different impairments inflicted by the channel and the transmitter and receiver imperfections. It is also to be noted that the positive difference is equivalent to the notion of guard interval time in conventional OFDM.

Working in the discrete-time domain at both transmitter and receiver, the sampled version of the transmitted signal is represented by the infinite vectorwhere is the transmitted signal sample at time , with . This vector can be written aswhere the function , used for the transmission of symbol , results from a time shift, by , and a frequency shift, by , of the transmit waveform vector , and , , are independent identically distributed modulated symbols, of zero mean and common mean transmit energy . As illustrated in Figure 1, the TF shifts, dictated by quincunx/hexagonal lattices, are determined by the generator matrix [7, 9]