Wireless Communications and Mobile Computing

Volume 2018, Article ID 4637181, 14 pages

https://doi.org/10.1155/2018/4637181

## How Much FBMC/OQAM Is Better than FBMC/QAM? A Tentative Response Using the POPS Paradigm

Mediatron Laboratory, Higher School of Communications of Tunis (SUP’COM), 2083 Ariana, Tunisia

Correspondence should be addressed to Wafa Khrouf; nt.mocpus@fuorhk.afaw

Received 28 July 2017; Revised 3 November 2017; Accepted 20 December 2017; Published 10 April 2018

Academic Editor: Malte Schellmann

Copyright © 2018 Wafa Khrouf 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

A major trend of the current research in 5G is to find well time and frequency localized waveforms, dedicated to non-orthogonal wireless multi-carrier systems. The ping-pong optimized pulse shaping (POPS) paradigm was proposed as a powerful technique to generate a family of waveforms, ensuring an optimal signal to interference plus noise ratio (SINR) at the receiver. In this paper, we derive, for the first time, the analytical expression of the SINR for FBMC/OQAM systems. We then adopt the POPS algorithm in the design of optimum transmit and receive waveforms for FBMC/OQAM, with respect to the SINR criterion. For relatively high dispersions, numerical results show that the optimized waveforms provide a gain of 7 dB, in terms of SINR, compared to the PHYDYAS waveform. They also show that the obtained waveforms offer better out-of-band (OOB) emissions with regard to those of the IOTA waveform. Furthermore, we notice that FBMC/OQAM systems present a gain of dB in SINR, compared to FBMC/QAM systems, when both operate at their time-frequency lattice critical densities. However, FBMC/QAM systems can guarantee, with a reduced computational complexity, a comparable performance to FBMC/OQAM systems, in terms of SINR, when their spectral efficiency is relatively reduced by less than 5%.

#### 1. Introduction

Orthogonal frequency division multiplexing (OFDM) systems have witnessed a considerable interest in the last decade [1]. However, in their present form, they are deemed inapt of guaranteeing the required quality of service (QoS) in several new challenging applications brought by 5G systems. This inaptitude is due to many factors. Among them, we can cite the strong spectral leakage, which can only be controlled with strict frequency synchronization. As a consequence, any lack of perfect frequency synchronization causes important intercarrier interference (ICI). Besides, this incapacity is a result of the intersymbol interference (ISI), when the actual channel delay spread exceeds the cyclic prefix (CP) duration, and/or strict time synchronization is relaxed to save signalling resources and delays [2].

To overcome OFDM limitations and meet 5G requirements, several European projects have been launched recently, such as “mobile and wireless communications enablers for twenty-twenty (2020) information society” (METIS) [3], “flexible air interface for scalable service delivery within wireless communication networks of the 5th generation” (FANTASTIC5G) [4], “enhanced multi-carrier techniques for professional ad-hoc and cell-based communications” (EMPHATIC) [5], and “5th generation non-orthogonal waveforms for asynchronous signalling” (5GNOW) [6]. In the 5GNOW project, for instance, various modulations have been suggested [7], namely, filter bank multi-carrier (FBMC), generalized frequency division multiplexing (GFDM), and universal filtered multi-carrier (UFMC). Among these multiple access techniques, FBMC seems to be a good candidate for 5G systems. In the literature, many researches shed light on the advantages of FBMC with offset quadrature amplitude modulation (FBMC/OQAM) systems compared to OFDM systems. Among these advantages, one can cite their robustness to channel time and frequency spreading [8], since their waveforms are well localized in time and frequency. However, FBMC/OQAM implementation and analytical derivation of the SINR are more complex than those of OFDM.

In this paper, we focus on FBMC/OQAM systems, which are also known as OFDM/OQAM and staggered multitone (SMT) [9]. They consist in transmitting separately and alternately, in each subcarrier, the in-phase and quadrature components of complex symbols used in quadrature amplitude modulation (QAM). We believe that there are a limited number of studies which have attempted to derive the SINR analytical expression of FBMC/OQAM systems [10–12]. In [10], the SINR is derived for known transmitter (Tx) and receiver (Rx) waveforms. On the other hand, in [11], the authors calculate the signal to interference ratio (SIR) for a channel with a carrier frequency offset (CFO), whereas in [12], the authors derive the SIR for a channel with a CFO and/or a time offset (TO). In contrast, in this paper, we derive the analytical expression of the SINR for multi-carrier transmissions in the case of highly time and frequency dispersive channels. Once the SINR analytical expression is established, we seek the appropriate Tx/Rx waveforms, which optimize the SINR. To this end, we extend the application of the POPS paradigm, adopted for FBMC/QAM systems in [13], to FBMC/OQAM systems. The POPS algorithm, firstly introduced in [14], has the merit of being very effective in the optimization of the transmit and receive waveforms. It is an iterative algorithm, which enables simple and offline optimization of the waveforms at the Tx/Rx sides, by maximizing the SINR.

The optimal Tx/Rx waveforms, resulting from POPS, are robust against the ISI and ICI incurred by mobile radio communication propagation channels. However, the imaginary part of the interference, known as intrinsic interference [15, 16], is inherent to FBMC/OQAM systems. To account for it, channel estimation is always required. The intrinsic interference, remaining after channel estimation, cannot be neglected unless the channel is not severely dispersive in single-input single-output (SISO) systems, or linear spatial equalization schemes are used in moderately dispersive channels in multiple-input multiple-output (MIMO) systems. Moreover, if spatial equalization other than linear schemes is used in MIMO, a complex equalization process is required to account for the intrinsic interference. Keeping this in mind, we exclusively focus, in this paper, on SISO systems and target the optimization of Tx/Rx waveforms in this context.

Our main contributions in this paper are the following:(i)We provide an analytical expression of the SINR, for FBMC/OQAM systems, in arbitrary channel conditions, for whatever Tx/Rx waveforms.(ii)We compare, theoretically, the performances of FBMC/QAM and FBMC/OQAM systems in an arbitrary propagation framework.(iii)We present and detail the performance of FBMC/QAM systems with lattice densities below or equal to .(iv)We quantify potential gains that can be realized by FBMC/OQAM systems with respect to FBMC/QAM systems, using identical and/or different lattice densities.(v)We compare the performances of POPS waveforms with respect to physical layer for dynamic access and cognitive radio (PHYDYAS) and isotropic orthogonal transform algorithm (IOTA) waveforms, in FBMC/OQAM systems.

This paper is organized as follows. In Section 3, we present the adopted system model for FBMC/OQAM and FBMC/QAM systems. Then, we focus on the derivation of the useful, interference, and noise powers in Section 4 and derive the SINR expression for FBMC/OQAM systems in Section 5. In Section 6, we describe the POPS algorithm suitable for the design of optimal waveforms. We dedicate Section 7 to the illustration of the obtained analytical results. Finally, we present conclusion and perspectives to this work in Section 8.

#### 2. Notations

Boldface lower and upper case letters refer to vectors and matrices, respectively, and refers interchangeably to the element of the matrix . The superscripts , , and denote the conjugate of a function, the transpose of a vector, and the inverse of a matrix, respectively. We denote by the expectation operator, by the real-part operator, by the absolute value, and by the set of integers between and , where . We denote by the Hermitian scalar product, by the real scalar product, and by the norm of .

#### 3. System Model

In this section, we consider a general FBMC system model in its continuous-time version. We denote by the FBMC (QAM or OQAM) symbol period, by the frequency separation between adjacent subcarriers, by the time-frequency occupancy of each transmitted symbol, and by the lattice density. To preserve the same spectral efficiency as in FBMC/QAM, FBMC/OQAM should use twice the lattice density, since real symbols are transmitted in place of complex symbols. Hence, to have a critical density, the lattice density should be equal to in FBMC/OQAM, when in FBMC/QAM.

The FBMC transmitted signal can be written under the following expressionwhere , is the data symbol transmitted at time and frequency , andrefers to the phase, time, and frequency shifted version of the transmitter prototype waveform, , used to transmit a symbol . The phase shift is used in FBMC/OQAM to guarantee the orthogonality between the in-phase and quadrature phase components of the complex symbols used in FBMC/QAM, with respect to the real scalar product . In practice, we take , for FBMC/OQAM systems, and , for FBMC/QAM systems.

The received signal is given bywhere is the channel-distorted version of , is the channel impulse response (CIR) at time , and is a base-band complex additive white Gaussian noise (AWGN) with zero mean and two-sided power spectral density (PSD), . For simplification reasons, we consider a channel with a finite number of paths, , and a CIR equal to , where , , and are, respectively, the amplitude, frequency Doppler shift, and time delay shift of the path. The paths amplitudes , , are assumed to be centered and uncorrelated random complex Gaussian variables with average powers .

#### 4. Useful, Interference, and Noise Powers

In this section, we evaluate the useful, interference, and noise powers of both FBMC/OQAM and FBMC/QAM systems, using the same propagation channel conditions.

##### 4.1. FBMC/OQAM System

In FBMC/OQAM systems, decision variables are calculated using a real scalar product. Therefore, in nondispersive channels, where the interference is purely imaginary, perfect orthogonality is achieved. Unfortunately, in the more general case of dispersive channels, as treated in this paper, the real part of the interference becomes an integral part of the decision variable on symbol , in (1), which is given bywhere is the phase, time, and frequency shifted version of the receiver prototype waveform, , used for the demodulation of the real symbol , and the phase is used to compensate, even partially, the phase shift incurred by the channel at the time-frequency position, , occupied by . Choosing a phase shift for , equal to , guarantees a quasi-orthogonality between the alternately transmitted in-phase and quadrature phase components of FBMC/QAM, whatever the considered real symbol, , to be demodulated. Therefore, the decision variables characteristics are invariant by time and frequency translations within the time-frequency lattice, keeping the time-frequency lattice unchanged. Making this necessary assumption, we can, without loss of generality, focus on the evaluation of the SINR for symbol . The decision variable on can be expanded into three terms, as where , , and are the useful, interference, and noise terms, respectively.

Conditional on a given realization of the CIR, , the average powers of the useful and interference terms are expressed asThe expression in (7) results from the uncorrelated nature of the real transmitted symbols, . The average transmitted energy corresponding to symbol is given by . Let be the mean energy of a complex symbol in the case of FBMC/QAM systems, then, for comparison purposes, we assume that , which means that .

To calculate the SINR, an averaging over the channel realizations is needed. This step is precisely the most challenging and complicated step in FBMC/OQAM systems, since, in addition to , the phase compensation term, , is also dependent on the same channel realizations. For an optimum choice of the compensation factor maximizing the SINR, we usewhich captures the phase shift incurred by the channel realization on the decision variable of symbol , prior to casting to the real part. This is the ideal choice of that allows a compensation with the exact phase shift experienced by the symbol, which is caused by the channel. However, this choice makes the optimization step intractable, since the expected form of the SINR will not be a generalized Rayleigh quotient, but the ratio of two quaternary forms on each of the searched transmitter and receiver waveforms. For a further simplification of the optimization problem, with an expected tractable form of the SINR, we can use the transfer function of the channel, , which is the Fourier transform of the CIR, , with respect to , and use the compensation factor,By averaging expressions (6) and (7) on the realizations of the channel, the useful and interference powers are, respectively, given byLet be the average energy received per complex symbol and let be the normalized multipath power profile of the channel, with . Using the results of Appendix 1, we can writeTo extend the obtained results to more general diffuse channels, obeying the wide sense stationary uncorrelated scattering (WSSUS) property [17], we consider the asymptotic configuration, where in (12). Denoting by the scattering function of the channel and by its normalized version, we can rewrite and in a more general form as