Low Complexity Modem Structure for OFDM-Based Orthogonal Time Frequency Space Modulation

1 Low Complexity Modem Structure for OFDM-based Orthogonal Time Frequency Space Modulation Arman Farhang, Ahmad RezazadehReyhani, Linda E. Doyle, and Behrouz Farhang-Boroujeny Abstract—Orthogonal time frequency space (OTFS) modula- signaling technique that is capable of handling the TV chan- tion is a two-dimensional signaling technique that has recently nels. OTFS was first introduced in the pioneering work of emerged in the literature to tackle the time-varying (TV) wireless Handani et al., [5], where the two dimensional (2D) Doppler- channels. OTFS deploys the Doppler-delay plane to multiplex the transmit data where the time variations of the TV channel are delay domain was proposed for multiplexing the transmit data. integrated over time and hence the equivalent channel relating OTFS modulation is a generalized signaling framework where the input and output of the system boils down to a time-invariant precoding and post-processing units are added to the mod- one. This signaling technique can be implemented on the top of ulator and demodulator of a multicarrier waveform allowing a given multicarrier waveform with the addition of precoding for taking advantage of full time and frequency diversity gain and post-processing units to the modulator and demodulator. In this paper, we present discrete-time formulation of an OFDM- of doubly dispersive channels. This process also coverts the based OTFS system. We argue against deployment of window TV channel to a time-invariant one. This modulation scheme functions at the OTFS transmitter in realistic scenarios and has two stages. First, a set of complex data symbols in the thus limit any sort of windowing to the receiver side. We study Doppler-delay domain are converted to the time-frequency the channel impact in discrete-time providing deeper insights domain through an inverse symplectic finite Fourier trans- into OTFS systems. Moreover, our derivations lead to simplified 1 modulator and demodulator structures that are far simpler than form (SFFT− ), [5]. In the second stage, the resulting time- those in the literature. frequency samples are fed into a multicarrier modulator to form the time domain transmit signal. The reverse operations are performed at the receiver to map the received signal back I. INTRODUCTION to the Doppler-delay domain. In addition, to further improve HILE orthogonal frequency division multiplexing the channel sparsity in the Doppler-delay space, application of 1 W (OFDM) modulation scheme achieves a performance proper window functions to the 2D signals, after the SFFT− near the capacity limits in linear time-invariant channels, it at the transmitter and before the SFFT block at the receiver, leads to a poor performance in doubly dispersive channels. is proposed in [5]. However, very little is said on the choice This is due to the large amount of interference that is imposed of such windows. Moreover, although the OTFS formulations by the channel Doppler spread. The common approach to cope presented in [5] are kept general, the numerical results pre- with this issue in the existing wireless standards such as IEEE sented there seem to be limited to the case where OFDM is 802.11a and digital video broadcasting systems is to shorten used for transmission of the generated time-frequency signals. OFDM symbol duration in time so that the channel variations In this paper, we present a discrete-time formulation of over each OFDM symbol are negligible. However, this reduces OTFS modulation while limiting ourselves to the case where the spectral efficiency of transmission since the cyclic prefix OFDM is used for time-frequency signal modulation. This (CP) length should remain constant. A thorough analysis of study reveals that in realistic scenarios this formulation leads to OFDM in such channels is conducted in [1]. Another classical highly simplified transmitter and receiver structures compared approach for handling the time-varying (TV) channels is to with the existing ones in [5]–[8]. We note that window arXiv:1710.00655v1 [cs.IT] 2 Oct 2017 utilize filtered multicarrier systems that are optimized for a functions (at the transmitter and receiver) that may be effective balanced performance in doubly dispersive channels, [2]. in converting the channel to a sparse one in the Doppler-delay Recently, new signaling techniques have emerged in the space require knowledge of channel variations on a given data literature to tackle the TV channels, [3], [4]. In [3], the packet at the transmitter which obviously is unknown. Hence, authors introduce a new waveform called frequency-division it is hard to say any effective window in this sense could be multiplexing with a frequency-domain cyclic prefix (FDM- applied at the transmitter. On this basis, we will ignore the FDCP). This waveform, at its current stage, outperforms windowing step of the OTFS at the transmitter side. At the OFDM in channels with high Doppler and low delay spread. receiver side, on the other hand, an iterative channel estimation However, its performance in channels with high Doppler and and equalization may allow the use of an effective window that medium to high delay spread is yet to be understood, [3]. leads to a sparse channel, e.g. [9], [10]. This is an interesting Orthogonal time frequency space (OTFS) modulation, topic that falls out of the scope of this paper and may be left which is the main focus of this paper, is another emerging for future studies. In this paper, we include the windowing of the time-frequency signal at the receiver side, but limit our This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) and is co-funded under the discussion to the case where this is a separable window along European Regional Development Fund under Grant Number 13/RC/2077. the time and frequency dimensions and emphasise on channel A. Farhang and L. E. Doyle are with Trinity College Dublin, Dublin 2, independent window functions with some reasonable impact. Ireland (e-mail: {farhanga, ledoyle}@tcd.ie). A. RezazadehReyhani and B. Farhang-Boroujeny are with the University The rest of this paper is organized as follows. In Section II, of Utah, USA (e-mail: {Ahmad.Rezazadeh, farhang}@ece.utah.edu). we present OFDM-based OTFS system model in its general PSfrag replacements 2 OFDM OFDM-based OTFS Transmitter OFDM-based OTFS Receiver −1 ym,n OFDM OFDM zm,n Windowing xk,l SFFT LTV Channel + x˜k,l Modulator Demodulator SFFT Fig. 1. OTFS transmitter and receiver structure. form. Section III presents the channel impact in the Doppler- The received signal samples after transmission over a linear L 1 delay domain. In Section IV, we present our proposed OFDM- time varying (LTV) channel with the length L, h(κ,ℓ) ℓ=0− , based OTFS modem structure. We analyze the computational can be obtained as { } complexity of our proposed structure in Section V and finally, L 1 − conclude the paper in Section VI. r(κ)= X h(κ,ℓ)s(κ ℓ)+ ν(κ), (3) − Notations: Throughout the paper, matrices, vectors and ℓ=0 scalar quantities are denoted by boldface uppercase, boldface 2 where ν(κ) (0, σν ) is the channel noise. Assuming lowercase and normal letters, respectively. [A] , and A 1 ∼ CN mn − MCP L, the received OFDM symbols are free of inter- th represent the mn element, and the inverse of A, respectively. symbol≥ interference. Thus, the received OFDM symbol n at The function vec A vectorizes A by stacking its columns the output of the OFDM demodulator can be written as on the top of one{ another} in a column vector. The functions H ν circ a , and diag a form a circulant matrix whose first zn = FM HnFM yn + n, (4) { } { } column is the vector a and a diagonal matrix with the diagonal th where yn is the n column of Y, Hn = RCPH˘ nACP, elements in a, respectively. Im and 0m n are the identity ˘ × RCP = [0M MCP , IM ] is the CP removal matrix, and Hn matrix of size m and an all zero m n matrix, respectively. The is the LTV channel× matrix realizing the linear convolution T H × superscripts ( ) , ( ) and ( )∗ indicate transpose, conjugate operation, (3), on the samples of the nth OFDM symbol. transpose and· conjugate· operations,· respectively. Finally, the The vector νn contains the noise samples at the OFDM operators , , ⊛, and (( ))N represent elementwise mul- T ⊙ ⊗ · demodulator output and zn = [z0,n,...,zM 1,n] . As the tiplication, Kronecker product, 2D circular convolution, and second step of the OTFS demodulator, the− time-frequency modulo-N operations, respectively. samples, zm,n, are windowed using a receive window function wm,n, and then the result is converted back to the Doppler- delay domain through a SFFT operation. This process can be II. OFDM-BASED OTFS SYSTEM represented as We consider an OTFS system transmitting a block of M N M 1 N 1 × − − Doppler-delay quadrature amplitude modulated (QAM) data x˜k,l = X X wm,nzm,nbk,l∗ (m,n), (5) symbols xk,l where k =0,...,M 1 and l =0,...,N 1. m=0 n=0 − − In OTFS modulation, the data symbols, xk,l, are first converted where k = 0,...,M 1, and l = 0,...,N 1. It is shown to the time-frequency domain through the SFFT 1 operation, − − − in [5] that the OTFS receiver output samples, x˜k,ls, can be M 1 N 1 obtained as the 2D circular convolution of the QAM data sym- − − bols xk,l and a time-invariant channel impulse response in the ym,n = X X xk,lbk,l(m,n), (1) k=0 l=0 Doppler-delay domain. The OFDM-based OTFS modulation and demodulation process is shown in Fig.

Load more