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[3], a called channels, with waveform new TV a the introduce authors tackle to literature [2]. fo channels, optimized dispersive to are doubly in is that performance channels systems balanced (TV) multicarrier time-varying filtered classical the utilize Another of handling [1]. analysis in for conducted thorough approach is A channels pre such constant. in cyclic remain OFDM the should since length transmission (CP) of efficiency reduces variations this spectral However, channel negligible. the the are symbol shorte that OFDM so to each time over is in systems duration IE broadcasting symbol as OFDM video such standards digital wireless and existing 802.11a the cope to in impose approach issue is common this that The channels. with interference spread. dispersive of Doppler channel amount doubly the large by the in to channels performance due time-invariant is poor This linear a in to limits leads capacity the near W fUa,UA(e-mail: USA Utah, of .Rzzdhehn n .FragBrueyaewt h U the with are Farhang-Boroujeny B. and RezazadehReyhani A. .FragadL .Dyeaewt rnt olg uln Du Dublin, College Trinity with are Doyle E. L. and Farhang A. hspbiainhseaae rmrsac odce with conducted research from emanated has publication This Abstract rhgnltm rqec pc OF), (OTFS) space frequency time Orthogonal the in emerged have techniques signaling new Recently, rhgnlTm rqec pc Modulation Space Frequency Time Orthogonal OD)mdlto ceeahee performance multiplexing a achieves division scheme modulation frequency (OFDM) orthogonal HILE Otooa iefeunysae(TS modula- (OTFS) space frequency time —Orthogonal { ahna ledoyle farhanga, ra ahn,AmdRzzdhehn,LnaE ol,and Doyle, E. Linda RezazadehReyhani, Ahmad Farhang, Arman { ha.eaae,farhang Ahmad.Rezazadeh, .I I. 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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 process is shown in Fig. 1. More details and 1 j2π( mk nl ) where b (m,n) = e M N . Due to the discus- derivations on the baseband channel response in the Doppler- k,l √MN − − sion in Section I, we ignore transmit windowing step of OTFS. delay domain are presented in the following section. Hence, the time-frequency samples ym,n are directly fed into the OFDM transmitter to form the OTFS transmit signal, III. CHANNEL IMPACT To study the channel impact, in this section, we expand the H S = ACPFM Y, (2) formulation presented in Section II. In addition to providing a deeper insight into OTFS, this paves the way for the derivation where Y is the M N matrix containing the time-frequency of a novel modem structure in Section IV. The SFFT 1 × th − samples ym,n on its mn elements. FM is the normalized operation in (1) can be represented as M-point discrete Fourier transform (DFT) matrix with the − 1 j2πpq H elements [F ] = e M , for p, q = 0,...,M 1. Y = FM XFN , (6) M pq √M T T T − ACP = [GCP, IM ] is the cyclic prefix (CP) addition matrix where X is an M N matrix containing the data symbols xk,l × where the MCP M matrix GCP is formed by taking the on its klth elements. From (6), y F Xf where f × n = M N,n∗ N,n∗ last MCP rows of the identity matrix IM and MCP is the CP is the nth column of the N-point DFT matrix. Consequently, length. Finally, the (M + MCP) N matrix S includes the (4) can be expanded as OFDM time domain transmit signals× on its columns. After H ν parallel to serial conversion of S, the OTFS transmit signal at zn = FM HnFM FM XfN,n∗ + n the baseband can be formed as s = vec S . = FM HnXf ∗ + νn. (7) { } N,n 3

Forming an M N matrix Z = [z0,..., zN 1], the receiver N-point −PSfrag replacementsdκ M P/S windowing as well× as the SFFT operation shown in (5) can IDFT − be rearranged as Z 1 H N-point X˜ = F (W Z)FN , (8) M P/S M ⊙ IDFT . CP where the M N matrix X˜ contains the samples x˜k,l at the . . s × . Addition OTFS receiver output and the 2D window matrix W includes . . − . th Z 1 wm,n on its mn elements. As mentioned in Section I, here, . . we consider the general class of separable receiver window N-point c r c M P/S functions, i.e. wm,n = wmwn, where the functions wm and IDFT r wn window the columns and rows of Z, respectively. In this case, (8) can be written as Fig. 2. Our proposed OFDM-based OTFS modulator structure. ˜ H c r X = FM (W ZW )FN , (9) where the columns of the M N matrix HDD,w include × H c c c r the first columns of the matrices n that are windowed by where W = diag [w0,...,wM 1] , and W = c { − } the window function wm in the delay domain. In particu- diag [wr ,...,wr ] . c 0 N 1 lar, the elements kl of H are equal to [W H ] for To{ derive a clearcut− } representation of the channel effect on DD,w l k0 k =0,...,M 1 and l =0,...,N 1. the transmit data symbols x s, we expand the nth column of k,l As a final note− here, we note that− (13) is a linear system the matrix X˜ as of equations that relates d˜ to d. It, thus, may be used as a H c r x˜n = FM (W ZW )fN,n basis to derive linear detectors such as zero-forcing (ZF) and N 1 minimum mean square error (MMSE) ones or to develop soft 1 H c − r j2πni = F W z w e− N , (10) detectors/equalizers. √ M X i i N i=0 IV. PROPOSED MODEM STRUCTURE Substituting zi from (7) into (10), we have As discussed in the previous sections, our focus in this paper N 1 1 c − j2πni r N ν is on the general class of channel independent and separable x˜n = W X HiXfN,i∗ wi e− + ˜n √ window functions that are only applied at the receiver. In prac- N i=0 N 1 N 1 tical systems, without application of a window, the Doppler- 1 c − − j2π(n−k)i r N ν delay channel impulse response along the delay dimension = W X X Hixkwi e− + ˜n N remains sparse and has the support of L M. Any channel i=0 k=0 ≪ N 1 independent window along the frequency dimension, before − c H ν application of the SFFT, translates into a circular convolution = W X ((n k))N xk + ˜n, (11) − k=0 along the delay dimension. Most likely, this increases the

1 H c N 1 r j2πni c support of the channel impulse response, and hence, degrades where ν˜ = F W − ν w e N , W = n √N M Pi=0 i i − the channel sparsity. In contrast, abrupt start and end of H c FM W FM , and the subcarrier signals along the time dimension in the time- N 1 frequency signal matrix Z translates to a non-sparse impulse − j2πki 1 r H = H w e N . (12) response along the Doppler dimension. On these bases, we k N X i i − i=0 propose utilization of a rectangular window along the fre- quency domain, while using a window with smooth corners Next, defining an MN MN block circulant matrix HBC = HT HT T× along the consecutive OFDM symbols in time. circ [ 0 ,..., N 1] , the MN MN block diagonal c c 1 { − } × Next, we recall that the SFFT− operation in (1) can be windowing matrix W = IN W , the set of equations (11), ⊗ H for n =0, 1,...,N 1, can be combined together to obtain represented as Y = FM XFN . Inserting this into (2), one − may realize that the IDFT operation at OFDM modulator c 1 d˜ = (W HBC)d + vec V˜ , (13) and the M-point DFT operation in SFFT block cancel out. { } − Consequently, (2) boils down to where d˜ = vec X˜ , d = vec X , and V˜ = [ν˜0,..., ν˜N 1]. { } { } H H− H From (13), one may realize that if the submatrices n in BC S = ACPXFN . (15) are circulant, i.e. the channel can be assumed time-invariant c X FH over each OFDM symbol, the multiplication of W H In (15), the operation times N can be implemented BC N to d realizes the 2D circular convolution of the windowed through a set of M IDFT operations of size that are applied to the rows of X. CPs are then added to the columns of the Doppler-delay channel impulse response HDD,w with the data c resulting matrix. Fig. 2 presents a system structure that takes matrix X. This is due to the fact that the matrices W , the QAM symbol elements of X (denoted by the sequence d ), and H are block circulant with circulant submatrices and κ BC distributes them as inputs to the M IDFT blocks, serializes and their multiplication result in a matrix preserving their circulant interleaves the results, and finally adds a CP prior to each block property. Consequently, X˜ can be written as of M samples at the output. The output here is the vectorized X˜ = HDD,w ⊛ X + V˜ , (14) form of the matrix S. 4

TABLE I COMPUTATIONAL COMPLEXITYOF DIFFERENT MODEM STRUCTURES

Structure Number of modulator CMs Number of demodulator CMs MN M MN N MN M MN N OTFS mod./demod. in [5] log2 + 2 log2 log2 + 2 (1 + log2 ) MN M MN M OFDM mod./demod. 2 log2 2 log2 MN N MN N Our proposed mod./demod. 2 log2 2 (1 + log2 )

wr PSfrag replacements n the complexity by combining the inverse SFFT and SFFT CP N-point r M P/S blocks with the OFDM modulator and demodulator blocks, Removal DFT respectively. Moreover, noting that in practical systems the Z−1 number of OFDM symbols, N, in each data packet is much N-point smaller than the FFT size, M, one will observe that our M P/S DFT proposed OTFS modulator and demodulator blocks may have a . . . d˜ significantly lower complexity than their OFDM counterparts. . κ . . − . Z 1 . . VI. CONCLUSION N-point M P/S In this paper, we studied OFDM-based OTFS modulation DFT scheme. We presented the discrete-time formulation of such Fig. 3. Our proposed OFDM-based OTFS demodulator structure. systems while investigating the channel impact on the transmit data symbols. An interesting finding of our study is that the 1 To derive our proposed OTFS demodulator structure, we SFFT− /SFFT and OFDM modulator/demodulator blocks in expand Z as OTFS systems can be combined. This led us to a design of low complexity modulator and demodulator structures. Our Z = [FM H0XfN,∗ 0,..., FM HN 1XfN,N∗ 1]+ V − − complexity analysis revealed that our proposed OTFS modem = FM [H0XfN,∗ 0,..., HN 1XfN,∗ 0]+ V, (16) structure offers a great amount of savings in computational − c complexity compared to the existing structures. where V = [ν0,..., νN 1]. Replacing W with IM in (9), while using (16), one realizes− that similar to the transmitter, REFERENCES the DFT operation at the OFDM demodulator and the IDFT in the SFFT block cancel out. As a result, (9) can be expanded [1] T. Wang, J. G. Proakis, E. Masry, and J. R. Zeidler, “Performance degra- dation of OFDM systems due to Doppler spreading,” IEEE Transactions as on Wireless Communications, vol. 5, no. 6, pp. 1422–1432, June 2006. ˜ r r ˜ [2] B. Farhang-Boroujeny, “Filter bank multicarrier modulation: A wave- X=[w0H0XfN,∗ 0,...,wN 1HN 1XfN,N∗ 1]FN + V, (17) form candidate for 5G and beyond,” Advances in Electrical Engineering, − − − pp. 1–25, 2014. r where the M 1 vectors wnHnXfN,n∗ are the OFDM received [3] T. Dean, M. Chowdhury, and A. Goldsmith, “A new modulation tech- symbols after× the CP removal and application of the window nique for Doppler compensation in frequency-dispersive channels,” in r ˜ H ¯ Submitted to IEEE PIMRC, 2017. coefficients wn. Also, V = FM VFN . Equation (17) shows [4] A. Monk, R. Hadani, M. Tsatsanis, and S. Rakib, “OTFS - Orthogonal that after CP removal, OTFS demodulator only requires scaling time frequency space,” arXiv preprint arXiv:1608.02993, 2016. r [5] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. the OFDM symbols with the scalar wn and application of Molisch, and R. Calderbank, “Orthogonal time frequency space modu- DFTs (of size N) to the rows of the result. This leads to the lation,” in IEEE Wireless Communications and Networking Conference OTFS demodulator structure that is presented in Fig. 3. Here, (WCNC), March 2017, pp. 1–6. the output sequence d˜κ is the serialized elements of X˜ . [6] S. Rakib and R. Hadani, “Orthogonal time frequency space communication system compatible with OFDM,” Jan. 5 2017, WO Patent App. PCT/US2016/039,662. [Online]. Available: V. COMPLEXITY ANALYSIS http://www.google.com/patents/WO2017003952A1?cl=en [7] ——, “Orthogonal time frequency space modulation system for the This section focuses on the computational complexity of Internet of Things,” Feb. 2 2017, US Patent App. 15/143,323. [Online]. the modulator and demodulator structures that were presented Available: https://www.google.com/patents/US20170033899 [8] R. Hadani and S. Rakib, “Compatible use of orthogonal time frequency above and compares them with their counterparts in [5] as space modulation within an LTE communication system,” Mar. 23 well as those of OFDM. Table. I presents a summary of the 2017, WO Patent App. PCT/US2016/052,524. [Online]. Available: computational complexity of different modulator/demodulator https://www.google.com/patents/WO2017049303A1?cl=en [9] P. Raviteja, K. T. Phan, Q. Jin, Y. Hong, and E. Viterbo, “Low- structures in terms of the respective number of complex complexity iterative detection for orthogonal time frequency space multiplications (CMs). Clearly, the original OTFS structures modulation,” arXiv preprint arXiv:1709.09402, 2017. that have been presented in [5] and elsewhere, [6]–[8], are sig- [10] L. A. Li, H. Wei, Y. Huang, Y. Yao, W. Ling, G. Chen, P. Li, and Y. Cai, “A simple two-stage equalizer with simplified orthogonal time nificantly more complex than their OFDM counterparts. This is frequency space modulation over rapidly time-varying channels,” arXiv because of the separation of the inverse SFFT and SFFT blocks preprint arXiv:1709.02505, 2017. from the modulator and demodulator blocks, respectively. The OTFS structure proposed in this paper substantially reduces