I Introduction
Future wireless systems including 5G systems need to operate in dynamic channel conditions, where operation in high mobility scenarios (e.g., highspeed trains) and millimeter wave (mm Wave) bands are envisioned. The wireless channels in such scenarios are doublydispersive, where multipath propagation effects cause time dispersion and Doppler shifts cause frequency dispersion [1]. OFDM systems are usually employed to mitigate the effect of intersymbol interference (ISI) caused by time dispersion [2]. However, Doppler shifts result in intercarrier interference (ICI) in OFDM and degrades performance [3]. An approach to jointly combat ISI and ICI is to use pulse shaped OFDM systems [4][6]. Pulse shaped OFDM systems use general timefrequency lattices and optimized pulse shapes in the timefrequency domain. However, systems that employ the pulse shaping approach do not efficiently address the need to support high Doppler shifts.
Orthogonal time frequency space (OTFS) modulation is a recently proposed multiplexing scheme [7][10] which meets the highDoppler signaling need through a different approach, namely, multiplexing the modulation symbols in the delayDoppler domain (instead of multiplexing symbols in timefrequency domain as in traditional modulation techniques such as OFDM). OTFS waveform has been shown to be resilient to delayDoppler shifts in the wireless channel. For example, OTFS has been shown to achieve significantly better error performance compared to OFDM for vehicle speeds ranging from 30 km/h to 500 km/h in 4 GHz band, and that the robustness to highDoppler channels (e.g., 500 km/h vehicle speeds) is especially notable, as OFDM performance breaks down in such highDoppler scenarios [9]. When OTFS waveform is viewed in the delayDoppler domain, it corresponds to a 2D localized pulse. Modulation symbols, such as QAM symbols, are multiplexed using these pulses as basis functions. The idea is to transform the timevarying multipath channel into a 2D timeinvariant channel in the delayDoppler domain. This results in a simple and symmetric coupling between the channel and the modulation symbols, due to which significant performance gains compared to other multiplexing techniques are achieved [7]. OTFS modulation can be architected over any multicarrier modulation by adding preprocessing and postprocessing blocks. This is very attractive from an implementation viewpoint.
Recognizing the promise of OTFS in future wireless systems, including mmWave communication systems [10], several works on OTFS have started emerging in the recent literature [11][16]
. These works have addressed the formulation of inputoutput relation in vectorized form, equalization and detection, and channel estimation. Multipleinput multipleoutput (MIMO) techniques along with OTFS (MIMOOTFS) can achieve increased spectral/energy efficiencies and robustness in rapidly varying MIMO channels. It is shown in
[7] that OTFS approaches channel capacity through linear scaling of spectral efficiency with the MIMO order. We, in this paper, consider the signal detection and channel estimation aspects in MIMOOTFS.Our contributions can be summarized as follows. We first present a vectorized inputoutput formulation for the MIMOOTFS system. Initially, we assume perfect channel knowledge at the receiver and employ an iterative algorithm based on message passing for signal detection. The algorithm has low complexity and it achieves very good performance. For example, in a MIMOOTFS system, a bit error rate (BER) of is achieved at an SNR of about 14 dB for a Doppler of 1880 Hz (500 km/hr speed at 4 GHz). For the same system, MIMOOFDM BER performance floors at a BER of 0.02. Next, we relax the perfect channel estimation assumption and present a channel estimation scheme in the delayDoppler domain. The proposed scheme uses impulses in the delayDoppler domain as pilots for MIMOOTFS channel estimation. The proposed scheme is simple and effective in highDoppler MIMO channels. For example, compared to the case of perfect channel knowledge, the proposed scheme loses performance only by less than a fraction of a dB.
The rest of the paper is organized as follows. The OTFS modulation is introduced in Sec. II. The MIMOOTFS system model and the vectorized inputoutput relation are developed in Sec. III. MIMOOTFS signal detection using message passing and the resulting BER performance are presented in Sec. IV. The channel estimation scheme in the delayDoppler domain and the achieved performance are presented in Sec. V. Conclusions are presented in Sec. VI.
Ii OTFS Modulation
OTFS modulation uses the delayDoppler domain for multiplexing the modulation symbols and for channel representation. When the channel impulse response is represented in the delayDoppler domain, the received signal is the sum of reflected copies of the transmitted signal , which are delayed in time (), shifted in frequency (), and multiplied by the complex gain [8]. Thus, the coupling between an input signal and the channel in this domain is given by the following double integral:
(1) 
The block diagram of the OTFS modulation scheme is shown in Fig. 1. The inner box is the familiar timefrequency multicarrier modulation, and the outer box with a pre and postprocessor implements the OTFS modulation scheme in the delayDoppler domain. The information symbols (e.g., QAM symbols) residing in the delayDoppler domain are first transformed to the familiar timefrequency (TF) domain signal
through the 2D inverse symplectic finite Fourier transform (ISFFT) and windowing. The Heisenberg transform is then applied to the TF signal
to transform to the time domain signal for transmission. At the receiver, the received signal is transformed back to a TF domain signal through Wigner transform (inverse Heisenberg transform). thus obtained is transformed to the delayDoppler domain signal through the symplectic finite Fourier transform (SFFT) for demodulation.In the following subsections, we describe the signal models in TF modulation and OTFS modulation. Let denote the TF modulation symbol time and denote the subcarrier spacing. Let , , be the information symbols transmitted in a given packet burst. Let and denote the transmit and receive windows, respectively.
Iia Timefrequency modulation

Let and denote the transmit and receive pulses, respectively, which are biorthogonal with respect to time and frequency translations. Signal in the TF domain , , is transmitted in a given packet burst.

TF modulation/Heisenberg transform: The signal in the timefrequency domain is transformed to the time domain signal using the Heisenberg transform given by
(2) 
TF demodulation/Wigner transform: At the receiver, the time domain signal is transformed back to the TF domain using Wigner transform given by
(3) where is the cross ambiguity function given by
(4)
and is related to by (1). The relation between and for TF modulation can be derived as [9]
(5) 
where is the additive white Gaussian noise and is given by
(6) 
IiB OTFS modulation

Let be the periodized version of with period . The SFFT of is given by
(7) and the ISFFT is , given by
(8) 
Information symbols , , , are transmitted in a given packet burst.

OTFS transform/preprocessing: The information symbols in the delayDoppler domain are mapped to TF domain symbols as
(9) where is the transmit windowing square summable function.

thus obtained is in the TF domain and it is TF modulated as described in the previous subsection, and is obtained by (3).

OTFS demodulation/postprocessing: A receive window is applied to and periodized to obtain which has the period , as
(10) The symplectic finite Fourier transform is then applied to to convert it from TF domain back to delayDoppler domain , as
(11)
The inputoutput relation in OTFS modulation can be derived as [9]
(12) 
where
(13) 
where is the circular convolution of the channel response with a windowing function , given by
(14) 
where is given by
(15) 
IiC Vectorized formulation of the inputoutput relation
Consider a channel with signal propagation paths (taps). Let the path be associated with a delay , a Doppler , and a fade coefficient . The channel impulse response in the delayDoppler domain can be written as
(16) 
Assume that the windows used in modulation, and are rectangular. Define and , where and are integers denoting the indices of the delay tap (with delay ) and Doppler tap (with Doppler value ). In practice, although the delay and Doppler values are not exactly integer multiples of the taps, they can be well approximated by a few delayDoppler taps in the discrete domain [19]. With the above assumptions, the inputoutput relation for the channel in (16) can be derived as [12]
(17) 
where . The above equation can be represented in vectorized form as [12]
(18) 
where , , the th element of , , , and the same relation holds for and as well. In this representation, there are only nonzero elements in each row and column of the equivalent channel matrix () due to modulo operations.
Iii MIMOOTFS Modulation
Consider a MIMOOTFS system as shown in Fig. 2 with equal number of transmit () and receive antennas (), i.e., . Each antenna transmits OTFS modulated information symbols independently. Let the windows , used for modulation be rectangular. Assume that the channel corresponding to th transmit antenna and th receive antenna has taps as in (16). Therefore, the channel representation can be written as
(19) 
, . Thus, we can use the vectorized formulation in Sec. IIC for each transmit and receive antenna pair to describe the inputoutput relation.
Iiia Vectorized formulation of the inputoutput relation for MIMOOTFS
Let denote the equivalent channel matrix corresponding to th transmit antenna and th receive antenna. Let denote the transmit vector from the th transmit antenna and denote the received vector corresponding to th receive antenna in a given frame. Then, similar to the system model in (18) for a SISOOTFS, we can derive a linear system model describing the input and output for the MIMOOTFS system as given below
(20) 
Define
Then, (20) can be written as
(21) 
where , . Thus, in this representation, each row and column of has only nonzero elements due to modulo operations.
Iv MIMOOTFS Signal Detection
In this section, we present a MIMOOTFS signal detection scheme using an iterative algorithm based on message passing and present a performance comparison between MIMOOTFS and MIMOOFDM in highDoppler scenarios.
Iva Algorithm for MIMOOTFS signal detection
Let the sets of nonzero positions in the th row and th column of be denoted by and , respectively. Using (21), the system can be modeled as a sparsely connected factor graph with variable nodes corresponding to the elements in and observation nodes corresponding to the elements in . Each observation node is connected to the set of variable nodes {}, and each variable node is connected to the set of observation nodes {}. Also, . The maximum a posteriori (MAP) decision rule for (21) is given by
(22) 
where is the modulation alphabet (e.g., QAM) used. The detection as per (22) has exponential complexity. Hence, we use symbol by symbol MAP rule for for detection as follows:
The transmitted symbols are assumed to be equally likely and the components f are nearly independent for a given due to the sparsity in . This can be solved using the message passing based algorithm described below. The message that is passed from the variable node , for each , to the observation node for , is the pmf denoted by of the symbols in the constellation . Let denote the element in the th row and th column of . The message passing algorithm is described as follows.
and variance
of the interference term are passed as messages from to .can be approximated as a Gaussian random variable and is given by
(23) 
(24) 
(25) 
(26) 
(27) 
IvB Vectorized formulation of the inputoutput relation for MIMOOFDM
In this subsection, in order to provide a performance comparison between MIMOOTFS and MIMOOFDM, we present the vectorized formulation of the inputoutput relation for MIMOOFDM. OFDM uses the TF domain for signaling and channel representation. We will first derive the vectorized formulation for a SISOOFDM and extend it to MIMOOFDM. For a fair comparison with the OTFS modulation, we will consider consecutive OFDM blocks (each of size ) to be one frame, i.e., the transmit vector , and message passing detection is done jointly over one frame. Consider the channel in (16). The timedelay representation is related to the delayDoppler representation by a Fourier transform along the time axis, and is given by
(28) 
Sample the time axis at . The sampled timedelay representation is given by
(29) 
Let denote the cyclic prefix length used in each OFDM block and let . The size of one frame after cyclic prefix insertion to each block will then be . Let denote the matrix that inserts cyclic prefix for one block, where contains the last
rows of the identity matrix
. Also, let denote the the matrix that removes the cyclic prefix for one block [18]. Let and denote the DFT and IDFT matrices of size . We use the following notations.
: cyclic prefix insertion matrix for consecutive OFDM blocks.

: cyclic prefix removal matrix for consecutive OFDM blocks.

: DFT matrix for consecutive OFDM blocks.

: IDFT matrix for consecutive OFDM blocks.

The channel in the timedelay domain for a given frame can be written as a matrix using (29) and has size .
Using the above, the endtoend relationship in OFDM modulation can be described by the following linear model:
(30) 
where , .
IvB1 MimoOfdm
The vectorized formulation of the inputoutput relation for SISOOFDM derived above can be extended to MIMOOFDM in a similar fashion as was done for the MIMOOTFS system described in Sec. IIIA. Let denote the equivalent channel matrix corresponding to th transmit antenna and th receive antenna. Let denote the transmit vector from the th transmit antenna and denote the received vector corresponding to th receive antenna in a given frame. Define
The inputoutput relation for MIMOOFDM can be written as
(31) 
where and .
IvC Performance results and discussions
In this subsection, we present the BER performance of MIMOOTFS and compare it with that of MIMOOFDM. Perfect channel knowledge is assumed at the receiver. Message passing algorithm is used for both MIMOOTFS and MIMOOFDM. A damping factor of 0.5 is used. The maximum number of iterations and the value used are 30 and 0.01, respectively. We use the channel model in (19) and the number of taps is taken to be 5. The delayDoppler profile considered in the simulation is shown in Table I. Other simulation parameters used are given in Table II.
Path index  

Delay (), s  
Doppler (), Hz 
Parameter  Value 

Carrier frequency (GHz)  4 
Subcarrier spacing (kHz)  15 
Frame size  
Modulation scheme  BPSK 
MIMO configuration  11, 22, 33 
Maximum speed (kmph)  507.6 
Figure 3 shows the BER performance of MIMOOTFS for SISO as well as and MIMO configurations. The maximum considered speed of 507.6 kmph corresponds to 1880 Hz Doppler frequency at a carrier frequency of 4 GHz. Even at this highDoppler value, MIMOOTFS is found to achieve very good BER performance. We observe that, a BER of is achieved at an SNR of about 14 dB for the 22 system, while the SNR required to achieve the same BER reduces by about 2 dB for the 3 system. Thus, with the proposed detection algorithm, MIMOOTFS brings in the advantages of linear increase in spectral efficiency with number of transmit antennas and the robustness of OTFS modulation in highDoppler scenarios.
Figure 4 shows the BER performance comparison between MIMOOTFS and MIMOOFDM in a MIMO system. The maximum Doppler spread in the considered system is high (1880 Hz) which causes severe ICI in the TF domain. Because of the severe ICI, the performance of MIMOOFDM is found to break down and floor at a BER value of about . However, MIMOOTFS is able to achieve a BER of at an SNR value of about 14 dB. This is because OTFS uses the delayDoppler domain for signaling instead of TF domain. Thus, the BER plots clearly illustrate the robust performance of MIMOOTFS and its superiority over MIMOOFDM under rapidly varying channel conditions.
V Channel Estimation for MIMOOTFS
In this section, we relax the assumption of perfect channel knowledge and present a channel estimation scheme in the delayDoppler domain. The scheme uses impulses in the delayDoppler domain as pilots. Figure 5 gives an illustration of the pilots, channel response, and received signal in a MIMO system with the delayDoppler profile and system parameters given in Tables I and II. Each transmit and receive antenna pair sees a different channel having a finite support in the delayDoppler domain. The support is determined by the delay and Doppler spread of the channel [8]. This fact can be used to estimate the channel for all the transmitreceive antenna pairs simultaneously using a single MIMOOTFS frame as described below.
The OTFS inputoutput relation for th transmit antenna and th receive antenna pair can be written using (12) as
(32) 
If we transmit
(33) 
as pilot from the th antenna, the received signal at the th antenna will be
(34) 
We can estimate from (34), since, being the pilots, and are known at the receiver a priori. From this, we can get the equivalent channel matrix using the vectorized formulation of Sec. IIC. From (34) we also see that, due to the 2Dconvolution inputoutput relation, the impulse at is spread by the channel only to the extent of the support of the channel in the delayDoppler domain. Thus, if we send the pilot impulses from the transmit antennas with sufficient spacing in the delayDoppler domain, they will be received without overlap. Hence, we can estimate the channel responses corresponding to all the transmitreceive antenna pairs simultaneously and get the estimate of the equivalent MIMOOTFS channel matrix using a single MIMOOTFS frame. This is illustrated in Fig. 5 for a MIMOOTFS system with frame size at an SNR value of 4 dB. The first antenna transmits the pilot impulse at and the second antenna transmits the pilot impulse at in the delayDoppler domain. We observe that the impulse response and are nonoverlapping at the receiver. Thus, they can be estimated simultaneously using a single pilot MIMOOTFS frame.
Va Performance results and discussions
In this subsection, we present the BER performance of the MIMOOTFS system using the estimated channel. We use the MIMOOTFS channel estimation scheme described above, for estimating the equivalent channel matrix and use the message passing algorithm for detection. The delayDoppler profile and the simulation parameters are as given in Table I and Table II, respectively.
In Fig. 6, we plot the Frobenius norm of the difference between the equivalent channel matrix () and the estimated equivalent channel matrix () (a measure of estimation error) as a function of pilot SNR for a MIMOOTFS system with system parameters as in Tables I and II. We observe that, as expected, the Frobenius norm of the difference matrix decreases with pilot SNR. Figure 7 shows the corresponding BER performance using the proposed channel estimation scheme for the MIMOOTFS system. It is observed that the BER performance achieved with the estimated channel is quite close to the performance with perfect channel knowledge. For example, a BER of is achieved at SNR values of about 12.5 dB and 13 dB with perfect channel knowledge and estimated channel knowledge, respectively. At the considered maximum Doppler frequency of 1880 Hz, channel estimation in the timefrequency domain leads to inaccurate estimation because of the rapid variations of the channel in time. On the other hand, the sparse channel representation in the delayDoppler domain is timeinvariant over a larger observation time. This, along with the OTFS channelsymbol coupling (2D periodic convolution) in the delayDoppler domain, enables the proposed channel estimation for MIMOOTFS to be simple and efficient.
Vi Conclusions
We investigated signal detection and channel estimation aspects of MIMOOTFS under highDoppler channel conditions. We developed a vectorized formulation of the inputoutput relationship for MIMOOTFS which enables MIMOOTFS signal detection. We presented a low complexity iterative algorithm for MIMOOTFS detection based on message passing. The algorithm was shown to achieve very good BER performance even at high Doppler frequencies (e.g., 1880 Hz) in a MIMO system where MIMOOFDM was shown to floor in its BER performance. We also presented a channel estimation scheme in the delayDoppler domain, where delayDoppler impulses are used as pilots. The proposed channel estimation scheme was shown to be efficient and the BER degradation was small as compared to the performance with perfect channel knowledge. The sparse nature of the channel in the delayDoppler domain which is timeinvariant over a larger observation time enabled the proposed estimation scheme to be simple and efficient.
References
 [1] W. C. Jakes, Microwave Mobile Communications, New York: IEEE Press, reprinted, 1994.
 [2] A. Goldsmith, Wireless Communications, Cambridge Univ. press, 2005.
 [3] T. Wang, J. G. Proakis, E. Masry, and J. R. Zeidler, “Performance degradation of OFDM systems due to Doppler spreading,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 14221432, Jun. 2006.
 [4] T. Strohmer and S. Beaver, “Optimal OFDM design for timefrequency dispersive channels,” IEEE Trans. Commun., vol. 51, no. 7, pp. 11111122, Jul. 2003.
 [5] FM. Han and XD. Zhang, “Hexagonal multicarrier modulation: a robust transmission scheme for timefrequency dispersive channels,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 19551961, May 2007.
 [6] FM. Han and XD. Zhang, “Wireless multicarrier digital transmission via WeylHeisenberg frames over timefrequency dispersive channels,” IEEE Trans. Commun., vol. 57, no. 6, pp. 17211733, Jun. 2009.
 [7] R. Hadani and A. Monk, “OTFS: A new generation of modulation addressing the challenges of 5G,” online: arXiv:1802.02623 [cs.IT] 7 Feb 2018.
 [8] A. Monk, R. Hadani, M. Tsatsanis, and S. Rakib, “OTFS  orthogonal time frequency space: a novel modulation technique meeting 5G high mobility and massive MIMO challenges,” online: arXiv:1608.02993 [cs.IT] 9 Aug 2016.
 [9] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” Proc. IEEE WCNC’2017, pp. 17, Mar. 2017.
 [10] R. Hadani, S. Rakib, A. F. Molisch, C. Ibars, A. Monk, M. Tsatsanis, J. Delfeld, A. Goldsmith, and R. Calderbank, “Orthogonal time frequency space (OTFS) modulation for millimeterwave communications systems,” in Proc. IEEE MTTS Intl. Microwave Symp., pp. 681683, Jun. 2017.
 [11] L. Li, H. Wei, Y. Huang, Y. Yao, W. Ling, G. Chen, P. Li, and Y. Cai, “A simple twostage equalizer with simplified orthogonal time frequency space modulation over rapidly timevarying channels,” online: arXiv:1709.02505v1 [cs.IT] 8 Sep 2017.
 [12] P. Raviteja, K. T. Phan, Q. Jin, Y. Hong, and E. Viterbo, “Lowcomplexity iterative detection for orthogonal time frequency space modulation,” online: arXiv:1709.09402v1 [cs.IT] 27 Sep 2017.
 [13] A. R. Reyhani, A. Farhang, M. Ji, RR. Chen, and B. FarhangBoroujeny, “Analysis of discretetime MIMO OFDMbased orthogonal time frequency space modulation,” arXiv:1710.07900v1 [cs.IT] 22 Oct 2017.
 [14] T. Dean, M. Chowdhury, and A. Goldsmith, “A new modulation technique for Doppler compensation in frequencydispersive channels,” Proc. IEEE PIMRC’2017, Oct. 2017.
 [15] A. Farhang, A. Rezazadeh Reyhani, L. E. Doyle, and B. FarhangBoroujeny, “Low complexity modem structure for OFDMbased orthogonal time frequency space modulation,” IEEE Wireless Commun. Lett., doi: 10.1109/LWC.2017.2776942, Nov. 2017.
 [16] K. R. Murali and A. Chockalingam, “On OTFS modulation for highDoppler fading channels,” Proc. ITA’2018, San Diego, Feb. 2018.
 [17] Y.G. Li, J.H. Winters, and N.R. Sollenberger, “MIMOOFDM for wireless comunications: signal detection with enhanced channel estimation,” IEEE Trans. Commun., vol. 50, no. 9, pp. 14711477, Sep. 2002.
 [18] F. Hlawatsch and G. Matz, Wireless Communications Over Rapidly TimeVarying Channels, Academic Press, 2011.
 [19] A. Fish, S. Gurevich, R. Hadani, A. M. Sayeed, and O. Schwartz, “DelayDoppler channel estimation in almost linear complexity,” IEEE Trans. Inf. Theory, vol. 59, no. 11, pp. 76327644, Nov. 2013.
Comments
There are no comments yet.