LowComplexity Iterative Detection for
Orthogonal Time Frequency Space Modulation
Abstract
We elaborate on the recently proposed orthogonal time frequency space (OTFS) modulation technique, which provides significant advantages over orthogonal frequency division multiplexing (OFDM) in Doppler channels. We first derive the input–output relation describing OTFS modulation and demodulation (mod/demod) for delay–Doppler channels with arbitrary number of paths, with given delay and Doppler values. We then propose a lowcomplexity message passing (MP) detection algorithm, which is suitable for largescale OTFS taking advantage of the inherent channel sparsity. Since the fractional Doppler paths (i.e., not exactly aligned with the Doppler taps) produce the inter Doppler interference (IDI), we adapt the MP detection algorithm to compensate for the effect of IDI in order to further improve performance. Simulations results illustrate the superior performance gains of OTFS over OFDM under various channel conditions.
I Introduction
Fifthgeneration (5G) mobile systems are expected to accommodate an enormous number of emerging wireless applications with high data rate requirements (e.g., realtime video streaming, and online gaming, connected and autonomous vehicles etc.). While the orthogonal frequency division multiplexing (OFDM) modulation scheme currently deployed in fourthgeneration (4G) mobile systems achieves high spectral efficiency for timeinvariant frequency selective channels, it is not robust to timevarying channels, especially for channels with high Doppler spread (e.g., highspeed railway mobile communications). Hence, new modulation schemes/waveforms that are robust to channel timevariations are being extensively explored.
Recently, orthogonal time frequency space (OTFS) modulation was proposed in [1] showing significant advantages over OFDM, in delay–Doppler channels with a number of paths, with given delay and Doppler values. The delayDoppler domain is an alternative representation of a timevarying channel geometry due to moving objects (e.g. transmitters, receivers, reflectors) in the scene. Leveraging on this representation, the OTFS modulator spreads each information (e.g., QAM) symbol over a set of two dimensional (2D) orthogonal basis functions, which span across the frequency–time resources required to transmit a burst. The basis function set is specifically designed to combat the dynamics of the timevarying multipath channel. The general framework of OTFS was given in [1] and a coded OTFS system with forward error correction (FEC) and turbo equalization was compared with coded OFDM, showing significant gain.
In this paper, we analyze the inputoutput relation describing uncoded OTFS modulation/demodulation for delay–Doppler channels with a number of paths, with given delay and Doppler values. We then propose a lowcomplexity message passing (MP) detection algorithm, which is suitable for largescale uncoded OTFS taking advantage of the inherent channel sparsity. The MP detection algorithm, based on a sparse factor graph, uses Gaussian approximation of the interference terms to further reduce the complexity, similar to [8] which was applied to massive MIMO detection. Since the fractional Doppler paths (i.e., not exactly aligned with the Doppler taps) produce the inter Doppler interference (IDI), we adapt the MP detection algorithm to compensate for the effect of IDI in order to further improve performance. We show that the proposed MP detection algorithm can also be applied to OFDM systems to compensate for the Doppler effects. Through simulations, we show the superior performance gains of OTFS over OFDM under various channel conditions.
Ii OTFS modulation/demodulation
In this section, we describe OTFS modulation/demodulation [1].
Iia General OTFS block diagram
The OTFS system diagram is given in Fig. 1. OTFS modulation is produced by a cascade of two 2D transforms at both the transmitter and the receiver. The modulator first maps the information symbols in the delay–Doppler domain to symbols in the time–frequency domain using inverse symplectic finite Fourier transform (ISFFT). Next, the Heisenberg transform is applied to time–frequency symbols to create the time domain signal transmitted over the wireless channel. At the receiver, the received timedomain signal is mapped to the time–frequency domain through the Wigner transform (the inverse of the Heisenberg transform), and then to the delay–Doppler domain for symbol demodulation.
We first introduce the following notation:

The timefrequency plane is discretized by sampling time and frequency axes at intervals (seconds) and (Hz), respectively:

A packet burst has duration and occupies a bandwidth .

Modulated symbol set is transmitted over a given packet burst.

Transmit and receive pulses are denoted by and .
Moreover, the delay–Doppler plane is discretized as follows:
where and represent the quantization intervals of the Doppler frequency shift and time delay, respectively.
IiB OTFS modulation
Consider a set of information symbols from a modulation alphabet (e.g. QAM), which are arranged on a 2D delay–Doppler grid that we wish to transmit.
The OTFS maps to symbols in the time–frequency domain using inverse symplectic finite Fourier transform (SFFT) as follows:
(1) 
for .
Next, a time–frequency modulator maps on the grid to a transmitted waveform by
(2) 
Remark 1
The modulation operation in (2) generalizes OFDM which maps information symbols from frequency domain to time domain. If is a rectangle waveform with duration of then (2) reduces to conventional inverse discrete Fourier transform. When , the inner box in Fig. 1 is an OFDM system. Therefore, one OTFS symbol (packet burst) can be viewed as a SFFT precoding applied on consecutive independent OFDM symbols with subcarriers.
IiC Wireless transmission and reception
The signal is transmitted over a timevarying channel with complex baseband channel impulse response , which characterizes the channel response to an impulse with delay and Doppler [2]. The received signal is given by:
(3) 
where is the additive noise at the receiver.
IiD OTFS demodulation
IiD1 Sufficient statistics and channel distortion
The matched filter computes the crossambiguity function :
(4) 
The matched filter output can be obtained by sampling the function at and at :
(5) 
Remark 2
Wigner transform is a generalization of the OFDM receiver, which maps the received time domain signal to the frequency domain modulated symbols. When is rectangle waveform, it corresponds to the discrete Fourier Transform in OFDM.
The relationship between the matched filter output and the transmitter input was established in [1] as:
(6) 
where , and , and
given that has finite support bounded by , and the pulses and are ideal, i.e., they satisfy the condition for , , where , and are the maximum delay and Doppler values among channel paths. The condition on ideal pulses is called the biorthogonal property and it does not hold for practical pulses (for example, rectangular pulses). Nonetheless, we assume ideal pulses as in [1], and the practical pulses are discussed in [10].
Next we apply the SFFT on , yielding [1]:
(7) 
where is the additive white noise, and is a sampled version of the impulse response function:
for being the circular convolution of the channel response with a windowing function:
Here, we assume that the rectangular window is applied on the transmitter and receiver symbols and .
Iii OTFS under sparse delay–Doppler channel representation
Iiia Sparse representation of the delay–Doppler channel
A sparse representation of the delay–Doppler channel in (3) can be expressed as:
(8) 
where is the number of reflectors; , , and represent the channel gain, delay, and Doppler shift associated with reflector, respectively. Here, we assume that the delays for each reflector are different.
We now analyze the received symbols in (7) using the sparse presentation of the delay–Doppler channel in (8).
Let us consider the expression for by substituting the delay–Doppler channel in (8),
(9) 
where , , and .
Let us first evaluate at ,
(10) 
Here, we assume that , where is a positive integer, as the received signal is sampled at interval [4]. From (10), we see that
where represents mod operation. Hence, the function evaluates to only if , and .
Similarly, we have
(11) 
Here, we assume , with an integer and (i.e., Doppler frequencies are not necessarily at the sampling points in the delayDoppler plane). Specifically, and represents the indexes of the delay tap and Doppler frequency tap, corresponding to delay and Doppler frequency , respectively. The negative indexes of the Doppler frequency taps, where , can also be view as those of positive frequency taps considering mod operation.
We recall that the delay–Doppler channel can be seen as a discretized grid with and representing the indexes of the maximum Doppler and delay taps, respectively. We will refer to as the fractional Doppler since it represents the fractional shift from a Doppler tap in .
The magnitude of the function is
(12) 
The above lower bound is tight for small values of . When , the above function has the peak of the main lobe at and the peaks of the side lobes decay at rate of .
Therefore, for a given and , the function of in (12) has the following properties:

Twosided decreasing function with the peak at , for .

Twosided decreasing function with the peak at , for .
Therefore, we only consider a finite number () of significant values of in (11) for and , where (e.g., for ).
Then we can rewrite the receive signal in (7) as
(13) 
The above inputoutput expression simplifies for the following special cases.
i) Ideal channel – Assuming , the received signal becomes
and behaves as an AWGN channel.
ii) No fractional Doppler for – Assuming that Doppler frequencies are the exact integer multiples of Doppler taps, the received signal can be obtained by replacing in (13), i.e.,
For each path, the transmitted signal is circularly shifted by the delay and Doppler taps and scaled by the associated channel gain.
From (13), we can see that with the fractional Doppler, the transmitted signal not only shifts by the delay and Doppler taps but also affects the neighboring Doppler taps ( to ). We refer to this interference on the neighboring Doppler taps as inter Doppler interference (IDI).
Iv Message passing detection algorithm for OTFS
We now propose a message passing (MP) detection algorithm for OTFS using the inputoutput relation in (13).
Iva Lowcomplexity MP detection algorithm for OTFS
The received signal in vectorized form can be written as
(14) 
where and . The th element of is for , . The elements of and are similarly related to and , respectively. Due to mod and mod operations in (13), we observe that only elements out of are nonzero in each row and column of . Let and denote the sets of nonzero positions in the row and column, respectively, then .
Based on (14), we model the system as a sparsely connected factor graph with variable nodes corresponding to and observation nodes corresponding to . In this factor graph, the observation node is connected to the set of variable nodes . Similarly, the variable node is connected to the set of variable nodes .
The joint maximum a posteriori probability (MAP) detection rule for estimating the transmitted information is given by
which has a complexity exponential in . Since the joint MAP detection is intractable even for very small values of and , we consider the symbolbysymbol MAP detection rule for ,
(15)  
(16)  
(17) 
In (17), we assume all the transmitted symbols are equally likely and the components of are approximately independent for a given , due to the sparsity of . In order to solve the approximate symbolbysymbol MAP detection in (17), we propose a MP detector which has a complexity linear in . Similarly to [8], for each , a variable is isolated from the other interference terms, which are then approximated as Gaussian noise with an easily computable mean and variance.
In MP, mean and variance of the interference terms are used as messages from observation nodes to variable nodes. The message passed from a variable node , for each , to the observation node , for , is the probability mass function (pmf) of the alphabet symbols in . Fig. 2 shows the connections and the messages passed between the observation and variable nodes.
The MP algorithm operates as follows:
Step 1: Initialize iteration index and for
and .
Step 2: Messages are passed from the observation nodes to the variable nodes.
The message passed from to is a Gaussian pdf which can be computed form
(18) 
where the interferenceplusnoise term is approximated as Gaussian random variable with mean
(19) 
and variance
(20)  
Further, we assume that transmitted symbols are i.i.d. and independent from the noise.
Step 3: Messages are passed from variable nodes to the observation nodes. The new message from to contains the pmf vector with elements
(21) 
where is the damping factor ([7]) to improve the convergence rate, and
where
Note that this excludes the information of .
Step 4: Repeat Step 2 and Step 3 until
or a maximum number of iterations is reached.
Step 5: The final decisions about the transmitted symbols are obtained as
where
Complexity: The complexity of one iteration involves the computation of (19) and (20), where each computation has a complexity of the order . Therefore, the overall complexity per symbol is , where is the number of iterations. In simulations, we observed that the algorithm converges typically within 20 iterations. We conclude that the sparsity of the delayDoppler channel representation is a key factor in reducing the complexity of the decoder. The memory requirement is dominated by the storage of real values for and . In addition, we have the massages , requiring complex values and real values, respectively.
IvB Application of MP detection algorithm for OFDM over delay–Doppler channels
We now apply the above MP algorithm to OFDM to compensate the Doppler effects.
The OFDM system can be illustrated by the inner dashed box in Fig. 1, i.e., the TimeFrequency domain. Specifically, the Heisenberg Transform module is replaced by IFFT, cyclic prefix (CP) addition, serialtoparallel and digitaltoanalog conversion, and the Wigner Transform module is substituted with analogtodigital, paralleltoserial, CP removal and FFT operation. Also, as mentioned in Remark 1, for OFDM systems, is set to 1.
In OFDM, the received signal and noise in (3) are sampled at . Then, the frequencydomain signal after FFT operation is given by
(22) 
where denotes Hermitian transpose, is point FFT matrix, and is the transmitted information OFDM symbol. The elements of timedomain channel matrix are given as [9]
Using the frequencydomain channel matrix , we can rewrite (22) as:
(23) 
Since (23) has similar form as (14), the MP previously developed for OTFS can also be applied for OFDM symbol detection. We note that is diagonally dominant and the values of offdiagonal elements in each row decay as we move away from the diagonal entry. Hence, the matrix of OFDM is also sparse, which enables the use of the proposed low complexity MP detection algorithm.
Remark 3
From (13) and (22), we can observe the effects of channel gain on the transmitted symbols are different in OTFS and OFDM. In OTFS, all the transmitted symbols experience the same channel gain (independent of and ), whereas in OFDM, the channel gains are distinct at different subcarriers because of the FFT operation on .
V Illustrative Results
In this section, we simulate the uncoded biterrorrate (BER) performance of OTFS and OFDM over delayDoppler channels. All relevant simulation parameters are given in Table I. For both OTFS and OFDM systems, Extended Vehicular A model[3] is applied for the channel delay model, and the Doppler shift of the path is generated using
where is uniformly distributed.
Parameter  Value 

Carrier frequency  4 GHz 
No. of subcarriers ()  512 
No. of OTFS symbols ()  128 
Subcarrier spacing  15 KHz 
Cyclic prefix of OFDM  2.6 s 
Modulation alphabet  4QAM 
UE speed (km/h)  30, 120, 500 
Channel estimation  Ideal 
Fig. 3 shows the BER performance of OTFS system using the proposed MP detector for different number of interference terms () with QAM signaling at SNR dB over the delayDoppler channel, where UE speed is km/h. Here, we consider . We can see that there is a significant performance improvement till and saturation thereafter, due to the IDI caused by the fractional Doppler. Fewer neighboring interference terms are sufficient to consider in MP (e.g. ).
In Fig. 4, we illustrate the variation of BER and average number of iterations of OTFS using our MP detector over the delayDoppler channel, where UE speed is km/h. We adopt the damping factor for . We consider QAM signaling and SNR dB. We observe that, when , the BER of MP remains almost the same, but deteriorates thereafter. Further, when , MP converges with the least number of iterations. Hence, we choose as the optimum damping factor.
In Fig. 5, we compare the BER performance of OTFS and OFDM systems using QAM signaling over the delayDoppler channels of different Doppler frequencies (UE speeds of km/h). We observe that OTFS outperforms OFDM by approximately dB at BER of , thanks to the constant channel gain over all transmitted symbols in OTFS, whereas in OFDM, the error performance is limited by the subcarrier with the lowest gain. Moreover, OTFS exhibits the same performance for different Doppler frequencies thanks to the IDI reduction provided by the MP detector and the assumption on and . Similar behavior applies to OFDM, since the inter carrier interference (ICI) can be removed by the MP detector.
Vi Conclusion
In this paper, we have analyzed the input–output relation describing OTFS mod/demod in terms of sparse representation of the channel in the delay–Doppler domain. In particular, we have introduced the notion of inter Doppler interference caused by the fractional Doppler. We then proposed a linear complexity message passing (MP) detection algorithm which exploits the channel sparsity. Through simulations, we have shown that the effect of IDI can be mitigated by adapting the MP detection algorithm. We have also shown that OTFS has significant BER gains over OFDM under various channel conditions.
Acknowledgement
This research work is support by the Australian Research Council under Discovery Project ARC DP160100528. Simulations were undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government.
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