Coded Orthogonal Time Frequency Space Modulation

Figure 1 Grids in time frequency (TF) plane and delay-Doppler (DD) plane

The reciprocal delay-Doppler grid is represented as:

Λ^{⊥} = \{(\frac{k}{N T}, \frac{l}{M Δ f}), k = 0, . . ., N - 1, l = 0, . . ., M - 1\}

,(2)

where $\frac{1}{N T}$ and $\frac{1}{M Δ f}$ represent the sampling intervals along the Doppler axis and the delay axis, respectively.

The mapping between signals in the TF domain and DD domain depends on two-dimensional symplectic finite Fourier transform (2D SFFT) pairs, which can be exemplified as:

h (τ, ν) = \iint H (t, f) e^{- j 2 π (ν t - f τ)} d t d f

,(3)

H (t, f) = \iint h (τ, ν) e^{j 2 π (ν t - f τ)} d τ d ν

,(4)

where $h (τ, ν)$ and $H (t, f)$ are the responses of linear time-varying (LTV) wireless channels in the DD domain and TF domain, respectively. Without loss of generality, the DD domain representation of an LTV wireless channel can be expressed as:

h (τ, ν) = \sum_{i = 1}^{P} h_{i} δ (τ - τ_{i}) δ (ν - ν_{i})

,(5)

where $δ (\cdot)$ denotes the Dirac delta function, $P$ is the number of resolvable paths, and $h_{i}$ , $τ_{i}$ and $ν_{i}$ are the channel coefficient, delay and Doppler shift of the i-th path respectively. Here, $τ_{i}$ and $ν_{i}$ are defined as:

τ_{i} = \frac{l_{i}}{M Δ f}

ν_{i} = \frac{k_{i} + κ_{i}}{N T}

.(6)

In Eq. (6), $l_{i}$ represents the index of the delay with integer values, $k_{i}$ represents the index of the Doppler shift with integer values, and $κ_{i} \in (- 0.5, 0.5]$ is the real number, indicating the fractional shift from the nearest Doppler index $k_{i}$ , which is also called fractional Doppler^[5].

Specifically, the 2D SFFT pairs can be realized by simple discrete Fourier transform (DFT) pairs or fast Fourier transform (FFT) pairs. For example, in view of the DFT, the inverse symplectic finite Fourier transform (ISFFT) can be regarded as the M-point DFT along the delay axisand the N-point inverse DFT (IDFT) along the Doppler axis for the two-dimensional signal with the size of $M \times N$ in the DD domain, resulting in the corresponding TF domain signal.

2.2 Coded OTFS System Model

Fig. 2 shows the proposed coded OTFS system model in this paper. Suppose that the information bit sequence $u$ of length $K$ is encoded using a forward error correction (FEC) code, resulting in the codeword $c$ of length $N_{c}$ . After interleaving, the interleaved sequence $v$ is then mapped to an M-ary signal constellation $𝔸$ , such as M-ary phase shift keying (MPSK) or M-ary quadrature amplitude modulation (MQAM), and the modulated symbol vector $x_{D D}$ of length $M N$ is arranged as a two-dimensional signal matrix $x_{D D} \in 𝔸^{M \times N}$ in the DD domain, where $M$ is the number of the sub-carriers, and $N$ is the number of time slots for each OTFS symbol. The element $x [l, k]$ of the $x_{D D}$ represents the modulated signals in the l-th Delay and k-th Doppler grid, for $l \in \{0,1, . . ., M - 1\}$ and $k \in \{0,1, . . ., N - 1\}$ . Then the symbol $X [m, n]$ in the m-th frequency and n-th time grid is obtained by the ISFFT, which is given as:

Figure 2

Figure 2 Proposed model of coded OTFS system

X [m, n] = I S F F T (x [l, k]) = \frac{1}{\sqrt[]{N M}} \sum_{l = 0}^{M - 1} \sum_{k = 0}^{N - 1} x [l, k] e^{j 2 π (\frac{n k}{N} - \frac{m l}{M})}

,(7)

for $m = 0,1, . . ., M - 1, n = 0,1, . . ., N - 1$ . The two-dimensional signal matrix in the TF domain is denoted by $X_{T F} \in ℂ^{M \times N}$ . The early literature^[1–3] explained that the composition of the ISFFT and the windowing function in the TF domain are referred to as the OTFS transform. The window operations of the transmitter and receiver affect the cross-symbol interference of the effective impulse response^[1]. The window design has the potential to increase the effective channel sparsity in the DD domain, which is conducive to the channel estimation, as described in Refs. [6] and [7]. Furthermore, the influence of the design of the TF domain window on improving the performance of the channel estimation and data detection is discussed in Ref. [16]. Here, the rectangular window is considered.

The transmitted signal in the time domain is obtained from the TF domain symbols $X [m, n]$ using the Heisenberg transform parameterized by the pulse shaping filter $g_{t x} (t)$ , which can be written as:

s (t) = \sum_{m = 0}^{M - 1} \sum_{n = 0}^{N - 1} X [m, n] g_{t x} (t - n T) e^{j 2 π m Δ f (t - n T)}

.(8)

This can be regarded as a general form of the OFDM modulation^[3]. Moreover, OTFS modulation can be implemented as a cascade of a pre-coder (ISFFT) and a traditional OFDM modulator^[6], as shown in Fig. 2.

Assume that the channel is the LTV channel described in Eq. (5), the received signal can be expressed as:

r (t) = \iint h (τ, ν) s (t - τ) e^{j 2 π ν (t - τ)} d τ d ν + w (t)

,(9)

where $w (t)$ is the additive white Gaussian noise with zero mean and one-sided power spectral density of $N_{0}$ .

At the receiver, $r (t)$ is subject to the Wigner transform to obtain the received symbols $Y [m, n]$ in the TF domain, given by

Y [m, n] = \int r (t) g_{r x}^{*} (t - n T) e^{- j 2 π m Δ f (t - n T)} d t

,(10)

where $g_{r x} (t)$ is the pulse shaping filter at the receiver. Eq. (10) can be further written as^[5]:

Y [m, n] = \sum_{m^{'} = 0}^{M - 1} \sum_{n^{'} = 0}^{N - 1} H_{m, n} [m^{'}, n^{'}] X [m^{'}, n^{'}] + \bar{w} [m, n]

,(11)

where $\bar{w} [m, n]$ is the noise sample in the TF domain, and $H_{m, n} [m^{'}, n^{'}]$ is the channel impulse response in the TF domain, i.e.,

H_{m, n} [m^{'}, n^{'}] = \iint h (τ, ν) A_{g_{r x}, g_{t x}} ((m - m^{'}) Δ f - ν, (n - n^{'}) T - τ) e^{j 2 π (ν + m^{'} Δ f) ((n - n^{'} T) - τ)} e^{j 2 π ν n^{'} T} d τ d ν

.(12)

In Eq. (12), $A_{g_{r x}, g_{t x}} (τ, ν)$ is referred to as the cross-ambiguity function, which represents the interference between symbols in the DD domain caused by the channel dispersion^[20], and can be expressed as:

A_{g_{r x}, g_{t x}} (τ, ν) = \int g_{r x}^{*} (t - τ) g_{t x} (t) e^{j 2 π ν Δ f t} d t

.(13)

The received symbols in the DD domain are given as:

y [l, k] = S F F T (Y [m, n]) = \frac{1}{\sqrt[]{N M}} \sum_{m = 0}^{M - 1} \sum_{n = 0}^{N - 1} Y [m, n] e^{- j 2 π (\frac{n k}{N} - \frac{m l}{M})} + w [l, k]

,(14)

for $l = 0,1, . . ., M - 1, k = 0,1, . . ., N - 1$ , where $w [l, k]$ is the noise sample in the DD domain. Upon the received symbols $y [l, k]$ , a signal detection algorithm is then performed. In addition, a joint iterative strategy between detection and decoding can also be considered.

It is generally known that the MAP detection is optimum for OTFS systems. However, the complexity of the MAP detection increases exponentially with the block size of each OTFS frame. As a compromise of the MAP detection, a lot of literature has studied the massage passing detection algorithm based on the factor graph, which can effectively reduce the detection complexity, such as Ref. [5]. For the coded system, the iterative signal processing of the detector and the decoder is usually considered at the receiver, as shown in Fig. 2. Correspondingly, soft decision detection algorithms should be adopted, such as MP, unitary approximate MP (UAMP), vector AMP (VAMP), sum-product algorithm (SPA) and other message passing algorithms. With Log-Likelihood Ratios (LLRs), the message $L_{e, d e t}$ passed from the detector to the decoder is calculated as:

\begin{array}{l} L_{e, d e t} (x_{l, k} = a_{j}) = L_{a p p, d e t} (x_{l, k} = a_{j}) - L_{a, d e t} (x_{l, k} = a_{j}) = \\ l n \frac{P (x_{l, k} = a_{j} |Y_{D D})}{P (x_{l . k} = 0 |Y_{D D})} - L_{a, d e t} (x_{l, k} = a_{j}), \end{array}

(15)

where $L_{a p p, d e t}$ and $L_{a, d e t}$ represent the a posteriori LLRs and the priori LLRs of the detector, respectively; $P (\cdot)$ denotes the priori symbol probabilities, $a_{j} \in 𝔸$ , $j = 1, . . ., |𝔸|$ , and the binary vector (signal label) corresponding to $a_{j}$ can be expressed as $v_{j}$ . Similarly, the extrinsic LLRs of the decoder $L_{e, d e c}$ are also obtained by subtracting priori LLRs $L_{a, d e c}$ from the a posterioriLLRs $L_{a p p, d e c}$ , and $L_{a, d e c}$ is updated by the extrinsic LLRs of the detector. The iterative process between the detector and the decoder can be described as follows.

Algorithm 1. Algorithm of the iterative process between the detector and the decoder

1: Initialization: Set the number of joint iterations $η = 0$ , the maximum number of joint iterations $η_{m a x}$ , the maximum number of detection iterations $η_{m a x}^{d e t}$ , the maximum number of decoding iterations $η_{m a x}^{d e c}$ , and the priori LLRs $L_{a, d e t} = 0$

2: while $η \leq η_{m a x}$ do

3: Perform detection until $η_{m a x}^{d e t}$ is satisfied.

4: Calculate the a posteriori LLRs $L_{a p p, d e t}$ and the extrinsic LLRs of the detector $L_{e, d e t} (x_{l, k} = a_{j})$ for all $j = 1, . . ., |𝔸|$ , $k = 0, \dots, N - 1$ , $l = 0, . . ., M - 1$

5: Convert the symbol form of $L_{e, d e t} (x_{l, k} = a_{j})$ into the bit form $L_{e, d e t} (c_{i})$ , $i = 0,1, . . ., N_{c} - 1$

6: Deinterleave the extrinsic LLRs $L_{e, d e t} (c_{i})$ , resulting in $L_{e, d e t}^{'} (c_{i})$

7: Set $L_{a, d e c} (c_{i}) = L_{e, d e t}^{'} (c_{i})$ , $i = 0,1, . . ., N_{c} - 1$

8: Perform decoding until $η_{m a x}^{d e c}$ is satisfied

9: Calculate the a posteriori LLRs $L_{a p p, d e c}$ and the extrinsic LLRs of the decoder $L_{e, d e c} (c_{i})$ for $i = 0,1, . . ., N_{c} - 1$

10: if $η < η_{m a x}$ then

11: Interleave the extrinsic LLRs $L_{e, d e c} (c_{i})$ , resulting in $L_{e, d e c}^{'} (c_{i})$

12: Convert the bit form of $L_{e, d e c}^{'} (c_{i})$ into the symbol form $L_{e, d e c}^{'} (x_{l, k} = a_{j})$

13: Set $L_{a, d e t} (x_{l, k} = a_{j}) = L_{e, d e c}^{'} (x_{l, k} = a_{j})$

14: end if

15: $η = η + 1$

16: end while

17: Make decisions of $c_{i}$ according to the a posteriori LLRs $L_{a p p, d e c}$ for $i = 0,1, . . ., K - 1$

2.3 Vectorization Representation of the System

With respect to the vectorization, the following definitions are given, $x_{D D} = v e c (x_{D D}) \in 𝔸^{M N \times 1}$ , $x_{T F} = v e c (X_{T F}) \in ℂ^{M N \times 1}$ ,

$y_{T F} = v e c (Y_{T F}) \in 𝔸^{M N \times 1}$ , and $y_{D D} = v e c (Y_{D D}) \in ℂ^{M N \times 1}$ , where $v e c (\cdot)$ denotes the vectorized version of the 2D matrix formed by stacking the columns of the one into a single column vector. Besides, the N-point DFT matrix and its inverse are represented by $F_{N}$ and $F_{N}^{H}$ respectively and are assumed to be normalized so that $F_{N} F_{N}^{H} = I_{N}$ . According to the introduction of the coded OTFS system in the above subsection, the relationship between the symbol matrix $X_{T F} \in ℂ^{M \times N}$ in the TF domain and $x_{D D} \in ℂ^{M \times N}$ in the DD domain can be described as:

X_{T F} = F_{M} x_{D D} F_{N}^{H}

.(16)

The vectorized form can be expressed as:

x_{T F} = v e c (X_{T F}) = (F_{N}^{H} \otimes F_{M}) x_{D D}

.(17)

Considering that the pulse shaping filter $g_{t x} (t)$ is the rectangular form, the output of the Heisenberg transform is given by

S = F_{M}^{H} X_{T F} = x_{D D} F_{N}^{H}

,(18)

where $S \in ℂ^{M \times N}$ represents the transmitted signal matrix in the time domain. Vectorize $S$ by stacking each column of $S$ into a vector, we have

s = v e c (S) = (F_{N}^{H} \otimes I_{M}) x_{D D}

.(19)

At the receiver, the received signal expressed by Eq. (9) in discrete form is

r (n) = \sum_{i = 1}^{P} h_{i} e^{\frac{j 2 π k_{i} (n - l_{i})}{M N}} s ({[n - l_{i}]}_{M N}) + w (n)

,(20)

where ${[\cdot]}_{M N}$ indicates mod MN operation, $r (n)$ , $s (n)$ and $w (n)$ are the corresponding discrete forms of $r (t)$ , $s (t)$ and $w (t)$ , respectively. Thus, the received signal can be written in the vector form as

r = H_{T} s + w

.(21)

In the above formula, $H_{T}$ is an $M N \times M N$ matrix, given by Ref. [10],

H_{T} = \sum_{i = 1}^{P} h_{i} Π^{l_{i}} Δ^{(k_{i})}

,(22)

where $Π$ is the permutation matrix (forward cyclic shift), and $Δ^{(k_{i})}$ is the diagonal matrix, as shown below.

Π = {[\begin{matrix} 0 & \dots & 0 & 1 \\ 1 & ⋱ & 0 & 0 \\ ⋮ & ⋱ & ⋱ & ⋮ \\ 0 & \dots & 1 & 0 \end{matrix}]}_{M N \times M N}

,(23)

Δ^{(k_{i})} = {[\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & e^{\frac{j 2 π k_{i}}{M N}} & \dots & 0 \\ ⋮ & ⋱ & ⋱ & ⋮ \\ 0 & 0 & \dots & e^{\frac{j 2 π k_{i} (M N - 1)}{M N}} \end{matrix}]}_{M N \times M N}

.(24)

The received signal vector $r$ is devectorized into an $M \times N$ matrix $R$ . Then, the Wigner transform and SFFT can be successive to obtain the received signal matrix $Y_{D D}$ with the size $M \times N$ in the DD domain as follows:

Y_{D D} = F_{M}^{H} (F_{M} R) F_{N} = R F_{N}

.(25)

The vector form can be obtained by

y = v e c (Y_{D D}) = (F_{N} \otimes I_{M}) r

.(26)

Substituting Eqs. (19) and (21) with Eq. (26), we can get the vector form of the input-output relation in the DD domain as follows:

\begin{array}{l} y = (F_{N} \otimes I_{M}) H_{T} (F_{N}^{H} \otimes I_{M}) x_{D D} + \tilde{w} = \\ H_{e f f} x_{D D} + \tilde{w} . \end{array}

(27)

The vectorized forms of each operation are simple and more vivid, which contribute to understanding the OTFS modulation more clearly and they are widely used in the research of the OTFS modulation.

3 Error Performance of the Coded OTFS System

3.1 Error Performance Analysis

PEP is commonly used in communication systems for analyzing the error performance of the system. In Ref. [8], the achievable diversity order of the OTFS system is analyzed based on the PEP under the maximum likelihood (ML) detection. Similarly, the PEP under the ML detection is also used to analyze the error performance of OTFS modulation in Ref. [21]. On this basis, the effective diversity (ED) is introduced from the perspective of PEP^[9]. In Ref. [20], the conditional PEP and the unconditional PEP are utilized to analyze the error performance of coded OTFS systems. And an approximate upper bound on the unconditional PEP for small P is derived by:

P r (x_{D D} \to {\hat{x}}_{D D}) ≲ {(d_{E}^{2} (e) / P)}^{- r} {(\frac{E_{s}}{4 N_{0}})}^{- r}

,(28)

where $e ≜ x_{D D} - {\hat{x}}_{D D}$ is the codeword difference vector, $d_{E}^{2} (e) = e^{H} e$ is the squared Euclidean distance between $x_{D D}$ and ${\hat{x}}_{D D}$ , and r is the rank of the positive semidefinite Hermite matrix $Ω (e)$ given by Eq. (18)^[20]. In Eq. (28), the exponent r and the term $d_{E}^{2} (e) / P$ are regarded as the diversity gain and the coding gain, respectively. According to the early works, e.g. Refs. [1] and [8], OTFS can achieve full diversity, whose order is the number of the separable multipath P. When the channel code is given, the term $d_{E}^{2} (e)$ is also fixed. Thus, as described in Corollary 1 in Ref. [20], the diversity gain increases and the coding gain decreases with the increase of P, which reveals an interesting trade-off between them. In addition, an approximate upper bound on the unconditional PEP for large P is also given by

P r (x_{D D} \to {\hat{x}}_{D D}) ≲ e x p (- \frac{E_{s}}{16 N_{0}} d_{E}^{2} (e))

,(29)

which only depends on the signal to noise ratio (SNR) and $d_{E}^{2} (e)$ , and demonstrates that channels with a large number of resolvable paths approach an AWGN model.

More detailed derivation and illustration can be found in Ref. [20]. It should be noted that the upper bound on the unconditional PEP shown in Eqs. (28) and (29) are approximate. Referring to the appendix A in Ref. [22], the upper bound on the unconditional PEP has a more accurate display.

Note that $Ω (e)$ is also a Gram matrix^[23] corresponding to vectors $\{u_{1}, u_{2}, . . ., u_{P}\}$ , where $u_{i} ≜ Ξ_{i} e$ , and $Ξ_{i} ≜ (F_{N} \otimes I_{M}) Π^{l_{i}} Δ^{(k_{i})} (F_{N}^{H} \otimes I_{M})$ . According to the appendix A in Ref. [22] , the determinant of Gram matrix $Ω (e)$ can be calculated by

\begin{array}{l} d e t (Ω (e)) = G D (\{u_{1}, u_{2}, . . ., u_{P - 1}\}) {‖{\tilde{u}}_{P}‖}^{2} \leq \\ G D (\{u_{1}, u_{2}, . . ., u_{P - 1}\}) {‖u_{P}‖}^{2} \\ ⋮ \\ \leq \prod_{j = 1}^{P} {‖u_{j}‖}^{2} = \prod_{j = 1}^{P} e^{H} e = {(d_{E}^{2} (e))}^{P}, \end{array}

(30)

where ${\tilde{u}}_{j}$ is the orthogonal projection of $u_{j}$ onto the orthogonal complement of span $(u_{1}, u_{2}, . . ., u_{j - 1})$ . Besides, the maximum value of the rank of the matrix $Ω (e)$ is the number of resolvable paths P. In particular, when matrix $Ω (e)$ is full-rank, we have $r = P$ . Then the upper bound on the unconditional PEP of the coded OTFS system can be rewritten as

P r (x_{D D} \to {\hat{x}}_{D D}) \leq {(\frac{d_{E}^{2} (e)}{P})}^{- P} {(\frac{E_{s}}{4 N_{0}})}^{- P}

,(31)

Furthermore, the equality holds if $Ω (e)$ is a diagonal matrix.

3.2 Design Issues of Channel Codes for OTFS Systems

In Ref. [20], the code design criterion for the coded OTFS is given based on the PEP analysis, which is to maximize the minimum squared Euclidean distance of all possible codeword pairs. Simulation results show the performance of the coded OTFS system under convolutional codes with different minimum squared Euclidean distances and verify the proposed code design criterion. At present, most channel codes are designed for AWGN channels. In Ref. [20], authors also reveal that the channel with a large number of diversity paths approaches an AWGN channel when the number of resolvable paths P is large enough. In this case, some good channel codes can be used in the OTFS system. However, the increase of P will bring about large ISI, making signal detection more complicated. Therefore, it is necessary to design channel codes according to the characteristics of the OTFS modulation. In particular, the joint iteration between decoding and detection is needed, when the channel conditions are poor. In a word, the design of the channel coding scheme is still an interesting challenge.

A simple and direct method to analyze coded OTFS systems is to use the extrinsic information transfer (EXIT) chart^[24], which is commonly used to aid the construction of good iteratively-decoded error-correcting codes. EXIT charts are especially popular in the analysis of low-density parity-check (LDPC) codes and Turbo codes. In the most works of coded OFDM systems, the tool, EXIT chart, is also commonly utilized to optimize the performance of iterative decoding, and parameters of the corresponding channel coding scheme and detection, such as Refs. [25–26].

Another possible method is to learn from the code construction method under ISI channels. In general, ISI channels can be conveniently represented by a trellis^[27] or a factor graph^[28]. Codes such as Turbo codes and LDPC codes can also be represented by a trellis or a factor graph. Note this, the channel factor graph and the code factor graph are considered together to obtain the joint channel/code graph in Ref. [29]. The limits of the performance of LDPC codes over binary linear ISI channels are also studied in Ref. [29]. With the use of density evolution, the noise tolerance threshold is calculated. This may provide some reference for the design of coded OTFS system, because the received signals with ISI can also be represented by a trellis or a factor graph.

4 Numerical Results

Numerical results of the considered coded OTFS system are provided in this section. The 5G LDPC code is used, whose code rate and length of information sequence are $R = 1 / 2$ and $K = 1 024$ , respectively. Without loss of generality, quadrature phase shift keying (QPSK) and 16QAM are chosen as the traditional modulation schemes, whose corresponding OTFS frame sizes are $M \times N = 64 \times 16$ and $M \times N = 32 \times 16$ , respectively. In all simulations, the LTV channel with path number $P = 4$ is used, where the path gain follows the Rayleigh distribution with respect to the exponential power delay profile. For the DD domain channel, the indices of delay and Doppler shifts are integers. Moreover, according to 4G LTE and 5G NR, the carrier frequency and subcarrier interval are selected as 4 GHz and 15 kHz, respectively. Thus, we consider the maximum delay index $l_{m a x} = 5$ and the maximum Doppler shift index $k_{m a x} = 1, 2, 3$ , corresponding to the cases in which relative speeds are around 275 km/h, 500 km/h, and 750 km/h, respectively. It should be noted that the delay and Doppler shift indices are generated uniformly at random. At the receiver, the near-optimal symbol-by-symbol MAP detection algorithm^[19] is used, unless otherwise specified. In order to accelerate the iterative convergence between the detector and the decoder, the offset min-sum algorithm (MSA) with an offset factor of 0.5 is adopted by the decoder. The maximum iteration number of the detection is 10, while that of the decoding is 50.

Fig. 3 shows the frame error rate (FER) performances of the uncoded OTFS and OFDM systems with 16QAM and different relative speeds, such as 275 km/h, 500 km/h and 750 km/h. For a fair comparison, we also apply the near-optimal symbol-by-symbol MAP detection^[19] for OFDM systems, which is designed to exploit all the interference (including both ISI and ICI). Unless otherwise specified, this detection algorithm will be used in subsequent simulations of uncoded/coded OFDM systems. As shown in Fig. 3, we first observe that both uncoded OTFS and OFDM systems have good robustness at different relative speeds. This is because the channel coherence time ( $T_{c} = 1 / f_{d} = 0.981, 0.539, 0.360$ ms corresponding to relative speeds of 275 km/h, 500 km/h and 750 km/h, respectively) is longer than the OFDM symbol time ( $T_{s} = 1 / Δ f = 0.067$ ms). The channel variation is slow at considered relative speeds, and the interference between adjacent subcarriers demonstrates similar property. It is also assumed that the channel state information is perfectly known to the receiver. Thus, with the use of the near-optimal symbol-by-symbol MAP detection, all the ISI and ICI can be effectively cancelled. Furthermore, the error performances for OFDM transmission with considered relative speeds are similar. On the other hand, for OTFS transmission, different Doppler shifts caused by different relative speeds do not change the 2D convolution nature of the signal-channel interaction in the DD domain. Therefore, OTFS is insensitive to Doppler effects. In addition, we notice that the OTFS system has a better error performance than the corresponding OFDM system. Moreover, the slope of the FER curve for the OTFS system is greatly higher than that for the OFDM system, which indicates that OTFS enjoys a larger diversity advantage. Those observations align with the findings in Refs. [5] and [20].

Figure 3

Figure 3 FER performance of uncoded OTFS and OFDM systems with 16QAM, where relative speeds are 275 km/h, 500 km/h, and 750 km/h respectively

The FER performances of the coded OTFS and OFDM systems without the joint iteration are also shown in Fig. 4, where the relative speeds are 275 km/h, 500 km/h, and 750 km/h, respectively. The 16QAM modulated symbols are considered in the simulation. Similar to Fig. 3, we observe that the FER performances of both coded OTFS and OFDM systems do not change much with different relative speeds, thanks to the near-optimal MAP detection. Furthermore, compared with uncoded cases, both coded OTFS and OFDM systems enjoy an improved error performance. In addition, we also notice that the error performance of the coded OTFS is much better than that of the coded OFDM. However, it can be noticed that the coding improvement for the OFDM system is more significant compared with that of the OTFS system. Moreover, the FER curve of the coded OFDM system shares almost the same slope as that of the coded OTFS system. This is because OTFS has the potential to achieve the full channel diversity and consequently, channel coding cannot improve the diversity performance very much for OTFS systems. In contrast, OFDM systems rely deeply on the channel coding to achieve the larger diversity gain. Those observations are also consistent with the analysis in Ref. [20].

Figure 4

Figure 4 FER performance of coded OTFS and OFDM systems with 16QAM, where relative speeds are 275 km/h, 500 km/h, and 750 km/h respectively

The FER performance of the coded OTFS system with different joint detection and decoding iterations is compared in Fig. 5, as well as the uncoded OTFS system. The modulation type is QPSK and the relative speed is 500 km/h in the simulation. We can obviously observe that the channel coding significantly improves the error performance. In addition, we notice that the iterations between the detector and decoder do not improve the error performance very much. This is because the near-optimal symbol-by-symbol MAP detection algorithm used in the coded OTFS system exploits all possible interference patterns, and therefore the a priori information from channel decoding cannot improve the extrinsic information from the near-optimal detection for decoding. Consequently, the iterations between the detector and decoder cannot improve the error performance very much.

Figure 5

Figure 5 FER performance of coded OTFS system with different iterations between detection and decoding, where QPSK and relative speed 500 km/h are considered

5 Conclusions

In this paper, the coded OTFS system is introduced and the existing research work of the coded OTFS is summarized. Based on this, the upper bound on unconditional PEP for the coded OTFS system is supplemented, and the design issues of the channel coding scheme are discussed. Furthermore, the performance of the uncoded OTFS and OFDM systems is analyzed, as well as that of 5G LDPC coded systems. Simulation results show that the performance of the OTFS system significantly outperforms that of OFDM system under different speeds, modulation schemes and iterations, whether coded or uncoded.

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