2023.07.10 Although there are already superimposed pilot designs with higher spectral efficiency, this paper can be called a classic of OTFS embedded pilots and is often cited by other papers. After thinking about it, I feel that it is necessary to re-read and record it. learning process. (Note: The part about MIMO has not been explored in depth yet).
[OTFS and Signal Processing: Paper Reading] Embedded Pilot-Aided Channel Estimation for OTFS in Delay–Doppler Channel
I. Introduction
1.1 Written in front
论文题目：Embedded Pilot-Aided Channel Estimation for OTFS in Delay–Doppler Channels
论文来源：IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 5, MAY 2019
论文链接：https://ieeexplore.ieee.org/document/8671740
写在前面：起初拿到这篇论文，主要想要解决两个问题：
1. 从整数扩展到分数多普勒（特别针对论文中使用的阈值估计法而言）做了哪些相应的改进与调整？
2. 不同条件下（整数/分数多普勒以及不同导频结构下）信道估计与检测的表达式分别有何不同？其不同之处所对应的原因是什么？
1.2 中心思想
该论文提出了OTFS嵌入式导频辅助的信道估计方法。简单来看，大致有三个核心工作点：
a. 提出一个新的导频结构（嵌入式导频）。
b. 在接收端采用基于阈值方法实现信道估计。
c. 在b得到信道估计的基础上，引入消息传递算法实现信号检测。最后作者将所提出的方案扩展到了MIMO以及多用户上/下行链路场景。
1.3 INTRODUCTION
这里简要概括下该论文第一部分的基本内容：首先引入OFDM并简要介绍其不足（搞多普勒场景下不再具有鲁棒性并产生严重的性能下降），接着引入OTFS来解决这一问题。之后，介绍了OTFS的研究现状（OTFS信道估计、MIMO等）以及OTFS的调制/解调过程。最后，总结了本文的贡献点以及论文的基本结构。
二、系统模型
The most exciting part about the system model is that the author established the input-output-output relationship of OTFS under integer and fractional Doppler respectively, which can help us understand the differences in the system model after extending the problem to fractional Doppler. Before talking about it, let’s identify a few confusing symbols:
x[k,l] delay-Doppler domain signal, X[n,m] time-frequency domain signal, s(t) time-domain transmitted signal, r (t) Time domain received signal after passing through the channel, h(τ,ν) complex baseband channel impulse response.
2.1 Basic OTFS concepts/symbols
First, sampling is performed at intervals of T and Δf on the time and frequency domain axes to discretize the problem, so that the time and frequency domain plane is broken up into M×N grid points. Perform SFFT transformation on it to obtain a grid point with a resolution of M×N on the delay-Doppler plane, where MΔf is the bandwidth and NT is the duration of the entire data packet:
OTFS modulation/demodulation: This part does not have many formulas and can be summarized with a block diagram.
2.2 OTFS input and output analysis (the highlight is coming!)
Delay-Doppler domain channel: Since the delay-Doppler domain channel is sparse (this is also one of the advantages of OTFS, see the figure below), only a few
parameters are needed to establish it in the delay-Doppler domain. Out of the channel model, the sparse expression of the channel is:
h ( τ , ν ) = ∑ i = 1 P hi δ ( τ − τ i ) δ ( ν − ν i ) (1) \mathrm{h}\big(\tau ,\nu\big)=\sum_{i=1}^{P}\mathrm{h}_{i}\delta\big(\tau-\tau_{i}\big)\delta\big(\nu -\nu_{i}\big)\tag{1}h ( τ ,n )=i=1∑Phid ( t−ti) d ( n−ni)( 1 )
where P is the number of propagation paths,hi, τ i, ν i h_i, τ_i, ν_ihi、ti、niPart IIComplex gain, delay and Doppler frequency shift of i path,l τ i l_{τ_i}lti和k ν i k_{ν_i}kniIt will further be expressed in the form of time delay and Doppler tap, that is: τ i = l τ i M Δ f , ν i = k ν i + κ ν i NT \tau_{i}=\frac{l_{\tau_ {i}}}{M\Delta f},\nu_{i}=\frac{k_{\nu_{i}}+\kappa_{\nu_{i}}}{NT}ti=M D flti,ni=NTkni+ Mrni，− 1 / 2 < κ vi ≤ 1 / 2 −1/2<\kappa_{v_i}≤1/2−1/2<Kvi≤1/2 represents fractional Doppler, which is actually the nearest Doppler tapk ν i k_{ν_i}kniThe offset generated based on .目前认为发送和接收脉冲 g t x ( t ) g_{tx} (t) gtx(t)与 g r x ( t ) g_{rx} (t) grx(t)是理想的，即波形满足双正交性。接下来分别讨论整数和分数多普勒两种情况下的输入输出关系：
case1：整数多普勒频移
从时延-多普勒域来看，信号的输入输出关系为：
y [ k , l ] = ∑ k ′ = − k v k v ∑ l ′ = 0 l τ b [ k ′ , l ′ ] h [ k ′ , l ′ ] x [ [ k − k ′ ] N [ l − l ′ ] M ] + ν [ k , l ] (2) y[k,l]=\sum_{k'=-k_v}^{k_v}\sum_{l'=0}^{l_{\tau}}b[k^{\prime},l^{\prime}]h[k^{\prime},l^{\prime}]x[[k-k^{\prime}]_{N}[l-l^{\prime}]_{M}]+\nu[k,l]\tag{2} y[k,l]=k′=−kv∑kvl′=0∑ltb[k′,l′]h[k′,l′]x[[k−k′]N[l−l′]M]+n [ k ,l]( 2 )
Among them,b [ k ′ , l ′ ] b[k^{\prime},l^{\prime}]b[k′,l′ ]can be regarded as a path indicator,b [ k ′ , l ′ ] = 1 b[k^{\prime},l^{\prime}]=1b[k′,l′]=1 shows that there is a Doppler tapk ′ k'k′ The delay tap isl ′ l'l′ path, otherwise, it is considered that there is no such path, that is,b [ k ′ , l ′ ] = h [ k ′ , l ′ ] = 0 b[k^{\prime},l^{\prime}]= h[k^{\prime},l^{\prime}]=0b[k′,l′]=h[k′,l′]=0； h [ k ′ , l ′ ] h[k^{\prime},l^{\prime}] h[k′,l′ ]is the( k ′ , l ′ ) (k',l')(k′,l′ )paths; the total number of paths is the sum of the number of path indicators in the two dimensions of delay and Doppler, that is:
∑ k ′ = − k ν k ν ∑ l ′ = 0 l τ b [ k ′ , l ′ ] = P (3) \sum_{k^{\prime}=-k_{\nu}}^{k_{\nu}}\sum_{l^{\prime}=0}^{ l_{\tau}}b[k^{\prime},l^{\prime}]=P\tag{3}k′=−kn∑knl′=0∑ltb[k′,l′]=P(3)
case2: Fractional Doppler shift
Similarly, the input-output relationship of the DD domain under fractional Doppler is:
y [ k , l ] = ∑ k ′ = − k ν k ν ∑ l ′ = 0 lvb [ k ′ , l ′ ] ∑ q = − N / 2 N / 2 − 1 h ‾ [ k ′ , l ′ , k ′ , q ] x [ [ k − k ′ + q ] N , [ l − l ′ ] M ] + v [ k , l ] (4 ) {y[k,l]=\sum_{k^{\prime}=-k_{\nu}}^{k_{\nu}}\sum_{l^{\prime}=0}^{l_{ v}}b[k^{\prime},l^{\prime}]\sum_{q=-N/2}^{N/_{2}-1}\overline{h}[k^{\ prime},l^{\prime},k^{\prime},q]x[[kk^{\prime}+q]_{N},[ll^{\prime}]_{M}]+ v[k,l]}\tag{4}and [ k ,l]=k′=−kn∑knl′=0∑lvb[k′,l′]q=−N/2∑N/2−1h[k′,l′,k′,q]x[[k−k′+q]N,[l−l′]M]+v[k,l]( 4 )
Among them, path gain: h ˉ [ k ′ , l ′ , κ ′ , q ] :\bar{h}[k^{\prime},l^{\prime},\kappa^{\prime} ,q]:hˉ[k′,l′,K′,q ]作：
h ‾ [ k ′ , l ′ , κ ′ , q ] = ( ej ⋅ 2 π ( − q − κ ′ ) − 1 N ej 2 π N ( − q − κ ′ ) − N ) h [ k ′ , l ′ ] e − j ⋅ 2 π k ′ + κ ′ NT l ′ M Δ f = α ( q , κ ′ ) h [ k ′ , l ′ ] e − j ⋅ 2 π k ′ + κ ′ NT l ′ M Δ f ′ (5) \overline{\mathrm{h}}[k^{\prime},l^{\prime},\kappa^{\prime},q]=\left(\frac {e^{j\cdot2\pi(-q-\kappa^{\prime})}-1}{Ne^{j\frac{2\pi}N(-q-\kappa^{\prime}) }-N}\right)\mathrm{h}[k^{\prime},l^{\prime}]e^{-j\cdot2\pi\frac{k^{\prime}+\kappa^{ \prime}}{NT}\frac{l^{\prime}}{M\Delta f}}=\alpha\left(q,\kappa^{\prime}\right)\mathrm{h}[k^ {\prime},l^{\prime}]e^{-j\cdot2\pi\frac{k^{\prime}+\kappa^{\prime}}{NT}\frac{l^{\prime }}{M\Delta f^{\prime}}}\tag{5}h[k′,l′,K′,q]=(Yes _jN2 p.m( − q − k′)−Nej ⋅ 2 π ( − q − k′)−1)h[k′,l′]e−j⋅2πNTk′ +Mr′M D fl′=a(q,K′)h[k′,l′]e−j⋅2πNTk′ +Mr′M D f′l′( 5 ) It can be seen from the above formula thatthe path gain mainly has two effects under the action of fractional Doppler: the amplitude is approximated as a sinc function by the impulse response, and the phase is further shifted based on the integer Doppler. Generate IDI. Among them, let the change of amplitude be∣ α ( q , κ ′ ) ∣ |\alpha\left(q,\kappa^{\prime}\right)|∣α(q,K′)∣ , this term can be regarded as the equivalent sampling version of the Dirac sinc function, where q represents the number of additional taps required to approximate the inter-Doppler interference generated by fractional Doppler (simply understood as the Doppler shift quantity), the image is:
It can be seen from the image that when the Doppler offset q is zero, the maximum sampling value is obtained. The code to construct the image is as follows:
clc
clear
q = -10:1:10;
N = 128;
fra_1 = 0.1;
fra_2 = 0.5;
amp_alpha1 = abs((exp(1i*2*pi*(-q-fra_1))-1)./(N*exp(1i*2*pi/N*(-q-fra_1))-N)); %如果仅为“/”则无法表示成多个向量的形式而是一个数
amp_alpha2 = abs((exp(1i*2*pi*(-q-fra_2))-1)./(N*exp(1i*2*pi/N*(-q-fra_2))-N));
figure(3)
semilogy(q,amp_alpha1,'b-*','Linewidth',1);
xlabel('q','FontSize',12);
ylabel('|α(q,k)|','FontSize',12);
hold on
semilogy(q,amp_alpha2,'r-X','Linewidth',1);
grid on
3. Embedded channel estimation (SISO)
This section will answer the second question mentioned in the preface "What are the differences between the expressions of channel estimation and detection under different conditions (integer/fractional Doppler and different pilot structures)? What are the differences corresponding to what is the reason?"
case1: Integer Doppler shift
1. Pilot scheme design
The paper sets three symbols in pilot: pilot signal xp x_pxp, blank guard interval, signal to be sent xd x_dxd, where the area size of the guard interval is determined by the maximum delay and maximum Doppler, and the arrangement is as follows:
x [ k , l ] = { xpk = kp , l = lp , 0 kp − 2 k ν ≤ k ≤ kp + 2 k ν , lp − l τ ≤ l ≤ lp + l τ , xd [ k , l ] otherwise . (6) x[k,l]=\begin{cases}&x_p&k=k_p,l= l_p,\\&0&k_p-2k_\nu\leq k\leq k_p+2k_\nu,\\&&l_p-l_\tau\leq l\leq l_p+l_\tau,\\&x_d[k,l]&\text{ otherwise}.\end{cases}\tag{6}x[k,l]=⎩
⎨
⎧xp0xd[k,l]k=kp,l=lp,kp−2k _n≤k≤kp+2k _n,lp−lt≤l≤lp+lt,otherwise.( 6 )
Then the received symbols are divided into two parts, which are used for channel estimation and signal detection respectively. Thearea used for channel estimation is smaller than the guard interval because the dispersion value of the pilot symbol can only be within± kv ±k_v±kv与l τ l_\taultIn the range. Since the guard interval is set based on the maximum delay and Doppler, it can be guaranteed that data symbols and pilot symbols will not interfere with each other:
2. Threshold channel estimation
Express the received symbols used for channel estimation as (since there is no fractional Doppler and there is only one pilot symbol, there is no need for summation and accumulation):
y [ k , l ] = b [ k − kp , l − lp ] h ^ [ k − kp , l − lp ] xp + v [ k , l ] (7) y[k,l]=b[k-k_{p},l-l_{p}]\hat{h} [k-k_{p},l-l_{p}]x_{p}+v[k,l]\tag{7}and [ k ,l]=b[k−kp,l−lp]h^[k−kp,l−lp]xp+v[k,l]( 7 ) Basic idea:
If the received signal amplitude is higher than the threshold, that is,∣ y [ k , l ] ∣ ≥ Γ |y[k,l]|\geq\Gamma∣y[k,l]∣≥Γ , whereΓ \GammaΓ is the positive value detection threshold, we will consider the path indication itemb [ k − kp , l − lp ] = 1 b[k-k_p,l-l_p]=1b[k−kp,l−lp]=1 , the corresponding path amplitude is:h ^ [ k − kp , l − lp ] = y [ k , l ] / xp \hat{h}[k-k_p,l-l_p]=y[k,l] /x_ph^[k−kp,l−lp]=and [ k ,l]/xp. On the contrary, it is considered that b [ k − kp , l − lp ] = h ^ [ k − kp , l − lp ] = 0 b\big[k-k_{p},l-l_{p}\big]=\ hat{h}\big[k-k_{p},l-l_{p}\big]=0b[k−kp,l−lp]=h^[k−kp,l−lp]=0 . Therefore, results similar to the figure below can be obtained:
Since the determination of the threshold value is closely related to the accuracy of the channel estimation effect, in the following discussion, the threshold value is set to3 σ p 3\sigma_{p}3 pp(Otherwise, σ p = 1 / SNR p = σ 2 / ∣ xp ∣ 2 \sigma_{p}=1/SNR_p=\sigma^2/|x_p|^2pp=1/SNRp=p2/∣xp∣2 represents the effective noise power of the pilot signal). Regarding how to select the threshold value, the paper obtained it through experimental simulation. Figure 10 shows the effect of different fractional Doppler channel estimation threshold values on bit errors under the full guard interval. Performance impact:
3. Signal detection
When the channel information is known, we express the received symbols used for detection as equation (8) (8)( 8 ) , and introduce the MP algorithm for signal detection (the details of the algorithm are not detailed, interested students can refer to the literature [1])
y [ k ⋅ l ] = ∑ k ′ = − k V k V ∑ l ′ = 0 l T b [ k ′ , l ] h ^ [ k ′ , l ] x [ [ k − k ′ ] N ′ [ l − l ′ ] M ] + v [ k , l ] (8) y[k\cdot l]=\sum_{k^{\prime}=-k_{V}}^{k_{V}}\sum_{l^{\prime}=0}^{l_{T}}b[k^{ \prime}, l]\hat{h}[k^{\prime}, l]x\left[\left[kk^{\prime}\right]_{N^{\prime}}\left[ll ^{\prime}\right]_{M}\right]+v[k,l]\tag{8}and [ k⋅l]=k′=−kV∑kVl′=0∑lTb[k′,l]h^[k′,l]x[[k−k′]N′[l−l′]M]+v[k,l](8)
case2: Fractional Doppler shift
According to the difference in spectrum efficiency, two pilot designs are mainly considered under fractional Doppler frequency shift: full protection symbols and partial protection symbols.
⭐Fully protected symbols
1. Pilot structure design
Due to the existence of fractional Doppler, the paper sets full protection symbols in the Doppler dimension, but only sets partial protection symbols in the delay dimension. Different from integer Doppler, in order to ensure that there is no interference between signals used for channel estimation and symbol detection, the guard interval needs to be extended to a larger range in the Doppler dimension (compare pilot design under integer Doppler Scheme) Its arrangement is as shown in the figure below:
After passing through the channel, the received symbols are divided into two parts, the "field" part is used for channel estimation, and the "▲" part is used for signal detection.
2. Threshold channel estimation
Received symbols y [ k , l ] y[k,l] used for channel estimationand [ k ,l]为：
y [ k , l ] = ∑ k ′ = − k v k v b [ k ′ , l − l p ] h ˉ [ k ′ , l − l p , k ′ , [ k p + k ′ − k ] N ] x p + v [ k , l ] (9) y[k,l]=\sum_{k^{\prime}=-k_{v}}^{k_{v}}b\left[k^{\prime},l-l_{p}\right]\bar{h}\left[k^{\prime},l-l_{p},k^{\prime},\left[k_{p}+k^{\prime}-k\right]_{N}\right]x_{p}+v[k,l]\tag{9} and [ k ,l]=k′=−kv∑kvb[k′,l−lp]hˉ[k′,l−lp,k′,[kp+k′−k]N]xp+v[k,l](9)
其中，
y [ k , l ] = b ~ [ l − l p ] h ~ [ [ k − k p ] N , l − l p ] x p + v [ k , l ] (10) \begin{aligned}y[k,l]=\tilde{b}[l-l_p]\tilde{h}[[k-k_p]_N,l-l_p]x_p+v[k,l]\end{aligned}\tag{10} and [ k ,l]=b~[l−lp]h~[[k−kp]N,l−lp]xp+v[k,l]( 10 )
h ~ [ [ k − kp ] N , l − lp ] = ∑ k ′ = − k ν κ ν b [ k ′ , l − lp ] h ˉ [ k ′ , l − lp , κ ′ , [ kp + k ′ − k ] N ] (11) \begin{aligned}\tilde{h}[[k-k_p]_N,l-l_p]&=\sum_{k'=-k_\nu}^{\ kappa_\nu}b[k',l-l_p]\bar{h}[k',l-l_p,\kappa',[k_p+k'-k]_N]\end{aligned}\tag{11}h~[[k−kp]N,l−lp]=k′=−kn∑Knb[k′,l−lp]hˉ[k′,l−lp,K′,[kp+k′−k]N]( 11 )
The following formula can be understood as: summing multiple path indicators in Doppler due to fractional Doppler dispersion(which is a process of turning zero into whole):
b ~ [ l − lp ] = { 1 , ∑ k ′ = − k ν k ν b [ k ′ , l − lp ] ≥ 1 0 , otherwise (12) \tilde{b}[l-l_p]=\begin{cases}1,&\sum_{k '=-k_\nu}^{k_\nu}b[k',l-l_p]\geq1\\0,&\text{otherwise}\end{cases}\tag{12}b~[l−lp]={
1,0,∑k′=−knknb[k′,l−lp]≥1otherwise( 12 )
Comparative formula(4) (4)( 4 ) It can be found that there is less information aboutl τ l_{\tau}ltThe sum of terms is because the paper does not consider the impact of fractional delay, and there is no dispersion in individual pilot symbols in the delay dimension.
The basic idea is the same as integer Doppler. Try to analyze the tap indicator b = 1 b=1b=1 o'clock,The difference between fractional and integer Doppler: Under integer Doppler, we estimate whether there is a separate path under a given delay and Doppler tap; and at this time we estimate whether there is at least one path under a given delay tap (combined with the previous zero to integer Thought). The estimation of path indicator b and effective channel gain h still uses the threshold method: if the symbols y [ k , l ] within the channel estimation range ≥ τ y[k,l]≥\tauand [ k ,l]≥τ is considered to be b=1, h=y/x, and the detailed formula can be found in the figure below:
⭐Partial protection symbols
1. Pilot structure design
In order to further improve spectral efficiency, try to reduce the number of guard intervals in the Doppler dimension (where, k ^ \hat{k}k^ represents the parameter used to measure the reduction of the guard interval in this dimension of Doppler), so the pilot structure design in the figure below is produced:
2. Threshold channel estimation
Received symbols y [ k , l ] y[k,l] used for channel estimationand [ k ,l ] is (and formula(9) (9)( 9 ) The difference is that the interference term L [ k , l ] \mathcal{L}[k,l] isaddedL[k,l]）：
y [ k ⋅ l ] = ∑ k ′ = − k ν k ν b [ k ′ , l − l p ] h ˉ [ k ′ , l − l p , k ′ , [ k p + k ′ − k ] N ] x p + u [ k , l ] + L [ k , l ] (10) y[k\cdot l]=\sum_{k^{\prime}=-k_{\nu}}^{k_{\nu}}b\left[k^{\prime},l-l_{p}\right]\bar{h}\left[k^{\prime},l-l_{p},k^{\prime},\left[k_{p}+k^{\prime}-k\right]_{N}\right]x_{p}+u[k, l]+\color{red}{\mathcal{L}[k,l}]\tag{10} and [ k⋅l]=k′=−kn∑knb[k′,l−lp]hˉ[k′,l−lp,k′,[kp+k′−k]N]xp+u[k,l]+L[k,l]( 10 )
Among them,L [ k , l ] \mathcal{L}[k,l]L[k,l ] is the symbol from the surrounding dataxd x_dxdDefinitely, independently, define:
L [ k , l ] = ∑ k ′ = − k ν k ν ∑ l ′ = 0 l π b [ k ′ , l ′ ] ∑ q ∉ [ kp − 2 k ν − 2 k ^ , kp + 2 k ν + 2 k ^ ] ] h ˉ [ k ′ , l ′ , x ′ , q ] x μ [ [ k − k ′ + q ] N , [ l − l ′ ] M ] (11) \mathcal{L}[k,l]=\sum_{k^{\prime}=-k_{\nu}}^{k_{\nu}}\sum_{l^{\prime}= 0}^{l_{\pi}}b[k^{\prime},l^{\prime}]\sum_{q\not\in\left[k_{p}-2k_{\nu}-2\ widehat{k},k_{p}+2k_{\nu}+2\widehat{k}\right]]}\bar{h}[k^{\prime}, l^{\prime},x^{ \prime},q]x_{\mu}\left[[kk^{\prime}+q]_{N},[ll^{\prime}]_{M}\right]\tag{11}L[k,l]=k′=−kn∑knl′=0∑lpb[k′,l′]q∈[kp− 2 kn−2k
,kp+ 2k _n+2k
]]∑hˉ[k′,l′,x′,q]xm[[k−k′+q]N,[l−l′]M](11)
q ∉ [ k p − 2 k V − 2 k ^ , k P + 2 k V + 2 k ^ ] q\notin\left[k_{p}-2k_{V}-2\widehat{k},k_{P}+2k_{V}+2\hat{k}\right] q∈/[kp−2k _V−2k
,kP+2k _V+2k^ ]shows that the interference comes from signal points outside the blank guard interval. When the Doppler dimension guard interval decreases, that is,k ^ \hat{k}kWhen ^ is small, the interference between symbols will increase, soL [ k , l ] \mathcal{L}[k,l]L[k,l ] increases. The above-mentioned interference is similar to the interference caused by the pilot symbols to the received symbols during full protection symbol signal detection.
case3: OTFS under rectangular waveform
So far, it is believed that both the sending and receiving pulse waveforms are ideal, that is, they satisfy bioorthogonality. However, this does not exist in the actual scenario. Therefore, next we will study the input and output of OTFS when the sending and receiving waveforms are both rectangular waves. relation. In the integer Doppler scenario, the expression of the rectangular pulse input and output symbols adds phase shift on the previous basis and can be rewritten as:
y [ k , l ] = ∑ k ′ = − k ν k ν ∑ l ′ = 0 l π b [ k ′ , l ′ ] h ^ [ k ′ , l ′ ] β [ k , l ] x [ [ k − k ′ ] N ′ [ l − l ′ ] M ] + v [ k , l ] (12) y[k,l]=\sum_{k^{\prime}=-k_{\nu}}^{k_{\nu}}\sum_{l^{\prime}=0}^{ l_{\pi}}b[k^{\prime},l^{\prime}]\hat{h}[k^{\prime},l^{\prime}]\beta[k,l]x \left[\left[kk^{\prime}\right]_{N^{\prime}}\left[ll^{\prime}\right]_{M}\right]+v[k,l] \tag{12}and [ k ,l]=k′=−kn∑knl′=0∑lpb[k′,l′]h^[k′,l′ ]b[k,l]x[[k−k′]N′[l−l′]M]+v[k,l]( 12 )
According to the integral functional equation:
β [ k , l ] = { ej 2 π ( 1 − 1 ′ M ) k ′ N l ′ ≤ l < MN − 1 N ej 2 π ( 1 − 1 ). ′ M ) k ′ N e − j 2 π ( [ k − k ′ ] NN ) 0 ≤ l < l ′ (13) \mathcal{\beta}[k,l]=\begin{cases}\mathrm{e}^{\mathrm{j}2\pi\left(\frac{1-1^{\prime}} {M}\right)\frac{k^{\prime}}{N}}&\mathrm{l'\le l<M}\\\frac{\mathrm{N}-1}{\mathrm{N }}\mathrm{e}^{\mathrm{j}2\pi\left(\frac{1-1^{\prime}}{M}\right)\frac{k^{\prime}}{N }}\mathrm{e}^{-j2\pi\left(\frac{[kk^{\prime}]_{N}}{N}\right)}&\mathrm{0\le l<l' }.\end{cases}\tag{13}b [ k ,l]=⎩
⎨
⎧ej 2 π (M1−1′)Nk′NN−1ej 2 π (M1−1′)Nk′e− j 2 π (N[k−k′]N)l′≤l<M0≤l<l′.( 13 )
Since this part of the phase shift is known, the threshold channel estimation and symbol detection methods mentioned earlier in the paper can be directly used.
4. Simulation results
4.1 Parameter settings
| carrier frequency | subcarrier spacing | Modulation | Channel model | size |
|---|---|---|---|---|
| 4GHz | 15KHz | 4-QAM | EVA[2]Jakes | M=512,N=128 |
4.2 Experimental design
In terms of experiments, the paper mainly conducts experiments in integer/fractional Doppler scenarios:
4.2.1 Integer Doppler
1. Pilot power
Figure 6 compares the impact of different pilot powers on bit error performance under integer Doppler, and designs the maximum delay tap l τ l\taul τ is 20, maximum Doppler tapkv k_vkvis 4 (corresponding to a maximum speed of 120Kmph). It can be seen from the image that when the pilot signal-to-noise ratio is 40dB, its bit error performance can approximate the ideal situation (in addition, it can also be seen that the power of the pilot is compared with the signal power is larger; in subsequent experiments, the author also specifically indicated the set pilot power value, which is necessary).
In addition, since the ISFFT transform enables OTFS to propagate each delay-Doppler domain symbol in the entire time-frequency plane, while the FFT operation of OFDM mainly unfolds in the time domain rather than the frequency domain, OTFS has lower performance than OFDM. When the pilot signal-to-noise ratio of MPAPR (approximately 4dB) is 18dB, the specific results of OTFS MPAPR are shown in the following table (this text is a translation of the original text of the paper. Due to the lack of derivation of the formula, I have no special understanding):
2 . Moving speed
Figure 7 considers different moving speeds of 30/120/500Kmph (corresponding to the maximum Doppler tap kv k_vkvFor the bit error performance of OTFS (1/4/16), it should be noted that the channel overhead at different speeds is different (refer to the table below). It can be seen from the simulation results that OTFS exhibits almost the same bit error performance under different Doppler frequencies.
3. Threshold
The previous experiments on threshold comparison have been supplemented. The optimal threshold Γ = 3 σ p \Gamma=3\sigma_{p} was found through simulation.C=3 pp, and this value is selected as the standard in all experiments of the paper.
4.2.2 Fractional Doppler
Taking fractional Doppler into consideration, the paper also designed experiments similar to the previous ones. Figures 9 and 10 respectively compare the effects of different pilot powers, different thresholds and different moving speeds on bit error performance under fully protected symbols. The conclusions are as follows Similar things as before will not be expanded here. Here is a special introduction. Figure 11 compares the coverage of the guard intervals in the Doppler dimension under some guard intervals (using the parameter h ^ \hat{h}h^ to measure), and the impact of window function on bit error performance.
It can be seen from the simulation results that the bit error performance will increase withh ^ \hat{h}hImproved by the increase of ^ , h ^ \hat{h}hThe bit error performance when ^ =5 is almost the same as the case of fully protected symbols (this conclusion is meaningful for improving spectral efficiency), so we need to consider the tradeoff between spectral efficiency and bit error performance. In addition, in terms of window functions, since the prolate window will cause noise correlation, the rectangular window has better bit error performance.
4.2.3 Low-latency communication
There is an urgent need for low-latency communication in future wireless communications. This paper designs simulation experiments in a low-latency scenario (that is, reducing the size of M and N).
It can be seen from the simulation results that the channel estimation scheme designed in this paper is effective in low-latency scenarios.
5. Summary
The paper mainly talks about the design of the pilot scheme under integer/fractional Doppler and derives the respective received signal expressions. According to the difference in spectral efficiency, two pilot schemes are designed in the Doppler dimension: full symbol protection and partial symbol protection. frequency plan. By comparing the received signal expressions of integer/fractional Doppler and thinking about why special considerations are needed in these places, it can effectively help us further understand the basic characteristics of fractional Doppler. About how to remove or design corresponding receiving methods to avoid the interference term L \mathcal{L}L did not elaborate in detail (maybe I didn’t read carefully enough). All in all, this paper is worth reading and studying carefully. It is recommended that you assist in learning the original text for better results.
references
[1] P. Raviteja, K. T. Phan, Q. Jin, Y. Hong, and E. Viterbo, “Low-complexity iterative detection for orthogonal time frequency space modulation,” in Proc. IEEE WCNC, Barcelona, Spain, Apr. 2018, pp. 1–6.
[2] “LTE; “Evolved Universal Terrestrial Radio Access (E-UTRA); Base Station (BS) radio transmission and reception,” 3GPP TS 36.104 version 8.6. 0 Release 8, Jul. 2009,” ETSI TS.
