Estimation of Doubly-Dispersive Channels in Linearly Precoded Multicarrier Systems Using Smoothness Regularization Andreas Pfadlerz Tom Szollmannz Peter Jungyzand Slawomir Stanczakyz

2025-04-29 0 0 3.08MB 13 页 10玖币

侵权投诉

Estimation of Doubly-Dispersive Channels in Linearly Precoded

Multicarrier Systems Using Smoothness Regularization

Andreas Pfadler∗‡, Tom Szollmann∗‡, Peter Jung†‡ and Slawomir Stanczak†‡

∗Volkswagen Commercial Vehicles, Wolfsburg, Germany

{andreas.pfadler, tom.szollmann}@volkswagen.de

†Fraunhofer Heinrich Hertz Institute, Berlin, Germany

{peter.jung, slawomir.stanczak}@hhi.fraunhofer.de

‡Technical University of Berlin, Berlin, Germany

Abstract—In this paper, we propose a novel channel estimation

scheme for pulse-shaped multicarrier systems using smooth-

ness regularization for ultra-reliable low-latency communication

(URLLC). It can be applied to any multicarrier system with

or without linear precoding to estimate challenging doubly-

dispersive channels. A recently proposed modulation scheme

using orthogonal precoding is orthogonal time-frequency and

space modulation (OTFS). In OTFS, pilot and data symbols are

placed in delay-Doppler (DD) domain and are jointly precoded

to the time-frequency (TF) domain. On the one hand, such

orthogonal precoding increases the achievable channel estimation

accuracy and enables high TF diversity at the receiver. On the

other hand, it introduces leakage effects which requires extensive

leakage suppression when the piloting is jointly precoded with

the data. To avoid this, we propose to precode the data symbols

only, place pilot symbols without precoding into the TF domain,

and estimate the channel coefﬁcients by interpolating smooth

functions from the pilot samples. Furthermore, we present a

piloting scheme enabling a smooth control of the number and

position of the pilot symbols. Our numerical results suggest that

the proposed scheme provides accurate channel estimation with

reduced signaling overhead compared to standard estimators

using Wiener ﬁltering in the discrete DD domain.

Index Terms—channel estimation, smoothness, pulse-shaping,

precoding, OTFS, URLLC

I. INTRODUCTION

Future mobile multicarrier systems have to meet a large

variety of requirements. They are driven by increasingly

demanding applications. Especially, the connectivity of high

mobility devices such as automated vehicles poses a challenge.

Automated vehicles have very strict requirements regarding

the quality of the communication which is commonly referred

to as quality of service (QoS) [1]. In particular, ultra-reliable

low-latency communication (URLLC) plays an important role

in this context [2]. It is essential for automated vehicles that

sufﬁcient QoS parameters, such as latency and data rate, are

reliably provided and this even in high mobility scenarios.

In these scenarios, the wireless channel is considered to be

doubly-dispersive, i.e., varying in both time and frequency. In

addition, efﬁciency plays an essential role due to limitations

of the available spectrum as it is already foreseen that the

5th generation wireless system (5G) cannot fulﬁll future

spectrum needs [3]. For this reason, it is important to aim at

improved efﬁciency during the development of future mobile

multicarrier systems. It does not sufﬁce to focus exclusively

on improvements at higher layers; the physical layer must

also be addressed. For example, it is desirable to reduce

signaling overhead, e.g., the number of pilot signals, and to

increase the reliability of the multicarrier system to avoid

packet retransmissions. In this paper, we focus on physical

layer enhancements by proposing a novel channel estimation

scheme and utilizing linear precoding.

To address those challenges, we need to improve the

transceiver structure of multicarrier systems taking pulse-

shaping ﬁlters into account. Nowadays, orthogonal frequency-

division multiplexing modulation (OFDM) is broadly used,

e.g., in the 4th generation wireless system (4G), 5G, and

wireless local area network (WiFi). OFDM uses rectangular

pulses at the transmitter and receiver ﬁlterbank.With this setup,

time-invariant channels reduce to convolution operators which

are easily manageable, but OFDM suffers signiﬁcant perfor-

mance losses, when the channel is time-variant [4], [5]. In

this context, orthogonal time-frequency and space modulation

(OTFS) has been introduced by Hadani et. al. [6]. It uses the

discrete symplectic Fourier transform (DSFT) as orthogonal

precoding transform to precode symbols over the entire time-

frequency (TF) domain. This approach is very distinct as

data and pilot symbols are both placed in the delay-Doppler

(DD) domain and are jointly orthogonal precoded [7]. Several

studies show that OTFS signiﬁcantly outperforms OFDM in

terms of bit error rate (BER) performance [8]–[11]. This is due

to the fact that the joint orthogonal precoding enables high

TF diversity. In particular the achievable channel estimation

accuracy is increased, since a pilot symbol placed in the DD

domain probes each TF coefﬁcient [12], [13]. However, it also

comes with some disadvantages. Firstly, channel estimation

suffers under leakage effects when it is done in the discrete

DD domain [14]. Secondly, resource allocation becomes less

ﬂexible regarding multiuser aspects [15]. Thirdly, the overhead

for piloting in the uplink grows proportionally to the number

of users [16]. This motivates the approach followed in this

paper, which is to apply precoding to the data but not the

pilot symbols. Although we loose some TF diversity this

way, we gain the ﬂexibility to choose any precoding for the

data symbols without affecting the piloting scheme. In [17],

it is shown that aside from the DSFT any other orthogonal

precoding, i.e., 2D orthogonal transform, yields the same

high TF diversity, e.g., the low-complexity 2D fast Walsh-

Hadamard transform (2D-FWHT).

arXiv:2210.05233v1 [cs.IT] 11 Oct 2022

In particular, the estimation of doubly-dispersive channels

is a very important aspect for future multicarrier systems

especially when TF symbols are precoded. Since the provi-

sion of an accurate channel state information (CSI) and the

usage of an appropriate equalizer is essential to enable high

TF diversity gains. Vehicular channels are considered to be

doubly-dispersive, underspread, and often also to be sparse in

the continuous DD domain following the wide-sense stationary

uncorrelated scattering (WSSUS) model [18]. A channel is

underspread if all delay shifts and Doppler shifts are contained

within a small region, i.e., both are relatively small. The

channel is sparse when only a few point-like scatterers in the

continuous DD domain exist. For pulse-shaped multicarrier

ﬁlterbanks, the inherent sparsity of the channel cannot be

harnessed using any form of discrete Fourier transform (DFT)

for channel estimation in the discrete DD domain [14], [19],

[20]. A common way to estimate the channel is to get the least-

squares (LS) estimator from the pilot samples and to smooth

them by means of the Wiener ﬁltering in the discrete DD

domain, which is commonly referred to as linear minimum

mean square error (LMMSE) estimator or Markov estimator

[21], [22]. This approach however suffers under leakage effects

[20], [23]. Leakage effects are caused by the presence of both

fractional Doppler shifts and fractional delay shifts which are

not consistent with the discrete nature of the DSFT [14].

To cope with leakage effects and to promote sparsity,

more complex estimation schemes are commonly followed.

In this scope, compressed sensing or even super resolution

are possible schemes, see for example [23], [24], respectively.

In [25], Rasheed et al. propose a compressed sensing based

algorithm using orthogonal matching and modiﬁed subspace

pursuit to estimate the time-varying channels. A framework

for sparse Bayesian learning with Laplace priors and a new

piloting scheme has been introduced by Zhao et al. in [26],

where they consider fractional Doppler shifts but not fractional

delay shifts. An off-grid sparse signal recovery to estimate

the original channel rather than the effective discrete channel

in the DD domain is proposed in [27]. In [28], an iterative

optimization method is presented by Liu et. al., where a

message passing signal recovery algorithm is utilized for

channel estimation which takes fractional Doppler shifts but

not fractional delay shifts into account. The listed schemes are

rather complex, require high computing power, and consider

longer time intervals, e.g., are computed adaptively over multi-

ple frames, which does not suite well to URLLC in the context

of rapidly changing vehicular channels, as it is known that

the WSSUS assumption only holds for a limited duration and

bandwidth [29]. This makes channel estimation challenging

and requires channel estimation on a per frame basis [10].

Computationally complex and iterative optimization methods

are therefore not considered in the presented paper.

Focusing on low-complexity estimators for URLLC, a com-

mon choice is the estimation of the channel main diagonal

(CMD) on a per frame basis. This can be done by using

an LMMSE estimator which however suffers from leakage

effects. In this paper, we propose a novel CMD estimator

in the TF domain in contrast to the estimation in the DD

domain used for OTFS and DFT based schemes for OFDM.

We place pilot symbols in the TF domain to enable higher

ﬂexibility and reduced overhead for pilot signaling. However,

the pilot and data symbols still need to be properly arranged

within a rectangular frame. To apply fast orthogonal precoding

transformations, we typically require the input dimension to be

to the power of two which equals the number of data symbols.

Therefore, the placement of the pilot symbols is not obvious.

To control the number and position of the pilot symbols,

we propose an algorithm and a so called accordion pilot

placement to place pilots in between the precoded symbols

in the TF domain. The main contributions of this paper can

be summarized as follows:

•We study pulse-shaped multicarrier systems with linear

precoding for URLLC over doubly-dispersive channels,

•we numerically compare different linear precoding trans-

formations,

•we propose a novel smoothness optimized estimation

scheme of the CMD coefﬁcients which minimizes the

energy of the discrete Hessian and takes the ratio between

the delay spread and Doppler spread, the self-interference

power, and receiver noise into account, and

•we introduce a pilot placement scheme, i.e., accordion

pilot placement, which enables a smooth control of the

number and position of the pilot symbols.

A. Paper Organization

In Section II, the Gabor signaling and doubly-dispersive

channel model is introduced. Linear precoding transforms and

their diversity gain are discussed in Section III. In Section

IV, we detail channel estimation, leakage effects, equalization,

data recovery, and the proposed channel estimation scheme.

The accordion pilot placement is presented in Section V.

In Section VI, we show our numerical results. Finally, we

summarize our conclusions in Section VII.

B. Notational Remarks

Random variable vectors, 2D-arrays and matrices are de-

noted with bold letters. Superscripts (·)∗and (·)Hdenote the

complex conjugate and the Hermitian transpose, respectively.

Let ∗denote the non-cyclic 2D convolution which only returns

the valid part. The column-wise vectorization operator, the

absolute value, the euclidean norm, and the Frobenius norm is

denoted as vec{·},|·|,k·k2, and k·kF, respectively. We denote

δ(·)as the Dirac distribution, as the Hadamard product,

E{·} as expectation operation, and j2=−1. We denote the

indices of down-converted received signal by (¯

·).

II. SYSTEM MODEL

In this section, we introduce the system model which

includes the doubly-dispersive channel and the input-output

mapping of the information resources. We use a time-

continuous Gabor (Weyl-Heisenberg) signaling to derive a dis-

crete system model for the pulse-shaped multicarrier scheme.

We deﬁne the Gabor grid Λ = FZM×TZNwith frequency

step size F > 0and time step size T > 0.The indices

I=ZM×ZNrun over the cyclic groups ZM=Z/M Z

(integers of modulo M) and ZN=Z/NZ(integers of modulo

N) taking in total Mfrequency steps and Ntime steps into

account. The overall frame duration Tfand bandwidth Bare

given by the products T N and F M, respectively. Regular

Gabor grids can be categorized into three types depending

on their TF product T F : Oversampling if T F < 1, critical

sampling for T F = 1, and undersampling if T F > 1.

Let us denote the complex-valued pulse-shaping ﬁlters for

synthesis and analysis as γ(t)and g(t), respectively. We design

the pulses to be biorthogonal to obtain a perfect reconstruction

in the absence of noise and channel distortions, i.e.,

Zg(t)∗γ(t−nT )e2πjmF t dt=(1, m =n= 0

0,else ,(1)

At the receiver the orthogonality is typically lost due to

channel dispersion which in turn causes self-interference [4],

[10], [30]–[34].

At the transmitter, the Gabor ﬁlterbank uses the synthesis

pulse γ(t)to synthesize the transmit signal, i.e.,

fTx(t):=X

(m,n)∈I

xm,nγ(t−nT )e2πjmF t,(2)

where x={xm,n}(m,n)∈I is the 2D-array of the TF symbols

containing data and pilot symbols. The data symbols are

modulated and encoded sequences of letters from a given

alphabet generated by an information source. In contrast to

the data symbols, the pilot symbols are known at the receiver

and are coming from a different alphabet.

The doubly-dispersive channel model in the continuous DD

domain with a total of Rmultipaths can be expressed as

η(τ, ν):=X

r∈J

ηrδ(τ−τr)δ(ν−νr),(3)

where the index set J={1, . . . , R}associated with each path

corresponds, respectively, to the delay shifts τr, the Doppler

shifts νr, and the complex-valued attenuation factors ηr. The

assumption of the channel being underspread implies that all

tuples (τr, νr)are contained within a small region referred to

as spreading region U ⊂ [0, τmax]×[−νmax, νmax]such that

|U| = 2τmaxνmax 1, where τmax and νmax correspond to the

largest delay spread and largest Doppler spread, respectively

[18]. In the time domain, the channel in (3) acts on the transmit

signal in (2) as a time-varying convolution. Hence the received

signal yields

fRx(t):=X

r∈J

ηrfTx(t−τr)e2πjνrt.(4)

The receiver analyzes the signal using another Gabor ﬁlter-

bank. We assume it uses the same Gabor grid as the transmitter

and can only differ in the choice of the analysis pulse g(t).

Then, we can describe the measured 2D-array of the TF

symbols y={y¯m,¯n}( ¯m,¯n)∈I by

y¯m,¯n=Zg∗(t−¯nT )e−2πj ¯mF tfRx(t) dt+w¯m,¯n,

=ZZZg∗

(t−¯nT )ej2πt(ν−¯mF )η(τ, ν)fTx(t−τ)dτ dνdt+w¯m,¯n,

r∈J

ηrZg∗(t−¯nT )e2πjt(νr−¯mF )fTx(t−τr) dt

| {z }

=:y¯m,¯n(τr,νr)

+w¯m,¯n,

(5)

where y(τ, ν) ={y¯m,¯n(τ, ν)}( ¯m,¯n)∈I is the 2D-array of the

receiver response to a single unit amplitude scatterer where τis

the delay shift, νis the Doppler shift, and w={w¯m,¯n}( ¯m,¯n)∈I

is the 2D-array of the noise. In our system model, we assume

that the measured noise samples w¯m,¯nare uncorrelated zero-

mean random variables with variance σ2>0. The unit

receiver response in (5) further evaluates to

y¯m,¯n(τ, ν) = Zg∗(t−¯nT )ej2πt(ν−¯mF )

×X

(m,n)∈I

xm,nγ(t−τ−nT )e2πjmF (t−τ)dt,

(m,n)∈I

xm,n

×Zg∗(t−¯nT )γ(t−τ−nT )e2πj(tν−¯mF t+mF t−mF τ )dt

| {z }

=:φ(m,n),( ¯m,¯n)(τ,ν)

(6)

where φ(τ, ν) ={φ(m,n),( ¯m,¯n)(τ, ν)}(m,n),( ¯m,¯n)∈I is the ef-

fective channel matrix corresponding to a single unit amplitude

scatterer. It can be written as

φ(m,n),( ¯m,¯n)(τ, ν)=e2πj(¯nT ν−mF τ+T F ¯n∆m)

×Zg∗(t)γ(t−τ−∆nT )e2πjt(ν+∆mF )dt, (7)

where ∆n=n−¯nand ∆m=m−¯mfor convenience.

Observe that the integral in (7) corresponds to the cross

ambiguity function of γand gwhich we deﬁne as

Aγ,g(τ, ν):=Zg(t)∗γ(t−τ)e2πjνtdt. (8)

The 2D-array of CMD coefﬁcients

h(τ, ν) ={h¯m,¯n(τ, ν)}( ¯m,¯n)∈I with respect to a single

unit scatterer for ∆m= 0 and ∆n= 0 is given as

h¯m,¯n(τ, ν):=φ( ¯m,¯n),( ¯m,¯n)(τ, ν).(9)

Due to the assumption of an underspread channel, the diag-

onal elements of the effective channel matrix are dominant

[31]–[34]. This motivates the use of CMD estimation which

is signiﬁcantly less complex than maximum-likelihood esti-

mation or iterative interference cancellation methods. Exact

orthogonality of the pulses in the integral of (7) would imply

that the effective channel matrix reduces to a diagonal matrix.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

EstimationofDoubly-DispersiveChannelsinLinearlyPrecodedMulticarrierSystemsUsingSmoothnessRegularizationAndreasPfadlerz,TomSzollmannz,PeterJungyzandSlawomirStanczakyzVolkswagenCommercialVehicles,Wolfsburg,Germany{andreas.pfadler,tom.szollmann}@volkswagen.deyFraunhoferHeinrichHertzInstitute,Berlin,...

展开>> 收起<<

Estimation of Doubly-Dispersive Channels in Linearly Precoded Multicarrier Systems Using Smoothness Regularization Andreas Pfadlerz Tom Szollmannz Peter Jungyzand Slawomir Stanczakyz.pdf

共13页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Estimation of Doubly-Dispersive Channels in Linearly Precoded Multicarrier Systems Using Smoothness Regularization Andreas Pfadlerz Tom Szollmannz Peter Jungyzand Slawomir Stanczakyz

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: