Abstract
OFDM suffers from intercarrier interference (ICI) when the channel is time varying. This article seeks to quantify the amount of interference resulting from wideband OFDM channels, which are assumed to follow the multiscale multilag (MSML) model. The MSML channel model results in full channel matrices both in the frequency and time domains. However, banded approximations are possible, leading to a significant reduction in the equalization complexity. Measures for determining whether a timedomain or frequencydomain approach should be undertaken are provided based on the interference analysis, and we propose to use the conjugate gradient (CG) algorithm to equalize the channel iteratively. The suitability of a preconditioning technique, that often accompanies the CG method to accelerate the convergence, is also discussed. We show that in order for the diagonal preconditioner to function properly, optimal resampling is indispensable.
Introduction
With many desirable properties such as a high spectral efficiency and inherent resilience to the multipath dispersions of frequencyselective channels, the orthogonal frequency division multiplexing (OFDM) technology shows attractive features to wireless radio applications [1]. OFDM relies on the assumption that the channel stays constant within at least one OFDM symbol period. When Doppler effects due to temporal channel variation cannot be ignored, this assumption does not hold any more since the communication channel becomes time varying. The Doppler effects corrupt the orthogonality among OFDM subcarriers by inducing nonnegligible intercarrier interference (ICI) [2], and can therefore severely deteriorate the system performance. For traditional terrestrial radio systems, compensation of ICI in channel equalization has widely been researched for narrowband systems. Due to the small relative signal bandwidth (actual bandwidth divided by the center frequency) of narrowband systems, the Doppler effects can be modeled primarily by frequency shifts [3,4], in which case it is reasonable to assume that each OFDM subcarrier experiences a statistically identical frequency offset [2]. Consequently, the effective channel matrix of a narrowband OFDM system in the presence of Doppler can be approximated as banded. Efficient equalization schemes for such a banded channel matrix have been studied in, e.g., [57].
In a wideband system, where the relative signal bandwidth is large, the Doppler effects should be more appropriately modeled as scalings of the signal waveform [3,4]. Wideband systems arise in, e.g., underwater acoustic (UWA) systems or wideband terrestrial radio frequency systems such as ultra wideband (UWB). Due to multipath, a wideband linear timevarying (LTV) channel can be more accurately described by a multiscale multilag (MSML) model [3,8]. Many signaling schemes have been studied for wideband systems. For instance, [9,10] consider directsequence spread spectrum (DSSS). Recently, the use of OFDM for UWA or UWB has received considerable attention. To counteract the scaling effect due to Doppler, [11] proposes a multiband OFDM system such that within each band, the narrowband assumption can still be valid. More commonly, many works assume a singlescale multilag (SSML) model for the wideband LTV channel. Based on the SSML assumption, after a resampling operation the channel can be approximated by a timeinvariant channel but subject to a carrier frequency offset (CFO) [12,13]. However, since the channel should be more accurately described by an MSML model, determining the optimal resampling rate is not trivial [14].
In this article, we consider OFDM transmission based on an MSML model. The resulting channel, which is a full matrix in the presence of Doppler, will be equalized by means of the conjugate gradient (CG) algorithm [15], whose performance is less sensitive to the condition of the channel matrix than, e.g., a leastsquares approach. On the other hand, the convergence rate of CG is inversely proportional to the channel matrix condition number. This is especially of significance if a truncated CG is to be used in practice, which halts the algorithm after a limited number of iterations in order to reduce the overall complexity. Therefore, it is desired that the channel matrix is wellconditioned to ensure a fast convergence. To this end, preconditioning techniques can be invoked to enforce the eigenvalues of the channel matrix to cluster around one [16]. To achieve a balance between performance and complexity, we restrict the preconditioner to be a diagonal matrix, whose diagonal entries can be designed by following the steps given in [17]. We notice that a circulant preconditioner in the time domain was introduced in [18], which is equivalent to a diagonal preconditioner in the frequency domain. This preconditioner is introduced based on a basis expansion model (BEM), which is often used to approximate the channel’s timevariation for a narrowband system. For a wideband system as considered in this article, it can be shown that this preconditioner in the frequency domain is equal to the inverse of the diagonal entries of the frequencydomain channel matrix.
What is not considered in [17,18] is the resampling operation at the receiver, which is an indispensable and crucial step for wideband LTV channels. Different from the trivial resampling scheme for SSML channel models, an optimum resampling method is proposed in [14] for MSML channels, which aims at minimizing the average error of approximating the MSML channel by an SSML model. This article studies the resampling from a preconditioning point of view. It is observed that if the major channel energy is located on the offdiagonals of the channel matrix, a diagonal preconditioner will deteriorate the channel matrix condition rather than improve it, thereby reducing the convergence rate of CG instead of increasing it as opposed to the claim of [17]. The energy distribution of the channel matrix is governed by the resampling. Different from [14], which only considers rescaling the received signal, and [19], which considers both rescaling and frequency synchronization, this article will show that for OFDM systems, all these three resampling parameters can have a significant impact on the system performance (i.e., rescaling, frequency synchronization and time synchronization). More specifically, we will extend the results of [19,20] by jointly optimizing these three resampling parameters both in the frequency domain and the time domain.
Notation: Upper (lower) boldface letters stand for matrices (vectors); superscripts T, H, and ∗ denote transpose, Hermitian transpose and conjugate transpose, respectively; we reserve j for the imaginary unit, <k> and ⌈k⌉ for integer rounding and ceiling of a number k, ∥x∥_{2} for the two norm of the vector x, ∥A∥_{Fro} for the Frobenius norm of the matrix A, [A]_{k,m} for the (k,m)th entry of the matrix A; diag(x) for a diagonal matrix with x on its main diagonal, and ⊙ for the Hadamard product of two matrices.
System model based on an MSML channel
Continuous data model
Suppose that the baseband transmit signal s(t) consists of K subcarriers, and can be written as
where the data symbol b_{k} is modulated on the kth subcarrier f_{k }=kΔf, for k = 0,1,…,K−1, with Δf being the OFDM subcarrier spacing. With T = 1/(kΔf), KT is the effective duration of an OFDM symbol. The cyclic prefix and postfix are given
as T_{pre} and T_{post}, respectively. The cyclic prefix is assumed to be longer than the delay spread and
the cyclic postfix is long enough to ensure signal completeness in case of scaling,
which will be defined later on. The rectangular pulse u(t) is defined to be 1 within t∈[−T_{pre},KT + T_{post}] and 0 otherwise. Prior to transmission, s(t) is upconverted to passband, yielding
The considered signal is transmitted over a wideband LTV channel, which is assumed
to comprise multiple resolvable paths. The lth path can mathematically be characterized by the following three parameters:
With a collection of L + 1 paths, the actual received signal
where
The equivalent complex baseband received signal of
where w(t) stands for the baseband noise. By substituting (1) in the above, we can rewrite y(t) as
where
which stands for the timevarying channel frequency response seen by the kth subcarrier. From the definition of h_{k}(t), we notice that the kth subcarrier experiences a frequency offset of (α_{l}−1)(f_{c} + f_{k}) over the lth path.
Remark 1
The cyclic prefix is assumed to be longer than the delay spread and the cyclic postfix
has a duration long enough to ensure signal continuity in the observation window for
t∈[0,KT]. Specifically, it is required that u(α_{l}t−α_{l}τ_{l})=1 within this window for all paths. In other words, because u(α_{l}t−α_{l}τ_{l}) gives a time support on
When the above conditions are satisfied, we are allowed to drop the notation of the rectangular pulse u(t) embedded in h_{k}(t) in the sequel for the sake of notational ease.
Discrete data model
For MSML channels, discretizing the received signal and achieving time/frequency synchronization is not trivial [12,14]. We illustrate such difficulty in Figure 1, where we assume the transmit signal propagates via three paths. Since the received symbol is the summation of these three paths, it invites the following questions:
1. Which point should we consider as the starting point of the OFDM symbol (time synchronization)?
Figure 1. Illustration of the synchronization and resampling problem; α_{l }stands for the scaling factor due to the lth path, and β for the rescaling factor adopted by the receiver during resampling.
2. What sampling rate should we adopt to discretize the received signal over MSML channels (rescaling)?
3. What frequency shift should we apply to remove the residual carrier frequency offset (frequency synchronization)?
These problems can mathematically be described by determining β, ϕ and σ in the following expression
where β is a positive number within [1,α_{max}] and βT represents the sampling rate at the receiver; σis the time shift factor, which is used to represent time synchronization; and likewise,
ϕ is the phase shift factor used for frequency synchronization.
After resampling, the noiseless sample obtained at the nth time instance in the time domain is given by (see Appendix 2 for the detailed derivation)
where we use
to denote the normalized carrier frequency and
to denote the normalized delay of the lth path; and the discrete channel coefficient is given by
In (9), the term
Let us now stack the received samples
where F_{α }denotes a fractional normalized discrete Fourier transform (DFT) matrix, whose (m,k)th entry is defined as
Obviously, F_{1 }reduces to a regular normalized DFT matrix. In addition,
and
where the superscript (β,ϕ) in
Interference analysis
Normally speaking, equalization of an OFDM channel is implemented in the frequency
domain. To this end, the received signal
where
with
where
It is obvious from (16) that in the absence of Dopper effects, i.e., α_{l }= 1 for l = 0,1,…,L, no rescaling and frequency synchronization is necessary, hence β = 1 and ϕ = 0, which leads to a diagonal
Clearly, such an offset is not only dependent on the Doppler spread α and the carrier frequency f_{c}, but also on the subcarrier frequency f_{k }= kΔf. The dependence of the signal energy offset on the subcarrier index is unique to wideband channels, and is also referred to as nonuniform Doppler shifts in [13]. In contrast, the frequency offset for narrowband channels is statistically identical for all the subcarriers [2].
The Dirichlet kernel in (16) also suggests that the signal energy is mostly concentrated
in subcarrier
where γ is a positive threshold no larger than 1. In the left plot of Figure
2, the relationship between
Figure 2. Bandwidth of
Since each
Figure 3. Illustration of the FD matrix
which is independent of σ. We refer the reader to Figure
3 for the physical meaning of the notations. It is important to underscore that since
the bandwidth
as the banded approximation of
With the banded approximation, let us rewrite (14) as
where
The above analysis can also be applied in the time domain in an analogous manner.
See Appendix 3 for the details. Here we only want to highlight that, different from
the energy distribution in the FD channel matrix which is influenced by the rescaling
factor β and the phaseshift factor ϕ[c.f. ξ_{l,F1} and ξ_{l,F2}in (16)], the energy distribution in the TD channel matrix is affected by the rescaling
factor β and the timeshift factor σ[c.f. ξ_{l,T1} and ξ_{l,T2} in (39)]. However, similarly as the FD channel matrix, we can also understand from
the right subplot of Figure
2 that
Channel equalization scheme
Let us now focus on the channel frequencydomain equalization, which is depicted in
Figure
4. In this figure, it is clear that, prior to the equalization, we propose an optimum
resampling operation to achieve
Figure 4. Depiction of our equalization scheme.
Additionally, we would like to highlight that just as a singlecarrier channel can be equalized in the frequency domain, it is also possible to equalize an OFDM channel in the time domain. Due to the similarity, we again refer the reader to Appendix 3 for a detailed mathematical derivation of the timedomain method. The question in which domain the wideband channel should be equalized, shall be addressed in the following section.
Iterative equalization
To better motivate the other components of our equalization scheme, we first introduce the channel equalization method itself. A zeroforcing equalizer in the frequency domain is considered, given by
where
Besides, the matrix inversion in (22) will be implemented iteratively using the CG
algorithm. An advantage of using CG rather than inverting the matrix directly is that
the resulting data estimates yielded by CG are always constrained in the Krylov subspace,
making its performance less susceptible to the spectral distribution of
from which an estimate of
The optimal design of C_{F}can be exhaustive
[23]. Inspirited by
[17], we find our preconditioner by minimizing a cost function based on the Frobenius
norm, which clusters most of the eigenvalues of
Diagonal preconditioning
In this section, we will show that the normal approach to design the diagonal preconditioner as described in [17] will not necessarily cluster eigenvalues around one. To realize this, let us consider the diagonal preconditioner C_{F,⋆} that minimizes the cost function in the Frobenius norm [17] given by
which leads to
where e_{k} is the kth column of the identity matrix.
One problem of the above diagonal preconditioner designed by (25) is that the eigenvalues may, in some situations, tend to cluster around zero instead of one, with the consequence that the condition number of the preconditioned channel matrix increases considerably. To understand this, assume there exists a ε_{1}>0 such that
for k={0,1,…,K−1}. At the same time, assume there exists a ε_{0}>0 such that
for k∈{0,…,K−1}.
If we denote the kth eigenvalue of the preconditioned channel matrix
which means that all μ_{k}’s lie inside a disk of radius
which implies that all μ_{k}’s at the same time lie inside a disk of
With c_{F,k,⋆}defined in (25), we can show that
and
Obviously, if
Figure 5. Left plots: eigenvalues with and without preconditioning; Right plots: convergence performance with and without preconditioning; FD matrix for top two plots corresponds to the original channel, FD matrix for bottom two plots is obtained after our optimum resampling; The MSML channel is set according to Table 1.
Table 1. Channel I: a frequencydomain case
To evaluate the impact of such a preconditioner on the convergence of CG, we compute the mean squared error (MSE) as
with
To alleviate this problem, we adapt the diagonal preconditioner in (24) and (25) as follows
where
In Section ‘Optimal resampling’, we will show how to enhance (32) with a higher probability by means of optimal resampling.
Optimal resampling
From the previous subsections, we understand that the effectiveness of a diagonal preconditioner depends on the energy distribution of the channel matrix. It is desired that the channel matrix should have most of its energy concentrated on the main diagonal. The analysis in Section ‘Discrete data model’ learns that the resampling operation (βϕσ) plays an important role in governing the energy distribution of the channel matrix, and so far we have left (βϕσ) open for choice. Recall that resampling is a standard step taken in many wideband LTV communication systems to compensate for the Doppler effect. For example, optimizing β is considered in [14], while βand ϕ are jointly optimized in [21]. In this sense, the optimal resampling proposed in this article can be considered as a generalization of [14,21].
Next, we shall discuss how to jointly optimize the resampling parameters (β,ϕ,σ). Focusing on the FD matrix
To this end, let us denote the diagonal energy ratio as
and define our resampling operation by solving
which leads to the maximal ratio
Since the energy governing mechanism is determined by the sinc function as indicated
in (16), we can equivalently rewrite (34) by only maximizing the diagonal energy of
where again
To illustrate our resampling approach in the frequency domain, we consider the channel
example specified in Table
1, where we also compare the properties of the resampled FD channel (i.e., the condition
number and diagonal power ratio of the channel matrix) with the original MSML FD channel.
A geometric interpretation may help to understand our resampling operation since β rotates the FD matrix through
In the lower plots of Figure 5, we show the effectiveness of diagonal preconditioning applied to the resampled channel in Table 1. It is clear that, after our resampling procedure, the diagonal preconditioner clusters the eigenvalues of the preconditioned FD channel matrix closer to one than without preconditioning, which further reduces the condition number from 23.36 to 7.17. In contrast, without optimal resampling, the preconditioner “wrongly” pushes the eigenvalues closer to zero. In this case, the matrix condition number increases from 4.26×10^{5} to 1.19×10^{6}, and hence the CG equalizer performs even worse than without preconditioning as shown in the top two plots of Figure 5.
Similarly, we can show that optimal resampling can also improve the performance of the CG in the time domain, for which we just provide Table 2 and Figure 6 here due to space limitations. From them, we can make the same observations as from Table 1 and Figure 5 for the frequency domain case.
Figure 6. Left plots: eigenvalues with and without preconditioning; Right plots: convergence performance with and without preconditioning; TD matrix for top two plots corresponds to the original channel, TD matrix for bottom two plots is obtained after our optimum resampling; The MSML channel is set according to Table 2.
Table 2. Channel II: a timedomain case
Frequencydomain or timedomain equalization?
In the previous sections, we showed that the equalization of an OFDM channel can be
implemented in either the frequency or the time domain. With the CG algorithm specified
in Appendix 4, it is clear that the cost of equalization in the frequency domain will
be upperbounded by
However, the evaluation of
where we reasonably assume
which indicates that the Doppler scale spread is welllimited, it follows that
which suggests that if the maximum difference between the Doppler shifts of each path
(i.e.,
To illustrate the above idea, we again use the channel examples specified in Tables 1 and 2, respectively. We use B_{rul}=5 to roughly capture γ=98% of the channel energy in both domains where γ is introduced in (18). In this way, we have ε≈0.10<1 for the channel in Table 1, while for the channel in Table 2, we have ε≈2.00>1.
For both channels, we compare the equalization performance in different domains. OFDM
with K=128 subcarriers using QPSK is transmitted and the receiver is assumed to have perfect
channel knowledge. We examine the bit error rate (BER) results of our CG equalization
with a fixed CG iteration number (e.g., i_{F,max}=i_{T,max}=100). We use different bandwidths for the banded approximation
The left subplot of Figure
7 plots the BER performance as a function of signaltonoise ratio (SNR) for Channel
I. Note that (β_{F,⋆},ϕ_{F,⋆},σ_{F,⋆})=(1.015,−0.015,−15) and (β_{T,⋆},ϕ_{T,⋆},σ_{T,⋆})=(1.015,−0.016,0.00) for this channel. It can be seen that the performance of the
FD equalizer (FDE) based on
The BER performance for Channel II is illustrated in the right subplot of Figure 7, where the optimal resampling parameters are (β_{T,⋆},ϕ_{T,⋆},σ_{T,⋆})=(1.016,−0.021,−1) and (β_{F,⋆},ϕ_{F,⋆},σ_{F,⋆})=(1.016,−0.016,−3). In this case, it is evident that the TD equalizer is more appealing.
These observations made for the channels in Tables 1 and 2 confirm our metric ε for determining which domain is more suitable for channel equalization. Additionally, we like to point out that, in either domain, with a larger bandwidth the BER performance of our CG equalization will be increased.
Numerical results
In this section, we randomly generate two different types of wideband channels as
specified in Table
3: ε<1 (Case I) represents wideband LTV channels where the Doppler differences among the
multipath are more pronounced than the delay differences; and ε>1 (Case II) is the case where the Doppler differences among the multipath are less
pronounced than the delay differences. For all simulations, OFDM with K=128 subcarriers is considered with QPSK. The wideband channels are assumed to have
L=5 paths, whose channel gains (i.e.,
Table 3. Channel parameters
In Figure 8, the convergence of the CG equalization is plotted in terms of the bit error rate (BER) against the number of iterations at SNR=30 dB for Case I. Since ε<1, frequencydomain equalization (FDE) is carried out. It is clear that the receiver, which simply adopts a diagonal preconditioner in (25) without resampling, performs worst. The performance is already considerably improved if optimal resampling is applied. Moreover the use of our preconditioner given by (31) boosts the performance even further.
Figure 8. BER versus number of iterations for Case I channels at SNR = 30 dB.
The proposed resampling and preconditioning method can also benefit from other Krylovbased algorithms. For instance, the LSQR algorithm exploiting a full channel matrix is studied in [18]. Note that [18] focuses on a narrowband LTV system where no resampling is required. Further, the preconditioner given in [18] is based on a truncated basis expansion model (BEM) which is usually used for the approximation of a narrowband timevarying channel. Because it is not clear whether such a truncated BEM is still suitable for a wideband LTV channel, in order to emulate a similar approach as in [18] for constructing the preconditioner, we utilize a (trivial) fullorder criticallysampled complex exponential BEM (the CCEBEM [25]) in the simulation. The preconditioner in [18] then boils down to the inverse of the diagonal of the frequencydomain channel matrix, which is obviously suboptimal in the Frobenius norm sense. Consequently, it is no surprise that directly applying the equalizer of [18] to wideband LTV channels yields a bad performance as shown in Figure 8. In comparison, the LSQR algorithm benefiting from the optimal resampling and our preconditioner renders the fastest convergence rate and lowest BER amongst all the equalization schemes. Of course, such an improved BER performance is achieved by leveraging the full channel matrix at the cost of a higher complexity, compared to our proposed method using banded matrices.
Figure 9 exhibits the BER versus SNR for the CGbased equalization schemes, where a truncated CG is used which halts at the 5th iteration. It can be seen in the figure that the equalizer leveraging the full channel matrix gives the best BER performance but inflicts more complexity. When using a banded channel matrix approximation, the frequencydomain approach performs much better than the timedomain approach because we have ε<1 for this type of channel. Additionally, the equalization approach in [14] is carried out and its performance is also shown in Figure 9. As we discussed earlier, the resampling operation in [14] is solely focused on the rescaling parameter ignoring the impact of frequency and time synchronization, which is therefore suboptimal. Besides, the equalizer in [14] approximates the channel matrix to be diagonal (i.e., using a bandwidth of one for the banded matrices), and thus its performance becomes inferior in the presence of higher scale differences among the multipath as in the tested channel here.
Figure 9. BER versus SNR for Case I channels.
The performance of the equalizers for Case II is depicted in Figure 10, where the significance of optimal resampling and our adapted preconditioner is again illustrated just like in Figure 8. Similarly, we can see that the LSQR algorithm in [18] also works well for this type of channel if optimal resampling and preconditioning are included.
Figure 10. BER versus number of iterations for Case II Channels at SNR = 30dB.
Different from Case I, the channels of Case II are subject to a larger delay spread than a Doppler spread (i.e., ε>1). In this case, a timedomain equalizer will be more effective than its frequencydomain counterpart as validated in Figure 11. The equalizer in [14] yields a much worse performance than ours since the Doppler scale spread differences in this case are even higher than for Case I.
Figure 11. BER versus SNR for Case II channels.
Conclusions
In this article, we have discussed iterative equalization of wideband channels using the conjugate gradient (CG) algorithm for OFDM systems. The channel follows a multiscale multilag (MSML) model, and suffers therefore from interferences in both the frequency domain and time domain. To lower the equalization complexity, the channel matrices are approximated to be banded in both domains. A novel method of optimal resampling is proposed, which is indispensable for wideband communications. A diagonal preconditioning technique, that accompanies the CG method to accelerate the convergence, has also been adapted to enhance its suitability. Experimental results have shown that our equalization scheme allows for a superior performance to those schemes based on a singlescale resampling method, without any resampling operation, or using a traditional preconditioning procedure. In addition, we gave a simple criterion to determine whether to use a frequencydomain or timedomain equalizer, depending on the channel situation, to obtain the best BER performance with the same complexity. Such a criterion is also validated by experiments.
Appendix 1
Detailed derivation of the discrete data model
Here we give the derivation of (9), assuming no noise is present. We start from (8) given by
where h_{k}(t) is defined in (5) and the embedded u(t) in h_{k}(t) is considered to be one for the concerned observation window as clarified in Remark 1.
Now, we substitute h_{k}(t) to obtain
where the channel coefficient is given by
Now, if we denote
for the normalized carrier frequency and
for the normalized delay of the lth path, we have
which gives (9).
Appendix 2
System model in the time domain and timedomain equalization
To derive the timedomain model, let us rewrite (10) as
where
with
where
Observing the analogy between (16) and (39), a similar interference analysis can be made on H_{T}. By defining
we can introduce the symbol
which determines the index set of the data symbols that contribute the most to the
mth received signal
Similarly as in the frequency domain, we obtain a banded approximation of
and a selection matrix
We can then rewrite (37) as
where
The timedomain equalization can be presented in an analogous manner as in the frequency domain. Similar to its FD counterpart in (23), we here rewrite the noiseless case for (44) as
where
We highlight that the adopted diagonal preconditioner C_{T,⋆}=diag{[c_{T,0,⋆},c_{T,1,⋆},…,c_{T,K−1,⋆}]^{T}} is defined in a similar manner as in the frequency domain. Specifically, we use
where
To enhance the suitability of the preconditioner, the optimal resampling operation is needed as given by
Appendix 3
Equalization using the conjugate gradient algorithm
If we consider to solve the preconditioned system in (23) in a similar manner as (22), we have
where
Its implementation using CG is described in the frequency domain as follows
1. Define
2. Perform the following iterations:
where a^{(0)}=g^{(0)}=d_{F},
3. Perform
Notably, the optimal stopping criterion for CG can be case dependent, e.g., as discussed
in
[23], and is not included in this article. When our CG iterations stop, we finally have
It is worthy to note that the computational complexity of each CG iteration above
is determined by the complex multiplication (CM) of
One can also repeat the above derivations using the TD notations for the TD CG equalization.
Appendix 4
Eigenvalue locations
We consider the diagonal matrix C_{F}=diag{[c_{F,0},c_{F,1},…,c_{F,K−1}]^{T}}, and denote the eigenvalues of
Let UWU be a Schur decomposition of
Note that
Similarly, we can also prove that
Endnote
^{a} As a matter of fact, the case where α_{l}<1 or τ_{l}<0 can be converted to the current situation by means of proper resampling and timing at the receiver. This justifies the assumption of a compressive and causal scenario without loss of generality.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
The first author wants to thank the National University of Defense Technology, China, and also the China Scholarship Council for the financial support. This work was supported in part by NWOSTW under the VICI program (project 10382). The work of Z. Tang is also supported in part by the European Defence Agency (EDA) project RACUN (Robust Acoustic Communication in Underwater Networks). In addition, we would like to thank Dr. Magnus Lundberg Nordenvaad from the Lulea University of Technology, Sweden, Prof. Urbashi Mitra from the University of Southern California, U.S., and Prof. Huihuang Chen from the Xiamen University, China, who participated in valuable discussions.
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