To overcome the problems with the existing algorithms in **channel** state inference in WSNs, such as high complexity, poor generalization, impracticability and so on, we develop a new **channel** **estimation** method with variational tempering for MIMO-OFDM scenario. A ground truth information extraction algorithm based on variational tempering is proposed and implemented. Our CEVTI provides insights that account for multiple factors in inferring truth and can generalize to higher dimensions and finds better local optimum. As a variation of SVI, CEVTI can find out the optimal hyper-parameters of channels with fast convergence rate, and can be applied to the case of CDMA and uplink massive MIMO easily. As can be seen in Section 5 , CEVTI can iteratively minimize the objective function with multiple dimensions. We demonstrate the performance of CEVTI through numerical simulation. The BER, convergence rate, and mutual information comparisons with the five existing CE algorithms show that CEVTI outperforms others under different noise variance and signal-to-noise ratio. Furthermore, the results show that the more parameters that are considered in each iteration, the faster the convergence rate and the lower the non-degenerate bit error rate with CEVTI. Analysis shows that CEVTI has satisfying computational complexity, and guarantees better local optimum. Therefore, this paper has contributed to the quest for developing efficient algorithms in artificial advanced sensor networks. Possible future research directions include investigation of how the graph structure can impact the performance of CEVTI, and constructing inference algorithms that can suit more complex situations.

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The problem to be addressed in this paper is the impact of residual **channel** **estimation** errors on recent TDMA based RA methods. The main issue is to be able to estimate the **channel**
parameters in the case of multiple superimposed signals and to achieve performance close to the perfect knowledge case. This challenge has already been addressed in part in the existing literature. In [5] a method based on the Expectation- Maximization (EM) algorithm is presented to estimate **channel** parameters simultaneously. In [6], another approach uses the autocorrelation to derive **channel** amplitude and frequency offsets from packets that did not experience collision. In [7], **channel** **estimation** using EM is evaluated for a network coded diversity protocol (NDCP). We have also presented a first con- tribution of our work in [8], where we have used an EM based **channel** **estimation** method and evaluated experimentally the effect of imperfect interference cancellation on the decoding of the remaining packet.

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transmitter employs Alamouti Coding. Regarding the proposed methods, Table I shows that the set of ambiguity matrices after **channel** **estimation** is Θ = {M 1 (𝜃), M 2 (𝜃)}. Figure 5 displays the NMSE versus SNR for a receiver composed of 𝑛 𝑟 = 3 antennae. Without multistart initialization, the geodesic SD clearly outperforms the classical SD since the latter exhibits an error floor at SNR greater than 4dB. This error floor is due to the fact that the Euclidean SD can lead to undesired suboptimal solutions even at high SNR [47], [48]. It should be observed that the multistart initialization strategy removes the error floor and improves the NMSE performances of the two proposed algorithms. Figure 6 compares the SER with the one obtained with a coherent ML receiver. As previously discussed, without multistart initialization, the performances of the Euclidean SD lead to an error floor at SNR greater than 4dB. However, it should be observed that algorithms 1 and 2 achieve near-optimal performance when a multistart ini- tizalization is used. A comparison of the average computation times is shown in Table III. It should be noted that classical SD is less computationally demanding than the geodesic SD at low-SNR, but this trend is reversed at high SNR.

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ABSTRACT
In this paper, a new fast adaptive blind **channel** **estimation** method is proposed using the subspace information from the correlation matrix. The algorithm is fully adaptive in the sense that both the subspace information and the optimization which leads to the **channel** **estimation** are computed adaptively. It is based on the re- cently proposed YAST subspace tracker which has been shown to outperform other methods both in terms of speed of convergence and computational complexity. A discussion on the convergence properties of the proposed algorithm is presented. We also propose a hybrid method which makes use of the YAST subspace tracker for initial fast convergence and the subspace information is then updated using the numerically stable OPAST subspace tracker.

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The main objective of this thesis is to propose techniques for **channel** estima- tion and information recovery in multiple-input-multiple-output (MIMO) Volterra communication systems. This kind of MIMO model is able of modeling nonlinear communication channels with multiple transmit and receive antennas, as well as multi-user channels with a single transmit antenna for each user and multiple re- ceive antennas. **Channel** **estimation** and equalization techniques are developed for three types of nonlinear MIMO communication systems: OFDM, ROF and Code division multiple access (CDMA)-ROF systems. According to the considered com- munication systems, different kinds of MIMO Volterra models are used. In the case of OFDM systems, we develop receivers that exploit the diversity provided by a proposed transmission scheme. In the case of time and space division multiple access (TDMA-SDMA) systems, a set of orthonormal polynomials is developed for increasing the convergence speed of a supervised adaptive MIMO Volterra es- timation algorithm. Moreover, in order to develop signal processing techniques for MIMO Volterra communication channels in a blind scenario, we make use of tensor decompositions. By exploiting the fact that Volterra models are linear with respect to their coefficients, blind **estimation** and equalization of MIMO Volterra channels are carried out by means of the Parallel Factor (PARAFAC) tensor decomposition, considering TDMA-SDMA and CDMA communication systems.

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Avenue de la Boulaie, 35576 Cesson - S´evign´e Cedex, France. Emails: yves.louet@centralesupelec.fr, and faouzi.bader@supelec.fr
Abstract—In this paper, the authors describe and compare two low-complexity approximations of the linear minimum mean square error (LMMSE) **channel** **estimation** method for orthogo- nal frequency division multiplexing/offset quadrature amplitude modulation (OFDM/OQAM) systems. Simulations reveal that we are able by proposed approximations to reduce the complexity of the LMMSE estimator without degrading the overall BER system performance.

Fig. 1: Packets transmission with collision on slot 1
II. S YSTEM M ODEL
In order to illustrate the main issues raised by imperfect **channel** **estimation** in interference cancellation based RA meth- ods, we consider the following example (see Fig. 1). Each user (1 and 2) sends two replicas (a and b) of the same packet on two different time slots on the frame. We suppose that the receiver first detects packet 1b as it is a clean packet, decodes it correctly and removes its corresponding signal from Slot 4. Then, using the known decoded bits of packet 1b, the signal corresponding to packet 1a is reconstructed and suppressed from Slot 1. Thus, packet 2a becomes collision free, and has a bigger probability to be decoded successfully.

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linear N ⊥ -carrier FSK modulation [6]. Efficient digital imple- mentation can be realized by inverse Fast Fourier Transform (FFT) followed by a cyclic prefix insertion as for Orthogonal Frequency Division Multiplexing (Fig. 1). At the receiver, a soft FSK-detector estimates the probabilities of each possible Turbo-FSK codeword. These probabilities are then fed to the decoder, which uses them as **channel** observation, while output of the other decoders will be used as a priori information. A modified version of the algorithm proposed by Bahl, Cocke, Jelinek and Raviv (BCJR) [7] is used to decode the trellis, and derive the a posteriori probabilities of the information bits. The association of the encoding with a non-linear modulation (FSK) allows to operate at very low levels of SNR. It has been demonstrated that the choice of the PSK is the optimal modulation minimizing **channel** capacity gap to Shannon’s limit [8]. Practical implementations of the FSK-detector consist of a FFT combined with a frequency domain equalizer that relies on accurate **channel** **estimation**.

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and 4, σ H = 0 and 0.2.
inaccuracies, and fixing the correlation factor to ρ = 0.1. First comparing the curves with N t = 8, it is verified that the larger
the RA array, the stronger the degradation of the performance. Indeed, as the number of RAs increases, the amount of inter- antenna interference becomes higher at the receiver side and the defect of interference cancellation of the ZF scheme due the **channel** **estimation** errors becomes more significant. This fact corroborates the analysis made on Eq. (26). Reversely, analyzing the curves with N r = 2 but changing N t leads to the

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In this paper, we are interested in the **estimation** of the aeronautical multipath **channel**. Based on parametric models for the impulse response of the multipath **channel**, we propose simple but efficient **channel** **estimation** methods that benefit from a known position of the mobile user. The proposed meth- ods exploit both the particular form of the **channel** impulse response and a priori knowledge of some parameters (mainly delays) that can be inferred from geometrical considerations based on geolocation. The first **estimation** method is based on a parametric multipath **channel** model while the second method tries to exploit the relative sparsity of the **channel** impulse response. In both cases, this reduces the number of variables to be estimated and it provides better performance compared to a direct classical least-square **estimation** of the discrete equivalent **channel** impulse response.

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In terms of PER, performance is even closer when LS filtered and LMMSE are compared (Fig. 8). For these simulation scenarios, the pilot sequence is repeated 64 times and averaged to improve the performance of the estimator. Except the LS estimator that has a loss of 1.4 dB in comparison to the reference (perfect **channel** **estimation**), the other estimators have a very limited loss between 0.2 dB (LMMSE) and 0.3 dB (LMMSE with uniform assumption). Consequently, the LS filtered solution is chosen, as it is a good compromise between complexity and performance.

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Fig. 7. BER for different **channel** models using BICM(23,35) at rate 1/2
with K = 3 in non perfect CSI.
VI. C
The EM algorithm has been studied to estimate the **channel** parameters of an M-PPM UWB communication. The parameters are composed of the noise signal and energy coefficients corresponding to K inter-symbol interferences caused by high data-rate communica- tions in dispersive channels. The **channel** **estimation** is used iteratively and jointly with a probabilistic equalization and a **channel** decoder. At 100 Mbits/s the EM is capable of a good **estimation** of parameters

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and α RZF .
The advantage of the proposed robust MMSE-RZF can be observed in Fig. 2. Note that MF, MF-RZF and ZF are all special cases of MMSE-RZF, which are not optimized with the system condition. The poor performance of ZF comes from the inverse Wishart distribution term in its power control factor at the relays, especially when M = N . We find that the ergodic capacities still satisfy the scaling law in [3], i.e., C = (M/2) log(K) + O(1) for large K with **channel** **estimation** errors, which is also consistent with the asymptotic capacities of robust MMSE-RZF. Fig. 3 compare the ergodic rate capacities versus the power of CSI error. we set K = 3 and PNR= 10dB, QNR= 10dB. MMSE-RZF outperforms others as the power of CSI error changes.

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Stopping criterion is essential for CS based non- sample spaced sparse **channel** **estimation**. Similar with the stopping criterion for the traditional LS or DFT based sparse **channel** **estimation** and sample spaced sparse **channel** **estimation** with CS, the **channel** statistics (power delay profile of the **channel**, **channel** sparsity et al), noise standard deviation (STD) or signal to noise ratio (SNR) can be employed as the basic parameters for obtaining effective stopping criteriaon[7-8].

of a specific filter bank applied on the sub-carriers. As a consequence, there exists intrinsic imaginary interference (IMI) among the neighboring sub-carriers (frequency-domain) and symbols (time-domain). Being interfered by the IMI, the **estimation** and compensation of **channel** dispersions for CO- OFDM/OQAM systems becomes more complicated compared to the conventional CO-OFDM ones, especially when polar- ization division multiplexing (PDM) is taken into account. Therefore, the quest for an optimal **channel** **estimation** and equalization for PDM CO-OFDM/OQAM systems is crucial. Recent researches in **channel** dispersion compensation for CO-OFDM/OQAM mainly focus on the equalizer design while assuming that the **channel** transfer function is known [7], [8] or perfectly estimated [9]. In [10] and [11], the **estimation** of **channel** response by using preamble was investigated. Both methods focus on the preamble design to minimize the effect of IMI by allocating zeros around the target pilots and count on interpolation to find the **channel** response at the sub-carriers associated with the zero pilots. In contrast, there exists a **channel** **estimation** method, called Interference approximation method (IAM), that even exploits IMI as an advantage. Indeed, IAM is a well-known **channel** **estimation** method for OFDM/OQAM, which was originally proposed for radio communications in [12], [13]. This method was introduced for PDM CO-OFDM/OQAM in [14]–[16]. In fact, the shifted versions of the CO-OFDM/OQAM prototype filter in frequency and time domains can be viewed as a grid of filters for different frequency-time (FT) positions. To recover the symbol at a given FT position, the filter at that position must be slided and multiplied with all filters in the filter grid before summed up. Since the filters are designed to be orthogonal in real field only, the multiplication between filters in different FT locations result in a imaginary value called IMI coefficients. Fortunately, the pattern of IMI coefficients in the FT grid is unchanged for any reference FT position. The IAM method relies on this characteristic to design the preamble.

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A novel efficient time domain threshold based sparse **channel** **estimation** technique is proposed for orthogonal frequency division multiplexing (OFDM) systems. The proposed method aims to realize effective **channel** **estimation** without prior knowledge of **channel** statistics and noise standard deviation within a comparatively wide range of sparsity. Firstly, classical least squares (LS) method is used to get an initial **channel** impulse response (CIR) es- timate. Then, an effective threshold, estimated from the noise coefficients of the initial estimated CIR, is proposed. Finally, the obtained threshold is used to select the most significant taps. Theoretical analysis and simula- tion results show that the proposed method achieves better performance in both BER (bit error rate) and NMSE (normalized mean square error) than the compared methods, has good spectral efficiency and moderate compu- tational complexity.

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In this paper, we are interested in the **estimation** of the aeronautical multipath **channel**. Based on parametric models for the impulse response of the multipath **channel**, we propose simple but efficient **channel** **estimation** methods that benefit from a known position of the mobile user. The proposed meth- ods exploit both the particular form of the **channel** impulse response and a priori knowledge of some parameters (mainly delays) that can be inferred from geometrical considerations based on geolocation. The first **estimation** method is based on a parametric multipath **channel** model while the second method tries to exploit the relative sparsity of the **channel** impulse response. In both cases, this reduces the number of variables to be estimated and it provides better performance compared to a direct classical least-square **estimation** of the discrete equivalent **channel** impulse response.

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2a 2b
Fig. 1: Packets transmission with collision on slot 1
II. S YSTEM M ODEL
In order to illustrate the main issues raised by imperfect **channel** **estimation** in interference cancellation based RA meth- ods, we consider the following example (see Fig. 1). Each user (1 and 2) sends two replicas (a and b) of the same packet on two different time slots on the frame. We suppose that the receiver first detects packet 1b as it is a clean packet, decodes it correctly and removes its corresponding signal from Slot 4. Then, using the known decoded bits of packet 1b, the signal corresponding to packet 1a is reconstructed and suppressed from Slot 1. Thus, packet 2a becomes collision free, and has a bigger probability to be decoded successfully.

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τ ∈T |< f τ , r l−1 >| is not sufficient
for accurate delay tracking and **channel** **estimation** with high precision. In other words, the delay points within a delay sub- set where the corresponding bases have high coherence with the residual vector, should be considered. With these delay points, the reference delay grid (RDG) guided RNM method is proposed in this paper to effectively fight against the non- uniform pilot arrangement and realize the near optimal delay searching of the l th **channel** tap, which will be discussed after the DT method in this section.

A common BEM is the (O)CE-BEM only based on complex exponentials. This model does not require any knowledge of the **channel** statistics. In this case, the BEM **channel** **estimation** suffers from the unknown sparsity of the **channel**. Indeed, we try to estimate a null path by a sum of weighted complex exponentials. Even the LS **estimation** with positioning a priori exhibits a large BEM modeling error as shown in Fig. 4. We can also observe that for the same number of basis functions, the KL-BEM outperforms the chosen OCE-BEM, which can be explained by the optimally of the KL-BEM. However, we

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