 Research
 Open Access
 Published:
Performance limits of channel parameter estimation for joint communication and positioning
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 178 (2012)
Abstract
Abstract
Recently, a system concept for joint communication and positioning has been proposed by the authors. Channel parameter estimation (CPE) is the core part of this system proposal. Parameters of the physical channel, which can be exploited for positioning, are estimated based on the assumption that a priori knowledge about pulse shaping and receive filtering is available. At the same time, channel estimates of the equivalent discretetime channel model, which are needed for data detection, are obtained inherently. This article focusses on the positioning part of the system proposal. Performance limits for CPE in terms of CramerRao lower bounds are determined for different channel models. The influence of oversampling and of different channel characteristics is investigated. Oversampling proves especially helpful in dense multipath scenarios, which are most challenging. Based on the presented results, oversampling with a factor of two is recommended in order to improve the positioning accuracy. Excessive oversampling like in conventional global positioning system receivers is not necessary.
Introduction
Interest in joint communication and positioning is steadily increasing[1–4]. The combination of communication and positioning offers a wide range of advantages and synergistic effects like enhanced resource allocation or improved power control in cellular networks. Furthermore, applications such as locating emergency calls, tracking, and guiding firefighters or policemen on a mission, or locationbased services become feasible. Communication and positioning can be combined in different ways: existing systems can be combined in a hybrid receiver, existing systems can be extended to provide additional services, or new systems with a unified signal structure can be designed. The latter approach is considered in this article. The main aim of such a joint communication and positioning system is to provide high data rates with low bit error rate for the communication part, and a high localization accuracy for the positioning part. Since it is quite challenging to fulfil both conditions at the same time, a flexible system configuration is desirable in order adjust the tradeoff between communication and positioning to the recent needs: in case of an emergency call, emphasis can be laid on the positioning part, whereas the communication payload can be increased in case of data transmissions like file downloads or video streaming. Furthermore, a flexible configuration allows adapting the transmission scheme to changing channel conditions in order to fulfil a certain quality of service.
Recently, a system concept for joint communication and positioning has been proposed by the authors of[5, 6]. The system proposal is based on multilayer interleavedivision multiple access (MLIDMA)[7] in combination with pilot layeraided channel estimation (PLACE)[8]. MLIDMA is a combination of interleavedivision multiplexing (IDM) and IDMA. The application of IDM as multiplexing scheme is essential for the system proposal since it provides the desired flexibility, while the choice of the multiple access scheme is less crucial. IDM is a special codedivision multiplexing scheme, where different data streams (called layers) are separated by layerwise interleaving. The key idea for joint communication and positioning is to allocate one layer with a known training pattern employed for positioning and channel estimation, whereas the remaining layers are informationbearing layers employed for data communication. This article focusses on the positioning part of the proposed system, which is based on the timeofarrival (ToA) concept: Radiolocation is typically performed in two steps[9]. First, parameters of the physical channel like the received signal strength, the angleofarrival, or the ToA are estimated (parameter estimation). Based on these parameters, the position of the mobile station is determined in a second step (position estimation). The parameter estimation error translates into a positioning error via the geometric dilution of precision (GDOP)[10]. Given a certain GDOP, the positioning accuracy increases if the parameter estimation error decreases. Thus, accurate parameter estimation is a prerequisite for precise radiolocation.
Channel parameter estimation (CPE) is the core part of the proposed system concept. Based on the assumption that a priori information about pulse shaping and receive filtering is available, parameters of the physical channel including the ToA are estimated and exploited for positioning. At the same time, estimates for the channel coefficients of the equivalent discretetime channel model, which are needed for data detection, are obtained inherently. Thus, CPE enables positioning and data detection. Typically, only one aspect of CPE is considered: On the one hand, CPE is well known in the context of channel sounding[11–13]. In this case, the parameters of the physical channel are of interest, while the channel coefficients of the equivalent discretetime channel model, which are available as well, are not further processed. On the other hand, the usage of a priori information about pulse shaping and receive filtering has already been suggested in[14–16] for improved channel estimation in communication systems. In this case, the information about the physical channel is discarded. For joint communication and positioning, both aspects of CPE are exploited.
The channel parameter estimator considered in this contribution is based on the maximumlikelihood principle. There are two equivalent approaches: On the one hand, the channel parameters can be estimated directly from the received samples as it is often the case for channel sounding[11–13]. Similar estimators have been investigated for example in the context of Rake receivers in codedivision multiple access[17] or for pure navigation purposes[18]. On the other hand, CPE can be performed in two steps: First, standard channel estimation (without a priori information about pulse shaping and receive filtering) is applied in order to obtain a preliminary estimate of the channel coefficients. Based on this prestage channel estimates, the parameters of the physical channel are estimated and enhanced channel estimates are obtained. This twostep approach is for example considered in[19]. Similar to the channel estimation approaches in[14–16], the channel estimates which are available after CPE are more reliable than the prestage channel estimates due to the exploitation of a priori information about pulse shaping and receive filtering. Since the approach based on the received samples and the twostep approach based on the prestage channel estimates are equivalent, the latter approach is recommended due to complexity reasons.
In this article, performance limits for parameter estimation in terms of CramerRao lower bounds (CRLBs) are determined for different channel models. Especially, the impact of oversampling is analyzed. Three different kinds of channel models are considered: A singlepath channel, several twopath channels, and different wireless world initiative new radio (WINNER) channels with numerous multipath components. The singlepath channel is taken into account since it is the best possible case for positioning and, thus, provides a lower bound for all other channel models. By means of the twopath channel models, the influence of different channel characteristics such as the excess delay, power ratio, and phase offset of the propagation paths can be investigated. The results obtained for the twopath channels are the basis for more realistic channel models with an arbitrary number of propagation paths like the WINNER channel models that are considered in this article. It is observed that oversampling provides a performance gain compared to symbolrate sampling. Oversampling proves especially helpful in dense multipath scenarios, which are most challenging with respect to positioning. Since the performance for all oversampling factors larger than two is about the same, oversampling with a factor of two is recommended.
Many applications of joint communication and positioning are located in urban or indoor areas including hotspots like train stations, airports, or shopping malls. In these environments, multipath components are typically dense, i.e., these environments are very demanding concerning radiolocation. However, the required positioning accuracy is quite high in urban or indoor areas. Often, it is not possible to meet the required accuracy with a single radiolocation method. Therefore, several radiolocation methods should be combined via sensor fusion[20–22] in order to improve the positioning accuracy. Keeping in mind that a system can always be extended by assisting concepts, the system proposal should be understood as a single contribution concerning positioning, that can be combined with other radiolocation methods via sensor fusion.
The remainder of this article is organized as follows: In Section “System concept”, the system and channel model is introduced and the joint communication and positioning system proposed by the authors is presented. CPE is explained in Section “CPE”. The basic estimation problem is introduced and the twostep maximumlikelihood approach is presented. In Section “Performance limits—CRLB”, performance limits for the derived channel parameter estimator are determined in terms of CRLBs. Numerical results for different channel models and for different oversampling factors are presented. Furthermore, the impact of the obtained results on the overall positioning process are discussed. Finally, conclusions are drawn in Section “Conclusion”.
System concept
Throughout this article, the discretetime complex baseband notation is used. Let x^{′}[ κ ], 0 ≤ κ < K^{′}, denote the κ th symbol of a coded and modulated burst of length K^{′}. If oversampling is applied, this sequence is upsampled to a burst of length K = J K^{′}, where J is the oversampling factor. The symbols of the upsampled sequence are given according to
In case of symbolrate sampling (J = 1), both sequences are the same (x[ k] = x^{′}[ κ] and K = K^{′}). Assuming a linear modulation scheme, the received sample y[ k] at time index k is given by
where h_{ l }[ k] is the l th channel coefficient of the equivalent discretetime channel model with channel memory length L and n[ k] is a zero mean Gaussian noise sample. The equivalent discretetime channel model comprises all continuoustime elements of a transmission link, namely the pulse shaping filter g_{Tx} (τ), the timevariant physical channel c(τ,t), additive white Gaussian noise (AWGN), the receive filter g_{Rx}(τ), and sampling. This means that the channel coefficients h_{ l }[ k] are the samples of the overall channel weight function h(τ,t), which is given by the convolution of g_{Tx} (τ), c(τ,t), and g_{Rx}(τ). Due to the associative and commutative properties of the convolution, pulse shaping and receive filtering can be combined: g(τ) = g_{Tx} (τ) ∗ g_{Rx}(τ). The physical channel c(τ,t) is typically modeled by a weighted sum of delayed Dirac impulses. In this case, the channel coefficients after sampling at$t=k\mathcal{T}+\epsilon $ are given by
where${f}_{i}\left[k\right]\in \u2102$ and${\tau}_{i}\left[k\right]\in {\mathbb{R}}_{\ge 0}$ are the complex amplitude and the propagation delay of the i th propagation path, respectively. Furthermore, I denotes the number of propagation paths,$\mathcal{T}={T}_{s}/J$ denotes the sampling period, which is given as a fraction of the symbol duration T_{ s }, and ε is the sampling phase, that accounts for sampling time offsets. The noise process n[ k] in (2) is generally colored because white Gaussian noise with zero mean and variance${\sigma}_{n}^{2}$ is added to the continuoustime signal before receive filtering, i.e., the white Gaussian noise is filtered by g_{Rx}(τ). Thus, the sampled autocorrelation function of n[ k] is given by
with Δk = k_{1} − k_{2} and where ψ_{Rx} (τ) = g_{Rx} (τ) ∗ g_{Rx}( −τ ) denotes the autocorrelation function of the receive filter. If a squareroot Nyquist pulse is applied at the receiver, the noise remains white for symbolrate sampling.
The channel coefficients in (3) depend on propagation delays τ_{ i } [ k] of the physical channel. For positioning based on the ToA, the propagation delay of the first arriving path, τ_{1}[ k], needs to be estimated. In contrast, perfect synchronization is often assumed for the simulation of communication systems. This means that the propagation delay of the first arriving path is known and eliminated perfectly such that excess delays ν_{ i } [ k] = τ_{ i }[ k] − τ_{1}[ k] are considered only. The sampling phase ε is zero in this case. Consequently, the leading channel coefficients with zero values are eliminated and a shorter channel memory length can be taken into account. The assumption of perfect synchronization is not applicable in this contribution since the positioning part of the proposed system concept is based on the ToA. Therefore, a coarse synchronization is considered subsequently, that eliminates the propagation delay of the first arriving path only approximately:
This means that excess delays in combination with a nonzero sampling phase are taken into account. In this case, the propagation delays in (3) are replaced by excess delays:
Based on the above assumptions, the estimation of the ToA${\widehat{\tau}}_{1}\left[k\right]$ corresponds to the estimation of the sampling phase$\widehat{\epsilon}$. The final ToA estimate, that can be exploited for positioning, is given as
It should be noted here that the requirements concerning synchronization differ for communication and positioning purposes. Assuming that a correlationbased synchronization is performed, the highest correlation peak should be chosen for communication purposes in order to maximize the signaltonoise ratio (SNR) at the receiver side. In contrast, the first correlation peak, which might not be the highest, is important for positioning via the ToA. With the proposed system concept both requirements can be met: First, a coarse synchronization is performed that maximizes the SNR. Then, the ToA is determined more accurately by estimating the sampling phase using CPE. Thus, CPE corresponds to a finetuning of synchronization for positioning purposes.
Now, it becomes clear why CPE is the core part of the proposed joint communication and positioning system. Channel estimation is mandatory for communication purposes since the channel coefficients of the equivalent discretetime channel model need to be known for data detection. If the parameters of the physical channel are estimated, positioning is enabled and estimates of the channel coefficients are available inherently. The relationship in (6) is the basis for CPE. From (6), it is obvious that the channel coefficients are known if the parameters of the physical channel (f_{ i }[ k], ν_{ i }[ k], 1 ≤ i ≤ I) and the shape of the filter g (τ) are known. Training symbols should be inserted into the transmission burst in order to simplify CPE. All multiplexing techniques including timedivision multiplexing (TDM) and frequencydivision multiplexing can be applied for that purposes. According to the system proposal in[5, 6], IDM[7] in combination with PLACE[8] is considered in this article. The main idea of IDM is to linearly superimpose several data streams of a user, which are called layers in the following. In case of PLACE, a pilot layer containing training symbols is additionally superimposed onto the data layers for CPE purposes as shown in Figure1. Each data layer is either dedicated to communication purposes (e.g., speech or video transmission) or it may carry auxiliary information for localization purposes (e.g., time of departure or positions of reference objects). The layers are distinguished by layerspecific interleavers: Let u_{ m }[ n], 0 ≤ n < N, 1 ≤ m ≤ M, denote the n th bit of the m th data layer. Each bit sequence is encoded with code rate R = N/K^{′} (ENC), interleaved by a layerspecific interleaver (π_{ m }) and mapped onto the complex plane via binary phase shift keying (BPSK), which leads to the layerwise symbols${x}_{m}^{\prime}\left[\kappa \right]$. Before all data layers and the pilot layer with training symbols${x}_{0}^{\prime}\left[\kappa \right]$ are summed up, an adequate power and phase allocation with complex weighting factors a_{ m }e^{j}ξ_{ m } is performed. Thus, the κ th symbol of the transmission burst of length K^{′} is given by
Each symbol x^{′}[ κ] carries$\mathcal{\mathcal{B}}=\mathit{RM}$ bits, where$\mathit{\mathcal{B}}$ is called bit load[7]. Since all layers employ the same encoders in combination with BPSK mapping, the transmitter structure of IDM is very simple. However, IDM offers a flexible configuration, which is desirable for joint communication and positioning, because the data rate can be easily adapted by changing the number of data layers M instead of changing the modulation scheme[23]. Furthermore, layerwise unequal error protection can easily be achieved by assigning different amplitude levels to different layers[7]. Similarly, the tradeoff between communication and positioning purposes can be regulated via an adequate power allocation. The ratio of the pilot layer power to the total power,
can be varied between 0 and 1, where ρ = 0 and ρ = 1 correspond to no training at all and pure training, respectively.
CPE
For the purpose of CPE, the channel model in (6) is reformulated by combining the excess delays with the sampling phase to auxiliary parameters ϱ_{ i }[k] = ν_{ i }[k] − ε 1 ≤ i ≤ I, which are termed coarse excess delays in the following. Furthermore, block fading is assumed, i.e., the parameters of the physical channel do not change over the transmission burst. In this case, the channel coefficients do not depend on the time index k anymore:
Block fading can be assumed for 0 ≤ B_{ D } · K^{′}T_{ s } < 0.01, where B_{ D } is the Doppler spread of the physical channel, K^{′}is the burst length and T_{ s } is the symbol duration. The Doppler spread depends on the mobility of the investigated scenario and can be equated with the maximum Doppler shift B_{ D } = f_{D, max} = v/c c · f_{0}. In this case, v represents the maximum possible velocity, c denotes the speed of light, and f_{0} is the carrier frequency. For joint communication and positioning, mainly indoor and urban areas are of interest. Hence, the maximum velocities, that typically occur, lie between 7 and 70 km/h. Thus, a wide range of reasonable parameter combinations (K^{′}, T_{ s }, f_{0}) exists, for which the block fading assumption is valid. If the parameters are fixed to specific values, that can not be changed, and the block fading assumption is violated, the transmission burst can be subdivided into smaller blocks, for which the block fading assumption is valid again. In this case, CPE can be performed block instead of burstwise (sliding window approach, see also[8]). Consequently, the assumption of block fading is adequate and hardly restricts the applicability of the proposed channel parameter estimator.
In order to emphasize the functional relationship between the parameters of the physical channel (f_{ i }, ϱ_{ i } 1 ≤ i ≤ I) and the channel coefficients of the equivalent discretetime channel h_{ l }, the channel parameters are stacked in a vector
of length P = 3I. Each propagation path is characterized by three parameters: the real and imaginary part of the complex amplitude, Re {f_{ i }} and Im {f_{ i }}, and the coarse excess delay ϱ_{ i }. The channel coefficients in (10) can be expressed as a function of the parameter vector in (11) according to
Based on the training symbols x_{0}k, CPE is performed: Inserting (8) into (2) leads to:
Only the first part in (13) is useful for CPE, while the second part (data layer interference) complicates CPE. Typically, turbotype iterative receivers are applied for data detection in case of IDM and related techniques[24–26], i.e., the receiver consists of a multilayer detector (MLD) and a bank of layerwise decoders. At the MLD, only the multiplexing and the channel constraint are taken into account ignoring the coding constraint. In contrast, the layerwise decoders consider the coding constraint only. Extrinsic information is exchanged iteratively between the MLD and the decoders. In this way, the quality of the incorporated estimates can be increased over iterations. In case of channel estimation, the feedback information from the decoders can be used to mitigate the data layer interference[8]: By means of data layer interference cancellation (DIC), improved channel estimates are obtained, which in turn lead to improved data estimates. Typically, the data layer interference can be cancelled nearly perfectly, i.e., the residual data layer interference after an adequate number of receiver iterations is negligible. Under certain conditions, the residual data layer interference is not negligible anymore. In this case, the residual data layer interference can be modeled as a Gaussian variable according to the central limit theorem because M and L are typically large. Then, the noise and the residual data layer interference can easily be combined to a single Gaussian distortion with increased variance compared to pure noise, i.e., nonnegligible residual data layer interference corresponds to a decrease in SNR. Hence, it is sufficient to consider perfect DIC for the derivation of performance limits for CPE. This means that CPE is first performed after an adequate number of receiver iterations. Previously, standard channel estimation is applied. The assumption of perfect DIC leads to
Without loss of generality, ξ_{0} = 0 can be assumed. The remaining phases ξ_{ m } are distributed equally between 0 and Π. Furthermore, the amplitude of the pilot layer can be expressed as${a}_{0}=\sqrt{\rho}$ due to power normalization$\left(\sum _{m=0}^{M}{a}_{m}^{2}\stackrel{!}{=}1\right)$. Based on the observations after DIC in (14), the parameter vector θ can be estimated via the maximumlikelihood (ML) approach[27, 28] exploiting a priori information about the pilot symbols x_{0}k and the overall pulse shape g(τ) (including pulse shaping and receive filtering). ML estimators are asymptotically optimal (unbiased and efficient), i.e., they achieve the best possible performance given by the CRLB for a large number of observations or at high SNR. Furthermore, the ML estimators correspond to the leastsquares (LS) estimators in case of Gaussian noise. For the derivation of the ML estimator for CPE, it is useful to express (14) in vector/matrix notation:
where y_{DIC} = [y_{DIC}[L],…,y_{DIC}[K−1]]^{T} is the observation vector containing the received samples after DIC and X_{0} is the training matrix with Toeplitz structure:
Furthermore, h(θ) = [h_{0}(θ),…,h_{ L }(θ)]^{T} and n = [n[L],…,n[k −1]]^{T} denote the channel coefficient vector and a zero mean Gaussian noise vector with covariance matrix C_{ n }, respectively. The entries of the noise covariance matrix are determined according to the autocorrelation function of the noise process given in (4):
As already mentioned above, there are two equivalent approaches to estimate the channel parameters θ based on the signal model in (15)–(17). The first approach is based on the received samples y_{DIC} and the second approach relies on socalled prestage channel estimates$\stackrel{\u030c}{\mathit{h}}$that are obtained via a standard channel estimation algorithm from the received samples y_{DIC}. Due to complexity reasons, only the second approach is considered subsequently. Since block fading is assumed, the prestage channel estimates can be obtained in closed form via (weighted) leastsquares channel estimation[28]:
where$\mathit{\eta}\sim \mathcal{C}\mathcal{N}\left(0,{\mathit{C}}_{\mathit{\eta}}\phantom{\rule{0.3em}{0ex}}\right)$ is the channel estimation error with covariance matrix
At this stage, the a priori information about pulse shaping and receive filtering is not exploited yet. The a priori information about pulse shaping and receive filtering is incorporated in a second step applying the ML principle: The parameters of the physical channel θ are estimated by maximizing the likelihood function$\mathit{p}\left(\stackrel{\mathit{\u030c}}{\mathit{h}}\mathbf{\theta}\right)$[28]:
Due to the pulse shape g (τ), the metric in (20) is nonlinear, i.e., the minimization cannot be solved in closed form. Exhaustive search is prohibitive because the search space (parameter space) is continuous and of high dimension (P = 3I). Hence, an optimization method needs to be applied, which typically performs the minimization/maximization in an iterative manner. Due to the superposition of several multipath components, there exist many local optima besides the global optimum. Hence, a global optimization method is required in general. A viable global optimization algorithm is for example particle swarm optimization (PSO)[29, 30] as demonstrated by the authors in[31]. Of course, other optimization methods are applicable as well. If a priori information about the approximate location of the global optimum is available, even local optimization methods like the LevenbergMarquardt algorithm[32, pp. 688–693] can be applied. For example, the parameter estimate of the preceding burst can be used as initial guess for the recent burst if the channel is changing slowly from burst to burst (tracking). However, a global optimization method like PSO is still required for acquisition. Even though the global optimization methods may be computationally complex in the acquisition phase, they can be reduced in complexity in the tracking phase, where realtime operation is more critical: The a priori information about the approximate location of the global optimum can be exploited by global optimization methods to narrow the search space and to simplify the global search. In this case, PSO needs much less computations on average compared to acquisition. In general, there is a tradeoff between performance and complexity. In order to choose an adequate optimization method for a specific application, the particular requirements of this application need to be taken into account. The complexity of prestage channel estimation is negligible since the prestage channel estimates$\stackrel{\u030c}{\mathit{h}}$are obtained in closed form. Furthermore, the pseudoinverse${\left({\mathit{X}}_{0}^{H}{\mathit{C}}_{\mathit{n}}^{1}\phantom{\rule{0.3em}{0ex}}{\mathit{X}}_{0}\phantom{\rule{0.3em}{0ex}}\right)}^{1}{\mathit{X}}_{0}^{H}\phantom{\rule{0.3em}{0ex}}{\mathit{C}}_{\mathit{n}}^{1}\phantom{\rule{0.3em}{0ex}}$ can be computed in advance since the pilot symbols x_{0}[k] and the receive filter g_{Rx} (τ) are known. Hence, the overall complexity is dominated by the optimization algorithm for CPE.
It should be noted here that the only parameter, that is needed for positioning, is${\widehat{\theta}}_{3}={\widehat{\varrho}}_{1}=\widehat{\epsilon}$. All remaining P−1 parameters are not relevant for positioning. However, the whole parameter estimate$\widehat{\theta}$ is needed to obtain enhanced channel estimates$\mathit{\u0125}=\mathit{h}\left(\widehat{\theta}\right)$, which are more reliable than the prestage channel estimates$\stackrel{\u030c}{h}$due to the usage of a priori information about pulse shaping and receive filtering.
Performance limits—CRLB
The CRLB corresponds to the best performance that any unbiased estimator can achieve. This means that the covariance matrix of the estimator,${\mathit{C}}_{\widehat{\theta}}\phantom{\rule{0.3em}{0ex}}$, is greater than or equal to the inverse of the Fisher information matrix I(θ)−^{1}[27, 28]:
i.e., the matrix${\mathit{C}}_{\widehat{\theta}}\phantom{\rule{0.3em}{0ex}}\mathit{I}{\left(\mathit{\theta}\right)}^{1}$ is positive semidefinite. If only a single parameter θ_{ p }, 1 ≤ p ≤ P, is considered, the variance of this parameter, which corresponds to the mean squared error (MSE) in case of an unbiased estimator, is greater than or equal to the corresponding diagonal entry of the inverse Fisher information matrix:
According to[28], each entry of the Fisher information matrix is defined as
Employing the Jacobian matrix of the channel function h(θ), which is given by
results in the following Fisher information matrix:
As already mentioned earlier, the estimation of the ToA corresponds to the estimation of the sampling phase and, thus, the only parameter of interest for positioning is${\widehat{\theta}}_{3}={\widehat{\varrho}}_{1}=\widehat{\epsilon}$. Given a certain parameter vector θ, only the corresponding CRLB according to (22) with p = 3 is considered:
Since there are many possible parameter sets θ, the CRLBs are determined semianalytically by means of Monte Carlo simulations. In each run of a Monte Carlo simulation, a different channel realization with a different parameter vector θ is generated and the corresponding Fisher information matrix is determined according to (25). The overall CRLB is given by the expectation of the inverse Fisher information matrices
where the expectation is taken with respect to the parameter vector θ. For all channel models examined below, the CRLBs are determined for different oversampling factors over the SNR in dB. To be more precise, the pilottonoise ratio (PNR),
is taken into account, which is only a fraction of the SNR,
The relationship between the PNR and the SNR is determined by the pilot layer power: γ_{ p } = ρ·γ_{ s }. The following simulation setup is applied if not stated otherwise: A burst length of K^{′} = 100 (K = J K^{′}) is assumed and a pseudorandom sequence of BPSK symbols is used as training. A Gaussian pulse shape
is applied. In order to obtain a causal pulse shape, the Gaussian pulse is shifted by a certain amount s. That means that the overall pulse shape is given by
which is equally distributed among the pulse shaping and the receive filter. Hence, the autocorrelation of each filter corresponds to a Gaussian pulse: ψ_{Tx} (τ) = ψ_{Rx} (τ) = p(τ). In this case, the noise covariance matrix is given by
An effective pulse width of T_{ g } = 8T_{ s } and a shift of s = 0.5 T_{ g } = 4T_{ s } are assumed. In each run of a Monte Carlo simulation, a uniformly distributed random sampling phase ε is generated in the interval [−0.5 T_{ s }, + 0.5 T_{ s }. The remaining parameters of θ are generated according to the applied channel model. In all figures below, the quantities concerning timing or delay measures are normalized with respect to the symbol duration, e.g., the CRLB of ε is normalized to${T}_{s}^{2}$. Three different kinds of channel models are considered: A singlepath channel, several twopath channels, and different WINNER channels according to[33]. The singlepath channel comprises only a lineofsight (LOS) path and is taken into account since it is the best possible case for positioning and, thus, provides a lower bound for all other channel models. The twopath channels comprise an additional propagation path beside the LOS path. By means of the twopath channel models, the influence of different channel characteristics such as the excess delay, power ratio, and phase offset between the two propagation paths can be investigated. The relationships observed for the twopath channels are the basis for more complex channel models with an arbitrary number of propagation paths. In this case, the mutual relationship between all paths determines the performance. This means that the results obtained for the twopath channels can be used to predict the performance for the WINNER channels, that model wireless radio propagation in urban and indoor environments in a realistic way. Due to the assumption of perfect DIC, the performance limits presented below are not only valid for the system concept proposed by the authors, but can also be applied to other multiplexing techniques like TDM.
Singlepath channel
The singlepath channel, which comprises a LOS path only, is taken into account since it is the best possible case for positioning and, thus, provides a lower bound for all other channel models. The channel coefficients of the singlepath channel are modeled by
with f = exp (j Φ), α_{l,J} = l/J + (ε −s)/T_{ s }. The starting phase Φ is generated randomly between 0 and 2Π. As there is only a singlepath, there are no excess delays and, thus, the channel memory length results in L^{′} = 9 (L = J L^{′}). In Figure2, the normalized CRLB of ε for the singlepath channel is plotted over the PNR. In Figure2a, the influence of the burst length K^{′} is shown for symbolrate sampling (J = 1), whereas in Figure2b the influence of the oversampling factor J is illustrated for a burst length of K^{′} = 100. In all cases, the CRLB decreases with the PNR and is much smaller than one, which corresponds to a small fraction of the squared symbol duration${T}_{s}^{2}$. This means that the estimation error is much smaller than the symbol duration T_{ s }. The larger the burst length, the better the performance: The CRLB improves by approximately 3 dB if the burst length is doubled (see Figure2a). The same influence of the burst length is observed in all other channel models. Hence, only burst lengths of K^{′} = 100 are considered in the following if not stated otherwise. Furthermore, oversampling provides a slight performance gain as shown in Figure2b: For all oversampling factors J ≥ 2, the CRLBs are improved by approximately 0.8 dB in comparison to symbolrate sampling. A similar behavior is observed for the remaining channel models as well, i.e., all oversampling factors J ≥ 2 lead to same performance gain. Hence, only J = 2 is considered subsequently. The question arises for what reason oversampling provides a performance gain? What is the difference between symbolrate sampling and oversampling? In order to answer these questions, the CRLB for the singlepath channel is examined in more detail. The starting point is the Fisher information matrix given in (25). In order to keep the following investigations manageable, two assumptions are applied which simplify the determination of the Fisher information matrix. First, white noise is assumed for all oversampling factors and, second, the burst length is assumed to be large. In this case, the inverse covariance matrix of the channel estimation error can be approximated by a scaled identity matrix
with scaling factor$\gamma =({K}^{\prime}{L}^{\prime})/\left(J{\sigma}_{n}^{2}\right)$. This leads to an approximate Fisher information matrix
Hence, only the Jacobian matrix needs to be determined. The partial derivatives of the channel coefficients are given by
Multiplying the Hermitian conjugate of the Jacobian matrix with the Jacobian matrix itself and taking the real part of the result leads to a Fisher information matrix of the following form:
with
Inverting the matrix in (37) results in
The CRLB of the sampling phase corresponds to the third main diagonal entry of the above matrix
Since f^{2} is always one in the singlepath channel, the difference between symbolrate sampling and oversampling must depend somehow on the sums over the exponential terms and, thus, on the sampling phase itself. In Figure3, the influence of the sampling phase on the approximate CRLB is illustrated: In Figure3a, the approximate CRLB is plotted over the sampling phase ε for symbolrate sampling and oversampling with J = 2 for γ_{ p } = 10 dB. For oversampling with J = 2, the approximate CRLB is constant and does not depend on ε. In contrast, the approximate CRLB varies with the sampling phase for symbolrate sampling. Thus, there is a considerable difference between symbolrate sampling and oversampling. The same behavior can be observed for the average channel power divided by the oversampling factor,
which corresponds to the numerator in (43). The average channel power according to (44) is plotted in Figure3b: For oversampling with J = 2, the power of the channel is constant and, thus, independent of the sampling phase. For symbolrate sampling, in contrast, the power of the channel is a function of the sampling phase: The highest power is obtained for ε = 0 (perfect synchronization), while the smallest power occurs for ε = 0.5T_{ s }. This means that nonperfect synchronization leads to a power loss concerning symbolrate sampling. Similar results have already been reported by the authors in[6] for a rectangular and a root raised cosine pulse shape.
Now, it is clear that there is a considerable difference between symbolrate sampling and oversampling. But the question remains: How does this difference lead to the observed performance gain of oversampling? In order to answer this question, it is necessary to return to the exact Fisher information matrix described by (25) and the corresponding exact CRLB. How does the behavior of the CRLB change if the exact instead of the approximate version is taken into account? In Figure4a, both types of the CRLB are plotted over the sampling phase ε for K^{′} = 100, γ_{ p } = 10 dB and J = 1,2. The curves are labeled with “A” for approximate and “E” for exact. In order to have a closer look at the different behavior of the approximate and the exact CRLB, the normalized difference between the symbolrate sampled and the oversampled CRLB according to
is plotted for each type in Figure4b. For the approximate and the exact CRLB, the shape of the corresponding curves is basically the same. The difference between the approximate CRLB and the exact CRLB lies in the fact that the curves of the exact CRLB are shifted downward with the shift for symbolrate sampling being smaller than the shift for oversampling. This means that the curves of the normalized difference between the CRLBs are shifted upward. A slight deviation is observed in case of the exact CRLB for J = 2: In contrast to the approximate CRLB for oversampling, the exact CRLB for oversampling is not absolutely constant over the whole range of sampling phases, but it slightly increases for ε ≥ 0 (see Figure4a). This effect is due to the short burst length of K^{′}=100. For larger burst lengths (e.g., K^{′}=400), this effect is not present anymore and the exact CRLB for J = 2 is absolutely constant over the whole sampling phase range as well. The most interesting aspect is revealed by the mean of the normalized difference, which is shown as a straight line in Figure 4b. If white noise is assumed for all oversampling factors (“A”), then the mean of the normalized difference (45) equals zero in all cases. That means that on average there is no difference between symbolrate sampling and oversampling such that the CRLBs determined via Monte Carlo simulations (with a random sampling phase) are the same. In case of colored noise (“E”), the mean of the normalized difference (45) is approximately 0.16 for K^{′}=100, i.e., the exact CRLBs for symbolrate sampling and oversampling differ on average by a small amount. This difference leads to the oversampling gain which has been observed in Figure2b.
Last but not least, it is interesting to note that the mean of the normalized difference as well as the oversampling gain vary with the burst length K^{′}. In Figure5, the influence of the burst length K^{′} is examined in more detail: In Figure5a, the CRLB of ε is plotted for symbolrate sampling and oversampling with different burst lengths of K^{′} = 100 and K^{′} = 400. As already mentioned before, the performance gain of oversampling is approximately 0.8 dB for K^{′} = 100. For K^{′} = 400, this gain is increased to approximately 1 dB. For comparison, the mean of the normalized difference is plotted over the burst length K^{′}in Figure5b. The circles denote the simulated values, while the dashed line corresponds to a fitted curve. It is observed that the mean normalized difference increases with the burst length and saturates around 0.223, i.e., from approximately K^{′} = 800 upward the mean normalized difference is constant. The mean normalized difference for K^{′} = 400 is approximately 0.2. Interestingly, the ratio of the performance gains corresponds to the ratio of the mean normalized differences: 0.8/1 dB = 0.8 = 0.16/0.2. Thus, the mean of the normalized difference translates into the performance gain with a positive factor. As the mean normalized difference saturates around 0.223, the maximum achievable oversampling gain is approximately 1.115 dB in a singlepath channel.
Twopath channels
Twopath channels represent the simplest form of a multipath channel. Since block fading is assumed, a twopath channel can be fully characterized according to
By means of the twopath channel models, the influence of different channel characteristics such as the excess delay ν_{2}, power ratio$\mathit{P}={a}_{1}^{2}/{a}_{2}^{2}$, and phase offset ΔΦ = Φ_{2} − Φ_{1}between the two propagation paths can be investigated. The maximum possible excess delay is fixed to${\nu}_{2}^{\text{max}}=2{T}_{s}$, which leads to a channel memory length of L^{′} = 11 (L = J L^{′}). At first, the CRLB of ε is examined over PNR, where the power ratio and the phase offset are generated randomly in the intervals [0.1, 10] and [0,2Π], respectively. Concerning the excess delay, two different scenarios are considered: one with small excess delays (ν_{2}/T_{ s } ∈[0.05,1]) and one with large excess delays (ν_{2}/T_{ s } ∈[1,2]). The corresponding CRLBs for different oversampling factors are shown in Figure6. The overall performance decreases with respect to the LOS channel (as expected) but the oversampling gain increases. Both, the overall performance and the oversampling gain, depend on the channel model: The smaller the excess delay, the worse is the CRLB and the larger is the oversampling gain. For the channel with small excess delay, the gain is approximately 6.5 dB, whereas it is only 2.6 dB for the channel with large excess delay. In order to have a closer look at this dependence, the CRLBs are determined over the excess delay for a fixed PNR (γ_{ p } = 10 dB). Similarly, the influence of the phase offset ΔΦ and the power ratio P is investigated.
In contrast to the LOS channel, the approximate Fisher information matrix according to (35) and the corresponding approximate CRLB are not considered here. Due to the increased size of the Fisher information matrix (6 × 6 instead of 3 × 3), the resulting formula for the approximate CRLB consists of many different terms and is, therefore, less concise than in the case of the LOS channel. Two basic simulation setups are applied, whose characteristics are tabulated in Table1. The parameter of interest is varied, while the other parameters are fixed to the values given in Table1. The corresponding CRLBs and their normalized differences according to (45) are shown on the lefthand side and on the righthand side of Figure7, respectively. The curves are labeled with “S1” for the first setup and “S2” for the second setup. The mean normalized difference of the LOS channel (for K^{′} = 100) is given for comparison in Figure7b,d,f since it corresponds to a lower bound concerning the normalized difference of the twopath channels. In the first row of Figure7, the influence of the excess delay ν_{2} is illustrated for both setups, while the influence of the phase offset ΔΦ and the power ratio P are shown in the second and third row, respectively. The overall performance and the oversampling gain strongly depend on the simulation setup and the channel characteristic of interest. It is obvious that the first setup corresponds to a kind of worst case concerning parameter estimation, whereas the second setup represents a kind of best case scenario. For the first setup, oversampling can provide a significant gain over symbolrate sampling. This gain is not influenced by the power ratio (see Figure7f), while it mainly depends on the excess delay ν_{2}(see Figure7b): The smaller the excess delay, the larger is the oversampling gain. Hence, oversampling proves especially helpful in dense multipath scenarios. Furthermore, the phase offset ΔΦ has a significant impact on the oversampling gain, which is highest when both propagations paths have the same or the opposite phase (see Figure7d). As mentioned above, the normalized difference translates into the oversampling gain with a positive factor. Thus, the larger the normalized difference, the larger is the oversampling gain.
Figure7 does not only illustrate the dependence of the oversampling gain on the channel characteristics, but gives also an insight into the relation between these characteristics and the overall performance. With decreasing excess delay ν_{2}, the CRLB increases as it becomes more difficult to separate the two propagation paths. Similarly, the CRLB is worst if the paths are in phase (ΔΦ = 0°) or have an opposite phase (ΔΦ = 180°). The smaller the power ratio P , the smaller is the amplitude of the first path in comparison to the second path and the more likely it is that the delay of the first path, namely the sampling phase, is estimated wrongly. Thus, the influence of the channel characteristics on the overall performance can be summarized as follows: Dense multipath scenarios with similar or opposite phases and a small power ratio are most challenging. The more challenging it is to estimate the sampling phase, the higher is the gain due to oversampling. Hence, the application of oversampling (with J = 2) is highly recommended.
WINNER channels
The WINNER channels described in[33] represent realistic scenarios with a high number of propagation paths. Many different propagation scenarios are considered including rural, suburban and urban as well as indoor scenarios. Generally, the WINNER channel models are suited for the evaluation of multipleinput multipleoutput systems. However, only the case of singleinput singleoutput (SISO) is considered here. Two types of channel models are presented in[33]: The generic model, that is suited for system level simulations, and the clustered delay line (CDL) model, that is reduced in complexity for fast link level simulations. In this contribution, the CDL models are utilized. The parameters of the CDL models like the excess delay or the average power of a multipath component are fixed and tabulated. A single multipath component is called cluster in[33]. With the Gaussian pulse shape in (30) and the block fading assumption, the channel coefficients for WINNER channel models with I clusters are given as
where the complex amplitudes f_{ i } are determined by the superposition of R = 20 rays according to
If a LOS component is present, the first cluster consists of R = 21 rays. Each cluster is assigned a normalized amplitude A_{ i } that is computed from the tabulated cluster powers P_{ i }as
in order to achieve a power normalization. The starting phases Φ_{i,r} are determined randomly between 0 and 2Π for each ray in every cluster. Three different WINNER channel models are considered in this paper: typical urban microcell (B1LOS), large indoor hall (B3LOS) and typical urban macrocell (C2LOS). For an exact description of these scenarios and the corresponding parameter tables please refer to[33].^{a} The most important parameters of the WINNER channel models are summarized in the upper part of Table2. In addition to the number of clusters I and the number of parameters P, the Rician factor K_{ R } as well as the excess delays ν_{2} and ν_{ I } are listed. Furthermore, the minimum delay difference, min {ν_{ i }−ν_{i−1}}, and the average delay difference, E {ν_{ i }−ν_{i−1}}, between neighboring paths is determined. For the minimum delay difference, the cluster number for which this difference occurs is given. Until now, only normalized excess delays ν_{2}/T_{ s }have been considered. Thus, the preceding results are valid for any choice of the symbol duration T_{ s }. In case of the WINNER channel models, the excess delays are given as absolute values in ns. Hence, the simulation results depend on the choice of the symbol duration T_{ s }. Here, T_{ s } = 10 ns is considered. In the lower part of Table2, the channel memory length L^{′} for symbolrate sampling ($L=J{L}^{\prime}$) as well as the relative excess delays and relative delay differences for T_{ s } = 10 ns are tabulated.
In the WINNER channel models, parameter estimation is much more challenging in comparison to the twopath channel models since a much larger number of propagation paths I>2 is considered. Thus, the dimension of the estimation process P = 3I increases significantly. However, the relationships observed for the twopath channels are the basis for more complex channel models with an arbitrary number of propagation paths like the WINNER channel models. In this case, the mutual relationship between all paths determines the performance, where the relationship between the first and the second path is of special importance. A performance prognosis can be given based on the results obtained for the twopath channel models: Dense multipath scenarios with similar or opposite phases and a small power ratio are most challenging. Due to the superposition of several rays per cluster with random starting phases, the phase offsets and power ratios of all paths are random and can not be influenced. Nevertheless, the Rician factor is a valuable indicator since it is defined as the average power ratio of the LOS path and the scattered components. The smaller the Rician factor, the smaller is the average LOS power and the worse the performance should be. From this point of view, the B3LOS channel is most challenging. Furthermore, the relative excess delays and delay differences are of interest. Especially, the relative excess delay of the first multipath component ν_{2}/T_{ s } is important since the corresponding pulse overlaps the most with the LOS component. For T_{ s }=10 ns, the relative excess delays of the first multipath component are in a reasonable range. The same is valid for the relative minimum delay differences. Again, the B3LOS is most challenging with respect to delay differences because a very small delay difference of 3.5 ns occurs twice. Taking all parameters into account, the worst performance is expected for the B3LOS channel (large indoor hall), whereas the best performance is predicted for the B1LOS channel (urban microcell). The CRLBs of all three WINNER channels for symbolrate sampling and oversampling with J = 2 are shown in Figure8. The above performance prediction is met by all simulation results: The best and worst performance is obtained for the B1LOS and the B3LOS channel, respectively. For oversampling with J = 2 (dashed lines), the CRLBs are even better than the corresponding CRLB for the twopath channel with small excess delay. The performance of the B1LOS channel is even similar to the performance of the twopath channel with large excess delay. In contrast, a significant performance degradation is observed for symbolrate sampling. For the B3LOS channel, the normalized CRLB at γ_{ p } = 0 dB is very high with a value close to 10^{4}. This means that oversampling gains larger than 30 dB are possible in case of the WINNER channel models. Hence, it is highly recommended to apply oversampling with J = 2 because accuracies well below the symbol duration can only be obtained for oversampling.
The symbol duration of T_{ s } = 10 ns has been chosen for the simulations in order to achieve reasonable values for the relative excess delays. With increasing symbol duration T_{ s }, the relative excess delays and delay differences decrease. Very small delay differences lead to illconditioned (or even rankdeficient) Fisher information matrices, that are difficult to invert. In order to avoid a complete failure of the matrix inversion, singular value decomposition as described in[32, p. 62ff.] can be applied. However, the corresponding CRLBs degrade significantly if the relative delay differences become too small. In this case, the channel models are not adequate anymore and should be revised because neighboring clusters are not resolvable by any means and act like additional rays that contribute to a single cluster. This means, that clusters with nearly the same relative excess delay should be combined to a single cluster. In this way, the number of clusters is reduced, the new clusters are resolvable and meaningful CRLBs can be determined again.
Impact of the performance limits for CPE on the overall positioning process
As already mentioned in the introduction part of this article, positioning is typically performed in two steps, namely parameter estimation and position estimation. Until now, only the first step (CPE) has been examined. In the following, the impact of the performance limits for CPE on the second step is discussed. Hence, several links between different reference objects (ROs) and a mobile station (MS) in a certain geometrical setup have to be taken into account. For a better understanding, the positioning problem for localization based on the ToA is shortly introduced and a corresponding CRLB is derived. Twodimensional positioning is considered for that purpose, i.e., at least three ROs are needed to determine the position of the MS. An extension to threedimensional localization is straight forward.
The MS’s position, that shall be estimated from the ToAs, is denoted by p =[p_{1},p_{2}]^{T} =[x,y]^{T}, while the known locations of the ROs are denoted by p_{ b } = [p_{b,1},p_{b,2}]^{T} =[x_{ b },y_{ b }]^{T}, 1 ≤ b ≤ B, where B is the number of reference objects. The true distance between the MS and the b th RO is a nonlinear function of the current MS’s position p and can be determined according to
These distances are estimated via the ToAs${\widehat{\tau}}_{b,1}$ based on${\widehat{d}}_{b}={\widehat{\tau}}_{b,1}\xb7\text{c}$, where c is the speed of light. The estimated distances$\widehat{d}={[{\widehat{d}}_{1},\dots ,{\widehat{d}}_{B}]}^{T}$ are called pseudoranges since they consist of the true distances d(p)=[d_{1}(p),…,d_{ B }(p)]T and estimation errors e =[e_{1},…,e_{ B }]^{T} with covariance matrix${\mathit{C}}_{\mathit{e}}=\text{diag}\left({\sigma}_{{e}_{1}}^{2},\dots ,{\sigma}_{{e}_{B}}^{2}\right)$:
Again, an ML estimator can be applied to estimate the position of the MS:
Similar to the ML estimator for CPE, the metric${\Omega}_{\widehat{d}}\left(\stackrel{\mathit{~}}{\mathit{p}}\right)$ for position estimation is nonlinear due to the nonlinear distance function (50). Hence, an optimization algorithm has to be applied similar to the case of CPE. Often, a GaussNewton approach known as Taylor series algorithm[34, 35] is utilized for positioning. But for positioning, there are additionally some approximative, noniterative estimators like the weighted leastsquares algorithm available[36, 37].
As in the case of CPE, a CRLB can be determined for positioning, that corresponds to the best performance that any unbiased estimator can achieve. This means that the covariance matrix of the position estimator${\mathit{C}}_{\widehat{p}}$ is greater than or equal to the inverse of the Fisher information matrix I^{−1}(p)[27, 28]:
i.e., the matrix${\mathit{C}}_{\widehat{p}}{\mathit{I}}^{1}\left(\mathit{p}\right)$ is positive semidefinite. The CRLB for positioning is given by the trace of the inverse Fisher information matrix
Similar to the derivations for CPE, the following expression is obtained for the Fisher information matrix in case of positioning:
where J(p) denotes the Jacobian matrix of the distance function d(p) given by
Since the positions of all involved ROs are required in order to determine the Fisher information matrix, the positioning accuracy depends on the geometry between the ROs and the MS. In order to separate the influence of the geometry from the influence of the estimation errors e on the positioning accuracy, the GDOP is taken into account[10]. The GDOP is defined as the square root of the CRLB, given the assumption that all pseudoranges are affected by the same error variance${\sigma}_{{e}_{b}}^{2}=1$, 1 ≤ b ≤ B (i.e.${\mathit{C}}_{\mathit{e}}=\mathcal{\mathcal{I}}$):
The GDOP can be influenced by an adequate spatial distribution of the ROs. Given a certain GDOP, the CRLB is only influenced by the variances of the pseudorange errors${\sigma}_{{e}_{b}}^{2}$, which should be as small as possible. Since the pseudoranges${\widehat{d}}_{b}$ are determined from the ToAs${\widehat{\tau}}_{b,1}$, which in turn depend on the estimated sampling phase${\widehat{\epsilon}}_{b}$ according to (7), the performance limits for CPE are required in order to determine a CRLB for positioning:
Hence, the performance limits for CPE directly translate into the performance limits for positioning given a certain GDOP.
Conclusion
In this article, the positioning part of the joint communication and positioning system recently proposed by the authors is investigated. CPE is the core part of the system proposal, which is based on IDM in combination with PLACE, and positioning via the ToA. Based on the assumption that a priori information about pulse shaping and receive filtering is available, parameters of the physical channel, that are exploited for positioning, are estimated jointly with the channel coefficients of the equivalent discretetime channel model, which are needed for data detection. There are two equivalent approaches for CPE, which are based on the maximumlikelihood principle: One approach is based on the received samples and the other approach, which is performed in two steps, is based on preliminary channel estimates obtained via standard channel estimation. The latter approach is recommended since it is advantageous from a complexity point of view. Performance limits in terms of CRLBs are determined for a singlepath model, different twopath channel models, and several WINNER channel models. The influence of oversampling and different channel characteristics is investigated. It is shown that oversampling provides a performance gain because the channel power as well as the CRLB for symbolrate sampling depend on the sampling phase, while they are independent of the sampling phase for oversampling. The influence of the channel characteristics in the twopath channels (namely the excess delay, phase offset and power ratio) can be summarized as follows: Dense multipath scenarios with similar or opposite phases and a small power ratio are most challenging. The more challenging it is to estimate the sampling phase, the higher is the gain due to oversampling. Hence, the application of oversampling (with J = 2) is highly recommended in order to improve the positioning accuracy. The relationships observed for the twopath channels are the basis for more complex channel models with an arbitrary number of propagation paths. In this case, the mutual relationship between all paths determines the performance. Consequently, the performance for the WINNER channels can be predicted. A performance prognosis is derived that is met by the simulation results for the WINNER channel models. All results presented in this article are not limited to the proposed system concept but apply for other multiplexing techniques as well.
Endnote
^{a} The WINNER channel models applied in this contribution differ slightly from those defined in[33]. In[33], the two strongest clusters are divided into three subclusters: The first subcluster is composed of ten rays and has a zero delay offset, the second subcluster consists of six rays and has a delay offset of 5 ns and the last subcluster comprises four rays with a delay offset of 10 ns[33, p. 41]. This division is neglected here: All 20 rays have the same delay, where the original delay is offset by 3.5 ns (mean offset).
Abbreviations
 AWGN:

additive white Gaussian noise
 BPSK:

binary phase shift keying
 CDL:

clustered delay line
 CRLB:

CramerRao lower bound
 ENC:

encoder
 GDOP:

geometric dilution of precision
 IDM:

interleavedivision multiplexing
 IDMA:

interleavedivision multiple access
 LOS:

line of sight
 LS:

leastsquares
 ML:

maximumlikelihood
 MLIDMA:

multilayer interleavedivision multiple access
 MS:

mobile station
 MSE:

mean squared error
 PLACE:

pilot layer aided channel estimation
 PNR:

pilottonoise ratio
 RO:

reference object
 SNR:

signaltonoise ratio
 TDM:

timedivision multiplexing
 ToA:

timeofarrival.
References
 1.
HaebUmbach R, Peschke S: A novel similarity measure for positioning cellular phones by a comparison with a database of signal power levels. IEEE Trans. Veh. Technol 2007, 56: 368372.
 2.
Heinrichs G, Winkel J, Drewes C, Maurer L, Springer A, Stuhlberger R, Wicpalek C: System considerations for a combined UMTS/GNSS receiver,. Proceedings of 4th Workshop on Positioning, Navigation and Communication (WPNC), Hannover, Germany, 2007, pp. 189–198
 3.
Ali S, Nobles P: A novel indoor location sensing mechanism for IEEE 802.11 b/g wireless LAN,. Proceedings of 4th Workshop on Positioning, Navigation and Communication (WPNC), Hannover, Germany, 2007, pp. 9–15
 4.
Raulefs R, Plass S, Mensing C: The WHERE project—combining wireless communications and navigation,. Proceedings of 20th Wireless World Research Forum (WWRF), Ottawa, Canada, 2008, pp. 1–5
 5.
Hoeher PA, Schmeink K: Joint navigation communication based on interleavedivision multiple access,. Proceedings of 6th International Workshop on MultiCarrier Spread Spectrum (MCSS), Herrsching, Germany, 2007, pp. 97–106
 6.
Schmeink K, Hoeher PA: Multilayer interleavedivision multiple access for joint communication and localization,. Proceedings of 7th International ITGConference on Source and Channel Coding (SCC), Ulm, Germany, 2008, pp. 1–6
 7.
Hoeher PA, Schoeneich H, Fricke JC: Multilayer interleavedivision multiple access: theory and practice. Eur. Trans. Telecommun. (ETT) 2008, 19(5):523536. 10.1002/ett.1267
 8.
Schoeneich H, Hoeher PA: Iterative pilotlayer aided channel estimation with emphasis on interleavedivision multiple access systems. EURASIP J. Appl. Signal Process 2006, 2006: 115.
 9.
Gezici S: A survey on wireless position estimation. Wirel. Personal Commun. (Special Issue on Towards Global and Seamless Personal Navigation) 2008, 44(3):263282.
 10.
Langley RB: Dilution of precision. GPS World 1999, 10(5):5259.
 11.
Feder M, Weinstein E: Parameter estimation of superimposed signals using the EM algorithm. IEEE Trans. Acoust. Speech Signal Process 1988, 36(4):477489. 10.1109/29.1552
 12.
Fleury BH, Tschudin M, Heddergott R, Dalhaus D, Pedersen KI: Channel parameter estimation in mobile radio environments using the SAGE algorithm. IEEE J. Sel. Areas Commun 1999, 17(3):434450. 10.1109/49.753729
 13.
Richter A, Landmann M, Thomae RS: Maximum likelihood channel parameter estimation from multidimensional channel sounding measurements,. Proceedings of 57th IEEE Vehicular Technology Conference (VTC Spring), Vol, 2 Jeju, Korea, 2003, pp. 1056–1060
 14.
Khayrallah A, Ramesh R, Bottomley G, Koilpillai D: Improved channel estimation with side information,. Proceedings of 47th IEEE Vehicular Technology Conference (VTC Spring), Vol. 2 Phoenix, Arizona, USA, 1997, pp. 1049–1053
 15.
Liang JW, Ng B, Chen JT, Paulraj A: GMSK linearization and structured channel estimate for GSM signals,. Proceedings of IEEE Military Communications Conference (MILCOM), Vol. 2, Monterey, California, USA, 1997, pp. 817–821
 16.
Lee HN, Pottie GJ: Fast adaptive equalization/diversity combining for timevarying dispersive channels. IEEE Trans. Commun 1998, 46(9):11461162. 10.1109/26.718557
 17.
Fock G, SchulzRittich P, Schenke A, Meyr H: Low complexity high resolution subspacebased delay estimation for DSCDMA,. Proceedings of IEEE International Conference on Communications (ICC), New York, USA, 2002, pp. 31–35
 18.
Antreich F, Nossek J, Utschick W: Maximum likelihood delay estimation in a navigation receiver for aeronautical applications. Aerosp. Sci. Technol 2008, 12(3):256267. 10.1016/j.ast.2007.06.005
 19.
van der Veen AJ, Vanderveen M, Paulraj A: Joint angle and delay estimation using shiftinvariance techniques. IEEE Trans. Signal Process 1998, 46(2):405418. 10.1109/78.655425
 20.
Hoeher PA, Robertson P, Offer E, Woerz T: The softoutput principle—reminiscences and new developments. Eur. Trans. Telecommun 2007, 18(8):829835. 10.1002/ett.1200
 21.
Wendlandt K, Khider M, Angermann M, Robertson P: Continuous location and direction estimation with multiple sensors using particle filtering,. Proceedings of International Conference in Multisensor Fusion and Integration for Intelligent Systems (MFI), Heidelberg, Germany, 2006, pp. 92–97
 22.
Robertson P, Roeckl M, Angermann M: Advances in multisensor data fusion for ubiquitous positioning: novel approaches for robust localization and mapping. Proceeding of VDE Kongress Leipzig, Germany, 2010
 23.
Schoeneich H, Hoeher PA, Fricke JC: Adaptive 4G uplink proposal based on interleavedivision multiple access,. Proceedings of General Assembly of International Union of Radio Science (URSI GA), New Delhi, India, 2005, paper no. C03.5
 24.
Ping L, Liu L, Wu K, Leung WK: Interleavedivision multiple access. IEEE Trans. Wirel. Commun 2006, 55(4):938947.
 25.
Kusume K, Dietl G, Utschick W, Bauch G: Performance of interleave division multiple access based on minimum mean square error detection. Proceedings of IEEE International Conference on Communications (ICC), Glasgow, Scotland, 2007, pp. 2961–2966
 26.
Hao D, Hoeher PA: Superposition modulation with reliabilitybased hybrid detection. Proceedings of 6th International Symposium on Turbo Codes & Iterative Information Processing (ISTC), Brest, France, 2010, pp. 280–284
 27.
Scharf LL: Statistical Signal Processing: Detection, Estimation, and Time Series Analysis. (AddisonWesley, Reading, MA, 1991)
 28.
Kay SM: Fundamentals of Statistical Signal Processing: Estimation Theory. (PrenticeHall, Upper Saddle River, NJ, 1993)
 29.
Kennedy J, Eberhart R: Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks Perth, Australia, 1995, pp. 1942–1948
 30.
Bratton D, Kennedy J: Defining a standard for particle swarm optimization. Proceedings of IEEE Swarm Intelligence Symposium (SIS), Honolulu, Hawaii, 2007, pp. 120–127
 31.
Schmeink K, Block R, Knievel C, Hoeher PA, Joint channel and parameter estimation for combined communication and navigation using particle swarm optimizationDresden, Germany, 2010, pp. 4–9
 32.
Press WH, Teukolsky SA, Vetterling WT, Flannery BP: Flannery, Numerical Recipes in C++: The Art of Scientific Computing. Cambridge University Press, Cambridge, England, 2002
 33.
Kyoesti P, Meinilae J, Hentilae L, Zhao X, Jaemsae T, Schneider C, Narandzic M, Milojevic M, Hong A, Ylitalo J, Holappa VM, Alatossava M, Bultitude R, de Yong Y, Rautianinen T: IST4027756 WINNER II D1. 1.2 V1. 2, WINNER II Channel Models. 2008. [http://projects.celticinitiative.org/winner+/WINNER2Deliverables/D1.1.2v1.2.pdf] []
 34.
Understanding GPS: Principles and Applications, chap. 2. Artech House, Boston, MA, 1996
 35.
Shen G, Zetik R, Thomae RS: Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment,. Proceedings of 6th Workshop on Positioning, Navigation and Communication (WPNC), Hannover, Germany, 2008, pp. 71–78
 36.
Sayed AH, Tarighat A, Khajenouri N: Networkbased wireless location: challenges faced in developing techniques for accurate wireless location information. IEEE Signal Process. Mag 2005, 22(4):2440.
 37.
Guevenc I, Chong CC, Watanabe F, Inamura H: NLOS identification weighted leastsquares localization for UWB systems using multipath channel statistics. EURASIP J. Appl, Signal Process. 2008 Article ID 271984 2008, 114.
Acknowledgements
This study was partly funded by the German Research Foundation (DFG project number HO 2226/111).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Schmeink, K., Adam, R. & Hoeher, P.A. Performance limits of channel parameter estimation for joint communication and positioning. EURASIP J. Adv. Signal Process. 2012, 178 (2012). https://doi.org/10.1186/168761802012178
Received:
Accepted:
Published:
Keywords
 Channel Estimation
 Channel Model
 Fisher Information Matrix
 Channel Coefficient
 Physical Channel