Parametric Schemes for Prediction of Wideband MIMO Wireless Channels
Information on the future state of time varying frequency selective channels can significantly enhance the effectiveness of feedback in adaptive and limited feedback MIMO-OFDM systems. This paper investigates the parametric extrapolation of wideband MIMO channels using variations of the double directional MIMO model. We propose three predictors which estimate parameters of the channel using 4D, 3D and 2D extensions of the ESPRIT algorithm and predict future states of the channel using the models. Furthermore, using the vector formulation of the Cramer Rao lower bound for functions of parameters, we derive a bound on the prediction error in wideband MIMO channels. Numerical simulations are used to evaluate the performance of the proposed algorithms under different channel and transmission conditions, and a comparison is made with the derived error bound.
The combination of MIMO transmission with Orthogonal Frequency Division Multiplexing (OFDM)  is a spectrally efficient technique for achieving reliable high bit rate transmission over mobile wideband channels. It is employed in wireless standards such as 3GPP LTE and LTE Advanced , IEEE 802.16e (WiMAX) [3, 4]. Recent capacity approaching MIMO-OFDM based transmission schemes, such as adaptive MIMO precoding , adaptive coding and modulation, adaptive multiuser resource allocation and scheduling [6, 7] and various forms of codebook and non-codebook based limited feedback MIMO, require both at the transmitter and the receiver knowledge of the channel state information (CSI). In time division duplex (TDD) systems, channel reciprocity is used to obtain CSI. In frequency division duplex (FDD) systems however, CSI is estimated at the receiver and relayed in quantized form to the transmitter via a low rate feedback link. In practical MIMO sytems, due to delays in estimation, processing and feedback, the CSI may become outdated before its actual use at the transmitter, resulting in significant performance degradation especially in high mobility environments. Prediction of the CSI has been recognised as an effective technique for mitigating this performance degradation [8, 9].
The problem of channel prediction for single input single output (SISO) channels has been studied extensively. In [8, 10, 11], the narrowband SISO channel is modelled as an autoregressive (AR) process and a linear minimum mean squared error (MMSE) predictor is used to extrapolate the channel states. These schemes consider the time-varying channel as a stochastic wide sense stationary process and use the temporal correlation for prediction without accounting for the physical scattering phenomena that cause the fading. Other researchers [12, 13, 14] have considered a ray based sum of sinusoids model where the fading channel is modelled as a sum of a finite number of plane waves. Extensions of these algorithms to wideband SISO channels have also been studied [15, 16]. Analytical and simulation results on SISO prediction have proven that with dense scattering, SISO channels can only be predicted over a very short distance (on the order of tenths of a wavelength) depending on the environment and propagation scenarios. The bound on SISO channel prediction error  indicates that CSI is required over several wavelengths in order to accurately predict the channel. It has also been shown that prediction beyond a wavelength is not realistic, particularly in practical cases where the stationarity assumption does not hold for a time comparable to the duration of the observation.
The prediction of multi-antenna channels was first investigated in  through an evaluation of downlink beamforming with channel prediction. Improved MISO channel prediction was shown, as more structure of the wavefield is revealed through multiple sampling. Bounds on the prediction error of MIMO channels  and MIMO-OFDM channels  indicate that better prediction can be obtained by utilizing the channel spatial structure. The authors illustrated this using AR modelling for the prediction of beamspace transformed CSI, and argue that the transformation reduces the effective number of rays present in the channel, ultimately resulting in longer prediction. A similar approach based on ray cancelling was presented in .
MIMO prediction schemes can be broadly classified into codebook based precoder prediction and non-codebook based CSI prediction. The codebook based schemes [22, 23] predict the precoder for the next transmission frame using linear prediction. Others adopt Givens rotations to transform the precoding matrix and perform prediction of the Givens parameter . These schemes are limited to one step prediction and the channel model is often assumed to be independent and identically distributed (i.i.d). Non-codebook based schemes predict the actual CSI using either autoregressive modelling or parametric model based SISO approaches . These methods do not utilize the additional spatial information that is revealed by the presence of multiple sensors.
Motivated by the results in [20, 26], where it was shown using error bounds that schemes which incorporate both temporal and spatial channel information offer significant improvement in prediction performance, we make the following contributions in this paper.
Using a double directional spatial channel model, we derive three formulations of prediction models by progressively removing the dependence on array geometry. This allows the investigation of the effects of transmit and/or receive spatial dimensions on prediction performance, and the development of schemes applicable to systems with different antenna geometries.
We propose prediction schemes based on the above models and original adaptations of multidimensional ESPRIT (estimation of signal parameters via rotational invariance techniques) . We show that this approach allows for the resolution of many more paths than the number of antennas at both ends of the link.
The remainder of the paper is organized as follows. Section II presents different formulations of the double directional channel model for wideband MIMO systems. In Section III, we describe the data transformation and preprocessing required for the implementation of the proposed schemes. The channel predictors are presented in Section IV, followed by a derivation of the bound on prediction error in Section V. Performance evaluation results and discussion are given in Section VI. Finally, we draw conclusions in Section VII.
Ii Channel Models
We consider several formulations of a ray-based wideband spatial MIMO channel model for the development of the prediction schemes in this paper. The formulations are extensions of the continuous time impulse response of doubly selective SISO fading channels, defined as
where and are the time and delay variables respectively, is the number of paths, and and are the time-varying complex attenuation and delay of the th path, respectively. We assume that the scattering sources are in the far field of both the transmit and receive antenna arrays such that the propagating waves can be modelled as plane waves. The complex attenuation of the th path can thus be defined as
where is the number of rays in the th path, , and are the complex amplitude and Doppler frequency of the th ray in the th path, respectively. The model in (2) can be extended to a MIMO channel with transmit and receive antennas via the introduction of transmit and receive array structures, giving
where denotes the non-conjugate transpose of the associated matrix, and are the receive and transmit array response vectors and and are the directions of arrival and directions of departure, respectively. Summing (3) over the clusters and taking the Fourier transform in the delay domain, we obtain the frequency response of the MIMO channel as
where is the frequency variable. Assuming symbol duration and subcarrier spacing , the sampled frequency response is given by
where and are the time and subcarrier indices, respectively. and are the normalized radian Doppler frequency and delay, respectively. Combining indices in (5), we obtain
where is the total number of propagating rays. Each ray is characterized by the parameter set . We assume that no two rays share a common parameter set, but different rays may have one or more equal parameters. Note that in practical scenarios, (6) has a finite support in the transmit angular, Doppler, delay and receive angular domains since the multipath parameters are bounded.
We now describe different formulations of the model in (6), where we progressively remove restrictions on the array structure.
Ii-a DOD/DOA Model
The first model is based on the assumption that the array response vectors and are explicit functions of the directions of arrival (DOA) and directions of departures (DOD)  as shown in (6). Note that this model is valid for any array geometry. We will consider systems with uniform linear arrays (ULA) at both ends of the link. The receive steering vector for an element array is thus
is the receive array element spacing. The transmit array steering vector is defined analogously, with
Here, is the transmit array element spacing. Henceforth, the parameter set for the models will include the spatial frequencies and rather than the actual directions and , since the latter can be trivially obtained from , using (8) and (9).
Ii-B Transmit Spatial Signature Model (TSSM)
The DOD/DOA model depends on the specific array configurations and the angles of arrival and departure. Since the transmitter is often stationary
in mobile wireless systems, estimation of the angles of departure may be difficult and possibly not required for accurate prediction of the channel.
We therefore replace the product of the complex amplitude and the transmit array steering vector for each path in (6) by an unstructured
transmit spatial signature (TSS)
where is the TSS for the th propagating wave.
Ii-C Matrix Spatial Signature Model (MSSM)
Similar to the TSS model, the MSSM  replaces the product of the array steering vectors by an unstructured matrix spatial signature, , giving
Note that although the three models described in this section are derived from the double directional MIMO model, they differ in the parametrization and number of parameters required. A summary of the number of parameters required for each model and dependence on number of antenna elements is shown in Table I. The parameters of the channel are assumed quasi-stationary over the spatial distance for which channel observation/measurement and prediction is made. This assumption has been shown in the industry standard 3GPP/WINNER II SCM model [28, p. 55] to be valid for mobile movements up to . We also assume that temporal samples and frequency samples of the channel frequency response matrix are available by transmitting known training sequences or from other channel estimation approaches.
In practice, the estimated or measured channel will be imperfect due to the effects of noise and interference. The estimated CSI matrix at
time instant for the th subcarrier is therefore defined as
where is a matrix of complex Gaussian random variables that accounts for channel estimation errors.
|Model||Structural Param.||Amp.||Real Param.|
Iii Data Transformation
Having described the channel model variations for the development of the prediction schemes, we now present the data preprocessing necessary for extraction of the parameters from available channel observations.
Iii-a DOD/DOA Transformation
As shown in (6), the DOD/DOA model is characterized by structural parameters and complex amplitudes. Extraction of these parameters from the channel observations requires a four dimensional array data structure. Since we consider MIMO systems with a 1-D antenna array (i.e., a ULA) at both ends of the link, it is necessary to convert the channel matrices into a form that allows 4D parameter estimation to be performed. Let be a vector obtained by stacking the columns of . Using (6) and the properties of the Kronecker product, it can be shown that
where denotes the Kronecker product. Note that the transformation in (13) combines the receive and transmit spatial dimension of the channel. In order to introduce the temporal dimension, we define
and form a Hankel matrix by sliding an rectangular window through (14) to obtain
where . The frequency dimension of the channel is similarly introduced by forming a block Hankel matrix from such matrices to obtain
where and are the Hankel matrix size parameters with . The values of , , , and are selected such that . There is, however, a compromise in selecting these: large values of and increases the number of rows in and hence the number of sources that can be resolved, but this results in small values of and which affects the accuracy of covariance estimates. Using the model in (13) and the transformations in (15) and (16), the data in the columns of can be modelled as
Defining , (17) can be expressed as
where and is a Vandermonde structured steering matrix, with defined as vectors of their respective parameters. Clearly, (17) corresponds to a four dimensional array data model obtained by combining the transmit spatial, temporal, frequency and receive spatial dimensions of the wideband MIMO channel. A summary of the dimensions and corresponding parameters is shown in Table II.
Iii-B TSS Transformation
As shown in (10) and Table I, parametrizing the TSSM requires structural parameters . We will here derive a data structure that allows joint extraction of these parameters from (10). Using the temporal samples, we start by forming a block Hankel matrix for each frequency sample,
Note that (20) combines the receive spatial and temporal property of the channel into one dimension corresponding to the columns of . In order to include the frequency dimension of the wideband channel, we form another block Hankel matrix from (20) as
where is defined analogously to (III-A).
Iii-C MSS Transformation
The MSSM is parametrized by structural parameters and complex amplitude parameters. Extraction of these parameters from the channel requires a two-dimensional datum. Similar to (14), we form a Hankel matrix
Note that the columns of correspond to temporal measurements of the channel and can be modelled as
where is an dimensional vector defined in (III-A). The data in (24) provides information about the Doppler shifts of the channel. The frequency structure of the channel can be included by forming a block Hankel matrix analogous to with replaced with . The columns of can be shown using (24) to be
As shown in (III-C), corresponds to a two-dimensional datum obtained by combining the temporal and frequency structure of the channel. The Doppler shifts and delays of arrival can therefore be extracted jointly using appropriate parameter estimation algorithms.
Iv Prediction Algorithms
We now propose prediction algorithms using the models developed in Section II and the transformed data derived in Section III. We will henceforth refer to the algorithms as WIMEMCHAP: WIdeband Multidimensional Esprit based Mimo CHAnnel Predictor, and use the acronyms for the models as prefixes to distinguish the schemes. For example, the algorithm based on TSSM will be called TSSM-WIMEMCHAP and so on. Note that although the algorithms are based on the same idea of parametric modelling, they differ in the model, dimension of parameter estimation and number of amplitude and structural parameters to be estimated.
Consider the transformed data model in (17) and (III-A). Since the array steering matrix is equivalent to a product of four Vandermonde matrices, the invariance structure in can be utilized to estimate the parameters of the channel. Motivated by the accuracy and computational efficiency of the ESPRIT algorithm , we propose an adaptation of multidimensional extension of ESPRIT to jointly extract the parameter sets and apply the parameter estimates to extrapolate the channel. The prediction algorithm can be divided into the following stages:
covariance matrix estimation and subspace separation,
number of paths estimation,
joint parameter estimation,
We will now describe the different stages of the algorithm.
Covariance Matrix Estimation
In the presence of estimation or measurement noise, the model in (III-A) becomes
where denotes the Hermitian transpose and is the covariance matrix of the amplitude parameters. can be expressed in terms of in (16) as
Number of Paths Estimation and Subspace Decomposition
Estimation of the number of paths is typically a model order selection problem. The Minimum Description Length (MDL) is often used for this purpose due to its accuracy and consistency [29, 30]. We utilize a modified version of the MDL referred to as the Minimum Mean Square Error (MMSE)-MDL  defined as
where are the eigenvalues of . Once has been estimated, the eigenvalue decomposition of can be expressed as
where and are the signal subspace eigenvectors and the associated eigenvalues, respectively. The noise subspace eigenvectors and eigenvalues are contained in and , respectively.
We now outline the process of obtaining the parameter sets in Table I. In order to explore the invariance structure  in the Vandermonde structured space-time-frequency manifold matrix, , we define the following selection matrices:
where is an identity matrix and is an F-dimensional vector of zeros. Using the selection matrices in (IV-A3), we define the following invariance equations
where , , and are matrices, the eigenvalues of which contain information about the parameters:
where denotes the diagonal eigenvalue matrix of the associated matrix. We minimize the squared error of the equations in (IV-A3) to obtain
Estimates of the AOAs, AODs, Doppler shifts and delays could be obtained directly from the solutions of (34)–(37) followed by an additional pairing stage. In order to achieve automatic pairing of the estimates, we utilize a scheme similar to the mean eigenvalue decomposition (MEVD) pairing scheme . Defining
we perform eigenvalue decomposition of to obtain the common eigenvectors of the four matrices in the sum
The diagonal eigenvalue matrices are then obtained using
where , , and . Finally, estimates of the parameters are evaluated from (IV-A3) as
Complex Amplitude Estimation
We assume that the complex amplitude of each path is equal for all antenna pairs, which is reasonable considering the separation of gain from array dependent steering vectors and in (6). The complex amplitudes can therefore be estimated via a least square fit to the known channel. Using the Vandermonde structure of and (6), we form the following equation for the first entry of in (12) for the first subcarrier
which can be written in matrix form as
can be obtained via a regularized LS solution of (49) as
where is the regularization parameter choosen to minimize the effects of errors in on the estimation. For the rest of this paper is chosen emprically as . Note that although (50) gives an estimate of the complex amplitudes, our preliminary simulations show that improved estimates can be obtained by using more entries of in the estimation. We therefore generalize (49) as
with defined as
where denotes the Khatri-Rao product, and is defined as
is defined analogously. We combine the equations in (51) and solve for as
where and . It should be noted that the choice of using (50) or (54) is essentially a compromise between complexity and accuracy, since the improved amplitude estimates in (54) is achieved at the cost of increased computational complexity. We will utilize (54) for our analysis in this paper.
Once the parameters of the model have been estimated, the time-varying frequency selective channel is predicted via
Unlike the DOD/DOA-WIMEMCHAP approach, the TSSM-WIMEMCHAP involves 3D parameter estimation. The steps involved in the TSSM-WIMEMCHAP approach are as follow.
Covariance Matrix Estimation
The covariance matrix, containing the receive spatial, temporal and frequency correlations, is estimated from (21) as
Number of Paths Estimation and Subspace Decomposition
We again estimate using the MMSE-MDL criterion in (29) and eigendecompose as
Extraction of the parameter sets requires a 3D estimation procedure. Similar to (IV-A3), we form the following 3D selection matrices
The TSS can be similarly obtained via a least square approach. We assume that the TSS of the scattering sources are equal for all subcarriers and
use the channel for the first subcarrier in the estimation
where and is defined as
where is the receive array steering matrix defined for the ULA as
We solve (59) using the least square approach and estimate the TSS for each path as
Extrapolation of the channel using the TSSM-WIMEMCHAP is obtained from
This approach involves estimation of the Doppler shifts and delays, which can be achieved via a 2D estimation procedure. This method is essentially an extension of the SISO schemes in  to MIMO channels. A summary of the steps in the prediction is given below.
Covariance Matrix Estimation
The time-frequency covariance matrix is estimated using
Number of Paths Estimation and Subspace Decomposition
We estimate the number of paths using the MMSE-MDL criterion in (29) with the eigenvalues of . The eigendecomposition of can thus be expressed as
Channel prediction using the MSSM-WIMEMCHAP is achieved using
V Performance Bounds
A commonly used bound on the performance of an unbiased estimator is the Cramer-Rao lower bound (CRLB) . In this section, we derive the bound on the variance of prediction error in wideband MIMO systems. While similar results have been presented in [20, 26], we present an alternative, simpler formulation. Consider the vectorized model in (13). Using the properties of Kronecker products, can be written as