Limited Feedback Multiuser MIMO Techniques for Time-Correlated Channels

This work presents limited feedback schemes for closed-loop multiple-input multiple-output systems using frequency division duplex. The proposed methods employ compact feedback messages in order to (a) feed back and track a complete frequency-ﬂat channel matrix, to be used as input to multiuser multiplexing methods designed for full channel side information (CSI) at the transmitter, and (b) enable the receiver to command the transmit weight adaptation, in order to maximize the link reliability under strong intercell interference. Simulations show that the channel feedback accuracy provided by the proposed algorithms produces a negligible bit error probability (BEP) performance loss in low mobility scenarios compared to the full CSI performance, and that the proposed interference rejection techniques can e ﬀ ectively exploit an estimate of the interference statistics in order to enable multiple-stream communications under the permanent presence of intercell interference signals.


Introduction
Wireless communications with multiple antennas at transmitter and receiver ends have the potential of offering high data rates and spectral efficiency.On one hand, the data rates can be increased by transmitting several parallel streams.On the other, interference rejection techniques can be employed to enhance the link reliability, enabling communications under high interference levels.This can lead to an increase in the spectral efficiency of the system, for example, by tightening the reuse of the frequency spectrum.Similarly, multiuser multiplexing techniques can enable several users to share the same frequency resource, which also leads to a higher spectral efficiency of the system.
In order to realize the afore-mentioned benefits of MIMO systems in a computationally efficient manner, low computational complexity linear detectors may be employed that exploit full or partial CSI at the transmitter.The way in which the CSI is acquired depends on the system under consideration.For systems employing frequency division duplex (FDD), which are of interest here, the use of a feedback channel per user is necessary.Three main uses of the feedback channel can be found in literature.The first pertains the adaptation of the transmit antenna weights, commanded by the receiver.For example, single-user closedloop eigenbeamforming systems deal with the right singular vectors of a channel matrix, either by feeding back a quantized version or recursively tracking it, see, for example, [1][2][3][4][5][6].This type of feedback has typically considered only structured (e.g., orthonormal) matrices.The second use is intended to provide the transmitter with an approximation of the channel matrix estimated by the receiver, which is then used as an input to multiuser multiplexing algorithms such as [7][8][9].This type of feedback deals with unstructured matrices and has not received much attention in literature.The third type of feedback content, which is not treated, is the transmission of channel quality indicators (CQIs) for the purposes of multiuser scheduling.Recently published work in this area and related references can be found in [10].
This article is divided in the following two major sections.
(1) Closed-loop MIMO communications under strong interference conditions, which falls within the first category, but is different from the eigenbeamforming problem, whose scope is limited to the channel right singular vectors.In the proposed methods, the receiver informs the transmitter of the transmit weights that maximize the link reliability, conditioned on an estimate of the statistics of the noiseplus-interference signals.The general case of an arbitrary number of data streams is considered, and a specialized low computational complexity solution for the single stream case is also provided.Furthermore, the algorithms can employ either orthogonal or nonorthogonal transmit beams.
(2) Channel feedback algorithms that allow the reliable tracking of a complete frequency-flat channel matrix, which falls into the second category, and the goal is to provide the input to multiuser MIMO solutions designed for full CSI.To avoid excessive signaling of channel parameters, we propose channel feedback methods based on the principles of partial update.That is, only a small part of the channel matrix is updated at each feedback instant.Moreover, a static channel convergence analysis has been provided for the basic building block of the channel feedback algorithms.
High data rate transmissions in closed loop MIMO systems with limited feedback have been extensively studied, see, for example, [1,2,5].However, these solutions do not assume any external interfering signals, and are therefore not suitable for interference limited scenarios.In this work, we propose algorithms for multiple stream transmission and intercell interference cancellation, using a low rate feedback channel and a linear receiver.This constitutes an extension of the classical IRC receiver [11,12] to closed-loop MIMO systems, and differs from open-loop MIMO-IRC schemes such as [13], where no CSI is used.In the proposed algorithms, the receiver employs the feedback channel to instruct the transmitter on how to recursively adapt the beamforming weights in order to maximize the link reliability in the presence of intercell interference.The proposed tracking solution exploits both transmit and receive diversity, and a short-term estimate of the interference-plus-noise statistics.More specifically, the signal to interference-plus-noise ratios (SINRs) are computed for each stream, as a function of the transmit weights.These rates can then be used to compute a link performance metric and the weights can be adapted to optimize its value, with the feedback message conveying the weight update information to the transmitter.For the purpose of illustration, we use the total uncoded conditional BEP of the user as a link quality metric.The resulting algorithm can operate with any symbol constellation for which the uncoded performance of the detector is known.We stress that the formulation extends easily to other SINR to BEP mappings, including channel coding or laboratory measurements of actual receiver implementations.Furthermore, the proposed closed-loop MIMO-IRC algorithms can operate on streams with equal transmit power and equal bit load, giving a similar performance per stream.This can ease the design of the adaptive modulation and channel coding layer, when compared to a system using eigenbeamforming, where the gains per stream are intrinsically different due to the eigenvalue spread of the channel.
The closed-loop MIMO-IRC solutions presented can be implemented on both orthogonal and nonorthogonal transmit beams.This is a system design choice and will be reflected in the way that the precoding (beamforming) matrix will be updated, upon arrival of the feedback messages.For example, an orthogonal beamforming matrix can be updated based on increments to the real-valued angles that parameterize the matrix, while a nonorthogonal matrix can be updated via premultiplication with a matrix exponential.Orthogonal transmit beams have the advantage that the total transmit power is the sum of the individual beam powers, as opposed to the nonorthogonal beams case, where the total power varies with the nonorthogonality.This eases the dynamic range requirements of the power amplifier, compared to the usage of nonorthogonal beams.The use of orthonormal matrix decompositions to feed back or track the right singular vectors (eigenbeams) of a channel matrix has been considered in [1,3,4,6].In contrast, the MIMO-IRC algorithms presented here do not feed back the channel eigenbeams, but rather inform the transmitter of the weights that optimize the link performance metric, conditioned on the current channel and the estimate of the interference plus noise covariance matrix.For the particular case of a single user with only one data stream (single beam, single user system), a low computational complexity update arises as an extension of [5], where the update is based on a single complex-valued Givens rotor, which sequentially visits all the coordinate planes associated with the optimal beamformer.
In the second part of this article, channel feedback methods are presented, which allow reliable tracking of the complete channel matrix of a user, employing low rate feedback channels.The CSI so acquired can then serve as input to any MU-MIMO multiplexing solution designed for full CSI, for example, [7][8][9].This type of CSI is different from that considered in eigenbeamforming algorithms [1,4,5], where the main idea is to exploit the orthonormal structure of the right singular matrix of the channel, to enable an efficient representation.The feedback of the unstructured channel is based on a single-bit tracking of the real and imaginary parts of every element of the complex-valued channel matrix, where each scalar is tracked with the singlebit tracking structure presented in [14].Despite the simplicity of such solution, reserving two feedback bits for each channel coefficient may be prohibitive.To further reduce the feedback requirements, we propose an alternative approach based on partial updates, where only a reduced number of channel matrix elements are updated on each update instance.In particular, we consider a simple sequential strategy where the update proceeds taking groups from a circular list, which is shown to be sufficient in scenarios with moderate antenna array sizes and low fading rates.When the number of antennas or the fading rate increases, however, a more sophisticated selection rule to determine which elements of the tracked matrix will be updated is required.Thus, a selective or ranked partial-update approach sacrifices some feedback bits in order to signal which matrix elements are the most urgent to update.These partial update principles have been previously employed to decrease the computational complexity in adaptive filters [15,16], and to enable good tracking performance in low-rate closedloop eigenbeamforming [17,18].A further insight into the channel feedback problem is given in an accessory study, where a link to the closed-loop eigenbeamforming algorithms is made.Indeed, by vectorizing the channel matrix and normalizing the resulting vector, any method to track the dominant eigenbeam of a channel can be used, while the norm of the vector is tracked with a single bit tracking structure from [14].These methods, however, do not outperform the proposed partial update methods in the considered simulation campaign.
This paper is organized as follows.Section 2 provides a description of the system model under consideration.Section 3 describes a particular link performance metric used for linear receivers.This formulation will be used to optimize the transmission weights for the case of singleuser MIMO-IRC communications in Section 4. The channel feedback algorithms are described in Section 5, where both a sequential and selective partial update strategy will be applied to the recursive tracking of the whole channel matrix of a user.A static channel convergence analysis for the building block of the proposed channel feedback algorithms is given in Section 5.4.Simulations are provided in Section 6 and conclusions are given in Section 7. The appendix summarizes the different alternatives for tracking matrices and vectors employed throughout this work.

System Model
The system under consideration is illustrated in Figure 1 and consists of a multiuser MIMO system with N t transmit antennas, and N u slowly moving users.Each user has N r antennas and a feedback channel carrying binary messages of length n b , b i ∈ {0, 1} nb×1 with frequency f b = f x /L, where f x is the symbol frequency and L 1.The transmitter employs a fixed overall amount of N b beams, with one data stream per beam.Each user is allocated N bi out of the N b streams, and its beamforming weights and symbols are represented by W i ∈ C Nt×Nbi and x i ∈ C Nbi×1 , respectively.We assume that the symbols of each stream have an average power P ≡ E{|x| 2 }.The overall beamforming or linear precoding matrix contains all the per-user matrices as Similarly, all the per-user symbols constitute the symbol vector where H i (k) is the channel matrix between the transmitter and user i, n i (k) includes the Gaussian thermal noise ν i (k) with E{ν i (k)ν i (k) † } = σ 2 I, N I intercell interfering signals u mi received by user i, and the intracell interference from the beams belonging to the other n u − 1 users, and l denotes the update instant of the weights.
Let the noise-plus-interference covariance matrices Q i be defined as As mentioned earlier, we assume that there are L symbol periods, each representing an instance of (1), before the feedback message is transmitted.The feedback message b can convey information about the update of the transmit weights (single user case), or about the CSI of each user (multiuser case).The L symbol periods constitute a slot and the delay of the feedback message is neglected.We model the interfering signals as complex Gaussian signals, with a fading rate not necessarily equal to that of the user.
We consider linear receivers Ω i ∈ C Nr ×Nb i , upon which each user computes the following quantity for detection: where the user i receives N bi independent data streams, T i (k, l) and n i (k) define an equivalent linear system, and each user can compute its receiver Ω i independently.
The receiver matrix Ω i is in general a function of H i , W i , and Q i .For the multiuser case, there is coupling between the matrices Q i and W due to the intracell interference.This does not occur in the single-user case, where Q is merely due to intercell interference.
Conditioned on {H i } Nu i=1 , and given W and the receive filters {Ω i } Nu i=1 , the SINR for each stream of user i is obtained from (3) and (2) as where T i,pq is the element p, q of matrix T i given in (3), Q i is defined in (2), ω i,p is the column p of Ω i , and the term ω † i,p Q i ω i,p is the expected power of the filtered noise, for the stream p of user i.
In this paper, we will use the following receiver structure, referred to hereafter as the MIMO-IRC receiver: For the MIMO-IRC receiver (5), the matrix T i reduces to Similarly, the covariance matrix of the filtered noise becomes Therefore, the SINRs (4) for stream p under the IRC receiver can be written as These ratios can be used to define a link quality measure as a function of the transmit weights, given knowledge of the channel matrices and noise statistics.In the single user case, the receiver can acquire knowledge of H and Q, and use the feedback channel to command the adaptation of W to optimize its interference rejection capabilities.In the multiuser case, on the other hand, the receivers employ channel feedback methods to convey their matrices H i to the transmitter, which in turn uses the link performance metric to jointly determine all the per-user transmit weights.The use of an SINR to BEP mapping as a link performance metric is described in the following section, and channel feedback methods are described in Section 5.

Transmit Weight Optimization for Link Reliability
Adapting the transmit weights is a central part of closed-loop MIMO and closed-loop MU-MIMO systems.The weights typically optimize a link quality measure, given the channel conditions and noise statistics.For single user systems with linear receivers, the N b dominant right singular vectors of H are of interest if the noise is spatially white.Indeed, when the precoder is restricted to be an orthonormal matrix, it can be shown that the precoder formed by these vectors optimizes the mutual information, the SNR of the weakest stream under the ZF receiver, and the trace of the MSE matrix under the linear MMSE receiver [19].For MU-MIMO systems, the weights must optimize the simultaneous transmission of the N u users.For example, the sum-MMSE for all the users can be solved iteratively by the transmitter, assuming that it has knowledge of the channel matrices and the noise covariance matrices [7].
When the disturbance signals are not spatially white, however, the optimal MIMO precoder becomes a function of both the channel matrix and the covariance of the noise plus interference signals.Consider, for example, the mutual information for independent Gaussian source symbols and Gaussian noise with covariance Q, given H, and a fixed W: For spatially uncorrelated noise with Q = I Nr , P equals the transmit SNR divided by the number of streams, and the mutual information reduces to log 2 |I + P W † H † HW|, which is maximized by diagonalizing H † H.In any other case, using only the channel information to choose the precoder is suboptimal.Note that Q is only available to the receiver, which must compute and feed back the optimal matrix W, instead of the channel eigenbeams.Since the size of the matrix being fed back is the same, a properly designed feedback method has the potential of conveying the optimal precoder, without additional feedback requirements.
In this work, we will consider the optimization of the SINRs under the IRC receiver, as given in (8).For N u = N b = 1, optimizing the instantaneous SINR minimizes the expected uncoded BEP and maximizes the ergodic capacity.The optimal precoding vector can be easily solved by the receiver upon H, Q, and efficient tracking algorithms can be used to signal the precoder adaptation through the feedback channel.This is the subject of Section 4.1.
For the more general case of N b > 1, the statistics of each SINR determine the performance of the associated stream.However, a scalar function of the SINRs is needed as link quality metric for transmit weight adaptation.In the single user case, general-purpose stochastic search techniques are used in conjunction with an update rule of the transmit weights, to define a closed-loop IRC-MIMO system.This is described in Section 4.2.For the MU-MIMO case, the weight optimization provides the means to assess the performance of the channel feedback methods presented in Section 5. Simulation results are provided in Section 6, which quantify the performance loss incurred by the system when using the output of the channel feedback algorithms, instead of the true channel matrices H i .
The link quality measure considered in this article is computed upon multiple SINRs as the average conditional BEP, over all the streams.It is straightforward, however, to use other metrics such as mutual information or a weighted MSE.This is possible in the single user case, because the feedback mechanism presented in Section 4.2 can be used with any link quality measure.
Consider the SINRs per stream defined in (4) and a mapping P ip (•) between the SINR and the BEP, which represents the uncoded performance of the detector, depending on the symbol constellation employed on the data stream p by user i.The total conditional BEP across the data streams is a weighted sum of the BEP of the data streams of the user, depending on the bit load per stream: where b ip is the number of bits per symbol on stream p of user i and P ip (•) can be any suitable SINR to BEP mapping, including laboratory measurements.For the sake of simplicity, we present simulation results based on the AWGN BEP approximations for M-QAM and M-PSK given in [20].Furthermore, the total conditional BEP for the system, that is, the BEP across the streams of all the users, is the weighted sum of the individual BEP of the users where the BEP of user i from (10) weighs in the total BEP according to the ratio of its bit load to the total number of bits of the system.The total BEP of the system is therefore a function of W, when conditioned on the channel matrices and assuming that the statistics of the interference-plus-noise can be estimated.Note that orthogonal training sequences would be required in the MU-MIMO case, for the receivers to estimate their covariance matrices Q i and form the optimal combiners.

Interference Tolerant Closed-Loop MIMO Communications
This section describes novel closed-loop MIMO transmission schemes that enable single user communications in the presence of strong intercell interfering signals.Based on the SINRs of each data stream from (4) and a link quality measure defined upon the SINRs (as discussed in Section 3), the receiver commands the transmit weight adaptation through the feedback channel.Assuming that the receiver can acquire knowledge of H and Q, the link performance metric can be treated as a function of the transmit weights W.
For the case of a single data stream, the single SINR is used as metric, and the optimal weight vector can be computed in a simple manner.This enables the use of an efficient lowcomplexity tracking scheme to convey the optimal W to the transmitter, and is described in Section 4.1.For N b > 1, on the other hand, the optimal W needs to be computed iteratively, and a generic stochastic perturbation technique defines the update through the limited feedback channel.Furthermore, two update rules are considered, reflecting different orthogonality constraints for W.This is the subject of Section 4.2.
The following sections deal with the particular case of N u = 1, and we will drop the associated index i from matrices

Efficient Single Beam Algorithm Based on Jacobi Rotations.
The performance of the single data stream N b = 1 is determined by the statistics of the instantaneous SINR.In this section, we discuss the weight vector that maximizes the SINR under the IRC filter, and feedback mechanisms to enable the tracking of the optimal precoder at the transmitter.
Given the MIMO channel matrix H and the transmit weights W ≡ w, the equivalent channel at the receiver is a SIMO channel: Using the IRC combiner from ( 5), matrix T in (3) collapses to a scalar and the SINR gain becomes which is the SINR of the optimal SIMO combiner in [12].However, (13) shows that (1) the optimal weight vector w is an eigenvector associated to the dominant eigenvalue of H † Q −1 H, and (2) the dominant eigenvector of H † H produces ρ ≤ ρ opt , with equality only if Q = σ 2 I.This implies that to optimize the SINR, the feedback mechanism must track the dominant eigenvector of H † Q −1 H, rather than the dominant channel eigenbeam.
Let the eigenvalue decomposition of The optimal SINR can be written as In the SIMO case, the channel U † Q h is Gaussian, conditioned on Q.Furthermore, the joint statistics of the eigenvalues λ Q,m can be computed, and the expected BEP can be written as an iterated integral of the conditional BEP, where the first integral is taken over the channels and the second over the interferers.This approach has been explored in [11], where bounds for the symbol error probability are derived under the assumption of independent elements of h.In the MIMO case, however, the transformed channel U † Q h = U † Q Hw is not a linear combination of independent Gaussian variables because w is the dominant eigenvector of H † Q −1 H.This complicates the extension of the approach in [11] to the MIMO case.Furthermore, the exact PDF of ρ opt is difficult to obtain even in the SIMO case.To assess the best possible performance, we will compute an empirical PDF based on samples of users and interfering channels.This will be used as a reference for the performance of the feedback algorithms presented in what follows.
Assuming that both the channel matrix and the interference covariance matrix have some temporal autocorrelation, the dominant eigenvector of H † Q −1 H can be tracked at the transmitter with the use of the feedback channel.This can be accomplished, for example, by using the D-JAC algorithm [5] to track the dominant eigenvector of the modified channel correlation matrix The D-JAC algorithm [5] can track N eigenvectors of a generic Hermitian matrix R ∈ C M×M , with an update based on a single complex-valued Givens rotor.This rotor is associated to a coordinate plane that is chosen sequentially among all the MN − N(N + 1)/2 possible planes.In this case, we have M = N t , N = N b = 1 and, therefore, N t − 1 planes are considered.The combination of the IRC receiver with the D-JAC update operating upon H † Q −1 H for N b = 1 will be referred to as the IRC-D-JAC algorithm.Each plane is updated by one rotor, where the indices (p, q) defining the location of cosine and sine elements of the Givens rotor are taken circularly from the list {(2, 1), . . .(N t , 1)}.
The application of one rotor in plane (p, q) is defined as where J p,q is the complex-valued Givens rotor or Jacobi transformation in plane (p, q) [21], computed upon the matrix R and the auxiliary matrix Φ(l) as in [5], l denotes the update instant, and W 0 contains the first left column of the identity matrix.Alternatively, the use of vector codebooks can also be considered.Assuming that the interferers are independent of a spatially white user channel, we hypothesize that the optimal weight vector is isotropically distributed on the unit hypersphere, as in the case of spatially white noise.Then one can use, for example, a Grassmanian codebook [22] to feed back the optimal vector nonrecursively.On the other hand, the multiple beam algorithm presented in the following section can also be applied to the single beam case.However, the IRC-D-JAC algorithm has lower computational complexity and can achieve near-optimal performance at low mobile speeds (cf.Section 6), and is therefore preferred.

Multiple Beam Algorithm.
Consider a single user with N b > 1 data streams and the IRC combiner of (5).Because no other users are present, there is no intracell interference, and therefore no coupling between the matrices Q and W. The SINR of stream p follows directly from (8) and equals where the terms T pq represent interference between the streams.It can be seen that making While this can be accomplished efficiently by using the D-JAC algorithm to track the N b dominant eigenvectors of H † Q −1 H, it results in an SINR spread as large as the eigenvalue spread of H † Q −1 H.The SINR spread is performance detrimental if the symbol constellations of the streams are fixed and identical.This can be compensated by choosing fixed constellations with different bit loads, upon the statistics of the eigenvalues of H † Q −1 H.Alternatively, adaptive constellation switching can be used, at the expense of additional feedback overhead and computational complexity.This motivates the use of a link quality metric that can balance the stream performance while employing a single, fixed symbol constellation for all the streams.However, ( 6) and (17) determine how the matrix W can influence the effective SINR of each data stream, and therefore a link performance metric such as the total conditional uncoded BEP from (10) or the mutual information from (9).The goal of the algorithms presented in this section is to convey the optimal W to the transmitter through the feedback channel.
In order to allow the tracking of the optimal beamforming matrix based on short feedback messages, a stochastic perturbation search is performed based on the current knowledge of H, Q and the current beamforming weights W. The receiver tests 2 nb stochastic perturbations about the current W and chooses the one giving the best value of the cost function of choice, for example, the BEP across streams (10).Then the index of the chosen matrix is fed back to the transmitter, which updates W in the same way as the receiver.Two update formulas are considered, depending on whether or not the transmit beams are restricted to be orthogonal.In both cases, the candidate generation is controlled by a step size parameter μ > 0.
The update of a tall orthonormal matrix W is done by adding increments to the angles that parameterize the matrix through a cascade of complex-valued Givens rotors, as defined in (A.1).This defines the IRC-SCGAS algorithm, which can be considered an extension of the Stochastic Complex Givens-based search over the Angle Space (SCGAS) algorithm [6].Let θ(l) contain 2N t N b −N b (N b +1) real-valued angles associated to the current value of the beamforming matrix W, that is, W(l) = M(θ(l)) from (A.1).In this case, 2 nb−1 perturbations k n are generated as i.i.d.zero-mean real-valued Gaussian vectors and 2 nb candidate matrices are generated as Let m * ∈ {1, . . ., 2 nb } be the index of the candidate matrix giving the best value for the cost function.The update proceeds as where the updated parameters are kept within their nominal ranges by using the inverse mapping M −1 .Because the mapping M(•) only involves Givens rotors, the resulting W matrix is guaranteed to be orthonormal without explicitly enforcing the constraint.Alternatively, the candidates can be built based on left multiplication with unitary matrices.These matrices can be computed as cascades of complexvalued Givens rotors.This type of update has been used in the single-bit Incremental Givens Rotations Eigenbeamforming (IGREB) algorithm [6].Furthermore, the unitary matrices can also be built as matrix exponential of skew-Hermitian matrices [21].This has been exploited for singlebit eigenbeamforming in [23].
The update of a nonorthogonal W will be done by leftmultiplication with nounitary matrix exponential (expm) and is referred to as IRC-EXPM.The receiver generates 2 nb−1 matrices K n ∈ C Nt×Nt with i.i.d.zero-mean circular Gaussian entries.The 2 nb candidate matrices are then built as where the scaling restricts the squared Frobenius norm to N b and constrains the average transmit power, and e (•) is the matrix exponential [21].Note that due to the nonorthogonality of the transmit beams, a higher peakto-average power ratio (PAPR) of the transmitted signal is observed, compared to the case of the orthogonal beams.Let m * be the index of the chosen matrix, as in the IRC-SCGAS algorithm.The update is then As mentioned earlier, the matrices W produced in the IRC-EXPM algorithm are not constrained to have orthonormal columns, as those produced by IRC-SCGAS are.This can have an impact to the performance, because there are more degrees of freedom associated to the nonorthogonal beams, which allows the IRC-EXPM algorithm to find better solutions to the optimization of the total BEP from (10).The performance difference is discussed in Section 6 and illustrated in Figure 2.

Computational Complexity and Effect of the Fading
Rate.Recursive closed-loop MIMO algorithms can typically achieve better tracking performance than the nonrecursive solutions, over a range of low mobile speeds.The maximum speed up to which a recursive solution provides an advantage is algorithm-and system-specific, and depends on the convergence speed of the algorithm, the feedback frequency, the fading rate of the channel, and also on the operational SNR.Indeed, the errors associated to poor tracking performance are less severe in low SNR conditions, where noise and interference can partially mask the effects of outdated transmit weights.Moreover, the optimal step size also varies with the feedback rate and the mobile speed, see, for example, [24] for an analysis of the performance of signed stochastic gradient approximations in autoregressive channel models.A performance comparison between the use of static vector and matrix codebooks, and some recursive eigenbeamforming solutions can be found, for example, in [5,17,18].The use of switched codebook techniques such as [2] or hierarchical codebook structures such as the one described in [25] could improve the performance of the systems using static codebooks at low speeds.However, tuning the associated adaptation parameters for different mobile speeds can be time-consuming, and we have therefore restricted our choice of alternatives to static codebook techniques.
Simulation results are provided in Section 6, which illustrate the performance of the proposed algorithms under a fixed feedback frequency, and as a function of the mobile speed.In particular, it must be noted that the fading rate of the gain matrix T in the equivalent system (3) is a function of the relative motion speeds between transmitter and receiver, and between the receiver and the interferers.It is expected therefore that the performance degradation from increasing the mobile speed is less severe if the relative motion between the receiver and the interfering sources is slow.This can be the case, for example, when a dominant interferer moves EURASIP Journal on Advances in Signal Processing roughly in the same direction of the receiver, with a similar speed.
The single-beam algorithm IRC-D-JAC presented in Section 4.1 offers a computational complexity reduction, when compared to the cost of evaluating the optimal SINR over a vector codebook of size 2 nb .Both algorithms require computing R = H † Q −1 H.However, the codebook lookup implies 2 nb evaluations of the quadratic form w † Rw, which is O(N 2 t ), while the D-JAC has an update cost from (16), that is, O(N t ).
On the other hand, the multiple-beam algorithms presented in Section 4.2 incur in a higher computational complexity, when compared to the use of a fixed matrix codebook.Both the codebook lookup and the stochastic perturbations techniques require evaluating the cost function 2 nb times.However, generating the candidate matrices from the current weights W is an additional cost associated with the proposed algorithms.In the case of the IRC-EXPM algorithm, this can be alleviated partially by using precomputed matrix exponentials.In the case of the IRC-SCGAS method, however, the cost of building the candidate matrices from the perturbed angles cannot be avoided, albeit the Givens rotations operations can be implemented efficiently in hardware.

Limited Rate Channel Feedback Methods for MU-MIMO
In this section, we consider recursive channel feedback strategies for time correlated channels.These methods can provide the transmitter with the user channel matrices required by MU-MIMO solutions designed for full CSI, for example, [7,9,26].The proposed method is an alternative to predictive vector quantization (PVQ) schemes like [27,28], which can have a high computational complexity due to vector codebook lookup, codebook switching and the use of the vector predictor.Furthermore, the associated codebooks need to be trained for different mobility and channel correlation assumptions.In contrast, we propose the use of single-bit quantizers with adaptive step size, hereafter referred to as "trackers," to independently encode the realvalued components of each channel element.This results in a simplified design, which is shown to achieve good performance in low mobility scenarios and moderate antenna array sizes, with a low computational complexity (cf.Section 6).
Due to the limited-rate characteristic of the feedback channel, the information about the trackers update must be conveyed through messages of n b bits.This motivates the use of partial updates, where a group of trackers is updated on each slot.The ideas behind the tracker selection stem from partial update adaptive filters [15,16], and consist of both a sequential partial update (e.g., round-robin update), and a signal-dependent selective update, where a group of trackers is selected for update, that gives the best improvement of a given cost function.The single bit real-valued component trackers (SBRVTs) consist of a memory device, which holds the tracked value and the current value of the step size, and a fixed rule for step size adaptation.The single-bit quantizers are well-known components of linear and adaptive delta modulation (ADM) signal representation techniques [29,30], and the particular step size adaptation rule used here has been considered earlier in variable step size LMS filters [14].
The SBRVT tracking structure and the assignment to channel elements is described in the next section.Thereafter, the round-robin and selective updates are formulated in Sections 5.2 and 5.3, respectively.A convergence analysis of the SBRVT in static channels is provided in Section 5. 4, where a bound of the expected convergence time is derived, given the SBRVT parameters.Performance considerations about the impact of the mobile speed are given in Section 5.5.This section concludes with the formulation of an alternative approach to the channel feedback problem, where a connection to low feedback rate eigenbeamforming techniques can be formed.The resulting methods are described in Section 5.6.

Tracking Based on Single-Bit Update for Real-Valued
Components.The proposed channel feedback algorithms use a total of 2N r N t single-bit trackers, where each tracker follows a real-valued quantity defined as the real or imaginary part of an element of the channel matrix.Depending on the bit budget of n b bits and the update strategy, however, not all the 2N r N t trackers may be updated on a given feedback message.Let the real-valued components of the channel coefficients be denoted by h j , j = 1 . . .2N t N r and defined as that is, the real and imaginary parts of H mn are listed consecutively, and the enumeration proceeds along the rows of H, from the leftmost to the rightmost column.A full update will denote the feedback of 2N r N t bits, one for each tracker.Depending on the antenna array sizes and fading rates, a full update may not be necessary to enable good tracking of H and partial updates can be considered, as described in the following sections.We denote the tracked value of h j as h j .
The tracking function behind each h j is defined as follows.Let n e be the number of consecutive update bits with the same sign that have occurred prior to the current update instant.Similarly, let n d be the number of consecutive bits with different sign.Additionally, the step size adaptation is controlled by parameters Δ min , Δ max > 0 (bounds for the step size Δ), α u > 1, 0 < α d < 1 (multiplicative factors to vary the step size), and m 0 , m 1 , which are positive integers controlling the responsiveness of the adaptation rule.Both n e , n d are set to zero in the beginning, and the operation proceeds as follows.
(1) Compute the current error (l) = h(l) − h(l), with h(l) the true value of the channel component, assumed known to the receiver.
(5) Transmitter: upon receiving the feedback message, extract the single bit associated to sign[ (n)].
As mentioned earlier, this tracking function is similar to the continuously variable slope delta modulation techniques from early speech digital transmission works [29,30].A simplified version with Δ min = 0, Δ max = ∞ has been used in [1] to track each of the angles parameterizing the channel eigenbeams.We restrict our attention to the case α u = 1/α d ≡ α for simplicity.It will be shown in Section 5.4 that m 1 = α is a sufficient condition for convergence in static channels.For tracking applications, however, the parameters m 1 = m 0 = 1 can result in better performance due to faster adaptation of the step size [14].

Sequential Update Channel Feedback.
A simple partialupdate strategy updates groups of n b < 2N r N t trackers at each update instance.No priority is given to any tracker, and therefore all the trackers are visited sequentially in a circular manner, n b trackers on each feedback message.Due to the fixed update sequence, there is no need to include the indices of the trackers to be updated, in the feedback message.
Let I represent the last tracker updated on the previous slot.The update considers the n b trackers with indices That is, the indices are visited circularly in groups of n b trackers, and the feedback message b contains the n b bits destined to update the corresponding trackers.
Note that if n b is allowed to be larger than 2N t N r , some trackers are visited more than once on a given update instance, thus constituting a full update followed by a partial update of n b − 2N t N r trackers.This resembles a step in data reuse filtering [31], and can be necessary for fading rates higher than those of pedestrian speeds (cf. Figure 3).

Ranked Partial-Update for SBRVT.
As the dimensions of the channel matrix grow, the selection rule for choosing which trackers will be updated becomes important.Indeed, due to the limited feedback characteristics of the system, Tracking performance of channel feedback algorithms for a system with N t = N r = 4 at 10 km/h.The partial update alternative requires more than 40 bits to match the performance obtained with 13 bits at 3 km/h (Figure 4).The Givens rotor-based method for vectorized channels could improve the performance at n b = 40 provided that 8 rotor angles can be encoded reliably.This requires using (40 − 2)/16 ≈ 2.38 bits per angle, but the encoding method is still an open problem.a round-robin partial update may miss the trackers in the most urgent need for update, which will translate to a poor tracking performance.In this section, we describe a selective partial-update method that can ameliorate this effect.Such an approach employs part of the feedback message to signal a group of trackers that should be updated next, while the rest of the message contains the update bits for the selected trackers.This ranked partial update strategy has been applied before to closed-loop eigenbeamforming algorithms in [17,18] and is somewhat similar to the antenna selection (AS) strategy for transmit diversity, albeit AS requires only selecting which antennas are employed, and does not transmit any information associated with the selected antennas.
Consider a set {c n ∈ {0, 1} 2NtNr ×1 } Ng n=1 of binary vectors with Hamming weight N tr representing N g different groups of N tr trackers signaled for update.If a given vector c m is chosen, then the trackers h j with index corresponding to the nonzero entries of c m are to be updated.In order to ensure that every tracker can be updated, the binary addition n c n must be a vector containing only ones.
The receiver tests each tracker group c n and ranks them according to the total tracking error in the group, defined as (23) where c n,m is the element m of c n , h m is the current value of the tracker associated to h m , and δ(•, •) is one if both arguments are equal, and zero otherwise.The group with the largest error is then selected for update.The feedback message contains n b bits, out of which log 2 (N g ) bits signal the chosen group, and the remaining N tr bits contain the update information for each selected tracker.This algorithm will be referred to as the ranked single-bit per real-valued component tracking method (R-SBRVT).
The choice of n b , N tr , N g such that n b = N tr + log 2 N g is system-dependent and reflects a tradeoff between the signaling overhead and the benefit of the ranked update.A perfect ranking of the N tr most urgent trackers, on the other hand, can result in an excessive overhead and is in general not efficient.Such a scheme requires log 2 ( 2Nr Nt Ntr ) + N tr bits, and it can be outperformed by the sequential algorithm operating at the same feedback rate.
The problem of choosing the N g groups resembles a vector quantization problem over a binary space of dimension 2N r N t .A thorough treatment of the problem is beyond the scope of this article, and we have limited ourselves to finding some groups of indices providing good performance, through numerical search procedures over sets of N g binary vectors of size 2N r N t with a "large" minimum Hamming distance among them.As an example, Figure 4 shows the benefit of the selective update in a system with n b = 13, N t = 4, N r = 4, where 5 bits are used to signal out one of N g = 32 binary vectors of Hamming weight 8, each representing a group of trackers that can be updated.

Convergence of SBRVT in Static Channels.
In this section, we analyze the convergence properties of the SBRVT mechanism described in Section 5.1.First, we model the output of the tracker in response to a fixed input h, drawn from a known distribution F h (•).Let h(n) be the output of the algorithm at update instant n.We say that the algorithm converges to Δ min if there exists an integer v t > 0 such that Δ(n) = Δ min , for all n > v t .Without loss of generality, we assume that h > 0, and therefore the algorithm traces a monotonically increasing curve until it surpasses the value of h.The following three-branch function models the rise of the algorithm output under a stream of positive input bits, which is related to the aforementioned first segment of h(•): We characterize the learning curve h(•) by t monotonic segments and t vertices, where a vertex is a pair v, h(v) defined as In other words, the curve increases monotonically up to the first vertex, after which the sign of the error changes between vertices, up to vertex v t , after which the error is bounded by Δ min .In the following, propositions will summarize the results of the analysis.The proofs will be given in Appendix B.
The following stablishes a sufficient condition for convergence, assuming m 0 = 1.

Proposition 1. Given a static channel h, a sufficient condition for the SBRVT algorithm to reach a vertex
Assuming the sufficient condition for convergence m 1 ≤ α , m 0 = 1, the location of the first vertex can be computed as follows.
Proposition 2. Given a static channel h and the conditions m 1 ≤ α , m 0 = 1, the SBRVT algorithm reaches the first vertex at the update time v 1 given by , m 1 Δ min <h < f (Z) where Z denotes m 1 + p and where the function f (•) and the number p are given in (24).Now, the number of vertices and their locations can be determined.This gives the convergence time.

Proposition 3. The number of vertices
Furthermore, the vertices and their associated outputs are with v 1 computed from (26) and f (•), p defined in (24).
To conclude the analysis in static channels, we give an upper-bound for the convergence time, and the expected value of this bound, assuming that the cumulative distribution function (CDF) is known.

). Given a symmetric PDF of h with CDF F h (•), the expected value of the convergence time bound is given by
with the auxiliary function and number f (•), p defined in (24).
Note that the infinite summation term can be safely truncated at a value i max such that F h [g 3 (i max )] ≈ 1. Numerical examples of the expected convergence time are given in Section 6.3.

Performance Of SBRVT in Time-Varying Channels.
When the channel component h is time-varying, the tracking error h(l) − h(l) becomes a random variable, and one can attempt to compute its variance.One approach has been proposed in [32], where the average error power from tracking a linear segment h(n) = nξ is averaged over the PDF of the slope ξ.In particular, for Gaussian channels the slope ξ is also Gaussian, and its variance can be computed from the autocorrelation function, which in turn can be related to the maximum Doppler frequency experienced by the receiver.
The study in [32] employed a simulations-based function giving the average noise power for a given slope.Furthermore, the algorithm considered there was a discrete adaptive delta modulation scheme, where the possible values of the step size are a finite number of multiples of Δ min .On the other hand, the slope approach of [32] can be extended to the SBRVT case by using the formulation of the previous section.Indeed, one can use the learning curve h(n) = f (n ≤ v 1 ) to compute the average error in a linear segment.However, the error so computed depends strongly on the value assumed as initial step size, which in turn depends on the state of the SBRVT previous to the entering the linear segment.Thus, one needs to compute an expected initial step size, and subsequently use it to average the error power over the PDF of ξ.This calculation is complicated and beyond the scope of this paper.We therefore restrict ourselves to studying the performance of the sequential SBRVT-based channel feedback method of Section 5.2, assuming that the average error is known as a function of the mobile speed.
Let e(v) be the average tracking error of the SBRVT algorithm, for mobile speed v.The total average error for i.i.d.variables h i is defined as If using n b = 2N t N r feedback bits, every tracker is updated on every slot, and each tracker has an update frequency equal to the feedback frequency.More generally, each tracker experiences an effective speed v = 2N r N t v/n b , and the total error is This formula gives a rough prediction of the performance, as shown in Figure 5.

Methods Based on Normalized Column
Vectors.One of the main considerations when comparing the feedback of the whole matrix H with existing works in eigenbeamforming is that the matrix H has no structure, its power is arbitrary, and the columns are in general not orthogonal.In this section, however, we show that both problems can be related.Let h = vec(H)/ H F and n h = H 2 F , where vec(•) converts the matrix argument to a single column vector by stacking the columns vertically.Let n h be fed back to the transmitter by using a single-bit tracking structure [14], as any of the real-valued components h j of the previous sections.It only remains to define how to track and feed back h ∈ C NtNr ×1 , so that the transmitter can build its own version of H, namely, H t .
We consider two alternatives for tracking h, namely, (1) build R = h h † and use any closed-loop eigenbeamforming algorithm to follow the dominant eigenvector, which is trivially h, and (2) treat h as the N b = 1 case of the matrix W and feed it back as in the IRC-EXPM algorithm of Section 4.2, with cost function || h − h t || 2 , where h t is the tracked version of h.Some eigenbeamforming algorithms like [5,6] exploit a phase ambiguity in the eigenbeams.This comes from the fact that if v is an eigenvector of R, so is e jθ v, where θ is an arbitrary angle.When using these algorithms, therefore, a phase correction is required at the transmitter in order to track h.This angle can be tracked through a single bit tracker and is computed as − arg( h † h t ).
As the product N r N t increases, the computational complexity associated to updating h by premultiplying with matrix exponential also grows.Furthermore, the conver-gence speed of a solution based on the D-JAC algorithm [5] decreases with larger vector sizes, because the contribution of each rotor to the overall change in the vector (after update) becomes smaller.One possibility to escalate the algorithm with the antenna array size or the fading rate is to increase the number of coordinate planes per update to r > 1 rotors, that is, to apply (16) r times per update.This, however, poses a vector quantization problem, since all the 2r angles associated to the rotors must be fed back to the transmitter.The potential of this technique is illustrated in Figure 3, where the updates proceed using unquantized rotor angles, which constitutes the performance bound of the tracking method.

Simulations
The simulation scenario consists of a transmitter with N t = 4 transmit antennas, and users moving at 3 km/h with a carrier frequency of 2.1 [GHz], in spatially white Rayleigh fading channels.Each slot contains L = 160 symbols and has a length of 1/1500 [s].The interfering signals will be modeled as Gaussian SIMO interferers, with a fading rate that is either the same as that of the user, or is fixed at 3 km/h.A strong jamming situation is considered, where a single interfering signal arrives at the receiver with a signal-to-noise ratio equal to that of the transmitter.It is assumed that each receiver knows its channel matrix H i perfectly.
Whenever there is need to evaluate a conditional bit error probability (BEP), we use the two-terms approximations developed in [20] for M-QAM and M-PSK constellations.The Gaussian Q function involved in the BEP expressions can be evaluated with good accuracy with the approximation [33].

Interference Tolerant Single User Communications.
In order to provide a fair comparison between feedback methods, and restrict the effect of the channel variations to limited accuracy or limited convergence speed issues, we abstract from the covariance matrix estimation problem.Therefore, the algorithms will use a "near-perfect" estimate of Q(k, l), given by In other words, the true covariance is computed at the beginning of the slot l, and used throughout the L symbol periods.
Figure 6 shows the average BER of the single beam system, where 16QAM symbols are used, n b = 6 and the mobile speeds are 3 km/h.It can be seen that the IRC-DJAC reaches near-optimal performance.In comparison, the eigenbeamforming (EBF) system employing an IRC receiver suffers a penalty of several dB, because the interference statistics are not considered for weight adaptation.Moreover, the transmission without the IRC receiver is not feasible.The empirical PDF of SINR achieved by IRC-DJAC is shown in Figure 7  Single user, single beam system Permanent interfering signal  is also shown in Figure 6, where a difference of about 1 dB to the optimum is observed, at BER levels of 0.001.
The BER performance as function of the mobile speed is given in Figure 8.As discussed in Section 4.3, if the relative speed between the interferer and the receiver is small, then some slowness is retained, which yields a better tracking performance, compared to the case when the relative motion  between interferer and receiver is comparable to that between the transmitter and the receiver.It can be observed that the recursive techniques provide a performance advantage over the range 3-30 or 3-20 km/h, depending on the SNR.
On the other hand, when two parallel streams are used, a third receive antenna is necessary to cope with a single interferer.In general, the IRC algorithm requires diversity in order to cancel interference, so the requirement will be N r > N b .
Figure 2 shows the performance of the IRC-SCGAS and IRC-EXPM algorithms with two beams and 2 × 16QAM constellations.The performance of the fully jammed (the interferer has the same transmit power to noise power as the user) is shown to be feasible, and a loss of 4 dB at BEP 0.01 is observed, compared to the noise dominated scenario.Based on the results of Figure 2, it is clear that the transmission without the IRC receiver would be unfeasible.The nonorthogonal beams provided by the IRC-EXPM algorithm are shown to outperform the orthogonal beam configuration, and the Grassmanian codebook of size 64 from [19] performs comparatively poorly.To be fair, this codebook is meant for precoding in uncorrelated MIMO channels under the assumption of spatially white noise, and therefore the result presented here does not reflect negatively upon the codebook methodology.
Figure 9 shows the trade-off between the number of feedback bits for the IRC-SCGAS algorithm and its BEP performance.It can be seen that the algorithm is able to exploit additional receive diversity to further boost the performance.
The distributions of the instantaneous rates log 2 (1 + ρ i ) are shown in Figure 10, for the limited mobility scenario of 3 km/h for both transmitter-receiver and interferer-receiver channels.It can be seen that both streams have identical statistics.The average of the instantaneous rate log 2 (1 + ρ) is shown in Figure 11 for both streams.The sensitivity of the IRC-EXPM algorithm to the step size parameter μ is illustrated in Figure 12, and it is shown to depend on the SNR conditions.Finally, the performance degradation as a function of the speed is given in Figure 13.This degradation is also SNR-dependent, with smaller losses at lower SNR.

Effect of the Channel Feedback Accuracy in Multiuser
Multiplexing.parameterization of tall matrices (A.1).In order to update W, the transmitter minimizes the BEP (11) by iterating over the angle space with the random walk with direction exploitation technique [34].The well-known block diagonalization (BD) technique [9] has also been considered.The BD uses a factorized W: one factor eliminates the interference between users, and the other implements eigenbeamforming on the modified channel.For the BEP-based solution, it is assumed that the transmitter broadcasts the optimal combiners.Alternatively, orthogonal training sequences could be used for  estimation.The performance loss due to accuracy limitation of the channel feedback algorithms is shown to be small for the uncoded BEP range up to 0.01.It can be also observed that the impact of the inaccuracy of the channel feedback algorithms is more relevant when operating at very low BEP levels such as 0.0001.On the other hand, the minimum BEP solution achieves a performance gain of about 2 dB, compared to the BD scheme.This advantage comes from lifting the restriction of nulling the interference among users, which is a common approach in other MU-MIMO techniques, for example, [7].

Numerical Examples for the Analysis in Static Channels.
In this section, we apply the analysis given in Section 5.4, to study the effect of the different parameters on the expected convergence time.We assume a zero-mean Gaussian distribution of h with variance 1/2, such as that of the real-valued components of the Rayleigh channel.[9], and "IRC-orth" minimizes the total conditional BEP from (11) with orthonormal beams.The average uncoded BER of both is shown, when using different types of CSI: perfect, SBRVT from Section 5.2 and a normalized vector-based method from Section 5.6 with incremental rotations based on matrix exponentials.

6.3.2.
Influence of Δ max and α.Assuming m 0 = 1 and m 1 = α , the expected convergence time is a function of Δ max , α.The expected upper-bound can be computed in closed form from (29), where F h (x) = 0.5(1 + erf(x)) is the CDF of the zero-mean Gaussian variable with variance 1/2.The integrand for the expectation is in general not smooth for the exact convergence time v t + 1, as shown in Figure 15.
Figure 16 shows the different expected values of the convergence time bound, as a function of Δ max and α.It can be seen that the choice of α is the most critical of the two.On the other hand, the sample average of the exact convergence time v t + 1 computed from (28) for each sample h is shown in Figure 17.By comparing Figures 16 and 17, we can see that the average of the exact convergence time is more sensitive to Δ max than the expected value of the bound.The best Δ max from both figures is Δ max = 0.17.Using this value, the convergence time as a function of α can be considered.As shown in Figure 18, both the bound and the exact formula give an optimal value of α = 3, for the current choice of Δ min , F h .It can be observed that both curves exhibit local minima at α = 2, 3.
In order to verify the validity of the exact convergence time 1 + v t computed from (28), the histogram of the convergence time is shown in Figure 19 times from (28) and from the actual algorithm run have been plotted.Additionally, the histogram for the bound of the convergence time, as given in Proposition 4. Furthermore, to quantify the tightness of the bound, the histogram of the difference between the bound and the true convergence time is shown in Figure 20.

Performance of the Channel Feedback Algorithms.
The accuracy of the channel feedback algorithms can be studied by defining the error between the true channel matrix H and the fed back channel matrix, H t as The error can be characterized statistically for a given scenario.For example, Figure 21 shows the empirical probability density function (PDF) of the errors for algorithms operating at n b = 7 for N t = 4, N r = 2.It is shown that a the sequential partial-update S-SBRVT outperforms the methods based on vectorized channels from Section 5.6 (premultiplication with matrix exponential and single Givens rotor update).Given the small number of bits in the feedback message, there is no gain to be achieved from the ranked partial update concept.On the other hand, the impact of to the BEP performance is shown in Figure 14, for MU-MIMO solutions computed upon the output of S-SBRVT and a vectorized channel with matrix exponential premultiplication.The BEP performance due to the channel   feedback methods is small in both cases, for levels of uncoded BEP down to 0.01 (1%).
The use of predictive vector quantization (PVQ) [27] can also be considered.However, choosing the classification criteria for codebook selection and determining the optimal number of codebooks complicates the design.We have restricted ourselves to the use of a third order predictor, concatenated with a scalar quantization scheme, which quantizes each real-valued component with 2 bits, thus requiring 2N r N t bits.The resulting error statistics are shown in Figure 22 for a system with N t = 4, N r = 2 and speed of 3 km/h, where the sequential SBRVT algorithm operating at the same feedback rate gives comparative performance.
Because the performance of the PVQ would surely improve if the proper codebooks and selection rules are implemented, we make no claim as to the relative merits, although we note that the complexity associated with the predictor is rather EURASIP Journal on Advances in Signal Processing high for vectors of size 16, compared to that of the sequential SBRVT tracking.The benefit of the channel feedback with ranked partial update is illustrated in Figure 4, where a larger receive array with N r = 4 antennas is used.A total budget of n b = 13 bits can be used for ranked partial update: 5 bits denote an octet selected from a set of 32 possibilities designed off-line, and the remaining 8 bits carry the corresponding updates for the chosen single-bit trackers.The benefit comes as an improved tracking performance: a faster decaying tail of the PDF of and a general shift to the left of the probability mass, for > 0.025.
When the mobile speed increases to 10 km/h, the channel tracking becomes more challenging.Figure 3 shows that even updating all the trackers on each update is not enough to have error levels comparable to the performance obtained at 3 km/h.Therefore, to improve performance, some trackers need to be updated twice.This is clear when comparing the performance of the S-SBRVT at 32 and 40 bits per update.On the other hand, by using several Givens rotors ( 16) per update, the performance can be improved.The performance bounds of this approach are shown in Figure 3, where the rotor angles are used in unquantized form.The joint encoding of the angles can be viewed a vector quantization problem.Considering that 2 bits are used for norm and overall phase tracking (see Section 5.6), then a budget of (n b − 2)/(2r) bits per angle would be available, where r is the number of rotors per update.Thus, n b = 40, 32 give 2.38 and 1.38 bits per angle, and have the potential of outperforming the S-SBRVT algorithm at the same feedback rate.However, the joint coding of the angles is still an open problem.One difficulty is that relatively large angles are required to adapt to sudden variations of the channel, while smaller angles are required to adapt to smaller fluctuations.

Conclusion
This paper studied the use of feedback channels in MIMOsystems, for the purposes of (a) enabling MIMO communications that are robust to strong intercell interference signals, through the use of transmit diversity-assisted IRC receivers and (b) recursive tracking of the complete channel matrix of a user, intended as input to multiuser multiplexing solutions designed for full CSI.
The MIMO-IRC algorithms are shown to enable reliable communications in the presence of strong jamming signals, where they outperform a classical IRC receiver in conjunction with a closed-loop eigenbeamforming solution.Moreover, the performance of the proposed recursive techniques has been compared to that of systems employing static Grassmanian codebooks, and are shown to give performance advantages in low mobility scenarios.
The channel feedback algorithms have been designed based on a combination of adaptive delta modulation and partial update filtering concepts.A convergence analysis for static channels has been provided.The performance of the channel feedback algorithms in time-varying channels has been assessed by simulations, where the impact on the uncoded BEP of the system is studied, and compared to the case of perfect CSI.It is shown for the selected system configuration, that the channel feedback algorithms are accurate enough to provide a small impact on the uncoded BEP of the system.Additionally, the total channel tracking error has been statistically characterized through an empirical probability density function in several system configurations.It is shown that the selective partial-update channel feedback can enhance the tracking performance, compared to the strictly sequential partial-update.

A. Summary of Update Styles for Vectors and Matrices
This section summarizes different update formulas used in tracking vectors and matrices, that are used throughout this paper.The tracking of vectors and matrices will be done by either (a) update of a real-valued vector of parameters that can be mapped to a vector or matrix, or (b) premultiplication by a square matrix.When premultiplying with a unitary matrix, the norm and orthogonality properties of the tracked vector or matrix are preserved.On the other hand, premultiplying with nonunitary matrices is used to track matrices not constrained to have orthogonal columns.This requires a further scaling step for normalization.
A.1.Update Based on Parameterization of Vectors and Matrices.Matrix and vector decompositions exist, which map real-valued parameter vectors into complex-valued matrices [21].The parameterization-based recursion builds the updated matrix from the updated parameters.Choosing an appropriate parameterization then guarantees that the constraints on the matrix are satisfied, for example, the norm or orthogonality of a tracked vector or matrix.Mappings based on a cascade of Givens rotors [35] have been used in different communications applications [1,6,36,37].Here, we consider the mapping employed in [6,36], summarized in what follows.
A real-valued vector θ ∈ R np×1 containing n p = 2MN − N(N +1) can be mapped to a tall matrix A ∈ C M×N through a cascade of MN −N (N +1)/2 complex Givens rotors G mn (•, •) as follows: where A 0 contains the N leftmost columns of the identity of size M and we have intentionally left out N parameters from the full parameterization (see, e.g., [6] for details), which are used in unit-modulus complex-valued scalings on each of the N columns.This is to exploit the fact that the performance of a system employing a beamforming matrix W = M(θ) is invariant to such scalings.On the other hand, if first step has no information about the previous bit.For the second branch, the edges are not uniformly spaced, but can be computed by solving for h such that the term under • in the expression for v 1 is an integer.Thus, from (26) we have ln h − m 1 Δ min Δ min + 1 (α − 1) + 1 1 ln(α) = i, (B.3) which gives stair edges g 2 (i) as given in the proposition.Note that h = m 1 Δ min gives i = 1, hence the term i + 1 in g 2 (•).Additionally, the last stair must have its right edge at the border of the second branch.Thus we can compute the integer associated to the last right edge before the branch end by evaluating the floor term at h = f (m 1 + p).After simplifying, this integer is p + 1, and the value of the integral for the last stair is added separately using left edge g 2 [p] and right edge f (m 1 + p).Note that we assume that the second branch has at least one stair, that is, p > 0 ↔ Δ max > αΔ min .
In the third branch, the number of vertices is constant and the edges are spaced by Δ max .One can define a number of stairs after which the integral is to be truncated, for example, such that the last right edge is larger than several standard deviations of the channel PDF.

Figure 1 :
Figure 1: System model.A fixed transmitter equipped with N t antennas and N u mobile users under intercell interfering signals.Each user has a limited-capacity feedback channel.

E 15 Figure 2 :
Figure 2: MIMO-IRC algorithms in transmission of two data streams, N t = 4, N r = 3, n b = 6 at 3 km/h.A strong jammer situation with SNR at the receiver equal to that of the transmitter, and one where the jammer has an SNR 30 dB below that of the user (noise dominated scenario).Both orthogonal and nonorthogonal transmit beams schemes are considered.

( 3 )
) Examine the sign change counters: if sign[ (l)] equals sign[ (l − 1)], increase n e by one and set n d to zero.Otherwise, increase n d by one and set n e to zero.Apply the step size control: if n e ≥ m 1 , then set Δ(l+1) to max{α u Δ(l), Δ max }.Otherwise, check if n d ≥ m 0 .If so, then set Δ(l + 1) to min{α d Δ(n), Δ min }.

1 FH − H t 2 Figure 3 :
Figure3: Tracking performance of channel feedback algorithms for a system with N t = N r = 4 at 10 km/h.The partial update alternative requires more than 40 bits to match the performance obtained with 13 bits at 3 km/h (Figure4).The Givens rotor-based method for vectorized channels could improve the performance at n b = 40 provided that 8 rotor angles can be encoded reliably.This requires using (40 − 2)/16 ≈ 2.38 bits per angle, but the encoding method is still an open problem.

10 EURASIPFH − H t 2 Figure 4 :
Figure 4: Tracking performance of channel feedback algorithms for a system with N t = N r = 4, n b = 13 at 3 km/h.The gain from reserving 5 bits for signaling the elements selected for update is shown.The reference for "perfect selection" has a very high equivalent feedback requirement of n b = 24 + 8 = 32 bits.

Figure 6 :
Figure 6: MIMO-IRC algorithms under permanent intercell interfering signal.Single beam system with N t = 4, N r = 2, n b = 6, mobile and interfering signals at 3 km/h.

Figure 7 :
Figure 7: Empirical PDF of SINR in N t = 4, N r = 2 single beam system under a strong interfering signal, for E b /N 0 values of 0, 8, 16 dB and 4 bits per channel use.The IRC-DJAC uses 6 feedback bits per slot.Its performance is compared to an unquantized eigenbeamformer, used in conjunction with an IRC receiver.The relative speeds Tx-Rx are identical to the relative speed interferer-Rx, and equal to 3 km/h.

Figure 8 :
Figure8: Impact of the mobile speeds on the performance of the single beam IRC-MIMO algorithm, using 6 feedback bits per slot.Solid curves represent scenarios where the relative speeds Tx-Rx are identical to the relative speed interferer-Rx.Dashed curves represent scenarios where the interferer-Rx relative speed is fixed at 3 km/h.

5 MIMO 2 nFigure 9 :Figure 10 :
Figure9: MIMO-IRC algorithms with two data streams and orthonormal transmit beams.The tradeoff between feedback bits per slot and uncoded average BER performance is shown for a system with N t = N r = 4.

Figure 11 :
Figure11: Ergodic capacity of each data stream for the IRC-Expm algorithm using 6 feedback bits per slot.The relative speeds Tx-Rx are identical to the relative speed interferer-Rx, and equal to 3 km/h.The step size is 0.05.

Figure 12 :
Figure 12: Impact of the convergence step on the performance of the IRC-EXPM algorithm with 2 data streams and a permanent interferer.Both Tx-Rx and interferer-Rx speeds assumed 3 km/h.

Figure 13 :
Figure13: Impact of the mobile speeds on the performance of the IRC-EXPM algorithm with two data streams and a permanent interferer.The relative speed interferer-receiver equals the abcissa value, unless otherwise stated.

Figure 15 :Figure 16 :
Figure15: Integrands for the expected convergence time calculation, in Gaussian-distributed static channels: using the bound (•), and using the exact convergence time (+).

Figure 17 :Figure 18 :
Figure 17: Average value of the exact time computed by 1 + v t from over 10 5 channel realizations.Curves are shown for different values of Δ max ∈ [0.05, 0.5] and Δ min = 0.00025, under convergence condition m 1 = α and m 0 = 1.

Figure 20 :
Figure 20: Differences between the exact convergence time and its upper bound, from Gaussian channel realizations.The average difference is 4.479.
Figure 14: Effect of the channel feedback algorithms in multiuser MIMO with N t = 4 and two users with N r = 2 each, both moving at 3 km/h and employing a single data stream with symbols from a 16QAM constellation.Two multiuser multiplexing solutions are considered: BD refers to block diagonalization Figure21: Tracking performance of channel feedback algorithms for a system with N t = 4, N r = 2, n b = 7 at 3 km/h.The sequential partial-update (SBRVT) outperforms the alternative methods based on norm-one vectorized channels.This scenario is not suitable for ranked partial update, due to the stringent feedback bit budget.