 Research
 Open Access
Multiuser MIMO downlink transmission with BEMbased limited feedback over doubly selective channels
 Hung NguyenLe^{1}Email author,
 Tho LeNgoc^{2} and
 Loïc CanonneVelasquez^{2}
https://doi.org/10.1186/168714992011138
© NguyenLe et al; licensee Springer. 2011
 Received: 22 December 2010
 Accepted: 26 October 2011
 Published: 26 October 2011
Abstract
This article studies the problem of limited feedback design for heterogeneous multiuser (MU) transmissions over time and frequencyselective (doubly selective) multipleinput multipleoutput downlink channels. Under a doubly selective propagation condition, a basis expansion model (BEM) is deployed as a fitting parametric model for capturing the timevariation of the MU downlink channels and for reducing the number of the channel parameters. The resulting dimension reduction in the timevariant channel representation, in turn, translates into a reduced feedback load of channel state information (CSI) to the base station (BS). To produce limited feedback information, vector quantization of the BEM coefficients is performed at mobile terminals under the assumption that perfect BEM coefficient estimation has been established by existing algorithms. Then, the output indices of the quantized BEM coefficient vectors are sent to the BS via errorfree, zerolatency feedback links. To assess the feasibility of using the BEMbased limited feedback design in a MU network with an arbitrary number of active users, the resultant sumrate performance of the network is provided by employing the blockdiagonalization precoding and greedy scheduling techniques at the BS. The relevant numerical results show that the BEMbased limited feedback scheme is able to significantly alleviate the detrimental effect of outdated CSI feedback which likely occurs as using the conventional blockfading assumption in MU transmissions over (fast) timevarying channels.
Keywords
 limited feedback
 greedy scheduling
 blockdiagonalization precoding
 vector quantization
 basis expansion model (BEM)
 time and frequencyselective channels
 heterogeneous multiuser network
1. Introduction
Besides the wellknown time, frequency and code divisions in wireless communications, spatial separation has been recently recognized as a new signal dimension for further system performance enhancement, especially in multiuser (MU) transmissions. In the socalled spatial division multiple access (SDMA), the use of multiantenna arrays allows the base station (BS) to simultaneously transmit multiple data streams to multiple users by exploiting the new signal dimension [1, 2]. Among several MU transmission techniques, it is well known that dirty paper coding (DPC) [3] is an optimal MU encoding strategy, whose performance achieves the capacity limit of MU broadcast channels. However, the optimal performance of DPC comes with the cost of impractically high complexity (in a large user pool). As an alternative lowcomplexity linear technique, blockdiagonalization (BD) precoding [4, 5] is a suboptimal MU encoding scheme with a realizable implementation.
In the literature, most of existing precoding techniques [2, 4–6] assume MU downlink channels to be homogeneous and timeinvariant within a transmission block/burst (i.e., the blockfading assumption). However, in a MU network with rapidly moving nodes (e.g., users in cars/trains in longterm evolution (LTE) systems), the resultant timeselectivity of the channel impulse response (CIR) introduces a large number of channel parameters. This induces a very high channel state information CSI feedback load for precoding and scheduling processes with consideration of timevarying channels at the BS. In addition, the presence of timeselective channels would give rise to the problem of outdated CSI feedback [7] that could severely degrade the system performance. To deal with the channels, [8, 9] has proposed a minimum mean squared errorbased beamforming algorithm for homogeneous MU transmissions over multipleinput singleoutput, spatially correlated, frequencyflat, timeselective channels. Specifically, the existing technique uses full feedback of channel distribution information and an iterative beamforming process to provide stable MU transmissions over the channels.
Unlike [8, 9], this paper is concerned with limited CSI feedback design for BD precoding and greedy scheduling over spatially uncorrelated, doubly selective, multipleinput multipleoutput downlink channels with heterogeneous users (i.e., mobile terminals with different numbers of receive antennas and different receiver noise powers). Over the doubly selective channels, a basis expansion model (BEM) [10, 11] is used as a fitting parametric model for capturing the timevariation of the channels and for reducing the number of the channel parameters. Specifically, to generate limited feedback information, vector quantization (VQ) of the BEM coefficients is performed at mobile terminals under the assumption that perfect BEM coefficient estimation has been established by existing algorithms. Then, the output indices of the quantized BEM coefficient vectors are sent to the BS via feedback links. To investigate the performance of the limited feedback scheme in a MU network with an arbitrary number of mobile terminals, BD precoding and greedy scheduling are deployed accordingly in the MU network.
The rest of the paper is organized as follows. Section 2 delineates the system and channel models. The suggested BEMbased limited feedback for BDbased heterogeneous MU transmissions is presented in Section 3. Simulation results and relevant discussions are located in Section 4. Finally, Section 5 provides some concluding remarks.
Notations: (X) ^{ T } and (X) ^{ H } denote the transpose and conjugate transpose (Hermitian operator) of the matrix X, respectively. $E\left(\cdot \right)$ stands for expectation operator. tr(X), X, and X denote the trace, determinant and Frobenius norm of the matrix X, respectively.
2. System Formulation
A. Transmitted Signal Model
where n ∈ {N_{ g } ,..., 0,..., N  1}, N_{ g } denotes the CP length, ${X}_{k,m}^{\left(p\right)}$ is the k th precoded (datamodulated) subcarrier.
B. Doubly selective channel model
where N_{ s } = N + N_{ g } denotes the OFDM symbol length after CP insertion and n = 0,..., N_{ s }  1. The mobile users' speeds are assumed to be unchanged within M OFDM symbols (in a duration of a number of LTE frames). L denotes the channel length. ${b}_{n+m{N}_{s},q}$ stand for the q th basis function values of the used BEM. ${c}_{q,l}^{\left({r}_{u},p\right)}$ are the BEM coefficients of the channel fitting. Q is the number of basis functions used in the basis expansion modeling.
In the simulation section of this article, the timevariant multipath channels ${h}_{l,n,m}^{\left({r}_{u},p\right)}$ are first generated by the modified Jakes model [12], and then fitted (approximated) by the DPSBEM [10], i.e., using a linear combination of Q basis functions as shown in (2).
C. Received signal model
where ${z}_{n,m}^{\left({r}_{u}\right)}$ is the additive white Gaussian noise with variance ${\sigma}_{u}^{2}$ at the u th user. It is assumed that different terminals may experience different receiver noise powers in the considered heterogeneous MU system.
where ${I}_{k,m}^{\left({r}_{u}\right)}=\sum _{p=1}^{{N}_{t}}{\sum}_{\begin{array}{c}i=0\\ i\ne k\end{array}}^{N1}{H}_{k,i,m}^{\left({r}_{u},p\right)}{X}_{i,m}^{\left(p\right)}$ is the intercarrier interference (ICI) induced by the timevarying channels. In particular, ${H}_{k,i,m}^{\left({r}_{u},p\right)}=\sum _{q=1}^{Q}\left(\sum _{l=0}^{L1}{c}_{q,l}^{\left({r}_{u},p\right)}{e}^{j2\pi il\u2215N}\right)\left(\frac{1}{N}\sum _{n=0}^{N1}{b}_{n+m{N}_{s},q}{e}^{j2\pi n\left(ik\right)\u2215N}\right)$ is the channel frequency response (CFR). ${Z}_{k,m}^{\left({r}_{u}\right)}=\frac{1}{\sqrt{N}}\sum _{n=0}^{N1}{z}_{n,m}^{\left({r}_{u}\right)}exp\left(j2\pi nk\u2215N\right)$ denotes receiver noise in the frequency domain.
where ${\mathbf{\text{Y}}}_{k,m,u}={\left[{Y}_{k,m}^{\left(1\right)},...,{Y}_{k,m}^{\left({R}_{u}\right)}\right]}^{T}$, ${\mathbf{\text{Z}}}_{k,m,u}={\left[{Z}_{k,m}^{\left(1\right)},...,{Z}_{k,m}^{\left({R}_{u}\right)}\right]}^{T}$, ${\mathbf{\text{H}}}_{k,m,u}={\left[{\left[{\mathbf{\text{H}}}_{k,m}^{\left(1\right)}\right]}^{T},...,{\left[{\mathbf{\text{H}}}_{k,m}^{\left({R}_{u}\right)}\right]}^{T}\right]}^{T}$, ${\mathbf{\text{H}}}_{k,m}^{\left({r}_{u}\right)}=\left[{H}_{k,k,m}^{\left({r}_{u},1\right)},...,{H}_{k,k,m}^{\left({r}_{u},{N}_{t}\right)}\right]$ and ${\mathbf{\text{X}}}_{k,m}={\left[{X}_{k,m}^{\left(1\right)},...,{X}_{k,m}^{\left({N}_{t}\right)}\right]}^{T}$. It is assumed that the BS has an average transmit power constraint tr(Δ) ≤ P_{Σ} where the covariance matrix of the transmitted signal is defined as $\Delta \triangleq E\left[{\mathbf{\text{X}}}_{k,m}{\mathbf{\text{X}}}_{k,m}^{H}\right]$.
3. Multiuser MIMO Transmission with BEMBased Limited Feedback
In this section, a limited feedback design over time and frequencyselective (doubly selective) channels is suggested to reduce the CSI feedback load and to alleviate the detrimental effect of outdated CSI feedback (that likely occurs as using the blockfading assumption in MU transmissions). More specifically, a BEM [10, 11] is used as a fitting parametric model of the doubly selective channels. The use of BEM helps to considerably reduce the number of timevariant channel representation parameters.
Unlike [5, 6] using BD precoding in MU transmission for a fixed number of homogeneous users, this section adopts the BD precoding and greedy scheduling to a MU network with an arbitrary number of heterogeneous users (supporting various types of terminals with different numbers of receive antennas and different SNRs).
A. BEMbased limited feedback
where ${\mathbf{\text{c}}}_{x,l}^{\left({r}_{u},p\right)}={\left[{c}_{\left(x1\right)V+1,l}^{\left({r}_{u},p\right)},...,{c}_{\left(x1\right)V+V,l}^{\left({r}_{u},p\right)}\right]}^{T}$, x = {1,..., Q/V} and V is the length of each partitioned BEM coefficient subvector.
where ${\mathbf{\text{g}}}_{x,l}^{\left(j\right)}={\left[{g}_{x,l,1}^{\left(j\right)},...,{g}_{x,l,V}^{\left(j\right)}\right]}^{T}$, l = 0,..., L  1 and B is the number of binary bits for representing each codevector in the used LBG codebook.
should be pregenerated for each possible target mobile speed using the LBG algorithm [16] with training vectors of BEM coefficients corresponding to that speed. Then, for each mobile terminal with a known speed, the LBG codebook G with target speed closest to the known speed should be deployed accordingly for the VQ of BEM coefficients.
In practice, it is very difficult to estimate exactly the actual speeds of mobile terminals. In addition, the memory capacity in the receiver of each mobile terminal also limits the number of pregenerated LBG codebooks G corresponding to different target speeds that can be prestored in the mobile terminal. Therefore, the speed mismatch between the actual mobile speed and the target speed of the used LBG codebook always exits in the VQ of BEM coefficient at mobile terminals. In particular, the effect of the speed mismatch problem on the performance of the considered MU network will be numerically investigated in Section IV.
where x = 1,..., Q/V and l = 0,..., L  1.
It is noted that the basis functions ${\left\{{b}_{n+m{N}_{s},q}\right\}}_{q=1}^{Q}$ are known at both the BS and the users.
As shown in (15), the channel response at each subcarrier in each OFDM symbol in the current LTE time slots can be determined using the quantized versions ${\hat{c}}_{q,l}^{\left({r}_{u},p\right)}$ of the BEM coefficients ${c}_{q,l}^{\left({r}_{u},p\right)}$. As aforementioned, these BEM coefficients are assumed to be perfectly estimated by existing BEMbased channel estimation algorithms [10, 11] using pilot signals from the previous LTE time slots.
After having the BEMbased limited feedback information at the BS, the quantized versions of the user channel responses ${\hat{\mathbf{\text{H}}}}_{k,m,u}$ are naively treated as perfect CSI in the BD precoding and greedy scheduling processes as presented in the next subsections.
Implementation steps of the BEMbased limited feedback
Step 1  BEM coefficients are estimated at mobile terminals using existing techniques [10, 11, 15] 

Step 2  The vector of the BEM coefficients is partitioned as in (7) 
Step 3  Limited feedback information is the indices of quantized vectors of BEM coefficients that are determined by (10) 
Step 4  With the limited feedback information, the BS can recover the CSI using (11) and (14) 
B. BD precoding
With the use of the BEMbased limited feedback, the BS only has quantized CSI versions ${\left\{{\hat{\mathbf{\text{H}}}}_{k,m,u}\right\}}_{u=1}^{U}$ in (14) and therefore the BD precoding matrices W_{k,m,u}will be determined by naively treating ${\left\{{\hat{\mathbf{\text{H}}}}_{k,m,u}\right\}}_{u=1}^{U}$ as perfect CSI ${\left\{{\mathbf{\text{H}}}_{k,m,u}\right\}}_{u=1}^{U}$.
where u = 1,..., U.
where Γ_{ u } contains the first ${n}_{u}={\sum}_{\begin{array}{c}{u}^{\prime}=1\\ {u}^{\prime}\ne u\end{array}}^{U}{R}_{{u}^{\prime}}$ right singular vectors corresponding to the nonzero singular values, and ${\Omega}_{u}\in {\u2102}^{{N}_{t}\times \left({N}_{t}{n}_{u}\right)}$ contains the last (N_{ t }  n_{ u } ) right singular vectors corresponding to zero singular values of ${\mathbf{\text{H}}}_{k,m,u}^{\perp}$.
where ${\Omega}_{{\phantom{\rule{0.1em}{0ex}}}^{u}}^{H}{\Omega}_{u}=\mathbf{\text{I}}.$ As a result, the columns of Ω_{ u } form a basis set in the null space of ${\mathbf{\text{H}}}_{k,m,u}^{\perp}$.
where ${n}_{u}={\sum}_{\begin{array}{c}{u}^{\prime}=1\\ {u}^{\prime}\ne u\end{array}}^{U}{R}_{{u}^{\prime}}$.
With the covariance matrix of the transmitted signals $\Delta \triangleq E\left[{\mathbf{\text{X}}}_{k,m}{\mathbf{\text{X}}}_{k,m}^{H}\right]$, one can deduce $\Delta ={\sum}_{u=1}^{U}{\mathbf{\text{P}}}_{k,m,u}{\mathbf{\text{C}}}_{k,m,u}{\mathbf{\text{P}}}_{k,m,u}^{H}$ since $E\left[{\mathbf{\text{s}}}_{k,m,u}{\mathbf{\text{s}}}_{k,m,u}^{H}\right]={\mathbf{\text{I}}}_{{R}_{u}}$. Under the sumpower constraint tr (Δ) ≤ P_{Σ} and the property tr(ABC) = tr(BCA), one can have the following condition ${\sum}_{u=1}^{U}tr\left({\mathbf{\text{C}}}_{k,m,u}\right)\le {P}_{\Sigma}$.
where (x)^{+} = max(x, 0).
Given a set of selected users, the above BD precoding process attempts to eliminate the interuser interference and maximize the system sumrate. As aforementioned, the feasibility of the suggested BEMbased limited feedback scheme will be investigated in a MU network with an arbitrary number of active users. In particular, the limited feedback links provide CSI to not only precoding but also scheduling at the BS. Under the use of sumrate performance metric, scheduling is to perform user selection with a reasonable complexity for maximizing the system sumrate. The considered scheduling technique will be addressed in detail in the next subsection.
C. Greedy Scheduling
where P_{k,m,u}and C_{k,m,u}are determined by (23) and (27), respectively.
 1)
Initialization: ${A}_{0}=\left\{1,2,...,{U}_{a}\right\}$ is the set of all active users' indices. ${S}_{0}=\left\{\varnothing \right\}$ is the set of selected users, initially assigned to a null set. v = 0 stands for the number of selected users, initially set to zero. R _{0} = 0 is the system sumrate of selected users, initially set to zero.
 2)
Repetition:

Let u* be the index of a selected user in the current iteration. Specifically, the index u* can be determined as follows:${u}^{*}=\underset{u\in {A}_{\upsilon}}{argmax}\left\{{\xi}_{k,m,BD}\left({S}_{\upsilon}\bigcup \left\{u\right\}\right)\right\}$(30)${\xi}_{max}={\xi}_{k,m,BD}\left({S}_{\upsilon}\bigcup \left\{{u}^{*}\right\}\right)$(31)

v = v + 1

If ξ_{max} < R_{v1}go to Step 3 otherwise do:

R_{ v } = ξ_{max}

${S}_{v}={S}_{v1}\bigcup \left\{{u}^{*}\right\}$ (select one more user)

${A}_{v}={A}_{v1}\backslash \left\{{u}^{*}\right\}$ (ignore the selected user in later consideration)

Go to Step 2 Repetition.
 3)
Stop the user selection process.
4. Simulation Results and Discussions
Following the 3GPPLTE system settings [13], the BDbased heterogeneous MU transmission using the suggested BEMbased limited feedback scheme over doubly selective MIMOOFDM downlink channels is simulated as follows. With the number of channel tap gains L = 5 and the exponentially decaying powerdelay profile [10], the timevariant multipath channels are first generated by the modified Jakes' model [12], and then fitted (approximated) by the DPSBEM [10] using Q basis functions. More specifically the realization of doubly selective channels ${h}_{l,n,m}^{\left({r}_{u},p\right)}$ is generated by using the exponentially decaying powerdelay profile of ${\lambda}_{l}^{2}=\frac{{e}^{l\u22154}}{{\sum}_{{l}^{\prime}=0}^{L1}{e}^{{l}^{\prime}\u22154}}$[10]. The discrete time indices n and l denote sampling at rate f_{ s } = 1.92 MHZ. The root mean square delay spread T_{ D } of the power delay profile can be determined by T_{ D } = 4/f_{ s } ≈ 2.1 μ s [10]. The autocorrelation function for every channel tap is given by ${R}_{hh}\left(\tau ,l\right)={\lambda}_{l}^{2}{J}_{0}\left(2\pi \frac{v{f}_{c}}{c}\frac{N}{{f}_{s}}\tau \right)$ which results in the classical Jake's spectrum where J_{0} (·)is the zerothorder Bessel function of the first kind, v and c denote the speeds of terminals and light, respectively. As a result, coherent bandwidth B_{ c } can be approximated to B_{ c } ≈1/T_{ D } = 0.48 MHz [[19], Sec. 3.3.2].
Unless otherwise stated, the considered heterogeneous MU network has U_{ a } = 4 active users with mobile speeds of 200 km/h and Q = 18, where U_{ a } /2 users are equipped with a single receive antenna (R_{ u } = 1 for u = 1,..., U_{ a } /2) and the remaining users have two receive antennas (R_{ u } = 2 for u = U_{ a } /2 + 1,..., U_{ a } ). The BS is equipped with four transmit antennas (N_{ t } = 4). As a frame format in the 3GPPLTE system settings [13], one LTE frame consists of 20 time slots and each of these contains seven OFDM symbols (i.e., 140 OFDM symbols in one LTE frame) in the simulated LTE transmission. In addition, 128point FFT and carrier frequency f_{ c } = 2 GHz is used for the simulated multicarrier transmissions. The CP length of each OFDM symbol is set to 10 samples [13]. Unless otherwise indicated, the average transmit power constraint is P_{Σ} = 10 and receiver noise variance ${\sigma}_{u}^{2}=1$. In the figures illustrating the simulation results, each plotted point of the sumrate performance is obtained by averaging over 500 independent channel realizations.
To pregenerate a LBG codebook G to be used at a given mobile speed, a set of 10^{5} training BEM coefficient vectors corresponding to the target speed is employed by the LBG algorithm [16]. Under an ideal scenario, each mobile terminal is assumed to know exactly the actual value of its mobile speed and uses the corresponding LBG codebook for the VQ process of BEM coefficient vectors. However, in practice, each mobile terminal may have only an estimated value of its mobile speed and chooses a LBG codebook with the target speed closest to the estimated speed value.
5. Conclusion
This article introduced a BEMbased limited feedback design for MU transmissions over doubly selective MIMO downlink channels. By employing a BEM to capture the channel's timevariation, the resulting feedback load of BEM coefficients is significantly smaller than that of CIR or CFR. Over timevarying channels, the BEMbased limited feedback helps to reduce the detrimental effect of outdated CSI feedback (as using the blockfading assumption in MU transmissions), and to provide stable sumrate performance for heterogeneous users with a wide range of mobile speeds.
Appendix
A. Proof of (21)
where ${\Psi}_{u}=diag\left({\lambda}_{1}^{2},...,{\lambda}_{{n}_{u}}^{2}\right)$ contains n_{ u } nonzero eigenvalues of ${\left[{\mathbf{\text{H}}}_{k,m,u}^{\perp}\right]}^{H}{\mathbf{\text{H}}}_{k,m,u}^{\perp}$.
B. Derivations of the waterfilling solution in (27)
where positive semidefinite matrices ${\Theta}_{u}\in {\u2102}^{{R}_{u}\times {R}_{u}}$ are the slack variables [17] to guarantee that C_{k,m,u}are positive semidefinite. The real nonnegative γ is a slack variable associated with the sumpower constraint.
where u = 1,..., U and r_{ u } = 1,... R_{ u } .
where (x)^{+} = max(x,0).
Declarations
Acknowledgements
The study presented in this article was partly supported by the NSERC CRD and Prompt Grants with InterDigital Canada.
Authors’ Affiliations
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