Optimized Time Splitting to Maximize the Lower Bound of Rate with Channel Estimation in an Interference Alignment Based Network

In this paper, for an interference alignment (IA) based network, a time splitting scheme for transmitting training and data symbols is optimized. The time allocated for transmitting training symbol will affect the precision of channel estimation (CE) and thus the achievable rate as well as the duration for data symbol transmission. With the least square (LS) and relaxed minimum mean square error (RMMSE) CE algorithm, the lower bounds of achievable rate are carefully derived, respectively. Then we formulate an optimization problem to maximize the lower bounds of achievable rate by optimizing the time splitting factor (TSF). The existence of optimum is first proved. Then, regarding the complexity of solution, Taylor expansion is adopted to find the approximated optimal TSF. Numerical results are presented to show the optimal TSF can achieve larger lower bound of achievable rate over other fixed TSFs due to its adaptivity to the channel characteristics and its statistics of CE errors. Numerical results also validate that the approximation just brings out some small and acceptable errors on the system rate. In addition, RMMSE CE algorithm shows better performance than LS CE because RMMSE considers noise statistics as modification.


Introduction
Interference alignment (IA) is a promising technology for interference management in the wireless communications [1].Different from interference avoidance of orthogonalizing the channel access, IA can limit all the interference to one half of the signal subspace and the other half can be used to transmit the desired signal without interference [2], [3].The study of IA has attracted a lot of attentions for its good performance in improving the system rate.
In a multiple-input multiple-output (MIMO) interference network, two easily implemented algorithms were proposed to obtain the solutions of IA iteratively under the assumption of perfect channel state information (CSI) in [4], which can maximize the total achievable rate.However, due to the estimation error, quantization error, or feedback delay in practical applications, the instant CSI used is always imperfect.With the imperfect CSI, the IA performance is deteriorating sharply and even not better than traditional orthogonal methods [5].Then, the robust design and performance evaluation of IA with imperfect CSI have been widely studied in [6][7][8][9][10].For example, a novel IA scheme with imperfect CSI based on antenna selection was proposed to improve the received signal to interference plus noise ratio (SINR) in [6].In [8], the average achievable rate of IA was provided considering channel estimation errors.Further in [9], the lower bound of capacity for IA with CSI errors are derived.
The above studies on IA with imperfect CSI are mostly based on the fixed duration training.In the information processing or communication system design, the system performance can be improved by optimizing some parameters [11][12][13].In this paper, we introduce the time splitting factor (TSF) as the optimization variable.We can adjust the duration allocated to the training and data symbols' transmission to maximize the achievable rate for the network.For a given coherent time, if the allocated training duration is too short, more errors will be generated for the estimated CSI, which degrades the achievable system rate.On the contrary, if the training duration is too long, the duration for data symbol transmission will be shortened, also resulting in the decrease of data rate [3,14,15].Therefore, how to perform the optimal time splitting for the training and data symbols transmission is quite important to achieve high rate.
In this paper, we evaluate the statistics property of the imperfect CSI due to practical channel estimation (CE) algorithm, such as least square (LS) and relaxed minimum mean square error (RMMSE) CE algorithm.With the different statistics of CE error for LS and RMMSE CE algorithm, the upper bound of interference leakage are derived respec-tively [16].Then, we propose an optimization problem of time splitting for the training and data symbols transmission to maximize the lower bound of achievable rate.For low complexity, Taylor expansion is applied to obtain the approximated optimal TSF.Finally, some numerical results are presented to validate the correctness of our proposed scheme.
The organization of the paper is as follows.In Sec. 2, the system model is presented and the received signal model is derived.Then we formulate the optimization problem and apply Taylor expansion to obtain the approximated optimal TSF in Sec. 3. In Sec. 4, numerical results are presented to validate the correctness and effectiveness of our scheme.Finally, Section 5 concludes the paper and schedules some future work.

System Model
In this paper, we consider an interference alignment based network, as shown in Fig. 1.The system includes K (K > 1) user pairs, sharing the same frequency band.Each user pair contains a transmitter, T i , equipped with m antennas and its intended receiver, R i , equipped with n antennas, for i = 1, • • • , K. Each transmitter transmits only one data stream, that is, d = 1.In this study, we consider a slow block fading channel model where a CE is valid throughout the duration of the coherence time, which is assumed to be T symbol periods.Let α ∈ (0, 1) be the TSF, such that the training time is αT symbol periods, and the signal transmission time is (1 − α)T symbol periods.
The received signal at the k-th receiver, y k ∈ C n×1 , can be expressed as where H kl ∈ C n×m , l, k = 1, • • • , K is the channel matrix from the l-th transmitter to the k-th receiver, n k ∈ C n×1 ∼ CN (0, N 0 I) represents the additive white Gaussian noise (AWGN) at the k-th receiver with N 0 being the noise power, and x l ∈ C m×1 denotes the transmitted signal vector.Let s l ∈ C d×1 be the transmitted symbol vector of the l-th transmitter.Let V l ∈ C m×d be the precoding matrix for the l-th transmitter based on IA technique.Then we have where s l ∈ C d×1 is the transmitted symbol vector of the l-th transmitter.Assuming that all transmitters have the same average power constraint, E s l where Ĥkl is the estimated channel matrix and ∆H kl denotes the channel errors.Substituting (2), ( 3) into (1) and considering the receiving filter matrix U k ∈ C n×d for IA, the received signal model in ( 1) is formulated as In the second equation in ( 4), all the signal from other user pairs, K l=1,l k U H k Ĥkl V l s l , can be removed due to the IA design based on Ĥkl .
To better analyze the impact of channel errors, as in [17], we introduce error bound for ∆H kl , δ2 kl , which can be defined as where (∆H kl ) i denotes the i-th row of ∆H kl .

Optimization Problem Formulation
As stated in Sec. 2, CE error has a large impact on the achievable system performance.Besides, the duration allocated to CE affects the precision of CE, i.e., CE error.The longer duration the CE procedure takes, the more precise CE we obtain, which helps to improve the achievable system rate.On the contrary, the longer CE duration causes shorter duration for data symbol transmission, which decreases the system rate instead.Therefore, we should optimize the TSF to trade-off the CE precision and the achievable system rate.
Note that, due to the randomness of CE error, we cannot easily evaluate the instantaneous achievable system rate.Generally, the ergodic rate [18] or lower bound of achievable rate are adopted.In this paper, the lower bound of achievable rate, R L k (α), is selected as the objective to optimize the TSF.Therefore, we formulate the optimization problem as Considering the characteristics of CE error in (5), the TSF and the received signal model in (4), the lower bound of the achievable rate for the k-th user pair is calculated as [ where 2 and ρ ∆ = ε N 0 denote the maximum error bound, the desired signal power and signal to noise ratio (SNR) of transmitted symbol, respectively.From (7), it is a challenge to estimate the value of δ 2 max , that is, δ 2 kl .In practical applications, LS or minimum mean square error (MMSE) CE algorithms are usually used in wireless communications [19].Then, in the following two subsections, we focus on the LS and RMMSE CE algorithm and derive the detailed expression of lower bound of rate, the objective function in (6), with LS and RMMSE CE, respectively.

Derivation of Lower Bound of Rate with LS CE
In this subsection, we adopt LS CE algorithm with the optimal training signal [20].In this case, the CE error under optimal training is given by [20] where ∆H LS kl and H LS kl represent the CE error matrix and estimated channel matrix based on LS CE algorithm, and ε t denotes the transmitted power for training symbol.Note that there is no error in feeding back the CSI to all users.
From the definition of δ 2 kl in (5), we easily have Further, combining (3) and ( 8), and with the definition of δ 2 max in (10), we have [21] where ρ t ∆ = ε t N 0 denotes the SNR for training symbol.
Substituting (10) into (7) and with some simple calculations, we can formulate the lower bound of achievable rate for the k-th user pair as where From ( 11), it can be observed that the lower bound of achievable rate, R L k (α), is related but not monotonic to the TSF α.

Derivation of Lower Bound of Rate with RMMSE CE
In this subsection, we focus on the RMMSE CE algorithm which represents a simplified and approximated version of MMSE method.Different from traditional MMSE algorithm, RMMSE algorithm requires only the knowledge of the trace of the correlation matrix and the received noise power.We adopt RMMSE CE algorithm with the optimal training signal and the CE error is given by [20] where and Substituting ( 15) into (7) and with some simple calculations, we can get the lower bound of achievable rate for the k-th user pair R L k (α) with RMMSE CE which has the same form as (11).However, the values of A, B, C and D are different from (12) and can be computed as

Existence of Optimal TSF
From ( 11), ( 12) and ( 16), it can be seen that the optimal solution for ( 6) is difficult to calculate.Before solving the optimization problem, we must first validate the existence of optimum.Consequently, we calculate the second order derivative of R L k (α) with respect to α as where From (11), we can easily prove that 0 < B < D and 0 < C < A for both LS and RMMSE CE algorithm.Therefore, we have With (19), we further have E < 0 and F < 0.Then, considering α ∈ (0, 1), we can prove Here, in order to find the maximum and the optimal TSF, we can take the first order derivative of R L k (α) with respect to α and set it to zero; and then we can obtain the following equation about α as From ( 20), we have and where When the SNR for training symbol ρ t is large enough, we can prove G < 0 for both LS and RMMSE CE algorithms; then we can further prove dα α=1 < 0. As for (22), because 0 < B < D and 0 < C < A for both LS and RMMSE CE algorithm, we have AD − BC > 0.

Solution for the Optimization Problem
The existence of optimal TSF is proven in previous subsection.And we can find α opt by solving (20).However, from (20), the equation includes a complicated logarithmic operation, which makes the equation more difficult to be solved.Moreover, due to the involvement of α 2 in (20), W-Lambert function [22], [23] cannot be applied to get the solution.The theoretical optimal TSF, α opt , can be determined by the root of (20), which can be solved easily using Matlab but difficultly for practical application.Actually, we can utilize the traversal searching method to obtain the optimal TSF in (0, 1) with a small step.Besides, we also want to propose another simplified algorithm to solve the equation in (20).In our previous work, we used Taylor expansion to reduce the complexity of the solution [24], [25].Here, we also introduce Taylor expansion to approximate the Logarithm function in (20) which effectively reduces the computational complexity and only brings some small and acceptable errors validated in the following simulations.The Taylor expansion can be express as (24) where k = Cα+D Aα+B , k−1 k+1 < 1 for all values of α and ρ .
Obviously, the term 1 2n−1 in ( 24) is rapidly decreasing with the increase of n.Thus, in the following analysis, ln (k) can be approximated with only the first term in (24) with small and acceptable error.Then, the logarithmic item in (20) is approximately represented as Substituting ( 25) into (20), we can obtain the detailed approximated representation for (20) as where In the following part, we will present some simulation results to verify the acceptance of small approximation errors compared to the theoretical computing results.

Numerical Results and Analysis
In this section, we present some numerical results to verify the correctness of the proposed optimization scheme.In the simulation, we set m = 2, n = 2, K = 3, T = 100T s (here T s denotes the symbol period), and σ 2 k = 1.Define the ratio of the transmitted SNRs for training and data symbols as µ = ρ/ρ t .And for simplification, assume that µ = 1 for equal power allocation for the training and data symbols except for Fig. 8 and Fig. 9.All elements in H kl are further assumed to be complex Gaussian distributed random variables with zeros mean and unit variance for k, l = 1, 2, • • • , K. All the derivations and simulations can be directly extended to other configurations of m, n, and K. Figure 2 and Figure 3 show the lower bound of achievable rate R L k (α) versus TSF α for different transmitted symbol SNR ρ with LS and RMMSE CE, respectively.From Fig. 2 and Fig. 3, it can be observed that, for the same TSF α, the lower bound of achievable rate, R L k (α), increases with the increase of ρ.This is consistent with the results we expected.However, from (11), For a certain ρ, the lower bound of achievable rate, R L k (α), first increases then decreases as α increases.When α is small, as the α increases, the estimated channel matrix Ĥkl becomes more precise and the interference leakage due to the CE error decreases.Thus, the lower bound of achievable rate first increases.However, when α increases up to a certain value, the increase of α can improve the precision of CE slightly but brings out the decrease of the lower bound of achievable rate due to shorter data transmission duration (1 − α)T.Consequently, R L k (α) decreases as α increases.
From Fig. 2 and Fig. 3, we also discover that, there is a maximum of the lower bound of rate and a corresponding optimal TSF for a given ρ.Moreover, with the increase of ρ, the channel becomes better, and shorter duration can generate precise enough CE.This means higher value of ρ results in smaller value of α opt .In Tab. 1 and Tab. 2, we compare the theoretical TSF directly calculated from (20), the simulated TSF from Fig. 3 and the approximated TSF calculated from (26) for different values of ρ with LS and RMMSE CE algorithms, respectively.From Tab. 1 and Tab. 2, we conclude that the theoretical and simulated α opt s are almost same for all values of ρ, where the small difference attributes to the simulation step for α; whereas the simulated and approximated α opt s have some small differences resulting from the limited items of Taylor expansion in (25).
Figure 4 shows the simulated optimal TSF α opt versus transmitted symbol SNR ρ.It can be seen that α opt decreases as ρ increases for both LS and RMMSE CE algorithm.As expected, the lower the SNR is, the longer training duration is required to provide a better estimation.As the SNR increases, shorter training duration is needed.In addition, we also discover that, the simulated optimal TSF α opt with RMMSE CE is smaller than α opt with LS CE, which indicates RMMSE CE algorithm needs shorter training duration than LS CE algorithm does for the same SNR in order to maximize the lower bound of achievable rate.This also contribute to the superiority of RMMSE CE over LS CE in terms of achievable rate.Figure 5 shows the optimized lower bound of achievable rate, R L k α opt , versus the transmitted symbol SNR ρ.For comparison, we plot the simulated and approximated lower bound of achievable rates by substituting the simulated α opt and the approximated α opt into (11), respectively.We can discover that the simulated and approximated rates coincide well with each other and with some acceptable and small gaps.Therefore, considering the difficulty to calculate the theoretical α opt from (20), the approximated α opt from (26) can be used to maximize the lower bound of achievable rate in practical applications.
From Fig. 5, it also can be seen that the optimized lower bound of achievable rate R L k α opt with RMMSE CE is higher than that with LS CE for all SNRs which indicates RMMSE CE algorithm is better that LS CE algorithm in the IA based network.
Figure 6 and Figure 7 illustrate the lower bounds of achievable rate under the optimal TSF α opt and some fixed TSFs with LS and RMMSE CE, respectively.We can discover that the optimal TSF α opt can achieve larger rate over other fixed TSFs due to its adaptivity to the channel characteristics and its statistics of CE errors.In addition, RMMSE CE algorithm shows better performance than LS CE because RMMSE considers noise statistics as modification.In the previous simulation, we consider the symmetrical case, where the training and data symbols' transmissions have the same power (µ = 1), i.e., ρ t = ρ.In the following two figures, we evaluate the optimal TSF α opt and lower bound of achievable rate for asymmetrical case, that is, µ 1.
Figure 8 shows the simulated optimal TSF α opt with RMMSE CE versus transmitted symbol SNR ρ under different values of µ.From Fig. 8, we can discover that, for a certain ρ, smaller µ means higher training symbol SNR ρ t ; as a result, shorter training duration is required to provide a better enough CE.
Figure 9 shows the optimized lower bound of achievable rate R L k α opt with RMMSE CE versus the transmitted symbol SNR ρ under different values of µ.As expected, we can discover that the smaller µ is, the higher optimized lower bound of achievable rate is for all values of ρ.This improvement attributes to better CE with higher training symbol SNR ρ t for smaller µ.

Conclusions
In this paper, we have studied an optimized time splitting scheme to improve the system performance in an IA based network with consideration of the practical CE algorithms.In order to lower down the optimization complexity, we introduce the lower bound of achievable rate as an optimization indicator and the Taylor expansion for approximation.With the numerical results and their correpsonding analysis, we discover that the optimized TSF can improve the system rate compared with the traditional fixed TSFs due to its adaptivity to the channel characteristics and its statistics of CE errors.In addition, the RMMSE CE algorithm shows better performance than the LS CE algorithm because the RMMSE CE considers noise statistics as modification.Furthermore, the approximation of optimal TSF just brings out some small and acceptable errors on the system rate.In our future work, we plan to consider to perform the optimization jointly combining power allocation and time splitting between the training and data symbols.

Fig. 6 .Fig. 7 .
Fig. 6.Comparison of the lower bounds of achievable rate under different TSFs with LS CE.

ρ in dB Theoretical Simulated Approximated
Tab. 2. Comparison of theoretical, simulated and approximated time splitting factors with RMMSE CE, α opt s, for different values of ρ.