MIMO precoding and mode adaptation in femtocellular systems
 Chenzi Jiang^{1}Email author,
 Leonard J CiminiJr^{1} and
 Nageen Himayat^{2}
https://doi.org/10.1186/16871499201312
© Jiang et al.; licensee Springer. 2013
Received: 2 August 2012
Accepted: 17 December 2012
Published: 23 January 2013
Abstract
Abstract
Hierarchical femtocellular architectures have become popular recently because of their potential to provide increased coverage and capacity in cellular systems. However, introduction of femtocells might reduce the overlay macrocellular system performance due to increased interference caused to macrocellular users. In this article, two MIMO precoding techniques are considered at the femtocellular base stations (FBSs) to control the interference to the macrocellular users: precoding matrix index (PMI), and least interference (LI). With MIMO precoding, the limited CSI at the transmitter is the index of the precoder chosen from the codebook fed back by the receiver. The LI technique can be employed at the FBSs to maximize the macrocellular throughput, but it also results in significant reduction in femtocellular throughput. The PMI approach can maximize the signal power at a desired receiver, with minimal feedback. In this article, we develop algorithms that adapt at the FBSs between the LI and PMI schemes to increase both the macrocellular and femtocellular throughputs. We show that allowing for mode adaptation at each FBS improves the system performance when compared with using the same mode across the system, and a simple binary choice at each FBS can nearly achieve the optimum modeadaptation performance. Analysis and simulation results in a multicell environment are presented to illustrate the improvement in system performance with the proposed techniques.
Keywords
Introduction
A femtocell is a lowpower, userdeployed base station designed for indoor use. Because of their potential to provide improvement in coverage and capacity[1–3], femtocells have attracted much attention recently. The introduction of femtocells into an existing cellular system, however, also brings new challenges[4–11]. One of the most important is the interference problem, and recent articles have addressed this issue from several different perspectives. In[12], the performance of twotier femtocellular networks with outage constraints is investigated considering cellular geometry and crosstier interference in the downlink. In[13], the use of OFDMA is considered to cope with this interference. Optimal power allocation for femtocells is discussed in[14], based on an analysis of the macrocellular interference in OFDMA systems with fractional frequency reuse. The use of frequency scheduling to manage the cochannel and intercarrier interference in OFDMA networks is studied in[15]. In[16], downlink carrier selection and transmit power calibration at the femtocells are proposed to manage interference for 3GPP systems. In[17], an uplink capacity analysis and interference avoidance strategy for a CDMAbased femtocell network is provided. Power control is used to mitigate cochannel crosslayer interference in[18, 19]. In[18], strategies for maximum transmit power adjustment at the femtocells to suppress interference at the macrocellular base stations (MBSs) are presented, and a downlink power control strategy at the femtocells, based on a distributed utilitybased, signaltointerferenceplusnoise ratio (SINR) adaptation, is proposed to alleviate the interference at the macrocell in[19]. Beam subset selection and codebook restriction are considered at the MBSs in[20, 21], respectively, to reduce crosslayer interference.
The motivation in this article is to improve the femtocellular system performance with MIMO precoding techniques applied at the femtocellular base stations (FBSs). MIMO precoding is one of the various closedloop techniques adopted by IEEE 802.16e[22]. For example, we can obtain the beamforming vector for any channel matrix by finding its singular value decomposition (SVD)[23]. A scheme that quantizes the unitary beamforming matrix was presented in[24]; the collection of quantized beamforming matrices is called a codebook. The codebook is obtained by optimizing over the chordal distance or mutual information between codewords. In[25], the authors improved upon the method in[24] by maximizing the minimum chordal distance between any pair of codewords; codebooks with four antennas and feedback sizes of 3 and 6 bits are given in[25, 26]. In this article, based on the design method in[25], codebooks for feedback sizes of 8 bits with four antennas are generated and applied in the simulation.
Also, MIMO precoding methods for interference mitigation in femtocellular systems will be studied in frequency division duplex (FDD) systems. With MIMO precoding, the receiver feeds back the index of the codeword in a codebook to the transmitter; this codeword is then applied as the precoder. With this limited CSI, it is difficult to achieve good performance for both macrocells and femtocells with the practical MIMO precoding schemes alone. Thus, we develop a mode adaptation approach at the FBSs to achieve better performance.
The transmission modes at the FBSs are adapted between least interference (LI) and precoding matrix index (PMI). The LI technique chooses the precoder at the FBS that generates the LI for the macrocellular user; this scheme maximizes the macrocellular throughput, but results in a reduction in the femtocellular throughput. The PMI approach chooses the precoder that generates the largest signal power to the femtocellular user; this maximizes the femtocellular throughput, but the interference generated to the macrocellular user is not considered.
Here, we develop a MIMO mode adaptation technique combining these two approaches to adapt the precoding mode at the FBSs and improve the system performance. A tuning factor is introduced to tune between the two modes. Two approaches, global and localized, according to the two different ways the tuning factor can be chosen, are studied and compared. With localized mode adaptation, the tuning factor is chosen independently at each FBS, while in the global approach, the same tuning factor is applied for all the FBSs. Note that the study in[27] also describes balancing the signal power at the desired receiver and the interference power at the other receivers. However, in[27], the channels are assumed to be known at the Tx and Rx, and the tuning factors are coefficients of the desired and interfering channel matrices.
System overview
where P is the average received power (including path loss), H is the N_{ r } × N_{ t } channel matrix, X is the N_{ s } × 1 transmitted signal vector (N_{ s } is the number of signal streams), n is an additive white Gaussian noise vector, and Q is a normalized N_{ t } × N_{ s } precoding matrix. In this article, we assume N_{ s } = 1, that is, a single stream is transmitted; so, Q is a vector. Also, it is assumed that no beamforming is done at the receivers. The channel coefficients are modeled as i.i.d. complex Gaussian random variables with zero mean and unit variance.
where${\mathbf{\text{Q}}}_{{\mathrm{M}}_{k}}$ and${\mathbf{\text{Q}}}_{{\mathrm{F}}_{k}}$ are the precoders at the k th MBS and FBS, and${\sigma}_{m}^{2}$ and${\sigma}_{f}^{2}$ are the noise powers at the MUE and FUE, respectively.
We consider the downlink (DL) performance of the femtocellular system, and the obtainable DL throughput is calculated using the SINR at the receiver and a fixed backoff δ from capacity. In this article, we assume that δ = 2[3], reflecting what can be achieved with practical coding schemes.
Codebook generation
Codebook parameters
L  u  b  minimum chordal distance 

2  [1,2,4,12]  [0.3536 + 0.3536j;  1 
0.3536 −0.3536j;  
−0.3536−0.3536j;  
−0.3536+0.3536j]  
3  [1,2,7,6]  [0.2895 + 0.3635j;  0.8282 
0.5287 −0.2752j;  
−0.2352−0.4247j;  
−0.4040+0.1729j]  
6  [1,45,22,49]  [0.3954 −0.0738j  0.3935 
0.0206 + 0.4326j;  
−0.1658−0.5445j;  
0.5487 −0.1599j]  
8  [1,10,102,177]  [0.4660 + 0.4660j;  0.1733 
0.2827 −0.2827j;  
−0.1964−0.1964j;  
−0.4054+0.4054j] 
MIMO precoding in femtocellular systems
Femtocells might generate significant interference to the MUEs. Although the transmit power of the femtocells is relatively low, the throughput of the MUE might degrade rapidly as the number of femtocells increases. In this section, two codebookbased methods of MIMO precoding schemes are considered at the FBSs: PMI, and LI.
Precoding matrix index
where Q_{PMI} represents the chosen codeword at the FBS using the PMI technique. The Lbit index of this codeword could be fed back to each FBS through a local connection. This method can maximize the signal power with a given codebook and, therefore, maximize the femtocellular throughput; on the other hand, the macrocellular throughput might be reduced significantly because the interference generated to the MUE is not considered in the optimization in (6). PMI can be compared to transmit beamforming (TXBF) which maximizes the signal power without the codebook constraint.
Least interference
where Q_{LI} represents the chosen codeword at the FBS using the LI technique. Similar to PMI restriction, the index of this codeword could be fed back to the MBS and then shared with the FBS through a local connection. This method can minimize the interference at the MUE from the FBSs and, therefore, maximize the macrocellular throughput; on the other hand, the femtocellular throughput might be reduced significantly because the signal power is not considered in the optimization in (7). LI can be compared to zeroforcing (ZF) which minimizes the interference power without the codebook constraint.
Mode adaptation with MIMO precoding
As stated above, the PMI and LI techniques each have their specific advantages, but each alone is not sufficient to obtain good system performance. PMI maximizes the femtocellular throughput but results in poor macrocellular performance; LI maximizes the macrocellular throughput but degrades the femtocellular performance severely. Therefore, here, we consider mode adaptation (MA) that combines PMI and LI to obtain the “best” system performance.
System performance
The motivation for combining PMI and LI at the FBSs is to improve the system performance. Note that different system requirements give different criteria for performance. When considering both the macrocellular and femtocellular performance, one possible system performance metric is the sumrate of all the users in the cell, including macrocellular and femtocellular users. However, in most cases, there will be many more FBSs and FUEs than MUEs at any instant in time; thus, maximizing the sumrate of all the users will lead to good femtocellular performance but poor macrocellular throughput. Another potential system performance metric is the weighted sum of the average macrocellular and femtocellular throughputs. But, maximizing the sum of the average throughputs does not guarantee fairness among users.
where T_{ m } and T_{ f } represent the average macrocellular and femtocellular throughputs, respectively, and 0 ≤ η ≤ 1 is the weight given to T_{ m }. T_{ g } represents the weighted geometric mean of the macrocellular and femtocellular throughputs. When η = 0, T_{ g } = T_{ f }, which means the femtocellular throughput will be maximized, and PMI will be applied at all the FBSs; when η = 1, T_{ g } = T_{ m }, meaning the macrocellular throughput will be maximized, and LI will be applied at all the FBSs.
Mode adaptation algorithm
where 0 ≤ c ≤ 1 is a tuning factor reflecting the relative importance of increasing the signal power at the desired user versus reducing the interference power at the undesired user.
 (i)
When $c=\frac{1}{2}$, $\lambda =\frac{1}{2}$;
 (ii)When $c\ne \frac{1}{2}$,$\lambda =\frac{\sqrt{c(1c)\text{Tr}{\left\{\Theta \right\}}^{2}+{(2c1)}^{2}}c\text{Tr}\left\{\Theta \right\}+2c1}{(2c1)(2\text{Tr}\{\Theta \left\}\right)},$(14)
where$\Theta ={\mathbf{\text{Q}}}_{\text{PMI}}{\mathbf{\text{Q}}}_{\text{LI}}^{H}+{\mathbf{\text{Q}}}_{\text{LI}}{\mathbf{\text{Q}}}_{\text{PMI}}^{H}$, and Tr{·} represents the trace of a square matrix. If λ < 0, set λ = 0; if λ > 1, set λ = 1.
With the same tuning function, the simplified MA algorithm reduces the complexity compared with the original one. When λ = 0, Q_{MA} = Q_{LI}; when λ = 1, Q_{MA} = Q_{PMI}; and when λ is between 0 and 1, the precoding mode at the FBS is tuned between LI and PMI. How to select the optimum λ will be addressed later based on the system performance metric.
 1.
The FUE determines the Q _{PMI} at the FBS which generates the largest signal power at the FUE, and feeds back the Lbit index of the Q _{PMI} to the FBS.
 2.
The MUE determines the Q _{LI} at the FBS which generates the LI at the MUE, and feeds back the Lbit index of the Q _{LI} to the MBS. The MBS shares this information with the FBS.
 3.
The FBS chooses the precoder according to (12) (with an appropriate choice of λ).
Note that after each FBS chooses the precoder, the FUE and MUE estimate the SINR and feed back the information to the base stations. Using this information, the base stations employ adaptive modulation and coding (AMC) to achieve a throughput close to the channel capacity.
Global MA
In the global MA approach, we assume that λ is determined on a systemwide level, and all the FBSs in the same macrocell apply the same tuning factor. Assume that the average SINRs at the MUE and the FUE can be estimated centrally and are known to the FBSs, then the average macrocellular (T_{ m }) and femtocellular throughputs (T_{ f }) can be estimated. The tuning factor λ can then be optimized over an objective function containing both T_{ m } and T_{ f }, which reflects the system requirement. Here, we use (8) as the system performance metric.
Localized MA
Global MA requires systemwide information, which might be difficult to obtain in practice. In addition, using the same λ at all the FBSs lacks flexibility. Therefore, we also consider a localized MA approach; in this case, each FBS independently chooses its tuning factor λ according to its specific situation.
The simplest form of mode adaptation at the FBSs is to apply binary MA. In this case, the value of the tuning factor λ is either 0 or 1, i.e., the precoding mode at each FBS is either LI or PMI. By observing (14), we can also find that the probability of λ = 0 or λ = 1 is quite high, which indicates that binary MA can also achieve good system performance. In the following, we consider two methods of localized binary MA, one is based on path loss, and the other one is based on distance.
Decision based on longterm performance observation
Which precoding mode is used by the FBS depends on the interference power it generates to the MUE. The average interference power received at the MUE from the FBS is determined by the transmit power and the longterm fading characteristics. We assume the transmit power of the FBSs is unchanged. Since the shadow and multipath fading are stochastic, the average path loss between the FBS and the MUE determines the average received interference power at the MUE from the FBS. Thus, here, we consider mode adaptation at the FBSs based on the average path loss between the FBS and the MUE.
We assume the uplink transmit power of the MUE is known to the FBS, and the FBS can measure the received uplink power from the MUE. Then, the average uplink (UL) path loss between the MUE and the FBS, PL_{ U L }, can be obtained. This average path loss reflects the distance between the FBS and the MUE. The average DL path loss, PL_{ D L }, is proportional to PL_{ U L }, so the UL path loss reflects the average interference power at the MUE from the FBS. Here, we consider this UL pathloss information to determine the value of λ and the precoding mode at each FBS. The FBS can apply PMI (λ = 1) if the average path loss is larger than a specified threshold; otherwise LI (λ = 0) is used. The optimal pathloss threshold can be estimated through longterm observation of the performance.
Note that we can also use a continuous value of λ, independently at each FBS, but, as we will show in the next section, using only a binary value for λ at the FBSs can achieve almost the same performance as continuous λ.
Decision based on distance
With the localized binary MA decision based on longterm performance observation, the optimal pathloss threshold needs to be estimated through longterm measurement and observation, which might be difficult to implement in practice. Therefore, here we consider another binary MA approach based on the distance between the FBS and the MUE, and assume that the location of the MUE can be obtained for example using GPS, which is widely available in many terminals, like smartphones. This method avoids the search for the optimum pathloss threshold, which may be difficult to estimate. Each FBS only needs to obtain the distribution of the distance between the MUE and itself. Since the location of each FBS is fixed, this distribution can be acquired at each FBS by longterm measurement.
Generally, longer distance results in larger path loss. We assume that the transmit power at each FBS is fixed, then the average interference power received at the MUE from the FBS can be determined by the distance between the FBS and the MUE. Given the distance threshold d_{th}, when the distance between the FBS and the MUE d < d_{th}, LI should be applied; otherwise, PMI is applied at the FBS. In order to achieve the optimum system performance, we need to determine the distance threshold d_{th} for each FBS.
The FBSs and the MUE are usually randomly distributed in the macrocell. Let p denote the probability of the LI scheme being applied at a FBS. Therefore, Pr{d < d_{th}} = Pr{LI} = p, and Pr{d ≥ d_{th}} = Pr{PMI} = 1 − p. So optimizing d_{th} is equivalent to optimizing p. The derivation of the value of p is given in Appendix Appendix 2.
The location of each FBS is fixed, and the location of the active MUE is random. Assume that the distribution of the distance between the FBS and the MUE is available at each FBS, the optimum distance threshold d_{th} at each FBS depends on the distribution of the distance between the FBS and the MUE. Since Pr{d < d_{th}} = Pr{LI} = p, the value of the optimum d_{th} for the k th FBS should be chosen so that${d}_{\mathit{\text{th}},k}={F}_{k}^{1}\left(p\right)$, where F_{ k } is the CDF of the distance between the MUE and the k th FBS.
 1.
By longterm observation, the MBS measures the average macrocellular and femtocellular throughputs when all the FBSs apply PMI; and then measures the average macrocellular and femtocellular throughputs when all the FBSs apply LI.
 2.
Each FBS obtains the distribution of the distance between the FBS and the MUE by longterm observation.
 3.
The MBS calculates the value of p according to (31) for a given η, and then shares this information and the location of the MUE with all the FBSs.
 4.
Each FBS calculates the distance threshold d _{t h,k}, using p and the distribution of the distance, then calculates the distance between the FBS and the MUE at each time instant, and chooses the appropriate value of λ.
Simulation results
Simulation parameters
Number of transmit antennas N_{ t }  4 
Number of receive antennas N_{ r }  1 
Carrier frequency f_{ c }  2.5 GHz 
Macro BS antenna height  
above mean rooftop level Δ h_{ B S }  20 m 
# of MUE per sector  1 
# of FUE per femtocell  1 
Macro BS transmit power  46 dBm 
Macrocell radius  1000 m 
Femtocell radius  10 m 
Power spectral density of noise  −174 dBm/Hz 
Bandwidth  10 MHz 
Performance of MIMO precoding algorithms
In this subsection, the performance of the MIMO precoding schemes PMI and LI will be illustrated with the macrocellular and femtocellular throughputs (in bits/sec/Hz) at the 50th percentile of the CDF (50% throughputs) as a function of the number of femtocells per macrocell sector.
Figure3 shows that the femtocellular performance does not degrade much as the number of FBSs increases because of the low transmit power of the FBSs. LI at the FBSs achieves the same femtocellular performance as not doing beamforming. This is because the signal power at the FUE is not considered in the LI approach, and the average signal power with LI is the same as that without BF. Better femtocellular performance can be obtained with PMI applied at the FBSs. The femtocellular throughput with PMI approaches that for unquantized TXBF as the size of the codebook increases. Performance with ZF applied at the FBSs is also fairly good, because here we assume each FBS is equipped with four transmit antennas and it is only necessary to eliminate the interference generated to the MUE in the same macrocell; thus the rest of the degrees of freedom can be utilized to increase the signal power at the FUE.
Similar trends are observed for the throughputs at the 10th percentile of the CDF (indicative of celledge performance). The figure is not shown here due to space limitation.
Performance of mode adaptation algorithms
Since a larger codebook is required for better tuning, an 8bit codebook is used. Here, we also assume that there are 10 FBSs in each macrocell sector.
Performance of global and localized ma
Performance comparison
The MA technique combines PMI and LI, and adapts the mode at the FBSs between the two; thus, MA has better system performance than either PMI or LI. The system performance is further improved using localized MA. With low FBS transmit power, the interference at the MUE from the FBSs is small, and the precoding modes at the FBSs are mainly PMI. As the FBS transmit power increases, T_{ g } initially improves because of the increased femtocellular throughput, and then degrades due to the reduced macrocellular throughput. At the high end of the FBS transmit power, LI outperforms PMI because LI avoids severe degradation of the macrocellular performance, and the femtocellular throughput is compensated by the high transmit power. PMI at the FBSs, in this case, leads to seriously degraded macrocellular throughput. The performance of the MA techniques will converge to that of LI as the FBS transmit power is further increased. This is because, with high FBS transmit power, the precoding mode at the FBSs reverts to LI in order to get better system performance.
Conclusions
In this study, codebookbased precoding methods of interference mitigation in femtocellular systems are considered. Different sizes of codebooks are generated and applied to these precoding methods. In general, LI maximizes the macrocellular throughput, but leads to low femtocellular throughput; and PMI achieves the maximum femtocellular throughput but poor macrocellular performance. Based on this, we considered mode adaptation at the FBSs in order to improve both the macrocellular and femtocellular throughputs and obtain better femtocellular system performance. The precoding mode is adapted at the FBSs between PMI and LI. Global and localized approaches were considered and compared. We showed that mode adaptation at the FBSs brings performance gain to the femtocellular system, and the system performance can be further improved by localized mode adaptation with more flexibility at each FBS. We also showed that a simple localized binary choice at each FBS can provide good performance for both the macrocellular and femtocellular users, and nearly achieve the upper bound on the performance of the mode adaptation approach.
In summary, we have the following points on precoding based mode adaptation in femtocells:

When the FBS is close to the MUE or the channel gain from the FBS and the MUE is high, LI should be applied at the FBS to minimize the interference power from the FBS to the MUE.

When the FBS is far from the MUE or the channel between the FBS and the MUE is in deep fading, the interference is small, so PMI should be applied at the FBS to maximize the signal power from the FBS to the FUE.

For all other cases, the tuning factor should be calculated using the mode adaptation algorithms to tune between PMI and LI.
More efficient and practical methods for choosing the optimum λ will be studied in the future. In addition, a comparison of the MA algorithm with other interference management schemes, e.g., enhanced intercell interference coordination (eICIC) in 3GPP, will also be investigated.
Appendix 1
Chordal distance d_{PMI}and d_{LI}
equality is achieved iff. Q_{ c } = Q_{PMI}. Given a channel matrix H_{ D } and a codebook, S_{max} is determined. However, in the case that H_{ D } is not known and only Q_{PMI} is available, if Q_{ c } ≠ Q_{ P M I }, we can improve the received signal power ∥H_{ D }Q_{ c }∥^{2} by choosing the Q_{ c } that maximizes$\parallel {\mathbf{\text{Q}}}_{c}^{H}{\mathbf{\text{Q}}}_{\text{PMI}}\parallel $, or minimizes${d}_{\text{PMI}}\left({\mathbf{\text{Q}}}_{c}\right)=1\parallel {\mathbf{\text{Q}}}_{c}^{H}{\mathbf{\text{Q}}}_{\text{PMI}}{\parallel}^{2}$.
equality is achieved iff. I_{min}=0 and Q_{ c }=Q_{LI}. Given a channel matrix H_{ I } and a codebook,$\sum _{i=1}^{{N}_{r}}\frac{{\sigma}_{i}^{2}}{\parallel {\mathbf{\text{Q}}}_{\text{LI}}^{H}{\mathbf{\text{v}}}_{i}{\parallel}^{2}}$ is determined. However, in the case that H_{ I } is not known and only Q_{LI} is available, if Q_{ c } ≠ Q_{ L I }, we can reduce the received interference power ∥H_{ I }Q_{ c }∥^{2} by choosing the Q_{ c } that maximizes$\parallel {\mathbf{\text{Q}}}_{c}^{H}{\mathbf{\text{Q}}}_{\text{LI}}\parallel $, or minimizes${d}_{\text{LI}}\left({\mathbf{\text{Q}}}_{c}\right)=1\parallel {\mathbf{\text{Q}}}_{c}^{H}{\mathbf{\text{Q}}}_{\text{LI}}{\parallel}^{2}$.
Appendix 2
Derivation of the value of p
where${T}_{f}^{\text{LI}}$ is the average femtocellular throughput when LI is applied at the FBS, and${T}_{f}^{\text{LI}}$ is the average femtocellular throughput when PMI is applied at the FBS.
is the instantaneous macrocellular throughput dependent on the channels, f({H}) is the distribution of all the channels related to the MUE, P_{S,m}({H}) and${\sigma}_{m}^{2}$ are respectively, the signal and noise power received at the MUE.
It is difficult to obtain f({H}), therefore, it is difficult to obtain a closedform expression for the average macrocellular throughput T_{ m } in (23). To simplify the problem, in the following, we will instead find the solution that maximizes a close upper bound of the average system performance T_{ g }. The upper bound can be obtained from the property of convex functions.
where${T}_{m}^{\text{LI}}$ represents the average macrocellular throughput when all FBSs apply LI, and${T}_{m}^{\text{PMI}}$ represents the average macrocellular throughput when all FBSs apply PMI. Denote${T}_{m}^{\text{UB}}=p\xb7{T}_{m}^{\text{LI}}+(1p)\xb7{T}_{m}^{\text{PMI}}$. Since both${T}_{m}^{\text{UB}}$ and T_{ m } are increasing functions of p, increasing${T}_{m}^{\text{UB}}$ by changing the value of p will also increase T_{ m }.
Since this upper bound is obtained from the property of convex functions, it is not a loose bound. By optimizing this upper bound, we can obtain a suboptimal value of p which can achieve similar performance as the optimal p. This can also be shown with simulation results.
Since 0 ≤ p ≤ 1, if p < 0, set p = 0; if p > 1, set p = 1.
Declarations
Acknowledgements
This study was supported by a grant provided by the Intel Corporation, and US National Science Foundation (NSF) under Grant No. 1017053.
Authors’ Affiliations
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