Chapter 3 Interference Alignment (IA) Aided Transceiver Design
3.4 Computer Simulations
The convergence behavior and sum-rate performance evaluations are presented for comparison between the UL CoMP transceiver scheme assisted with Min Leakage-IA, the UL CoMP transceiver scheme assisted with Max SINR-IA, and the proposed IA aided UL CoMP transceiver design which are called “Min Leakage-UL CoMP”, “Max SINR-UL CoMP”, and “IA aided UL CoMP”, respectively. The achievable sum-rate is calculated based on (2.7) mentioned in chapter 2 because a linear MMSE receiver is adopted in our work. ki involved in (2.7) is illustrated in Figure 3-2. Furthermore, sum capacity of UL CoMP with full BS cooperation also exhibits as a performance upper bound [15]. The simulation parameters chosen in this section are listed in Table 3-3.
Table 3-3: Simulation parameters
Parameter Value
Channel i.i.d. Rayleigh fading channel
Number of BSs / UEs (K) 3
Number of transmit antennas (Mt) 4
Number of receive antennas (Mr) 2, 4
Number of transmitted signal layers (d) 2, 3
Number of channel realizations 100 (sum-rate performance) 5000 (convergence behavior) Number of iterations for each algorithm 20 (sum-rate performance)
The convergence behavior are provided in Figure 3-4 and Figure 3-5 both of which are evaluated at SNR = 10 dB and at SNR = 30 dB. Figure 3-4 is simulated in the case with Mt=4, Mr=2, K=3, d=2; Figure 3-5 is simulated in the case with Mt=4, Mr=4, K=3, d=3. The simulation results show that Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP have superior convergence behavior in all cases especially for Min Leakage-UL CoMP. However, large rate degradation occurs when Min Leakage-UL CoMP is adopted which is consisted with the numerical evaluation in section 3.2. This is because Min Leakage-UL CoMP mechanism makes all its effort to minimize the interference leakage but assumes (3.2) is automatically satisfied; hence there is no assurance that the received power of desired signal can achieve an acceptable level as mentioned in section 3.2.
Figure 3-4: Rate convergence behavior of Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP with Mt=4, Mr=2, K=3, d=2, and SNR = 10/30 (dB)
Figure 3-5: Rate convergence behavior of Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP with Mt=4, Mr=4, K=3, d=3, and SNR = 10/30 (dB)
SNR = 30 dB
SNR = 10 dB SNR = 30 dB
SNR = 10 dB
In Figure 3-6 and Figure 3-7, sum-rate performance comparisons between Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP are displayed.
Figure 3-6 is simulated in the case with Mt=4, Mr=2, K=3, d=2; Figure 3-7 is simulated in the case with Mt=4, Mr=4, K=3, d=3. In all cases, it is found that Min Leakage-UL CoMP has substantial rate degradation compared to other two UL CoMP transceiver schemes due to its poor ability to maintain the received power of desired signal as mentioned above. The simulations also show that Max SINR-UL CoMP can achieve acceptable sum-rate performance as expected, and the proposed IA aided UL CoMP can reach even better sum-rate performance than Max SINR-UL CoMP. This is because that the proposed IA aided UL CoMP has the effect of balancing the SINR of each layer at the decoder output to improve the condition of the effective channel matrix. The sum capacity is provided here as a performance upper bound. We can find that the achievable sum-rate of IA aided UL CoMP is sufficiently close to the sum capacity especially for the case with Mt=4, Mr=2, K=3, d=2.
Figure 3-6: Sum-rate performance of Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP with Mt=4, Mr=2, K=3, and d=2
Figure 3-7: Sum-rate performance of Min Leakage-UL CoMP, Max SINR-UL CoMP, and IA aided UL CoMP with Mt=4, Mr=4, K=3, and d=3
3.5 Summary
Interference alignment assisted UL CoMP is discussed and evaluated comprehensively in this chapter. First, two popular interference alignment algorithms, Min Leakage-IA and Max SINR-IA [9], [12], developed in K-user interference channel are incorporated in the UL CoMP transceiver designs which are called Min Leakage-UL CoMP and Max SINR-UL CoMP, respectively. Their sum-rate performance are evaluated and it is demonstrated that Max SINR-UL CoMP has better sum-rate performance since a good compromise between interference and received power of desired signal can be preserved. Hence Max SINR-UL CoMP is regarded as a highly potential interference mitigation scheme. According to the study in the first stage, an IA aided UL CoMP transceiver scheme that incorporates the basic idea of Max SINR-UL CoMP and further balances the SINR of each layer at the output of decoder is proposed. The simulation results show that the proposed IA-aided UL CoMP transceiver can achieve superior sum-rate and convergence performance.
Chapter 4
Channel Condition Enhanced Transceiver Design
In the thesis, two centralized UL CoMP transceiver schemes are introduced and both of which are established based on the joint processing nature provided by full BS cooperation. One of the UL CoMP transceiver schemes aided by IA is presented in Chapter 3. The other one aiming at enhancing the condition of the effective channel is given in this chapter.
The proposed channel condition enhanced UL CoMP transceiver scheme involves an iterative procedure which is similar to the iterative procedure included in IA aided UL CoMP transceiver design mentioned in Chapter 3, and each iteration consists of two stages: 1) to calculate the joint decoder and 2) to compute the precoders. But unlike IA aided UL CoMP transceiver, the precoders in the channel condition enhanced scheme is obtained by exploiting the UL-DL duality where the centralized UL CoMP (MAC-like structure) is dual to the virtual centralized DL CoMP (broadcast channel-like structure, BC-like structure) by reversing the direction of communication.
The arrangement of this thesis is as follows. In section 4.1, the motivation of the proposed channel condition enhanced UL CoMP transceiver scheme is provided. Then the problem formulation and design procedure of the proposed channel condition
enhanced scheme are described in section 4.2. Next, we analyze the complexity behavior of the two proposed transceiver schemes in section 4.3, which is followed by the numerical simulations including evaluations of convergence behavior, achievable sum-rate performance, sensitivity to the initial value in the iterative procedure, and the fairness between different users in section 4.4. Last of all, we summarize this chapter in section 4.5.
4.1 Motivation
Due to full BS cooperation, a MAC-like structure and an effective channel matrix as shown in (2.5) are formed in centralized UL CoMP. The state of the effective channel matrix is a crucial factor in transceiver design and can significantly affect the system performance; hence we attempt to properly design the precoders and the joint decoder to induce a well-behaved effective channel matrix. In this thesis, two UL CoMP transceiver schemes for creating a well-behaved effective channel are proposed.
In Chapter 3, an IA-aided UL CoMP transceiver is proposed from the viewpoint of interference alignment and interference mitigation. The algorithm tries to reduce the residual interference and to increase the received power of the desired signal for each layer at the output of decoder. Then a near diagonal effective channel matrix is accomplished since the algorithm will somehow enlarge the ratio of diagonal terms to off-diagonal terms. On the other hand, in this chapter, we endeavor to develop a UL CoMP transceiver scheme which tries to enhance the effective channel condition.
It is well known that the effective channel matrix in a good condition must have small condition number and large singular values. Therefore, we try to develop a transceiver design criterion that can minimize the condition number and can maximize the singular values simultaneously. The proposed channel condition enhanced
transceiver is described in section 4.2.
4.2 Proposed Channel Condition Enhanced Transceiver
The proposed channel condition enhanced transceiver provided in this section attempts to properly design the precoders and the joint decoder such that a well-conditioned effective channel can be achieved by minimizing the condition number and maximizing the singular values of the effective channel matrix in the same time. The basic structure of the iteration process is similar to the one adopted in IA aided UL CoMP transceiver in section 3.3, where in each iteration, two stages are involved: 1) to calculate the joint decoder and 2) to compute the precoders. However, in the second stage, the precoders is obtained according to the virtual centralized DL following two criteria should be achieved:
1 2 d
diag
is a diagonal matrix with singular values on the diagonal;
A and B are unitary matrices, respectively.
which attempts to increase each of the singular value. In the meantime, the difference between the singular values will somehow be reduced due to the operation of product in (4.3). (4.3) can be equally expressed as
the eigenvalue decomposition of ΗΗ is given and can be shown as follows: H
1
According to the property that the product of the eigenvalues is equal to the
Based on (4.12) and the inequality in (4.14), the joint decoder can be obtained:
, ,
Following the same derivation in the joint decoder design, we can obtain the objective function shown below: According to Cauchy–Schwarz inequality and with the following assumption,
( ) ( ) ( ) 2
Table 4-1: Iterative procedure for the proposed channel condition enhanced transceiver Step 1. Start with arbitrary precoders Vk, k {1, 2, …, K}
Step 2. Compute the joint decoder U column by column using (4.15) to achieve better effective channel condition with Vk obtained from previous step,
k {1, 2, …, K} .
Step 3. Compute precoder Vk column by column by equation (4.20) based on the virtual DL CoMP system to achieve better effective channel condition with U obtained from previous step, k {1, 2, …, K} .
Step 4. Go back to Step 2 unless the number of iterations reaches a predefined limit. transceiver mentioned in section 3.3, the proposed channel condition enhanced transceiver mentioned in section 4.2, and the minimum sum mean square error (MSE) transceiver developed in [10], which are called “IA aided UL CoMP”, “Channel condition enhanced UL CoMP”, and “MMSE UL CoMP”, respectively.
Minimum MSE method is a typical approach in transceiver design. Similar to our proposed methods, the transceiver of MMSE UL CoMP is also calculated based on an iterative process, and UL-DL duality is adopted in the precoders design as well.
However, the criterion attempts to minimize the sum MSE at the output of the receiver which is different to our designs.
The major computation cost in IA aided UL CoMP involves matrix multiplication, matrix inversion, eigenvalue decomposition, and sort of sequence. In Channel condition enhanced UL CoMP, matrix multiplication, matrix inversion, and computation of determinant are needed. However, for MMSE UL CoMP, only matrix multiplication and matrix inversion are the most computation-consuming operation. The comparison of complexity is summarized in Table 4-2, in which the complexity of each algorithm is calculated for each iteration.
Table 4-2: Complexity comparison between different UL CoMP transceiver schemes Different UL CoMP transceiver
schemes condition enhanced UL CoMP achieve comparable computational complexity. The result also shows that MMSE UL CoMP leads to a lower computational complexity compared with the two proposed methods. However, the comparison is based on computation cost per iteration. The simulations in the next section (section 4.4) will show that both of the proposed methods achieve better convergence performance
compared to MMSE UL CoMP, although more calculations are needed per iteration.
Table 4-2 also reveals that more transmit antennas (Mt), receiver antennas (Mr), transmit layers (d), and number of elements in a cooperation group (K) will induce larger complexity per iteration for all cases.
4.4 Computer Simulations
In this section, the numerical evaluations of the proposed iterative “IA aided UL CoMP”, the proposed iterative “Channel condition enhanced UL CoMP”, and iterative
“MMSE UL CoMP” mentioned in [10] are provided. All of the following properties will be numerically simulated: convergence behavior, achievable sum-rate performance, sensitivity to the initial value in the iterative procedure, and the fairness between different users. For all cases, the achievable sum-rate is calculated based on the equation (2.7) mentioned in section 2.2. The simulation parameters chosen in this section are listed in Table 4-3.
Table 4-3: Simulation parameters
Parameter Value
Channel i.i.d. Rayleigh fading channel
Number of BSs / UEs (K) 3
The convergence behaviors are provided in Figure 4-1 and Figure 4-2, both of which are evaluated at SNR = 20 dB and at SNR = 30 dB, and are obtained by averaging over 1000 independent channel realizations. Figure 4-1 is simulated in the case with Mt=4, Mr=2, K=3, d=2; Figure 4-2 is simulated in the case with Mt=4, Mr=4, K=3, d=3. The simulation results demonstrate that the proposed Channel condition enhanced UL CoMP can achieve better convergence performance than the proposed IA aided UL CoMP, and the two proposed methods significantly outperform MMSE UL CoMP especially in the high SNR regime. This is because that the co-work of decoder and precoders facilitates the formation of a well-conditioned effective channel matrix.
This can improve the condition of effective channel within a few iterations and yield a more efficient iterative processing. Figure 4-1 and Figure 4-2 also reveals that although Channel condition enhanced UL CoMP has superior convergence performance, its sum-rate after convergence is a little poorer than the other two methods. This is because that Channel condition enhanced UL CoMP puts all its efforts on effective channel condition enhancing but ignores the effective noise term n in (2.3), where nU nH is a function of the joint decoder. On the other hand, IA aided UL CoMP and MMSE UL CoMP take both interference and noise terms into account and hence can attain better sum-rate after convergence.
Figure 4-1: Rate convergence behavior of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP with Mt=4, Mr=2, K=3, d=2, and SNR =
20/30 (dB)
Figure 4-2: Rate convergence behavior of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP with Mt=4, Mr=4, K=3, d=3, and SNR =
20/30 (dB) SNR = 30 dB
SNR = 20 dB SNR = 30 dB
SNR = 20 dB
[10]
[10]
In Figure 4-3 and Figure 4-4, sum-rate performance comparisons between IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP [10]
are displayed. Figure 4-3 is simulated in the case with Mt=4, Mr=2, K=3, d=2; Figure 4-4 is simulated in the case with Mt=4, Mr=4, K=3, d=3. In all cases, the simulation results are obtained by averaging over 100 independent channel realizations, and 10 iterations are executed for each iterative algorithm. Sum capacity of UL CoMP with full BS cooperation also exhibits as a performance upper bound [15]. According to the simulation results, we can find that in general Channel condition enhanced UL CoMP achieve better performance than IA aided UL CoMP, and the achievable sum-rate of the proposed two methods are much closer to the sum capacity compared with MMSE UL CoMP especially for the case with Mt=4, Mr=2, K=3, d=2, when 10 iterations are adopted. This is because the two proposed methods both achieve superior convergence performance, that is they can attain pretty good sum-rate performance within few iterations, like 10 iterations, as depicted in Figure 4-1 and Figure 4-2.
Figure 4-3: Sum-rate performance of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP with Mt=4, Mr=2, K=3, d=2, and no. of iterations=10
Figure 4-4: Sum-rate performance of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP with Mt=4, Mr=4, K=3, d=3, and no. of iterations=10
[10]
[10]
In Figure 4-5, the comparison of sensitivity to the initial value in the iterative procedure between IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP [10] is given. Each algorithm is simulated over 10 iterations under the scenario with Mt=4, Mr=2, K=3, d=2 and is simulated at SNR = 30 dB for a fixed channel realization. The probability density of the sum-rate are calculated based on the output from running different algorithms 100 times, and different initial value (Vk, k
1, 2, ,K
) is adopted each time. In our work, the probability density function is estimated by Parzen window method as the adoption in [14]. The simulation result reveals that the probability density function (pdf) associated with the proposed Channel condition enhanced UL CoMP and IA aided UL CoMP concentrate on the right hand side as shown in Figure 4-5, which means that they are more robust to the initial values, and can attain better sum rate performance. On the other hand, the pdf associated with MMSE UL CoMP spreads across a lower-value region. Namely, MMSE UL CoMP is sensitive to the initial values and has worse sum rate performance.Figure 4-5: Comparison of sensitivity to the initial value in the iterative procedure between IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL
CoMP with Mt=4, Mr=2, K=3, and d=2
In Figure 4-6, fairness between different users of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP [10] is provided. Each algorithm is simulated over 10 iterations under the scenario with Mt=4, Mr=2, K=3, d=2 and is simulated at SNR = 30 dB. In all cases, the simulation results are obtained by running 2000 independent channel realizations. The probability density function is also estimated by Parzen window method. In our work, the fairness between different users is defined as follows:
and minimum values among
R ii,
1, 2, ,K
, respectively. The simulation demonstrates that the proposed Channel condition enhanced UL CoMP exhibits the best fairness. The proposed IA aided UL CoMP is inferior to Channel condition enhanced UL CoMP. Compared to the proposed two methods, MMSE UL CoMP has the worst property of fairness.Figure 4-6: Fairness between different users of IA aided UL CoMP, Channel condition enhanced UL CoMP, and MMSE UL CoMP with Mt=4, Mr=2, K=3, and d=2
4.5 Summary
In this chapter, an UL CoMP transceiver scheme based on effective channel condition enhancing is proposed. The proposed Channel condition enhanced UL CoMP attempts to minimize the condition number and to maximize the singular values of the effective channel matrix by maximizing the product of singular values of the effective channel matrix. Next, the comparison of complexity between the proposed IA aided UL
[10]
CoMP, the proposed Channel condition enhanced UL CoMP, and MMSE UL CoMP [10]
is given, followed by the numerical evaluations of the two proposed methods and MMSE UL CoMP. The results show that the proposed IA aided UL CoMP and the proposed Channel condition enhanced UL CoMP have superior convergence behavior because the co-work of decoder and precoders facilitates the formation of a well-conditioned effective channel matrix, while more computation is needed for each iteration. The simulation results also show that the two proposed methods can achieve rather good sum-rate performance, provide robustness to the initial values in the iterative procedures, and lead to much fairer results, within few iterations, like 10 iterations.
Chapter 5
Conclusions and Future Works
To fulfill the increasing demands and to conduct effective communication in the next generation wireless systems, unity frequency reuses and multiuser transition scheme are adopted, which leads to an interference limited environment. Under such strict environments, coordinated multipoint (CoMP) transmission and reception and multiple input multiple output (MIMO) systems are developed and proposed as two key techniques to achieve advanced performance requirements. In this thesis, we consider uplink CoMP (UL CoMP) assisted with multiple antennas as our system model.
According to the potential for system performance improvement, we focus on the case called centralized UL CoMP which can provide full information exchange between BSs and support joint processing at CU. Based on this structure, two associated transceiver schemes are proposed to achieve improved system performance.
Our fist proposed centralized UL CoMP transceiver is developed in Chapter 3 through introducing the idea of interference alignment (IA), a recently emerged interference mitigation technique. The algorithm includes precoders at UEs and a joint decoder at CU followed by the processing of a linear MMSE receiver, and is iterative in
Our fist proposed centralized UL CoMP transceiver is developed in Chapter 3 through introducing the idea of interference alignment (IA), a recently emerged interference mitigation technique. The algorithm includes precoders at UEs and a joint decoder at CU followed by the processing of a linear MMSE receiver, and is iterative in