Summary

In this chapter, the concept of CoMP is introduced, which includes the classification of uplink and downlink CoMP [3-4] and the scenarios defined in [5].

Then, the centralized uplink CoMP system adopted in this thesis is presented, and its system architecture and mathematical formulation are introduced. For dealing with such an interference-limited wireless network, a technique recently proposed in the K-user systems (i.e., interference alignment) is discussed. Due to the difficulty in searching the optimal IA solutions, some popular iterative approaches for IA have been developed by exploiting the duality between the uplink and downlink transmissions.

Figure 2–10: Illustration of IA in K-user interference channel.

Chapter 3

Interference Alignment (IA) Aided Transceiver Design in Uplink CoMP Systems

The interference from other cells, which severely degrades the system performance, is a critical factor in modern wireless cellular communication systems and should be carefully managed. In such scenarios, the transceiver design based on centralized cooperation is a critical issue. To tackle this problem, a promising technique, CoMP transmission and reception, is developed.

In this section, the potential of the uplink CoMP systems and IA is first discussed.

Due to the desire for providing more DoFs in centralized uplink CoMP systems, the suitability of incorporating IA into the considered centralized uplink CoMP systems is first presented. Then, the optimal IA is adopted into the considered systems; since the irregular feasibility of the optimal IA, two iterative IA algorithms (i.e., max-SINR and min-leakage) are alternatively adopted. Finally, the performance of this two iterative algorithms is evaluated.

The organization of this chapter is shown below. The motivation of the proposed IA aided UL CoMP transceiver scheme is given in Section 3.1. In Section 3.2, the detailed definition of DoFs is presented; then, the DoFs of some communication

systems are also discussed. After the investigation in Section 3.2, the incorporation of IA in UL CoMP systems based on two popular iterative IA algorithms is introduced in Section 3.3; then, the performance of the two algorithms is evaluated. Finally, this chapter is summarized in Section 3.4.

3.1 Motivation

To achieve higher system capacity in an interference limited communication environment, many techniques are developed to cope with interference. Uplink CoMP is one of the techniques that aim to manage the interference caused by the UEs in other cells. However, the DoFs that can be provided by the centralized uplink CoMP systems are not fully obtained yet.

In the K-user MIMO interference channel, a recently proposed technique (i.e., IA) is suggesting to break the DoFs limitation, so each user can enjoy half the capacity of the interference free case [7]. From numerous researches, the ability of IA to break the DoFs limitation is also discussed under different interference channel models.

Furthermore, certain research works are dedicated to implementation-level considerations and challenges in applying IA techniques to existing cellular networks recently [10]-[12]. Due to the similar capability of uplink CoMP and IA, IA is adopted to the considered centralized uplink CoMP systems for providing more DoFs in the centralized uplink CoMP systems.

3.2 Degrees of Freedom

In this section, the definition of DoFs is first introduced. Then, the DoFs of different interference channel models are presented to claim the potential of IA and the centralized uplink CoMP systems. From the definition in [7], the number of DoFs

represents the rate of growth of network capacity with the log of the signal to noise

which is also equivalent to

( )

²

( ) (

²

( ) )

C r =hlog r +o log r , (3.2) where the ^{o log}

(

²

( )

^r

)

term becomes negligible in comparison with ^log2

( )

^{r at}

high SNR. Therefore, the number of DoFs is also called the “pre-log factor”. From the signal processing perspective, the DoFs can also represent interference free links per channel-use. In the following paragraphs, the DoFs of point-to-point MIMO channel, MAC channel, K-user interference channel, and K-user interference channel with full cooperation at the receiver side, are derived.

1. Point-to-point MIMO channel: The point-to-point MIMO channel shown in Figure 3–1 is considered, where the receiver and the transmitter are equipped with M and N antennas, respectively. The maximum number of interference free links

of the point-to-point MIMO channel can be the rank of the channel

( )

(

^{i.e., min M N,}

)

^. In addition, the maximum number of the DoFs is achieved by the subchannel decomposition [7].

Figure 3–1: Illustration of point-to-point MIMO channel.

2. MAC channel: Consider the MAC channel, comprised of K transmitters and one receiver. As shown in Figure 3–2, the receiver and each transmitter are equipped with M and N antennas, respectively. From the transmitter perspective, the number of transmitted layers of each transmitter cannot be larger than the number of transmit antennas. On the other hand, from the receiver perspective, the number of received layers cannot be larger than the number of receive antennas. To satisfy the above two constraints, the maximum number of the DoFs is ^min

(

^{M NK,}

)

^.

Figure 3–2: Illustration of MAC channel.

3. K-user interference channel: Consider the K-user interference channel, comprised of K transmitters and K receivers. As shown in Figure 3–3, each receiver and transmitter are equipped with M and N antennas, respectively. The number of interference free channels is only ^min

(

^{M N,}

)

(the number of DoFs is

( )

min M N, ) by orthogonalization in the time, frequency, or code domain to avoid multi-user interference. However, if spatial DoFs are used to mitigate interference, the number of DoFs could be ^K₂ ^min

(

M N asymptotically by IA [8]. ^,

)

Figure 3–3: Illustration of K-user interference channel.

4. K-user interference channel with full cooperation at the receiver side: The K-user interference channel shown in Figure 3–3 is considered. Due to the full cooperation at the receiver side, the channel model could be viewed as a MAC-like channel model, which is shown in Figure 3–4. From the result of the MAC channel, the number of DoFs is ^min

(

^{MK NK,}

)

^.

Figure 3–4: Illustration of equivalence between K-user channel with full cooperation at the receiver side and MAC-like channel.

From the above derivations, it can be seen that more DoFs could be provided by the incorporation of IA. Besides, because of the ability to utilize inter-cell interference, full cooperation at the receiver side (which is equivalent to the considered centralized uplink CoMP system) provides more DoFs. In the next section, we attempt to provide more DoFs based on the considered system to enhance the system performance.

3.3 Incorporation of Interference Alignment in Uplink CoMP Systems

For providing more DoFs to create more reliable links, IA is incorporated into the considered centralized uplink CoMP systems. IA can be adopted in either uplink CoMP systems or downlink CoMP systems to enhance the suppression of multi-user interference. As shown in Figure 3–5 (single cell is depicted for concise presentation), to employ IA in the uplink CoMP systems, the system should operate as depicted in Table 3-1.

Table 3-1: An information exchange procedure of UL CoMP systems with IA techniques Step 1: The BSs forward estimated CSI to the CU, and the CU evaluates

corresponding decoder and precoders.

Step 2: The BSs feed the precoder back to each UE.

Step 3: The UEs start to transmit signals.

Figure 3–5: Illustration of information exchange procedure of UL CoMP systems with IA techniques.

Alternatively, as shown in Figure 3–6 (single cell is depicted for concise presentation),

to employ IA in the downlink CoMP systems, the system should operate as depicted in Table 3-2.

Table 3-2: An information exchange procedure of DL CoMP systems with IA techniques Step 1: The UEs feed estimated CSI back to the BSs

Step 2: The BSs forward received CSI to the CU, and the CU evaluates corresponding decoders and precoder.

Step 3: The BSs feed the decoder back to each UE.

Step 4: The BSs start to transmit signals.

Figure 3–6: Illustration of information exchange procedure of DL CoMP systems with IA techniques.

Compared to the downlink CoMP systems, CSI feedback is not required in the uplink CoMP systems, and there is no modifications to the UEs. In this thesis, the uplink CoMP system is considered.

We here introduce the IA techniques into the interference channel model of cellular networks in the uplink CoMP scenario 3/4 involving intersite coordination between different RPs such as BSs/RRHs (i.e., the system model shown in Section 2.2), which is illustrated in Figure 3–7.

Figure 3–7: Centralized CoMP systems in heterogeneous networks.

Taking the advantages of the backhaul resource and centralized joint processing at the CU, more DoFs could be exploited for recovering the desired signals. Further, IA techniques can be incorporated to further improve the overall system throughput.

According to the considered system model, the optimal design criterion of IA in the uplink CoMP systems can be described as follows [8-9]:

( { }

^,,

)

, , corresponding to the dth layer belonging to the kth UE within U is denoted as

, =

å

₌^-₁¹ + .

 ^k

k d i i

d d d The main difference between IA in the K-user interference channel and IA in the centralized CoMP system is that the latter incorporates full cooperation between BSs for computing the decoder at the CU. Because the optimal design criterion would not be feasible, two iterative algorithms proposed in [9], min-leakage and max-SINR, are adopted.

3.3.1 Min-Leakage IA in Uplink CoMP Systems

In this section, the minimum leakage IA algorithm in the K-user interference channel is modified and adopted into the considered system model. The minimum leakage algorithm in [9] can be reformulated as:

( )

where Ik is the leakage of the kth UE at the output of the decoder, described as

( { }

^,,

) ^{{ }}

^,,

Then, the optimal decoding matrix corresponding to kth UE

{ }

^,

,1

can be formulated as

{ }

^,

( )

where eig

(

X,i

)

denotes the function of selecting eigenvectors corresponding to the 1st to ith smallest eigenvalues of X. To evaluate the precoding matrix V_k for the kth UE, Ik is reformulated by channel reciprocity [10] as follows:

( ( ) )

The optimal decoding matrix V minimizing I_k k can be formulated as

( )

The iterative procedure is summarized in Table 3-3. In this thesis, the algorithm provided in this section is denoted as the “min-leakage IA”.

Table 3-3: A procedure for min-leakage IA in UL CoMP systems

Initialization: Set an initial value for decoding matrix U; we suggest adopting partial FFT matrix as the initial point for faster convergence.

Step 1: Compute the precoders Vi, i=1,…,MP according to (3.11).

Step 2: Compute the decoder U according to (3.8).

Step 3: Go back to Step 1 till the constrained iteration number is achieved.

Step 4: Evaluate the equalizer and the achievable sum-rate according to (2.6) and (2.7).

3.3.2 Max-SINR IA in Uplink CoMP Systems

The maximum SINR IA algorithm in the K-user interference channel is modified and adopted into the considered system model in this section. The maximum SINR algorithm in [9] can be reformulated as:

(

, , ,

)

arg max, , , , ,

Then, the optimal decoding vector U^{(dk d,)} maximizing g_{k d,} can be formulated as

(^{dk d,})⁼

(

_{k d,}

)

^-1 _{k k}^{( )d}

(

_{k d,}

)

^-1 _{k k}^{( )d .}

F

U B H V B H V (3.15)

To evaluate the precoding vector V_k^{( )d} for the dth layer at the kth UE, g_{k d,} is reformulated by channel reciprocity [10] as follows:

( )

( )

( )

^{( )}

The iterative procedure is summarized in Table 3-1. In this thesis, the algorithm provided in this section is denoted as the “max-SINR IA”.

Table 3-4: A procedure for max-SINR IA in UL CoMP systems

Initialization: Set an initial value for decoding matrix U; we suggest adopting partial FFT matrix as the initial point for faster convergence.

Step 1: Compute the precoders Vi, i=1,…,MP according to (3.18).

Step 2: Compute the decoder U according to (3.15).

Step 3: Go back to Step 1 till the constrained iteration number is achieved.

Step 4: Evaluate the equalizer and the achievable sum-rate according to (2.6) and (2.7).

3.4 Computer Simulations

The convergence behavior and sum-rate performance evaluations are presented for the comparison between the uplink CoMP transceiver scheme assisted with and without IA. In the following simulation results, “min-Leakage IA” and “max-SINR IA”

represents the algorithms shown in Section 3.3.1 and Section 3.3.2. The achievable sum-rate is calculated based on (2.7) because a linear MMSE receiver is adopted in our work. In this thesis, “CoMP without IA” represents the approach that the precoder of each UE is formed by columns of identity matrix, and the received signal is directly equalized without the aid of the decoder. “Sum-Rate Ratio” is defined as the maximum eventual sum-rate with 4 10´ ⁴ iterations divided by the sum-rate with corresponding iterations to show the convergent behavior ( if the sum-rate ratio is closer to 1, the algorithm performs better). The channel matrices are set by e= 0.4 in all simulations in this thesis [13]. The simulation parameters chosen in this section are listed in Table 3-5.

Table 3-5: Simulation parameters

Parameter Value Channel i.i.d. Rayleigh fading channel

Number of BSs (M) 3 Number of UEs Per Cell (P) 1

Number of transmit antennas (Nt) 4

Number of receive antennas (Nr) 2

Number of transmitted signal layers (di) 2

Number of channel realizations 100

Firstly, the convergence behavior is shown in Figure 3–8. As shown, the min-leakage

IA algorithm has superior convergence rate in comparison with the max-SINR IA algorithm; however, the min-leakage IA algorithm converges to bad performance. From the observation, the max-SINR IA algorithm has nearly no sum-rate enhancement with more than 4 10´ ⁴ iterations. Therefore, the convergence condition is defined to be with a slight 3% sum-rate degradation compared with the sum-rate with 4 10´ ⁴ iterations.

Secondly, the average achievable sum-rates of different algorithms in convergence are shown in Figure 3–9. The results are observed as follows. Because the min-leakage IA algorithm only tries to suppress interference, desired signals might be suppressed simultaneously. The min-leakage IA algorithm performs worse than the max-SINR IA algorithm and the CoMP without IA. With enough iterations (8000 iterations), the max-SINR IA algorithm has superior performance in the comparison with the CoMP without IA; however, with insufficient iterations, the max-SINR IA performs nearly the same as the CoMP without IA.

From the above simulation results, incorporating IA into uplink CoMP systems is shown to achieve the promising performance in mitigating severe interference. However, tremendous number of iterations are required for better performance. In the next section, two algorithms will be proposed to boost the convergence rate.

10⁰ 10¹ 10² 10³ 10⁴

Figure 3–8: Convergence behavior of max-SINR IA and min-leakage IA in UL CoMP systems with P_UPW =30 and 40 dB.

10 15 20 25 30 35 40

Figure 3–9: Sum-rate performance of CoMP without IA, max-SINR IA, and min-leakage IA in uplink CoMP systems with 5 and 8000 iterations.

3.5 Summary

Interference alignment aided uplink CoMP is discussed and evaluated in this chapter. First, two popular interference alignment algorithms (i.e., min-leakage IA and max-SINR IA) [8-9], developed in the K-user interference channel are incorporated in the uplink CoMP transceiver design. Their sum-rate performance is evaluated, and it is demonstrated that the max-SINR IA algorithm has better sum-rate performance because of a good compromise between interference and received power of the desired signal.

Hence the max-SINR IA algorithm is regarded as a highly potential interference mitigation scheme. According to the observation, numerous iterations are required to guarantee that the max-SINR IA algorithm converges. In the next chapter, two IA algorithms are proposed to boost the convergence rate of IA for making IA easier to be implemented.

Chapter 4

Proposed Efficient Interference

Alignment in Uplink CoMP Systems

In Chapter 3, the max-SINR IA algorithm is considered as a candidate for uplink CoMP. However, it is found that numerous iterations are required to achieve the improved sum-rate performance. As a remedy, two new IA aided transceiver designs are proposed. One is to use the BQRD to eliminate the interdependency of precoders among UEs through the SIC technique. Due to interference pre-subtraction, it is expected that the proposed BQRD aided IA has a faster convergence rate. The other one, called the “two-stage IA”, is to directly optimize the structure of the effective channel defined in (2.4) and employ power loading. Different from previous numerical methods (e.g., the max-SINR IA algorithm proposed in Section 3.3.2), it is expected that the two-stage IA algorithm converges more quickly because the two-stage IA algorithm aims to get the characteristic of the effective channel in convergence. Due to the MAC-like nature of uplink CoMP, the iterative procedures of evaluating IA solutions are developed by exploiting the duality between multiple access and broadcast channels. [20-21].

The organization of this chapter is shown below. The motivation of the proposed IA aided UL CoMP transceiver schemes is given in Section 4.1. In Section 4.2, the proposed BQRD aided IA algorithm and its associated computer simulations are presented. To further minimize the performance gap at low SNR, a two-stage IA

algorithm is proposed to directly optimize the effective channel in Section 4.3. Then, the computational complexity of the two proposed transceiver schemes is analyzed in Section 4.4. Numerical simulation results including the convergence behavior and sum-rate performance are given in Section 4.5 to compare the merits and drawbacks of the two proposed transceiver schemes. Finally, this chapter is summarized in Section 4.6.

4.1 Motivation

To achieve a higher system capacity in an interference limited communication environment, many techniques have been developed to cope with interference. Uplink CoMP is one of the techniques that aim to manage the interference caused by the UEs in other cells. To obtain the available DoFs that can be provided by the centralized uplink CoMP systems, a recently proposed technique (i.e., the max-SINR IA algorithm) is suggested to provide more DoFs in Section 3.3.2.

However, numerous iterations are required to guarantee that the max-SINR IA algorithm converges, which makes IA difficult to be implemented. From our observations, one of the reasons is the interdependency between UEs in multi-cell joint transmission scenarios. In Section 4.2, the proposed BQRD aided IA algorithm attempts to eliminate this interdependency to boost the convergence rate. Next, from the observation of [15], the SINR can be improved through proper power allocation if the effective channel has the nearly diagonal structure leading to less interference.

Based on this idea, we further propose a two-stage IA algorithm to optimize the effective channel by mitigating different kinds of interference separately.

4.2 Proposed Block QR Decomposition Aided IA Algorithm

In this section, an efficient IA aided transceiver design algorithm for uplink CoMP systems is proposed to mitigate interference. The BQRD is used to resolve the interdependency of precoders among user UEs through the SIC technique. To further improve the efficiency, an additional constraint is employed by a projection operation.

In the following paragraphs, the detailed algorithm is depicted in Section 4.2.1. Then, some computer simulation are shown to confirm the performance and convergence rate in Section 4.2.2.

4.2.1 Proposed Algorithm

In this section, a new IA algorithm is proposed to pursue maximum sum-rate with small number of algorithmic iterations. The optimization problem is formulated as follows:

We consider the possibility to reduce iterations if the interdependency between UEs can be eliminated.

triangular matrix partitioned into sub-matrices R_{i j,} Î^{(N P Nr )´ t}, so that ( )

, _{r t}

i j = N P N´

R 0 for i > j. Second, by incorporating (4.2) with (2.8), the corresponding approximated SINRk,d value for each data layer can be evaluated as follows [16]:

(

( )

) (

^{( )}

)

dth layer belonging to the kth UE as

(

( )

) (

^{( )}

)

Next, the optimization problem of maximizing the achievable sum-rate can be approximated by minimizing the summation of the reciprocal of individual

(

^metric ,

)

^, satisfied in many practical scenarios, the problem in (4.1) can be approximated as

( ^metric )

where a and b are constants obtained by MMSE criterion within specific region but the values are invariant to the optimization problem. Next, the original optimization solution of (4.1) can be approximated as

( )

is the direct consequence of the block-wise upper triangular structure R. This is equivalent to the scenario where interference caused by the UE with lower index is pre-cancelled.

For further improving the sum-rate performance, we attempt to modify the

For further improving the sum-rate performance, we attempt to modify the

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