On the Energy Efficiency of Interference Alignment in the $K$ -User Interference Channel

Interference alignment (IA) is regarded as an important physical-layer interference management technique. Most research contributions on IA were focused on the analysis of its achievable spectral efficiency, either from the degrees of freedom or from the capacity perspective. Meanwhile, high energy-efficiency (EE) has become a key requirement for next-generation wireless communications. Hence, we focus our attention on the EE of IA in the fully connected $K$ -user interference channel, where each user has either a single antenna or multiple antennas. We consider both perfect and imperfect channel state information scenarios. New insights into the achievable EE of IA are obtained by investigating the impact of different precoding matrices, the number of users, the number of antennas, symbol extension values, channel estimation accuracy, total transmit power, and power allocation schemes. In particular, we demonstrate that in the single-antenna-user case, the IA schemes relying on unequal power allocation achieves higher EE than their equal power allocation counterparts. However, in the scenario where each user is equipped with multiple antennas, equal power allocation achieves higher EE than unequal power allocation for the IA. Furthermore, using non-uniform precoding-matrix-generating vector ${\boldsymbol {w}}$ is not necessarily beneficial for improving the achievable EE of IA. We also find that the EE performance of IA with smaller symbol extension values is higher than that with larger symbol extension values, the achievable EE of IA decays with the increase of the total transmit power, the EE performance of IA degrades as the channel estimation accuracy becomes low and that having a larger number of transmit/receive antennas on each user achieves a higher EE in IA.


I. INTRODUCTION
I NTERFERENCE alignment (IA) has attracted substantial research interests during the past decade [1]- [10], since it sheds light on a promising avenue for approaching the capacity of interference-limited wireless networks. Considering the fully connected K-user interference channel composed of K pairs of transmitters and receivers, Jafar et al. [5] showed that K/2 degrees of freedom (DoF) is achievable by IA under certain strict conditions, where the DoF is also known as the capacity pre-log characterizing the multiplexing gain [11].
By contrast, the DoF of the conventional orthogonal multiaccess based interference management schemes is 1, which is much lower than K/2 when K is high.
Meanwhile, increasing economical and environmental concerns are calling for a significant reduction of the carbonfootprint of the telecom industry. As a result, energy efficiency (EE) has become an important design metric of next-generation wireless communication systems [12]- [19]. However, most existing research contributions on IA were focused on the analysis and improvement of its spectral VOLUME xx, xxxx efficiency (SE) in various interference-limited systems [4]- [10], by invoking either the DoF or the capacity as the performance metric. By contrast, how to understand IA from the fundamental EE perspective remains rarely documented. There are a few treatises related to the EE of IA indeed. For example, Yoon et al. [20] proposed a distributed opportunistic IA scheme, which was shown to be more energyefficient than the opportunistic IA. Additionally, adaptive power allocation and transmission mode (sleeping or active) selection schemes were studied in [21] for optimizing the achievable EE of the K-user interference network that uses IA. In [22] the normalized-EE maximization problem was solved in a multi-cell multi-input-multi-output orthogonal frequency-division multi-access (MIMO-OFDMA) system that uses IA, where it was shown that the partial-IA scheme is more energy-efficient than the full-IA scheme. The authors of [23] proposed a new network architecture based on EE maximization for multi-cell MIMO interfering broadcast channels using IA, and different IA schemes were employed for the high and moderate signal-to-noise ratio (SNR) regions. These contributions provided valuable insights into the EE of several sophisticated systems, where different variants of IA were invoked. However, in these systems the IA schemes are entangled with other techniques, which limits our understanding of the EE of IA itself.
Owing to the importance of both IA and EE, the EE of IA techniques deserves a deeper and more comprehensive investigation. It is well-known that the IA technique was originally conceived for the fully connected K-user interference channel [5], where all the users are equipped either with a single antenna or with multiple antennas. For the IA schemes of [5], the transmitters have to know the perfect global channel state information (CSI) for constructing the precoding matrices. However, obtaining the global CSI is challenging. Moreover, the CSI obtained at transmitters of a practical system is usually imperfect due to the time-varying nature of channels, the estimation error, the quantization error, the feedback error, and so forth.
In this paper, we study the EE performance of representative IA schemes in the context of the fundamental fully connected K-user interference channel, for which the IA technique was originally invented. Our novel contributions are summarized as follows.
• Closed-form EE expressions of the IA schemes considered are formulated under both the perfect and imperfect CSI assumptions, and in both the single-antenna and multi-antenna systems. • In contrast to [5], [24], where the equal power allocation strategy and the precoding matrices determined by the all-one column vector w = 1 are used, we demonstrate that in the single-antenna-user case, the variant IA schemes relying on unequal power allocation achieves higher EE than their equal power allocation counterparts. However, in the scenario where each user is equipped with multiple antennas, equal power allocation achieves higher EE than unequal power allocation for IA. Furthermore, using non-uniform precoding-matrixgenerating vector w is not necessarily beneficial for improving the achievable EE of IA. • We find that in the symbol-extension based IA schemes, when the symbol extension values become sufficiently high, the "effective channel" at a given receiver becomes ill-conditioned and almost singular. This phenomenon will cause numerical accuracy and stability problems when using the zero forcing (ZF) receiver. We conceive a channel truncation method to address this issue. • We analyze the computational complexity of the IA schemes considered, and quantify how the number of users, the number of antennas, the symbol extension values, the channel estimation accuracy, the total transmit power and the power allocation schemes influence the EE of the IA schemes considered. The rest of this paper is organized as follows. The EE of IA in the K-user interference channel with single-antenna nodes and multi-antenna nodes is theoretically studied in Section II and Section III, respectively. The computational complexity of the IA schemes considered is analyzed in Section IV. Our simulation results and discussions are presented in Section V. Finally, our conclusions are offered in Section VI.
Notations: We use boldface uppercase and lowercase letters to denote matrices and vectors, respectively. I n represents the identity matrix of size n × n. (·) H and |·| refer to the conjugate transpose and the determinant of a matrix, respectively.

A. THE CANONICAL IA SCHEME
We first consider an interference channel having K singleantenna transmitters and K single-antenna receivers. For clarity, the canonical IA scheme of [5] in the single-antenna case is illustrated in Fig. 1, which is described in detail below.
Each transmitter, such as the transmitter j, is supposed to send information only to its associated unique receiver. The output at the kth receiver during the tth channel use is given as follows: where j, k = 1, 2, · · · , K is the user index; the integer t represents the time slot index, the frequency slot index, or the time-frequency block index; x j (t) denotes the input signal of the jth transmitter; and h kj (t) is the channel fading coefficient from transmitter j to receiver k during the tth channel use. To avoid degenerate channel conditions (e.g., the rankdeficient channel matrix, or the channel matrix that has zero or infinity entries), we assume that h kj (t) is constituted by independently and identically distributed (i.i.d.) random variables drawn from a continuous distribution, e.g., CN (0, 1), and has to satisfy: 0 < h min ≤ |h kj | ≤ h max < ∞. Additionally, z k (t) ∼ CN (0, 1) represents the additive white  Gaussian noise (AWGN) at the kth receiver, and all the noise terms across the receivers are assumed to be independent.
In the IA scheme of [5], upon defining it was shown that [(n + 1) N +(K −1)n N ] DoF is achievable over a symbol extension of time slots in the time-varying channel that has no intersymbol interference. More specifically, in this context, the first transmitter achieves (n + 1) N DoF, while each of the other transmitters achieves n N DoF, by transmitting the information-bearing signals with the help of their judiciously designed individual precoding vectors/matrices. Here n can be any positive integer. It is also assumed that all the transmitted signals arrive at the receivers simultaneously. Based on this multi-symbol extended channel, the signal vector at the kth receiver is expressed as where y k ∈ C Me×1 , and H kj is an M e ×M e diagonal matrix representing the CSI from transmitter j to receiver k over the M e -symbol extension of the channel, as shown in (5). In this paper, we consider both the perfect and the imperfect CSI scenarios. Additionally, x 1 ∈ C (n+1) N ×1 represents the input signal of the first transmitter, while x j ∈ C n N ×1 (j = 2, 3, · · · , K) is the input signal of the jth transmitter. Moreover, it is assumed that x 1 ∼ CN (0, P 1 I (n+1) N ) and x j (j = 2, 3, · · · , K) obeys CN (0, P j I n N ), where P j denotes the power of each transmitter. V 1 ∈ C Me×(n+1) N and V j ∈ C Me×n N (j = 2, 3, · · · , K) are the precoding matrices invoked at the individual transmitters, respectively. Finally, z k ∈ C Me×1 represents the AWGN, i.e. z k ∼ CN (0, I Me ).
To align the interference at each of the receivers, the precoding matrices are constructed in [5] as follows. The set of column vectors of V 1 is chosen to be equal to the set V 1 , which has (n + 1) N elements and is defined as the other precoding matrices are determined using where we have and the set of column vectors of B is chosen to be equal to the set B, which has n N elements and is defined as Herein w is an M e × 1 all-one column vector. Once the interference is aligned at the receivers, the following constraints are satisfied: where we have i, j = 2, 3, · · · , K, and the operator "≺" represents that the set of column vectors of the left-hand matrix is a subset of the set of column vectors composing the right-hand matrix.

B. A PRECODING-MATRIX OPTIMIZED IA SCHEME
The achievable DoF of the IA scheme in [5] is asymptotically optimal. To attain a higher DoF than the scheme of [5] at any given number of channel realizations, an IA scheme, which is capable of achieving the same DoF of [5] with smaller symbol extension values, was designed from the perspective of signal space for the case of K ≥ 3 based on an improved precoding matrix design criterion in [24]. Specifically, in [24] the length of symbol extension is given by where we have N = (K − 1)(K − 2) − 1. Additionally, for given q = 1, p = q, by defining . . , K, j = i, the set of column vectors of the precoding matrix V * 1 at the first transmitter is chosen to be the set V * 1 , which has n * +N +1 elements and is defined as (14). Furthermore, the set of column vectors of the precoding matrix V * q at the qth transmitter is chosen to be the set V * q , which has n * +N N elements and is defined as (15). Here n * can be any nonnegative integer. The other precoding matrices are determined using where we have k ∈ {2, 3, . . . , K}, k = q.

C. THE EE OF IA WITH PERFECT CSI
With the IA scheme of [5] in mind, we aim to design the precoding and receiving matrices for ensuring that the following conditions are satisfied: when the set of channel matrices and the set of precoding matrices are assumed to be entirely and perfectly known to all transmitters and receivers, the kth receiver's interference subspace spanned by the column vectors of H kj V j (∀j = k) can be eliminated by using the ZF technique at the kth receiver. We normalize the power of V k to 1.
For the first receiver, the interference arriving from transmitters 2, 3, . . . , K can be aligned according to: Therefore, the ZF filtering matrix at the first receiver is formulated as follows: where , while d 1 = (n + 1) N denotes the achieved DoF of the first transmitter.
For the other receivers k = 1, the condition H kj V j ≺ H k1 V 1 , j / ∈ {1, k} is satisfied. Then the ZF filtering matrix associated with receiver k can be expressed as where Applying the ZF receivers given by (17) and (18) to the received signal of (4) results in the following matrix-form SNR expression 2 at the receiver k: where P t denotes the total transmit power of all the transmitters, and χ k = P k Pt represents the percentage of P t that the kth transmitter's power dissipation P k accounts for, which must satisfy: Then the individual rate at the kth receiver is formulated as [bits per symbol extension block]. The average sum rate achieved by the IA scheme [5] in the perfect CSI scenario is then given byR [bits per channel use]. 1 We find that when the symbol extension values become sufficiently high, the channel H k in (17) and (18) become ill-conditioned and almost singular, if its entries are not truncated properly. Consequently, in numerical calculations the ZF filtering matrix that requires the inverse operation at each receiver cannot be obtained with a sufficiently high accuracy. On the other hand, although the channel coefficients following CN (0, 1) have a probability of taking any value, it is reasonable to truncate the extreme values that are far away from the mean zero, as it is well known that the probability of taking such values is very small. Therefore, to make the simulation results more accurate, we make such a truncated channel assumption as stated in Sec. II-A. 2 This is a diagonal matrix, in which the Me non-zero elements correspond to the SNR values associated with the Me symbol extension.
Moreover, the total power dissipation of the system designed for communicating over an interference channel can be modeled as where P c denotes the fixed circuit power consumption, and it significantly affects the system's EE performance. As a result, the EE of the IA system considered is formulated as As far as the IA scheme of [24] is concerned, by replacing V k in (17) and (18) with V * k of (14), (15) and (16), and using Equations (19) ∼ (23), a higher EE performance is achieved when K ≥ 4, 3 as demonstrated in Sec. IV. Additionally, it should be noted that both the IA scheme of [5] and the IA scheme of [24] employ the uniform precoding-matrixgenerating vector of w = 1 and the equal power allocation scheme characterized by χ k = 1/K. By contrast, we propose to improve the EE performance of IA by using non-uniform precoding-matrix-generating vectors, where the elements of w are not all equal to one, as well as unequal power allocation, where χ k can have different values for different k.

D. THE EE OF IA WITH IMPERFECT CSI
In realistic wireless environments, the perfect CSI is difficult to obtain at transceivers due to various factors, such as the time-varying nature of wireless channels, the channel estimation error, the quantization error and the feedback error/delay. Hence it is important to study the EE of IA in the imperfect CSI scenario, where the channel can be modeled as [25] where the M e × M e diagonal matrix H kj represents the estimated CSI available to network nodes, and the diagonal elements of E kj are i.i.d Gaussian noise components. Each diagonal element of H kj and E kj obeys CN (0, 1), and β controls the CSI accuracy. Substituting (24) into (4), the signal at the kth receiver can be expressed as The second term on the right-hand side of (25) is the additional interference imposed by the channel uncertainty.
Relying on (17) and (18), we have W ZF Applying the ZF receiver formulations mentioned above to the received signal given by (25), the matrix-form signal-to-interference-plus-noise ratio 3 Because the transmit beamforming design criterion [24] attained higher sum DoF using shorter symbol extension when K ≥ 4, V * k results in higher sum rate, as well as higher EE.  [5] where each node has an even number of multiple antennas.
(SINR) corresponding to M e symbol extension at the kth receiver is written as Thus (26) can be simplified as: where D j denotes the matrix composed of the diagonal elements of V j V H j , and the EE expression in the imperfect CSI scenario is given by replacing γ k of (19) with γ k of (27) and following Equations (20)∼(23).

III. THE EE OF IA IN THE MULTI-ANTENNA K-USER INTERFERENCE CHANNEL
Let us further investigate the EE of the canonical IA scheme [5] for the K-user interference channel, where each node has M > 1 antennas. In this multi-antenna scenario, the closedform EE expression is only available for K ≤ 3, hence we assume K = 3 in the following derivations. It has been shown [5] that each transmitter achieves M/2 DoF when the inter-symbol interference is eliminated and the channel matrices are constant during each transmission period.

A. THE EE OF IA FOR EVEN-VALUED M
The IA scheme for even-valued M is illustrated in Fig. 2. More specifically, the signal received by the kth receiver can be written as where y k ∈ C M ×1 , H kj (k, j ∈ {1, 2, 3}) is an M × M matrix representing the CSI from transmitter j to receiver k, V k ∈ C M × M 2 is the precoding matrix invoked by the individual transmitters, x k ∼ CN (0, P k I M 2 ), and z k ∈ C M ×1 represents the AWGN, i.e. z k ∼ CN (0, I M ).
To align the interference at each of the receivers, the following constraints must be satisfied: where span(A) represents the vector space spanned by the column vectors of matrix A. Since H kj has a full rank almost surely, the above equations can be equivalently written as where Let e 1 , e 2 , . . . , e M be the M eigenvectors of E. We set ]. Then V 2 and V 3 are found using (28)∼(30).
Similar to the derivation in Section II, the ZF filtering matrix at the first receiver is formulated as follows: where . For the other receivers k = 2, 3, the ZF filtering matrix can be expressed as where Applying the ZF receivers given by (38) and (39) to the received signal of (28), the average sum rate achieved by IA in the three-user multi-antenna interference channel is then given by [bits per channel use]. Correspondingly, the EE expression is given by replacingR in (23) with R * of (40).

B. THE EE OF IA FOR ODD-VALUED M
When M takes an odd value, to achieve a total of 3M/2 DoF per channel use in the three-user interference channel, a twosymbol extension of the channel assuming constant channel coefficients over the duration of two symbols is needed, as shown in Fig. 3. Specifically, we have . The IA scheme [5] where each node has an odd number of multiple antennas.
where y k and z k represent the two-symbol extension of the received signal y k and of the noise vector z k at the kth receiver, respectively. Furthermore, H kj is a 2M × 2M blockdiagonal matrix representing the extension of the channel, i.e., we have Still referring to (41), V k ∈ C 2M ×M is the precoding matrix invoked by the individual transmitters, and x k ∼ CN (0, P k I M ) is an M × 1 vector representing M independent data streams. Similar to the even-M case, the following equations ensure that the interference at each of the receivers is aligned: The above equations imply that where E, F and G are 2M × 2M block-diagonal matrices representing the two-symbol extension of E, F and G, respectively. Hence, E, F and G are obtained by replacing Again, the ZF filtering matrix at the first receiver is formulated as follows: where For the other receivers k = 2, 3, the ZF filtering matrix can be expressed as where Applying the ZF receivers given by (50) and (51) to the received signal of (41), the average sum rate achieved by IA in the three-user interference channel is then given by [bits per channel use]. The corresponding EE expression is given by replacingR in (23) with R k of (52).

IV. ANALYSIS OF COMPUTATIONAL COMPLEXITY
Most of the existing literature related to the computational complexity of IA, such as [26], [27], were focused on the distributed IA scheme proposed in [28]. It was proved in [26], [27] that the sum DoF maximization problem and the DoF achievability problem are both NP-hard if each transmitter/receiver has at least three antennas. By contrast, the IA schemes studied in this paper obtain the precoding matrices and the ZF filtering matrices through direct analysis and derivation, rather than through the iteration process of [28]. Thus, the computational complexity of the IA schemes studied herein is mainly related to matrix multiplication. In this paper, since the matrix multiplications from (6) to (10) use diagonal matrices, the computational complexity of this part is O(M e ). The computational complexity of (17) and (18) is O(D sum M 2 e ), where D sum denotes the sum of achievable DoF for all users. As a result, the complexity of the IA schemes considered in this paper is . Note that D sum and M e are obtained in different ways for the two IA schemes studied. In the canonical IA scheme of [5], D sum = (n + 1) N + (K − 1)n N and M e is given by (3). In the efficient IA scheme of [24], D sum = n * +N +1 and M e is given by (13). It is clear that the IA scheme of [24] has an advantage in terms of computational complexity over the IA scheme of [5].
In Fig. 4, we show the average achievable EE of the canonical IA scheme [5], the IA scheme [24], and their respective variants in the perfect CSI scenario with single-antenna nodes. We consider K = 3, 4, 5 users, different values of M e [calculated using (K, n) and (K, n * ) according to (3) and (13), respectively, and also given in the above paragraph for convenience], different precoding-matrix-generating vectors (indicated by w 1 ) and different transmit power allocation schemes (indicated by the values of χ 1 , · · · , χ K ).
We observe from Fig. 4 that the IA scheme [24] relying on an improved precoding matrix design criterion achieves higher EE than that of the canonical IA scheme [5] in the entire SNR Tx = P t /σ 2 region, when they use the same precoding-matrix-generating vector (e.g., w 1 = 1) and the same transmit power allocation scheme (e.g., equal power allocation) in the scenario of K ≥ 4.
Moreover, Fig. 4 shows that the EE performance changes when using different precoding-matrix-generating vectors. Specifically, comparing the green solid (or red dashed) curve having the marker of square with the curve of the IA scheme [5] (or [24]) in each subfigure of Fig. 4, we see that the variant schemes exhibit an inferior EE performance, when invoking the precoding-matrix-generating vector w 1 = 4 rather than w 1 = 1 used by the IA schemes [5] and [24]. However, this degradation is reduced when increasing K.
Furthermore, in Fig. 4 by comparing the curves of the IA scheme [5] (or [24]) with the green solid (or red dashed) curves having the marker of diamond, we find that the EE performance of IA with smaller symbol extension values (indicated here by lower n and n * ) is higher than that with larger symbol extension values, while using a certain transmit power allocation scheme in the whole SNR Tx region.
Additionally, by comparing the green solid (or red dashed) curves having the marker of triangle with that of the IA scheme [5] (or [24]) in Fig. 4, we observe that an unequal transmit power allocation strategy with χ 1 having a larger value results in a higher EE performance than the equal power allocation strategy in the IA schemes considered. It should also be pointed out that the EE decreases upon increasing the total transmit power (indicated by SNR Tx = P t /σ 2 ) in all scenarios considered. Notably, the EE ap-VOLUME xx, xxxx  proaches zero with sufficiently high SNR Tx values. The reason is as follows. With the increasing SNR Tx , the average sum rate achieved by the IA scheme grows in a slow "log" manner according to (20) and (21), while the total power dissipation P = P t + P c of the system grows in a faster "linear" manner. As a result, the EE approaches zero as the SNR Tx tends to a sufficiently large value, according to (23). This result implies that although increasing the SNR Tx can always increase the sum rate, the sum-rate return corresponding to per-Joule energy investment is diminishing from the EE perspective. The impact of channel estimation accuracy on the EE of the canonical IA scheme [5] is illustrated in Fig. 5 for the single-antenna case. We consider both 3-user and 4-user interference channels with M e = 3 and M e = 33 symbol extension at SNR Tx = 5 dB, respectively. Both cases are evaluated by using different power allocation schemes and different precoding-matrix-generating vectors. We observe that the average EE performance degrades as the channel estimation accuracy indicator β increases. The EE remains almost constant when β is lower than 10 −3 . When β approaches one (i.e., lg(β 2 ) approaches zero), the EE rapidly degrades. We also observe that the EE of K = 3 is higher than that of K = 4, assuming the same channel estimation accuracy, the same power allocation scheme and the same precoding-matrix-generating vectors. Fig. 6 shows the impact of the number of antennas M and different transmit power allocation schemes on the average achievable EE of IA. We observe that a larger M achieves a higher EE. However, it indicates that the IA scheme [5] relying on the equal power allocation of χ 1 = χ 2 = χ 3 = 1/3 outperforms the variant IA scheme relying on the unequal power allocation of χ 1 = 2/3, χ 2 = 1/6, χ 3 = 1/6 with the same value of M in terms of the EE. Note that this observation is different from that of the single-antenna-user case. The reason is as follows. In the MIMO case, all users equally share the DoF, whilst in the single-antenna-user case considered, the user 1 has a larger DoF than the other users. Similar to the results of Fig. 4, the achievable EE of the multiantenna scenario decays with the total transmit power.

VI. CONCLUSIONS
We have investigated the EE of IA in the fully connected single-antenna and multi-antenna K-user interference channels. We derive the EE expression of IA considering both perfect and imperfect CSI scenarios. New insights into the achievable EE of IA are obtained by investigating the impact of different precoding matrices, the number of users, symbol extension values, the number of antennas, channel estimation accuracy, total transmit power and power allocation schemes. In particular, we demonstrate that in the single-antenna-user case, the IA schemes relying on unequal power allocation achieves higher EE than their equal power allocation counterparts. However, in the scenario where each user is equipped with multiple antennas, equal power allocation achieves higher EE than unequal power allocation for IA. Furthermore, using non-uniform precoding-matrixgenerating vector w is not necessarily beneficial for improving the achievable EE of IA. We also find that the EE performance of IA with smaller symbol extension values is higher than that with larger symbol extension values, the EE performance of IA degrades as the channel estimation accuracy become low, and that having a larger number of transmit/receive antennas on each user achieves a higher EE in IA. Finally, the achievable EE of IA decays with the increase of the total transmit power. In our future work, we will study the design of optimal IA schemes that are capable of maximizing the achievable EE.