Macro-pico amplitude-space sharing with layered interference alignment

The deterministic channel model described above assumes that all SNRs and INRs are integer on the log scale, i.e., the signals fall exactly on the evenly spaced “levels” at the receiver. In Gaussian channels, however, this assumption is not valid. In the following, we will address the interference alignment problem taking into account arbitrary SNR and INR values.

Given the link quality matrix Γ, at each receiver, there is one collision pattern in amplitude space, i.e., part of the desired signal may collide with part of the interference. The collision pattern changes at different receivers because they experience different SNR and INRs. The layer partitioning requires to find all the possible collision areas in the amplitude space between the signals and interference.

The layer partitioning procedure is shown in Fig. 2. For each user k, the SNR of the direct channel is γ _k,k , which is represented by a bar with a height logγ _k,k at the transmitter side. At receiver k, there are K bars representing the K received signals. Their relative positions in the amplitude space are affected by the corresponding SNRs and INRs. For user i’s bar, the upper and lower boundaries are logγ _k,i and logγ _k,i − logγ _i,i , respectively. If the bar of user j,j≠k, overlaps with the bar of user k, both bars are split (separated into different layers) at the intersecting boundary positions. Repeating these processes at every receiver, we will obtain at most 2K(K−1) layers for each transmit signal. The results shown in Fig. 2 are obtained when a group of random values are set for Γ, i.e.,

$$ 10 \log_{10} \boldsymbol{\Gamma} = \left[ \begin{array}{llll} 7.43 & 3.47 & 1.57 & 8.69 \\ 7.42 & 6.51 & 4.16 & 3.92 \\ 3.45 & 7.18 & 9.40 & 2.53 \\ 8.84 & 9.58 & 4.50 & 8.33 \\ \end{array} \right] \text{dB}. $$

(6)

After layer partitioning, user k has L _k layers in the transmitter side, and the upper and lower boundaries of the l-th layer are respectively P _k,l and P _k,l−1. Thus, the l-th layer has an amplitude space represented by ρ _k,l =P _k,l /P _k,l−1. Assume now that we have a layered transmission scheme where the bits in the highest layers are decoded first. Thus, the decisions of the current layer can at most be disturbed by as-yet undecoded layers, i.e., the layers with a lower power than the current one. Thus, ρ _k,l can be viewed as a signal-to-interference ratio (SIR) of this layer. If Shannon capacity achieved transmission were used, R _k,l =1/2 log(1+ρ _k,l ) bits could be transmitted in this layer. Considering the encoding and decoding methods to be introduced in Section 4, which have a rate loss of at most 0.5 bit, we use R _k,l =1/2 log(ρ _k,l ).

where b _k,l denotes the transmit state of layer l of user k. When this layer is active, b _k,l =1, otherwise b _k,l =0. The second constraint is to avoid collision between the desired signal and interferences. If the l ^′-th layer of user j interferes with the l-th layer of user k through the cross-link, b _k,l and \(b_{j,l^{\prime }}\) can not be “1” simultaneously. Unlike in the deterministic model, the relation between l and l ^′ does not have an explicit expression here.

This optimization problem can also be solved by the LP-based branch-and-bound algorithm. The search result is shown in Fig. 3, where the contiguous active layers are combined. We can see that some interference layers overlap and occupy the same part of the amplitude space, but they may not be strictly aligned with respect to the upper and lower boundaries. Since we need to decode the superimposed interference, this kind of alignment complicates the encoding designs. We will discuss in detail the encoding and decoding schemes in Section 4.

Different from the deterministic channels, the addition of two layers of interference has a higher signal level than any of the two layers, i.e., the carry-over problem must be considered. The active layers assigned through (7) might be tightly connected, which means that at one receiver the lower boundary of an upper signal layer might be the upper boundary of a lower interference layer. If the interference layer is superimposed from two or more users, the carry-over part might collide with the upper signal layer. Therefore, at the bottom of the signal layer, we need to reserve some amplitude space for the carry-over interference. In practice, this can be done by retaining the transmit power and reducing the data rate of the upper signal layer.

where the first term inside the maximum operation is the upper boundary of the (l−1)-th layer of user k, and the second term is the sum power of two layers of interference divided by the square of the direct-link channel gain. If the second term is larger, at receiver k, the lower boundary of the l-th layer of user k is changed from \(h^{2}_{k,k} P_{k,l-1}\) to \(h^{2}_{k,j}P_{j,l^{\prime }} + h^{2}_{k,i}P_{i,l^{\prime \prime }}\). The reserved space is at most 3 dB and thus the data rate loss is at most 0.5 bit. When the lower layer is occupied by the superimposed interference from K users, the reserved space is at most 10 log10(K) dB and the data rate loss is at most 1/2 log(K) bits. This data rate loss can be neglected compared to the increasing sum rate when SNR goes to infinity.

In the layered interference alignment scheme, since the superimposed interference layers need to be decoded, random coding is no longer applicable. We thus present a coding scheme with multi-level nested lattice codes in this section. Lattice coding is a structural coding; if there are two codewords that are selected from a lattice, their sum and difference are also within the same lattice. Thus, we can directly decode the superposition of the aligned interferences, instead of decoding the interference signals one by one.

A lattice Λ is an n-dimensional discrete subgroup of the Euclidean space \(\mathbb {R}^{n}\) under vector addition. Thus, if λ ₁ and λ ₂ are in Λ, their sum and difference are also in Λ. A lattice Λ ₂ is said to be nested in a lattice Λ ₁ if Λ ₂⊆Λ ₁. The lattice Λ ₁ is often referred to as a fine lattice and Λ ₂ as a coarse lattice. A nested lattice code \(\mathcal {L}\) is the set of all points of the fine lattice that are within the fundamental Voronoi region \(\mathcal {V}_{2}\) of the coarse lattice, i.e., \(\mathcal {L} = \Lambda _{1} \cap \mathcal {V}_{2}\).

where \(|\mathcal {L}|\) is the cardinality of set \(\mathcal {L}\) and \(\text {Vol}(\mathcal {V}_{i})\) is the volume of Voronoi region \(\mathcal {V}_{i}\).

In [41], it is shown that nested lattice codes can achieve the capacity of point-to-point AWGN channels. In [42], a doubly nested lattice coding scheme was provided which can approach the capacity region of a two-way relay channel within 0.5 bit. Reference [43] provides a practical implementation scheme for nested lattice coding, where turbo coding and trellis shaping (multidimensional quantization) are involved. Due to space limitation, we refer the interested reader to [44, 45] for the detailed definitions and general construction methods of lattice codes.

For the transmitter k, define a sequence of nested lattice Λ _k,0, Λ _k,1, ⋯, \(\Lambda _{k,L_{k}}\), where \(\Lambda _{k,L_{k}} \subseteq \cdots \subseteq \Lambda _{k,1} \subseteq \Lambda _{k,0}\). The second moment of Λ _k,l is σ ²(Λ _k,l )=P _k,l .

However, it has decoding problem if we just use L _k layers of nested lattice code for each user k. As shown in Fig. 3, the received signals and interference are in layers and might be interlaced. If an interference layer is above a signal layer, we need first to decode and cancel the interference layer before decoding the signal layer. If the interference from two or more users exist above a signal layer, we need to decode the superimposed interference irrespective of whether the boundaries of interference layers are aligned or not.

For example, at receiver k, above the l-th signal layer there is a superimposed interference layer that is added by the l ^′-th layer of user j and the l ^′′-th layer of user i. The upper boundary of the signal layer is \(h^{2}_{k,k} P_{k,l}\), and the lower boundaries of two interference layers are respectively \(h^{2}_{k,j} P_{j,l'-1}\) and \(h^{2}_{k,i} P_{i,l^{\prime \prime }-1}\), where we have \(h^{2}_{k,j} P_{j,l'-1} \ge h^{2}_{k,k} P_{k,l}\) and \(h^{2}_{k,i} P_{i,l^{\prime \prime }-1} \ge h^{2}_{k,k} P_{k,l}\).

The received signal codeword is h _k,k c _k,l ∈h _k,k Λ _k,l−1, and the received interference codewords are \(h_{k,j}\boldsymbol {c}_{j,l^{\prime }} \in h_{k,j}\Lambda _{j,l^{\prime }-1}\) and \(h_{k,i}\boldsymbol {c}_{i,l^{\prime \prime }} \in h_{k,i}\Lambda _{i,l^{\prime \prime }-1}\). To decode the superimposed interference \(h_{k,j}\boldsymbol {c}_{j,l'}+h_{k,i}\boldsymbol {c}_{i,l^{\prime \prime }}\), we require that the lattices \(h_{k,j}\Lambda _{j,l^{\prime }-1}\) and \(h_{k,i}\Lambda _{i,l^{\prime \prime }-1}\) are aligned or nested. Through adjusting the transmit powers of related layers, we can make \(h_{k,j}\Lambda _{j,l^{\prime }-1}\) and \(h_{k,i}\Lambda _{i,l^{\prime \prime }-1}\) aligned at receiver k, but it is impossible to simultaneously make \(h_{k^{\prime },j}\Lambda _{j,l^{\prime }-1}\) and \(h_{k^{\prime },i}\Lambda _{i,l^{\prime \prime }-1}\) aligned at another receiver k ^′.

To solve this problem, we just require a relationship that \(h_{k,j}\Lambda _{j,l^{\prime }-1}\) and \(h_{k,i}\Lambda _{i,l^{\prime \prime }-1}\) are nested at receiver k. Since \(h^{2}_{k,j} P_{j,l'-1} \ge h^{2}_{k,k} P_{k,l}\) and \(h^{2}_{k,i} P_{i,l^{\prime \prime }-1} \ge h^{2}_{k,k} P_{k,l}\), we can add a level of nested lattice Λ _j,m with second moment \(h^{2}_{k,k}/h^{2}_{k,j} P_{k,l}\) at transmitter j, and add a level of nested lattice Λ _i,m with second moment \(h^{2}_{k,k}/h^{2}_{k,i} P_{k,l}\) at transmitter i. The lattice nesting relation becomes \(\Lambda _{j,l^{\prime }-1} \subseteq \Lambda _{j,m} \subseteq \Lambda _{j,l^{\prime }-2}\) and \(\Lambda _{i,l^{\prime \prime }-1} \subseteq \Lambda _{i,m} \subseteq \Lambda _{i,l^{\prime \prime }-2}\). Thus, both \(h_{k,j}\boldsymbol {c}_{j,l^{\prime }}\) and \(h_{k,i}\boldsymbol {c}_{i,l^{\prime \prime }}\) are nested in h _k,k Λ _k,l . Obviously, their sum is also nested in h _k,k Λ _k,l . The superimposed interferences can then be decoded. Similarly, at receiver k ^′, if there are interference layers above the signal layers and the interference layer is superimposed from multiple users, we can add more levels of nested lattice at the transmitter side to make the received interference codewords nested.

i.e., the received signal r _k is first quantized to lattice Λ _k,l−1 and then taken a modulus operation relative to lattice Λ _k,l . The quantization operation is to mitigate the interference of lower layers, and the modulus operation can mitigate the interference of the upper layers by decoding and cancelation.

The developed implementation scheme is based on the idea of nested alignment and can guarantee the superimposed interference decodable at every receiver with any kind of power relationships.

We first study the performance of the layered IA scheme in deterministic interference channels and compare it with the orthogonal transmission scheme, and a virtual IA scheme where each user can achieve 1/2 DoF of the channel.

The virtual IA scheme demonstrated here is not a real transmission scheme. Since currently there is only DoF result for the layered IA transmission with arbitrary channel coefficients, we use this result to provide a performance benchmark of the achievable sum rate. In the virtual IA scheme, we compute the achievable rate of each user based on its SNR and without taking into account the impact of interference, and the sum rate of K users is obtained by the summation of each user’s achievable rate and divided by 2. This method reflects the fact that each user can exploit 1/2 DoF of the channel. If there is only one user, this user will use the full DoF of the channel. The sum rate obtained in this way is labeled as “ K/2 DoF” in Figs. 4 and 5. When SNR approaches infinity, the DoF result can reflect the upper bound of channel capacity. However, in moderate SNR values the achieved rate obtained from the DoF may have large difference with the capacity.

In deterministic models, the logarithmic channel gains are integer. We set the logarithmic SNRs and INRs as random integer selected from 1 ∼6. In K-user orthogonal transmission scheme, each user occupies 1/K of the time or frequency resources no matter what the SNRs and INRs are. The corresponding results are shown in Fig. 4. We can see that the sum rate of the virtual IA scheme linearly increases with the number of users when K≥2, but its performance is not the best for all cases. In two-user or three-user interference channels, the proposed layered IA scheme can achieve higher sum rate. According to the optimization process in Section 2, the active level assignment is not an equal distribution among users. By contrast, the user with better channel conditions (e.g., large SNR and small INR) might be assigned more active levels to maximize the sum rate. Hence, when the number of users is not too much, the optimized level assignment scheme outperforms the scheme that each user exploits one half of the channel resource. In interference channels with more than three users, the virtual IA scheme will gradually dominate the performance due to the advantage that each user can exploit 1/2 DoF of the channel. No matter how many users exist, the achieved sum rate of the orthogonal transmission scheme always keeps unchanged.

The comparisons of the layered IA scheme and other two schemes in Gaussian channels are shown in Fig. 5, where the SNRs and INRs are all randomly selected from 0 ∼40 dB. Because of the carry-over effect, in moderate SNR levels, the achieved sum rate of the layered IA scheme grows not as fast as in deterministic channels. Yet with two and three users, the layered IA scheme still outperforms the virtual IA scheme. Similarly, with more users the virtual IA scheme will dominate the performance again since its achieved sum-rate keeps linearly increased.

We then investigate the performance of the layered IA scheme in cellular systems. Two network topologies, i.e., homogeneous networks and heterogeneous networks, are respectively studied. In homogeneous networks, every cell has similar coverage and the user is probably interfered by all other adjacent BSs. In heterogeneous networks, the macro-BS may interfere with a lot of pico-cell users, but the pico-BS only interferes with a few macro-cell users. Both kinds of cellular deployment are shown in Fig. 6.

where D is the distance between the BS and user. The cell-edge SNR is set as 5 dB, and small-scale Rayleigh fading is also considered.

We first observe the performance of layered IA scheme in three-cell homogeneous networks. Although the proposed scheme can be applied with any number of cells, for the homogeneous deployment, the interferences to one cell are mainly from the adjacent two cells. The outer cells with further distance only contribute noisy interference and will not affect the layer participation and alignment scheme. In this configuration, two kinds of user distribution are simulated. The first is symmetric distribution, where the macro-users are moved along the line connecting the macro-BS and the central vertex of the three cells, as the asterisks (“*”) shown in Fig. 6, and their distances to macro-BSs are the same. The second is random distribution, where the macro-users are randomly distributed in the cell-edge areas, as the dots (“ ·”) shown in Fig. 6, and the cell-edge boundaries are changed symmetrically. Conventionally, a FFR scheme is used to avoid inter-cell interference in the cell edge areas, where each cell uses 1/3 of the available bandwidth. In the layered IA scheme, we use frequency reuse factor of one, i.e., every cell uses all the bandwidth, and the interference are coordinated in the amplitude space. The sum rate performances are shown in Fig. 7. We can see that the layered IA scheme has approximately two times of the data rate over the FFR scheme both for the symmetric and random user distribution scenarios. The performance gain increases as the users approach the BSs. Although with small-scale fading, when the users are close to the BSs, the average SNRs are higher and the average INRs are lower; thus, there are more opportunities to assign active layers. The random distribution scenario is more practical; when the cell-edge boundary moves far away from the BS, the average sum rate achieved in this scenario can be greater than that in symmetric distribution scenario. The reason is that random user distribution diverges the SNR and INR values, which also creates more opportunities to assign active layers.

The performance of the layered IA scheme in heterogeneous networks is evaluated in Fig. 8, where 1 ∼3 pico-cells coexisting with one macro-cell are considered. In this kind of deployment, the macro-user is randomly distributed in one sector of the macro-cell, as shown in Fig. 6, and there is also one pico-user randomly distributed in each pico-cell. The layered IA scheme demonstrates great potential in this scenario, and the performance gain keeps increasing along with more coexisting pico-cells. When the distance between the pico-BS and macro-BS changes, the sum rate varies in different patterns with different number of pico-cells. With only one pico-cell coexists with the macro-cell, it constitutes a two-user interference channel. Although the position of the macro-user is random, the interference scenarios that the pico-user experienced have certain rules to follow. When the pico-cell moves from cell center to cell edge, the interference caused by the macro-BS to the pico-user will change from strong to weak. In this kind of two-user interference channel, the sum rate is higher both in strong interference or weak interference scenarios and is lower when the interference has a similar strength with the signal. From the explanation in Section 3, we also know that the strong interference can share the upper amplitude space above the signal layer, and the weak interference can share the lower amplitude space below the signal layer. If the interference has a similar strength as the signal, both the signal and the interference will reduce their occupied amplitude space for coexisting. In the simulation results of Fig. 8, the average sum rate for one pico-cell configuration is a concave curve, which subsidizes our analysis.

When the number of pico-cells increases to two and three, the sum rate curves are monotonically increasing. The reason behind this phenomenon is the interference among the pico-cells. In the simulations, the SNRs and INRs of all links are computed from the transmit power and the simulated channel gains, which include the large-scale path loss and small-scale Rayleigh fading. When the pico-cells are close to the macro-BS, as can be seen from Fig. 6, the distances between these pico-cells are shorter. In this scenario, not only the interference from macro-BS to pico-users is stronger but also the interference from pico-BSs to pico-users. When the pico-cells move away from the macro-BS, the interference from macro-BS and pico-BS are all weaker. As observed in Section 3, the layer partitioning and assignment are complicated in the multi-user case. Although we do not provide an explicit analysis, the simulation results show that the average sum rate will increase when we move the pico-cells towards the macro-cell edge.