On the Capacity and Generalized Degrees of Freedom of the X Channel

On the Capacity and Generalized Degrees of Freedom of the Channel

\authorblockNChiachi Huang, Viveck R. Cadambe, and Syed A. Jafar
\authorblockAElectrical Engineering and Computer Science
University of California Irvine
Irvine, California, USA
Email: {chiachih, vcadambe, syed}@uci.edu
Abstract

We explore the capacity and generalized degrees of freedom of the two-user Gaussian channel, i.e. a generalization of the user interference channel where there is an independent message from each transmitter to each receiver. There are three main results in this paper. First, we characterize the sum capacity of the deterministic X channel model under a symmetric setting. Second, we characterize the generalized degrees of freedom of the Gaussian channel under a similar symmetric model. Third, we extend the noisy interference capacity characterization previously obtained for the interference channel to the channel. Specifically, we show that the channel associated with noisy (very weak) interference channel has the same sum capacity as the noisy interference channel.

I Introduction

Recent research in multi-user information theory has been characterized by a surge of interest in the study of capacity regions of wireless Gaussian networks. Much of this interest has been fueled by significant recent progress in the search of the capacity region of wireless interference networks, a classical problem of multi-user information theory. In their seminal work [1], Etkin, Tse and Wang approximated the capacity region of the two-user Gaussian interference channel to within one bit. Further insight into the capacity of the two-user Gaussian interference network was revealed in [2, 3, 4]. These references found that the decoding strategy of treating interference as noise at each receiver in the interference network is capacity optimal for a class of interference channels, known as the “noisy interference” channels. Recent results have also found approximations to the capacity regions of certain classes of the -user interference channel in the high signal-to-noise ratio (SNR) regime. Reference [5] approximated the capacity region of the fully connected -user interference channel with time-varying channel coefficients as

where SNR represents the total transmit power of all nodes when the local noise power at each receiver is normalized to unity. In other words, it was shown that the time-varying -user interference channel has degrees of freedom. Similar capacity approximations of the -user () interference channel with constant channel coefficients (i.e., not time-varying or frequency-selective) are not known in general.

From the recent advances in the study of interference channels, many interesting and powerful tools related to the study of general wireless networks have emerged. Reference [1] introduced the notion of generalized degrees of freedom to study the performance of various interference management schemes in the interference channel. As its name suggests, the idea of generalized degrees of freedom is a generalization of the concept of degrees of freedom originally introduced in [6]. The idea of generalized degrees of freedom is powerful because in the multiple access, broadcast and two-user interference channels, achievable schemes that are optimal from a generalized degrees of freedom perspective also achieve within a constant number of bits of capacity [7]. A useful technique in the characterization of the generalized degrees of freedom of a wireless network is the deterministic approach, originally introduced in the context of relay networks [8]. The deterministic approach essentially maps a Gaussian network to a deterministic channel, i.e, a channel whose outputs are deterministic functions of its inputs. The deterministic channel captures the essential structure of the Gaussian channel, but is significantly simpler to analyze. Reference [7] showed that the deterministic approach leads to a characterization of the generalized degrees of freedom of wireless networks in the two-user interference network, which leads to a constant bit approximation of its capacity.

In this paper, we explore the two-user channel - a network with two transmitters, two receivers and four independent messages - one corresponding to each transmitter-receiver pair. The degrees of freedom of the Gaussian channel have been found in [9, 10]. This work pursues a more refined characterization in terms of the generalized degrees of freedom. Unlike the conventional degrees of freedom perspective where all signals are approximately equally strong in the scale, the generalized degrees of freedom perspective provides a richer characterization by allowing the full range of relative signal strengths in the scale. For example, consider the interference channel. The strong and weak interference scenarios are not visible in the conventional degrees of freedom perspective but become immediately obvious in the generalized degrees of freedom framework. Now consider the channel which is a generalization of the interference channel to a scenario where every transmitter has a message to every receiver. One of the key features of the channel is that, unlike the two-user interference channel, it provides the possibility of interference alignment [9, 10]. Interference alignment refers to the construction of signals such that they overlap at receivers where they cause interference, but remain distinguishable at receivers where they are desired. Interference alignment is the key to the degrees of freedom characterizations of the channel with or more users [11], and for the interference channel with or more users [5]. Since the potential for interference alignment does not arise in the user interference channel, the two-user channel provides the simplest possible setting for interference alignment, in terms of the number of transmitters/receivers and channel coefficients. It is shown in [10] that, due to interference alignment, the user channel has degrees of freedom (assuming time-varying channels), while the user interference channel has only degree of freedom. In this work, we explore this capacity advantage of the channel over the interference channel in the richer context of the generalized degrees of freedom. Specifically we quantify the benefits of interference alignment in terms of generalized degrees of freedom and identify operating regimes where alignment helps the channel outperform the interference channel. For simplicity, we will keep the number of channel parameters to a minimum by using the symmetric interference channel as our benchmark and presenting our main results for the corresponding symmetric channel.

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Fig. 1: The two-user Gaussian channel

Our approach to solving the generalized degrees of freedom of the channel follows the deterministic approach of [12]. We first introduce the deterministic channel, and find a tight outerbound and achievable scheme for the sum capacity of this channel in Section IV. In Section V, we extend the achievability and outerbound arguments of Section IV to the Gaussian channel yielding its generalized degrees of freedom. A second result we obtain is a generalization of the results of [2, 3, 4] to find the capacity of the Gaussian channel for a class of channel coefficients. We introduce the system model, formally define the notion of generalized degrees of freedom, and present the main results in the next section.

Ii System Model

Ii-a Deterministic Channel

Fig. 2: On the left is an example of the deterministic interference channel. On the right is the figure that shows only the signal levels observed at each receiver.

The deterministic channel is physically the same channel as the deterministic interference channel introduced in [7], except that the channel has independent messages where is the message that originates at transmitter and is intended for receiver . Note that the interference channel has only independent messages, e.g., or . The deterministic channel is shown is Fig. 2 and described by the input output equations

(1)
(2)

where , for , and is a shift matrix,

(3)

The message set and standard definitions and notations of the achievable rates are similar to those in the Gaussian setting. To avoid confusion, sometimes we add the subscript det to distinguish the notations for the deterministic channel from those for the Gaussian channel.

Ii-B The Gaussian Channel

The two-user Gaussian channel is described by the input-output equations

(4)
(5)

where at symbol index , and are the channel output symbol and additive white Gaussian noise (AWGN) respectively at receiver . is the channel input symbol at transmitter , and is the channel gain coefficient between transmitter and receiver for all . All symbols are real and the channel coefficients do not vary w.r.t symbol index. In the remainder of this paper, we suppress time index if no confusion would be caused. The AWGN is normalized to have zero mean and unit variance and the input power constraint is given by

(6)

There are four independent messages in the channel: where represents the message from transmitter to receiver . We indicate the size of the message by . For codewords spanning symbols, rates are achievable if the probability of error for all messages can be simultaneously made arbitrarily small by choosing an appropriate large . The capacity region of the channel is the set of all achievable rate tuples . We indicate the sum capacity of the channel by .

Ii-B1 Generalized Degrees of Freedom (GDOF)

To motivate our problem formulation, we briefly revisit the framework for the generalized degrees of freedom characterization of the symmetric interference channel. The interference channel is defined as:

(7)
(8)

and with the parameter defined as follows

(9)

the GDOF metric is defined as [1],

(10)

where is the sum capacity of the interference channel.

Since our main goal is to compare GDOF of the channel with the interference channel, we use the same symmetric interference channel model described above as the physical channel model for the channel. There is however, one notational difference. Since the terminology is not as appropriate for the channel, we instead use the parameter to substitute for these notions, resulting in the following system model for the channel GDOF characterization:

(11)
(12)

In other words, we have set , , and . Note that (11), (12) represent the same physical channel as (7), (8). However, as mentioned earlier, unlike the interference channel the channel has independent messages - one from each transmitter to each receiver. The GDOF characterization for the channel is defined as:

(13)

where is the sum capacity of the channel.

Note that we use to ensure that always exits. The half in the denominator is because all signals and channel gains are real.

Iii Main Results

Iii-a Sum Capacity of the Symmetric Deterministic Channel

The first main result of the paper is the characterization of the sum capacity of the symmetric deterministic channel in the symmetric setting where and . This result is given in the following theorem.

Theorem III.1

The sum capacity of the symmetric deterministic channel, i.e., the deterministic channel where and , is

(14)

Iii-B Generalized Degrees of Freedom of the Symmetric Gaussian Channel

The second main result of this paper builds upon the result of Theorem III.1 to find the generalized degrees of freedom characterization (shown in Figure 3) for the Gaussian channel.

Theorem III.2

The generalized degrees of freedom of the symmetric Gaussian channel can be characterized as

(15)
Fig. 3: Generalized Degrees of Freedom of the symmetric channel, and a comparison the the user interference channel

For comparison, Figure 3 also shows the generalized degrees of freedom characterization of the symmetric interference channel as obtained in [1]. For values of , characterization of is identical for both the symmetric two-user Gaussian channel and the symmetric two-user Gaussian interference channel (See [1] Figure IV.5). We prove this by showing that the Etkin-Tse-Wang (ETW) outerbound derived for the interference channel [1] holds for the channel as well (See Theorem V.3). The ETW outerbound is tight from a GDOF perspective in the interference channel for . Therefore, our extension of this outerbound implies that for a GDOF optimal achievable scheme is to set , so that the channel operates as an interference channel. For example, if , setting and treating interference as noise is GDOF optimal in the channel, since it is optimal in the corresponding interference channel [1]. Similarly, we show that for , it is GDOF optimal to set and operate the channel as an interference channel with messages and . It must be noted that for both and the GDOF optimal achievable scheme operates the channel as weak interference channel by setting the appropriate messages to null. For , we propose an interference alignment based achievable scheme for the channel. Thus, in this regime, the channel performs better than the interference channel by exploiting the possibility of interference alignment.

Iii-C Capacity of the “Noisy” Gaussian Channel

References [2, 3, 4] showed that in the interference channel, for a class of channel coefficients, encoding messages using Gaussian codebooks and decoding desired messages by treating interference as noise at each receiver is capacity optimal. Our last main result extends this conclusion to the channel as well. We show that if a user interference channel satisfies the noisy interference conditions obtained in [2, 3, 4] then the corresponding channel obtained by allowing all transmitters to communicate with all receivers, has the same sum capacity as the original noisy interference channel. This is a surprising result since it implies that for a class of channels, interference alignment has no capacity benefit. The result holds for the general (asymmetric) channel and is stated as such in Theorem VI.1 in Section VI. For simplicity we re-state the result here for the symmetric case ( in a notation consistent with [4], as follows.

Noisy “Symmetric” X Channel Result: If then the sum capacity of the Gaussian channel is given by . Similarly, if then the sum capacity of the Gaussian channel is given by .

The condition is the same as the noisy interference condition in [4]. It means that when the cross-links are too weak, there is no sum-capacity benefit in communicating messages over those links ( channel operation), even though it rules out interference alignment, and we are better off just communicating on the direct links while treating the weak interference as noise. Thus, in this case messages do not increase sum capacity of the channel.

The other condition refers to a strong cross-channel scenario. It says that when the cross-links are too strong relative to direct links, then sum capacity is achieved by communicating only over the strong cross-links and treating the weak interference received over the direct links as noise. In this case, messages do not increase the sum capacity of the channel.

Notation: In the rest of this paper, we use the notation

for any sequence .

Iv Sum Capacity of the Symmetric Deterministic Channel

The deterministic channel model is described in the symmetric setting by:

(16)
(17)

where .

To prove Theorem III.1, we use the following lemma.

Lemma IV.1
(18)

The lemma follows trivially from the symmetry in the channel. We now proceed to derive the converse argument for Theorem III.1.

Iv-a Upperbounds

In this section, we start from the capacity outerbounds for the (asymmetric) deterministic channel, and then we use the results to derive the capacity outerbounds for the symmetric setting. The following lemma provides a set of outerbounds for the achievable rate tuple of the (asymmetric) deterministic channel.

Theorem IV.2

The achievable rate tuple of the deterministic channel satisfies the following inequalities.

(19)
(20)
(21)
(22)
(23)
(24)
{proof}

The bound on , (19), is proved by a genie upperbound. Consider a genie-aided channel where a genie provides , , and to receiver . For a block length , we can bound as follows.

(25)
(26)
(27)
(28)
(30)
(31)

where (27) and (28) hold because all messages are independent of each other. (31) follows from the fact that is a function of . Using Fano’s inequality, can be bounded as follows.

(32)
(33)
(34)
(35)
(36)

Adding (31) and (36), we have

(37)
(38)

Letting , we prove (19). Similarly, we can prove (20), (21), and (22).

Next, the first bound on , (23), can be proved as follows. Consider a genie-aided channel where a genie provides and to receiver . For a block length , using Fano’s inequality, we can bound as the following.

(39)
(40)
(45)

Similarly, we have

(46)

Adding (45) and (46), we have

(47)
(48)
(49)

Letting , we prove (23). Similarly, we can prove (24).

After obtaining capacity outerbounds for the deterministic channel, we use them to derive sum-capacity upperbounds for the symmetric case.

Corollary IV.3

For any achievable scheme, the sum-rate can be bounded as

{proof}

Consider any reliable coding scheme achieving sum rate . Then we can write

(50)
(51)
(52)

Inequalities (51) and (52) are direct results of (23) and (24). To prove inequality (50), we do the following. Substituting and into (19) to (22), adding the resulting inequalities together, and dividing both sides by , we obtain (50).

Further, for the symmetric deterministic channel, if , then both receivers receive the same signals. Thus, the achievable sum rate is bounded by the multiple access channel bound.

(53)

The result of Corollary IV.3 follows from (50)-(53).

Iv-B Achievable Schemes

The following theorem gives the sum capacity of the symmetric deterministic channel.

Theorem IV.4

The sum-capacity upperbound given in (IV.3) is achievable. Equivalently,

(54)

Before we proceed to the proof, we will need the following lemma

Lemma IV.5

Let be positive integers such that . Then

  1. If is divisible by , then there exists a matrix whose entries are from such that

    where is a whose column vectors form a basis for the nullspace of

  2. There exists a matrix whose entries are from such that

    where

    and represents the matrix whose column vectors form a basis for the nullspace of

The proof of the lemma is placed in Appendix B. We now proceed to prove Theorem IV.4.
{proof} We only discuss the achievable scheme for the case that . The achievable schemes for can be obtained by using Corollary IV.1. For the case that , the achievable scheme is split into four different regimes viz. , , , and .

Achievability for is trivial, since an optimal achievable scheme sets and uses all the levels for at transmitter . We will treat the other cases below.

Case 1 :
We need to show that is achievable. Achievability follows by setting so that the channel operates as an interference channel. The capacity of the two-user deterministic interference channel found in [7, 13] implies that is achievable in this regime.

Case 2 :
We show that a sum rate of is achievable in this regime using interference alignment over the deterministic set up. The achievable scheme achieves a rate of for each of , and a rate of for and .

At transmitter , the top levels are used to transmit , the next levels are used to transmit , the next levels are kept zero, and the remaining levels are used to transmit for (See Figure 4). In other words, the achievable scheme transmits for , a column vector which can be represented as

where represents the identity matrix, are column vectors of sizes respectively. are used to encode message and is used to encode . As illustrated in Fig. 4, receiver can recover its intended messages without interference. Thus, we have . Note that at receiver , interference align at levels .

Fig. 4: Signal levels at receivers for .

Case 3 :
We first consider the case where is a multiple of . For this regime, we show that is achievable.

Fig. 5: Achievable scheme for the symmetric deterministic channel with

1) Transmit Scheme: We use linear precoding at the transmitters. Let be a times matrix whose column vectors form a basis for the null space of meaning that

At transmitter , we use, as precoding vectors for , column vectors of the matrix where has dimension . We will shortly explain how is chosen, but here we mention that the columns of are linearly independent of . Note that this implies that has a full rank of . For , we use