Degrees-of-Freedom Region of MISO-OFDMA Broadcast Channel with Imperfect CSIT

# Degrees-of-Freedom Region of MISO-OFDMA Broadcast Channel with Imperfect CSIT

Chenxi Hao, Borzoo Rassouli and Bruno Clerckx Chenxi Hao, Borzoo Rassouli and Bruno Clerckx are with the Communication and Signal Processing group of Department of Electrical and Electronics, Imperial College London, email: chenxi.hao10;b.rassouli12;b.clerckx@imperial.ac.ukThis paper was in part published in ”MISO Broadcast Channel with Imperfect and (Un)matched CSIT in the Frequency Domain: DoF Region and Transmission Strategies”, PIMRC’13.This work was supported in part by Samsung Electronics and by the Seventh Framework Programme for Research of the European Commission under grant number HARP-318489.
###### Abstract

This contribution investigates the Degrees-of-Freedom region of a two-user frequency correlated Multiple-Input-Single-Output (MISO) Broadcast Channel (BC) with imperfect Channel State Information at the transmitter (CSIT). We assume that the system consists of an arbitrary number of subbands, denoted as . Besides, the CSIT state varies across users and subbands. A tight outer-bound is found as a function of the minimum average CSIT quality between the two users. Based on the CSIT states across the subbands, the DoF region is interpreted as a weighted sum of the optimal DoF regions in the scenarios where the CSIT of both users are perfect, alternatively perfect and not known. Inspired by the weighted-sum interpretation and identifying the benefit of the optimal scheme for the unmatched CSIT proposed by Chen et al., we also design a scheme achieving the upper-bound for the general -subband scenario in frequency domain BC, thus showing the optimality of the DoF region.

## I Introduction

Channel State Information at the Transmitter (CSIT) is crucial to the DoF performance in downlink Broadcast Channel, but having perfect CSIT is a challenging issue. In practice, each user estimates, quantizes and reports its CSI to the transmitter. This process is subject to imperfectness and latency. Their impact on the DoF region has attracted a lot of attention in recent years. The usefulness of the perfect but completely outdated CSIT was studied in [1]. Literature [2] generalized the findings in [1] by giving an optimal DoF region for an alternative CSIT setting, where the CSIT of each user can be perfect, delayed or none. Moreover, authors in [3, 4] and [5] looked into the scenario with both perfect delayed CSIT and imperfect instantaneous CSIT, whose qualities are shown to make an impact on the optimal DoF region. [6] and [7] extended the results of [3] and [5] by considering the different qualities of instantaneous CSIT of the two users. Furthermore, the authors of [8] studied the scenario where both delayed CSIT and instantaneous CSIT are imperfect and the results were generalized to a scenario with multiple slots and evolving CSIT states in [9]. Recently, all the results found in two-user time-correlated MISO BC with delayed CSIT have been extended to the MIMO case in [10, 11, 12]. Other works, such as [13, 14, 15, 16, 17, 18, 19, 20, 21] have covered other related topics about the DoF region of time domain BC.

However, in practical systems like Long Term Evolution (LTE), the system performance loss of Multiuser MIMO (MU-MIMO) is primarily due to CSI measurement and feedback inaccuracy rather than delay [22]. Therefore, assuming the CSI arrives at the transmitter instantaneously, we are interested in the frequency domain BC where the CSI is measured and reported to the transmitter on a per-subband basis. Due to frequency selectivity, constraints on uplink overhead and user distribution in the cell, the quality of CSI reported to the transmitter varies across users and subbands.

The alternating CSIT state ( and 111 is the CSIT state of user in subband . means perfect CSIT for both users; stands for no CSIT for both users; refer to the CSIT states alternating between Perfect/None and None/Perfect.) can be interpreted as two users reporting their CSI in two different subbands. Those unmatched CSIT was shown still useful in benefiting the DoF region in [2]. A sum DoF of is achieved, outperforming that in the case without CSIT. The scheme proposed in [2], called , transmits two private symbols and one common message (to be decoded by both users) in two channel uses (subbands/slots). The key point lies in sending the common message twice in different subbands, so that the two users can decode it in turn due to the alternating CSIT in each subband. With the knowledge of the common message, the private symbols are recovered.

A more general scenario consists in having the channel state changing to and (where and represent the quality of the imperfect CSIT, both ranging from to ). Literature [23] was the first work investigating this issue. A novel transmission strategy integrating Maddah-Ali and Tse (MAT) scheme, ZFBF and FDMA was proposed. Recently, the DoF region found in [23] has been improved by the scheme proposed in [24], which combines scheme, ZFBF and FDMA. It outperforms [23] because no extra channel use is required to decode all the symbols. The DoF region in the alternating scenario has been conversed in our conference paper [25]. The optimal DoF region was interpreted as a weighted sum of the DoF region in the CSIT state , and . The weights are functions of the CSIT qualities of the two users, revealing an equivalence between the CSIT quality and the fraction of time when the CSIT is perfect as in [2].

So far, the literature addressing the problem of frequency domain BC (or time domain BC without delayed CSIT) focuses on two subbands and assumes that the CSIT states alternate. This assumption is relatively optimistic as in a more realistic wireless communication framework two users may be scheduled simultaneously on multiple subbands. The channels in different subbands may have weak correlation due to the frequency selectivity. The qualities of the CSIT can also vary across users and subbands. We aim at understanding whether the multiple and arbitrary CSIT state can synergistically boost the DoF region. In this paper, we generalize our results of [25] to an -subband scenario with arbitrary values of the CSIT qualities of both users (see Figure 1). In particular, we highlight the main contributions as follows:

1. We derive a tight outer-bound to the DoF region in the -subband frequency correlated MISO BC with arbitrary values of CSIT qualities. It is shown to be a function of the minimum average CSIT quality between the two users. The converse relies on the upper-bound in [26], the extremal inequality [27] and Lemma 1 in [3].

2. The DoF region is interpreted as the weighted sum of the DoF region in the subchannels with state , and , after we decompose the subbands into subchannels according to the qualities of the imperfect CSIT. The weights refer to the fraction of channel use of each type of the subchannels. For a given average CSIT quality but different distributions of the quality in each subband, we find the DoF region remains unchanged but the compositions of the region are varying. Besides, we find a similar expression of the DoF region as in [2], if we interpret the average CSIT quality as the fraction of channel use where the CSIT is perfect. This weighted-sum interpretation also provides an instructive insight into the achievable scheme.

3. By identifying the sub-optimality in the scheme proposed in [23] and the optimality of the scheme in [24] for a 2-subband scenario, we propose the optimal transmission strategy achieving the outer-bound of the DoF region in a -subband scenario with ( and are the qualities of user 1 and user 2 respectively in subband ). Also, we extend this scheme to the -subband scenario with . The key point lies in generating multiple common messages and sending them twice such that the two users can recover them alternatively and decode the private symbols afterwards.

4. Following the footsteps of the construction of the optimal scheme in the case with , we design an optimal transmission strategy for the -subband scenario with .

The rest of this paper is organized as follows. The system model is introduced in Section II, where the main results are also included. The converse of the DoF region is provided in Section III. A weighted-sum interpretation of the optimal DoF region is derived in Section IV. In Section V, by analyzing the achievability in the two-subband scenario, the optimal transmission strategy for -subband with is designed. In Section VI, we build the transmission strategy for -subband with . Section VII concludes the paper.

The following notations are used throughout the paper. Bold lower case letters stand for vectors whereas a symbol not in bold font represents a scalar. and represent the transpose and conjugate transpose of a matrix or vector respectively. denotes the orthogonal space of the channel vector . refers to the expectation of a random variable, vector or matrix. is the norm of a vector. refers to the set , if , otherwise . represents the cardinality of set , which equals to . If is a scalar, is the absolute value of . corresponds to , where is SNR throughout the paper and logarithms are in base .

## Ii System Model and Main Results

### Ii-a Frequency domain two-user MISO BC

In this contribution, we consider a system as shown in Figure 1, which has a transmitter with two antennas and two users each with a single antenna. Denoting the transmit signal as subject to , the observations at user 1 and 2, and respectively, are given by

 yj= hHjxj+ϵj1, (1) zj= gHjxj+ϵj2, (2)

where . and are unit power AWGN noise. and , both with unit norm, are respectively the CSI of user 1 and user 2 in subband . The CSI are i.i.d across users and subbands. In this contribution, the transmit signal can be made up of three kinds of messages:

• Common message I, denoted as hereafter, is broadcast to both users in subband . They should be recovered by both users, but can be intended exclusively for user 1 or user 2;

• Common message II, denoted as hereafter, should be recovered by both users, but can be intended exclusively for user 1 or user 2. Unlike , is broadcast twice, i.e. once in the subbands where the quality of CSIT of user 1 is higher than that of user 2, and once in the subbands where the quality of CSIT of user 2 is higher than that of user 1;

• Private message, is intended for one user only, namely for user 1 and for user 2 in subband .

### Ii-B CSI Feedback Model

Classically, in Frequency Division Duplexing (FDD), each user estimates their CSI in the specified subband using pilot and the estimated CSI is quantized and reported to the transmitter via a rate-limited link. In Time Division Duplexing (TDD), CSI is measured on the uplink and used in the downlink assuming channel reciprocity. We assume a general setup where the transmitter obtains the CSI instantaneously, but with imperfectness, due to the estimation error and/or finite rate in the feedback link.

Denoting and as the imperfect CSI of user 1 and user 2 in subband respectively, the CSI of user 1 and user 2 can be respectively modeled as

 hj=^hj+~hj,gj=^gj+~gj,j=1⋯L, (3)

where and are the corresponding error vectors, respectively with the covariance matrix and . and are respectively independent of and . The norm of and scale as at infinite SNR.

We employ the notation to represent the CSI of both users in subband . Similarly, is the set of the imperfect CSI, refers to the set of the CSI errors and . and respectively refer to sets of the imperfect CSI and the CSI error of user 1 while and are similarly defined. In addition, is available at both the transmitter side and the receiver side. and are only perfectly known by user 1 and user 2 respectively.

To investigate the impact of the imperfect CSIT on the DoF region, we assume that the variance of each entry in the error vector exponentially scales with SNR as in [3, 4, 6, 7, 8, 9, 10, 11, 12, 19, 21, 28, 23, 24, 25], namely and . and are respectively interpreted as the quality of the CSIT of user 1 and user 2 in subband , given as follows

 aj=limP→∞−logσ2j1logP,bj=limP→∞−logσ2j2logP. (4)

and vary within the range of . (resp. ) is equivalent to perfect CSIT because the full DoF region can be achieved by simply doing ZFBF. (resp. ) is equivalent to no CSIT because it means that the variance of the CSI error scales as , such that the imperfect CSIT cannot benefit the DoF when doing ZFBF. Besides, and vary across all the subbands. It is important to note the following quantities

 E[|hHj^h⊥j|2]= E[|(^hj+~hj)H^h⊥j|2] (5) = E[|~hHj^h⊥j|2] (6) = E[~hHj^h⊥j^h⊥Hj~hj]∼P−aj. (7)

as they are frequently used in the achievable schemes in Section V and VI. Similarly, we have .

It is worth noting that the CSIT pattern in Figure 1 is applicable to time domain. Specifically, the CSI report from each user arrives at the transmitter without latency, but it is imperfect due to the estimation error and/or finite rate in the feedback link. As the location of the users and their channel condition changes with time, the CSIT quality varies across users and transmission time-slots.

### Ii-C DoF Definition

Making use of the same notation as in [29] and [30], a rate pair is said to be achievable in an -subband BC with arbitrary imperfect CSIT qualities if there exists a code sequence such that

• Codebook construction: There is one message set for each user. To be specific, for user 1 is uniformly distributed in the set and , intended for user 2, is similarly distributed in the set .

• Encoding: The encoder randomly chooses a message from and generate according to the probability . At the same time with the probability is generated as a function of which is randomly chosen from . Finally, the codeword is generated with the probability .

• Decoding: Receiver 1 wishes to decode and declares message is sent if it is the unique message such that and are jointly typical. Similarly, receiver 2 declares message is sent if it is the unique message such that and are jointly typical. Otherwise, an error will occur. By the Law of Large Numbers, we have when .

The capacity region, , is formed by all the achievable rate pairs. The DoF region, , is accordingly defined on a per-channel-use basis as follows

 D≜{(d1,d2)|∀(w1,w2)∈N2,∀(R1,R2)∈C,w1d1+w2d2≤limP→∞supw1R1+w2R2rlogP}, (8)

where is the channel uses actually employed to achieve the rate pair .

### Ii-D Problem Model

The average CSIT quality of user 1 and user 2 are respectively expressed as and . Without the loss of generality, in the rest of this paper, we consider in subband 1 to () and for the remaining subbands. For convenience, we denote if (namely for ) while if (namely for ). Then, we have and .

For any positive integer , we define two classes of problems as follows

###### Definition 1.

Problem: Achieve the optimal DoF region in a scenario such that .

###### Definition 2.

Problem: Achieve the optimal DoF region in a scenario such that . Note that for (resp. ), it is called (resp. ) Problem hereafter.

A problem considers the general -subband scenario with . Specifically, if there exists a subset of the subbands, denoted as , such that , the problem can be solved as a combination of a and a problems. If no such subset is found, the problem considers the most complicated -subband scenario with . For instance, when and , there generally exist two possible CSIT patterns: 1) and ; 2) and . The first case refers to two problems and the optimal scheme is obtained by performing the solution to problem twice (separately and independently in subband 1 and 2). However, for the second case, the transmitted signal in each subband is correlated to each other (see Section V-B). Similarly, a problem considers the general -subband scenario with . In other words, for a and a problem, the transmitted signals vary according the actual CSIT quality pattern (formed by the frequency-user grid as shown in Figure 1). More details of the achievabilities in a problem and a are shown in Section V and VI respectively.

### Ii-E Main Results

###### Theorem 1.

The optimal DoF region, , in a -subband frequency correlated BC with imperfect varying CSIT is specified by

 d1+d2≤ 1+1Lmin(L∑j=1aj,L∑j=1bj), (9) d1≤ 1, (10) d2≤ 1, (11)

where and are respectively the quality of the CSIT of user 1 and user 2 in subband and can be any integer values.

Note that the optimal DoF region is bounded by the minimum average CSIT quality between user 1 and user 2. This result gives an affirmative answer to the conjecture in [2] that the sum DoF is in a two-user MISO BC with fixed CSIT state. The converse is provided in Section III. The achievability is discussed in Section V and VI, for () and () problem respectively. The following corollary provides an instructive insight into the formation of the optimal DoF region.

###### Corollary 1.

The optimal DoF region in the frequency correlated BC with imperfect CSIT can be interpreted as a weighted sum of three basis optimal DoF regions

 D=1L(¯r¯D+^r^D+~r~D), (12)

where , and refer to the optimal DoF region for a CSIT state of , and respectively and , and are the corresponding weights, given as

 ¯D: d1≤1,d2≤1, (13) ^D: d1+d2≤32,d1≤1,d2≤1, (14) ~D: d1+d2≤1, (15) ¯r= L∑j=1min(aj,bj), (16) ^r= 2min(l∑j=1q+j,L∑j=l+1q−j), (17) ~r= L−¯r−^r. (18)

## Iii Converse of Theorem 1

The objective of this section is to provide the converse of Theorem 1. Before going into the details, we highlight the key ingredients in the derivation as

• Nair-Gamal bound [26]: provides an upper-bound to the DoF region in a general BC;

• Extremal Inequality: maximizes a weighted difference of two different entropies;

• Lemma 1 in [3]: upper- and lower-bound the entropy.

Let us revisit the converse in previous works. In [15], the DoF region in the BC without CSIT is upper-bounded by considering one user’s observation is degraded compared to the other’s. In the BC with delayed CSIT [1][3][5], the outer-bound is obtained through the genie-aided model where one user’s observation is provided to the other, thus establishing a physically degraded BC to remove the delayed CSIT.

However, in this contribution, those methods are not adopted since the transmitter does not have delayed CSIT and the BC with imperfect CSIT cannot be simply considered as a degraded BC. Instead, we follow the assumption in [31]: We first consider that user 2 knows the message intended for user 1, which leads to an outer-bound denoted by ; Then by assuming that user 1 knows user 2’s desired message, we can have another region . The final DoF outer-bound results from the intersection of them, i.e. . This assumption is consistent with the derivation in Korner-Marton bound (Theorem 5 and Appendix I in [32]) and Nair-Gamal bound (Theorem 2.1 and 3.1 in [26], proof given in the Appendix of Lecture Notes 9 in [30]). Both of these two bounds provide an outer-bound to the general discrete memoryless broadcast channel and Nair-Gamal bound is said to be in general contained in Korner-Marton bound [26]. As a consequence, we aim at finding the following bounds

 R1+R2 ≤I(U;Y|V)+I(V;Z), (19) R1+R2 ≤I(U;Y)+I(V;Z|U). (20)

The key challenge lies in finding the auxiliary variables and .

Assuming user 2 has the knowledge of and according to Fano’s Inequality, we have

 nR1 ≤I(M1;Yn1|^Sn1,~Hn1) (21) =I(M1;Yn1|^Sn1,~Hn1,~Gn1) (22) nR2 ≤I(M2;Zn1|M1,^Sn1,~Gn1) (23) =I(M2;Zn1|M1,^Sn1,~Gn1,~Hn1) (24) n(R1+R2) ≤I(M1;Yn1|^Sn1,~Hn1,~Gn1)+I(M2;Zn1|M1,^Sn1,~Gn1,~Hn1) (25) =I(M1;Yn1|S)+I(M2;Zn1|M1,S), (26)

where (22) follows the fact that forms a Markov chain such that is independent of conditioned on . (24) follows similarly. In (26), is replaced by . (26) is bounded by

 n(R1+R2) ≤n∑j=1{I(M1,Yj−11,Znj+1,^Sn1;Yj|Sn1)+I(M2,Yj−11,Znj+1,^Sn1;Zj|M1,Sn1,Yj−11,Znj+1)} (27) =n∑j=1{I(Uj;Yj|Sn1)+I(Vj;Zj|Uj,Sn1)}. (28)

The derivation of (27) is provided in the Appendix. Now, we have found the auxiliary variables as

 Uj≜{M1,Yj−11,Znj+1,^Sn1}, (29) Vj≜{M2,Yj−11,Znj+1,^Sn1}. (30)

Continuing deriving (28), we have

 n(R1+R2)≤ n∑j=1h(Yj|Sn1)≤logP−h(Yj|Sn1,Uj)+h(Zj|Sn1,Uj)−h(Zj|Sn1,Uj,Vj)≤o(logP) (31) ≤ nlogP+n∑j=1{h(Zj|Sn1,Uj)−h(Yj|Sn1,Uj)}. (32)

Next, we focus on the terms in the summation of (32) and upper-bound them using a similar derivation as in [3]. For convenience, we give up the index . Consequently,

 h(Z|S,U)−h(Y|S,U)≤ maxPUPx|U{h(Z|U,S)−h(Y|U,S)} (33) ≤ maxPUEU{maxPx|Uh(Z|U=U∗,S)−h(Y|U=U∗,S)} (34) ≤ maxPUEU{maxPx|UES|U[h(Z|U=U∗,S=S∗)−h(Y|U=U∗,S=S∗)]} (35) = maxPUEU{maxPx|UES|^S[h(gHx+ϵ2|U=U∗)−h(hHx+ϵ1|U=U∗)]} (36) ≤ maxPUEU{max0⪯C,tr(C)≤Pmax\lx@stackrelPx|UCov(x|U)⪯CES|^S[h(gHx+ϵ2|U=U∗)−h(hHx+ϵ1|U=U∗)]} (37) ≤ maxPUEU{max0⪯C,tr(C)≤PES|^S[log(1+gHKg)−log(1+hHKh)]}, (38) ≤ E^S{max0⪯K,tr(K)≤PES|^S[log(1+gHKg)−log(1+hHKh)]}, (39)

where is the covariance matrix of (i.e. ) and is a semi-definite matrix, which is regarded as the constraint of . (38) is derived according to the fact 1) forms a Markov chain so that is independent of conditioned on ; 2) With a constrained covariance, a Gaussian distributed conditioned on is the optimal solution to the maximization of the weighted difference in (38) for any positive semi-definite , based on the proof of Corollary 6 in [27].

Using Lemma 1 in [3], we can respectively upper- and lower-bound the first and second terms in (39) as

 ES|^Slog(1+gHKg)≤ log(1+λ1E[||^g||2])+o(1), (40) ES|^Slog(1+hHKh)≥ log(1+e−γλ1E[||~h||2])+o(1), (41)

where is a constant, is the largest eigen-value of the covariance matrix . Substituting the terms in (39) with (40) and (41), we can have the th term in the summation of (32) upper-bounded by

 h(Zj|Sn1,Uj)−h(Yj|Sn1,Uj)≤ log1+λ1E[||^gj||2]1+e−γλ1E[||~hj||2] (42) ≈ ajlogP. (43)

Applying (43) to all the terms in (32), the sum rate is upper-bounded by

 n(R1+R2)≤ nlogP+n∑j=1ajlogP (44) R1+R2≤ logP+1nn∑j=1ajlogP. (45)

When tends to infinity, the -subband scenario defined in Figure 1 repeats infinite times. Consequently, the CSIT state in each subband appears times and (45) can be rewritten as

 R1+R2≤logP+1nL∑j=1nLajlogP=logP+1LL∑j=1ajlogP, (46)

Accordingly, the DoF region is specified as follows

 D1:d1+d2≤1+ae=1+∑Lj=1ajL. (47)

Switching the role of each user results in the sum rate and DoF region specified as

 n(R1+R2)≤ n∑j=1{I(Uj;Yj|Sn1,Vj)+I(Vj;Zj|Sn1)} (48) ≤ nlogP+n∑j=1h(Yj|Sn1,Vj)−h(Zj|Sn1,Vj) (49) ≤ nlogP+n∑j=1bjlogP, (50) R1+R2≤ logP+1LL∑j=1bjlogP, (51) D2:d1+d2≤ 1+be=1+∑Lj=1bjL. (52)

Taking the intersection of and results in (9). Together with the single-user constraint, Theorem 1 holds.

## Iv A Weighted-Sum Interpretation of the Optimal DoF Region

In this section, we provide an insight into the formation of the optimal DoF region. According to the particular CSIT setting, each subband is considered as composed of three parallel subchannels with different fraction of channel use. The DoF region of each subchannel has been found in previous work. We will show that the optimal DoF region stated in Theorem 1 can be calculated as a weighted sum of the DoF region of each subchannel.

### Iv-a Intuition: Channel Decomposition

In this part, we decompose the channel in each subband following the intuition that the imperfect CSIT with error variance can be considered as perfect for () channel use (i.e. the transmit power is reduced to ). We can see this by simply sending one private message per user using ZFBF precoding and with power . Since and , both users can recover their private messages subject to noise. Therefore, the rate is achieved per user. As only channel has been used, full DoF region (i.e. and ) is obtained according to (8). This is in fact a generalization of the fact that full DoF region can be obtained if the variance of CSIT error is scaled as [3].

Consequently, a subband with the CSIT error scaling as and for user 1 and 2 respectively, is decomposed as shown in Figure 2. The transmitter is assumed to have perfect knowledge of the CSI of user 1 for channel use while for the remaining channel use, no CSIT of user 1 is available. The same approach is employed for user 2. It results three subchannels, each of which can be interpreted using the same notation as in [2] (, and ).

• : channel, no CSIT of either user, with channel use ;

• : (resp. ) channel, perfect CSIT of user 1 (resp. 2) but no CSIT of use 2 (resp. 1), with channel use (resp. );

• : channel, perfect CSIT of both users, with channel use .

In this way, the original -subband scenario becomes the product of those parallel subchannels. The DoF region is obtained as the weighted-sum of that in each subchannel.

### Iv-B DoF Regions of the Subchannels

We split the rate of each user into three parts, namely and , where represents the rate pair achieved in the subchannel with state , refers to the rate pair achieved in the subchannel with alternating state while is the rate pair achieved in the subchannel with state . The message intended to user 1 and 2 is therefore generated from a set jointly formed by and respectively. The subsets (, and ) are independent of each other for . , and respectively result in the DoF region , and .

#### Iv-B1 Subchannel ¯j

When the transmitter has perfect CSI of both users, the optimal DoF region is expressed as follows

 ¯D: d1≤1,d2≤1, (53)

which can be achieved via ZFBF. The total amount of channel use of the subchannels with state is

 ¯r=L∑j=1min(aj,bj). (54)

#### Iv-B2 Subchannel ^j

In this class of subchannels, the transmitter has perfect knowledge of the CSI of user 1 or (exclusive) user 2. As a reminder, we assume in subband 1 to () and for the remaining subbands. Following the way where channels are decomposed (as in Figure 2), there are in total subchannels, each with channel use and subchannels, each with channel use . Literature [2] provides an optimal DoF region of the alternating scenario, which can be achieved by the simple scheme. This bound is denoted by and expressed as

 ^D: d1+d2≤32,d1≤1,d2≤1. (55)

However, this region is optimal for the alternating scenario where each and subchannel have the same amount of channel use, namely such that . In a general -subband scenario (Definition 1 and 2), this condition does not necessarily hold. Hence, we aim at showing that (55) still optimally bounds the DoF region of the and subchannels. To that end, we further decompose the subchannels in order to find the alternating scenario.

Figure 3 shows an example of the further decomposition. Firstly, subchannel is decomposed into two subchannels, namely and , respectively with fraction of channel use and . In this way, subchannel and form an alternating scenario where and states have equal amount of channel use. Secondly, subchannel is decomposed into two subchannels, namely and , respectively with fraction of channel use and