Opportunistic Network Decoupling With Virtual Full-Duplex Operation in Multi-Source Interfering Relay Networks

# Opportunistic Network Decoupling With Virtual Full-Duplex Operation in Multi-Source Interfering Relay Networks

Won-Yong Shin, Senior Member, IEEE, Vien V. Mai,
Bang Chul Jung Senior Member, IEEE, and Hyun Jong Yang, Member, IEEE
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2054577) and by the Ministry of Science, ICT & Future Planning (MSIP) (2015R1A2A1A15054248). The material in this paper was presented in part at the IEEE International Conference on Communications, Sydney, Australia, June 2014 [38].W.-Y. Shin is with the Department of Computer Science and Engineering, Dankook University, Yongin 448-701, Republic of Korea (E-mail: wyshin@dankook.ac.kr).V. V. Mai was with Dankook University, Yongin 448-701, Republic of Korea. He is now with KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden (E-mail: mienvanmaidt2@gmail.com).B. C. Jung is with the Department of Electronics Engineering, Chungnam National University, Daejeon 305-764, Republic of Korea (E-mail: bcjung@cnu.ac.kr).H. J. Yang (corresponding author) is with the School of Electrical and Computer Engineering, UNIST, Ulsan 689-798, Republic of Korea (E-mail: hjyang@unist.ac.kr).
###### Abstract

We introduce a new achievability scheme, termed opportunistic network decoupling (OND), operating in virtual full-duplex mode. In the scheme, a novel relay scheduling strategy is utilized in the channel with interfering relays, consisting of source–destination pairs and half-duplex relays in-between them. A subset of relays using alternate relaying is opportunistically selected in terms of producing the minimum total interference level, thereby resulting in network decoupling. As our main result, it is shown that under a certain relay scaling condition, the OND protocol achieves degrees of freedom even in the presence of interfering links among relays. Numerical evaluation is also shown to validate the performance of the proposed OND. Our protocol basically operates in a fully distributed fashion along with local channel state information, thereby resulting in a relatively easy implementation.

{keywords}

Degrees of freedom (DoF), half-duplex, interference, channel, opportunistic network decoupling (OND), relay, virtual full-duplex (FD).

## I Introduction

### I-a Previous Work

Interference between wireless links has been taken into account as a critical problem in wireless communication systems. Recently, interference alignment (IA) was proposed for fundamentally solving the interference problem when there are two communication pairs [1]. It was shown in [2] that the IA scheme can achieve the optimal degrees of freedom (DoF), which is equal to , in the -user interference channel with time-varying channel coefficients. Since then, interference management schemes based on IA have been further developed and analyzed in various wireless network environments: multiple-input multiple-output (MIMO) interference networks [3, 4], X networks [5], and cellular networks [6, 7, 8, 9].

On the one hand, following up on these successes for single-hop networks, more recent and emerging work has studied multihop networks with multiple source-destination (S–D) pairs. For the 2-user 2-hop network with 2 relays (referred to as the interference channel), it was shown in [10] that interference neutralization combining with symbol extension achieves the optimal DoF. A more challenging network model is to consider -user two-hop relay-aided interference channels, consisting of source-destination (S–D) pairs and helping relay nodes located in the path between S–D pairs, so-called the channel. Several achievability schemes have been known for the network, but more detailed understanding is still in progress. By applying the result from [11] to the channel, one can show that DoF is achieved by using orthogonalize-and-forward relaying, which completely neutralizes interference at all destinations if is greater than or equal to . Another achievable scheme, called aligned network diagonalization, was introduced in [12] and was shown to achieve the optimal DoF in the channel while tightening the required number of relays. The scheme in [12] is based on the real interference alignment framework [7]. In [10, 12], however, the system model under consideration assumes that there is no interfering signal between relays and the relays are full-duplex. Moreover, in [13], the interference channel with full-duplex relays interfering with each other was characterized and its DoF achievability was shown using aligned interference neutralization.111The idea in [13] was later extended to the 2-user 3-hop network with 4 relays, i.e., the interference channel [14].

On the other hand, there are lots of results on the usefulness of fading in the literature, where one can obtain the multiuser diversity gain in broadcast channels: opportunistic scheduling [15], opportunistic beamforming [16], and random beamforming [17]. Such opportunism can also be fully utilized in multi-cell uplink or downlink networks by using an opportunistic interference alignment strategy [9, 18, 19, 20]. Various scenarios exploiting the multiuser diversity gain have been studied in cooperative networks by applying an opportunistic two-hop relaying protocol [21] and an opportunistic routing [22], and in cognitive radio networks with opportunistic scheduling [24, 23]. In addition, recent results [25, 26] have shown how to utilize the opportunistic gain when there are a large number of channel realizations. More specifically, to amplify signals and cancel interference, the idea of opportunistically pairing complementary channel instances has been studied in interference networks [25] and multi-hop relay networks [26]. In cognitive radio environments [27], opportunistic spectrum sharing was introduced by allowing secondary users to share the radio spectrum originally allocated to primary users via transmit adaptation in space, time, or frequency.

### I-B Main Contributions

In this paper, we study the channel with interfering relays, which can be taken into account as one of practical multi-source interfering relay networks and be regarded as a fundamentally different channel model from the conventional channel in [12]. Then, we introduce an opportunistic network decoupling (OND) protocol that achieves full DoF with comparatively easy implementation under the channel model. This work focuses on the channel with one additional assumption that half-duplex (HD) relays interfere with each other, which is a more feasible scenario. The scheme adopts the notion of the multiuser diversity gain for performing interference management over two hops. More precisely, in our scheme, a scheduling strategy is presented in time-division duplexing (TDD) two-hop environments with time-invariant channel coefficients, where a subset of relays is opportunistically selected in terms of producing the minimum total interference level. To improve the spectral efficiency, the alternate relaying protocol in [28, 29, 30] is employed with a modification, which eventually enables our system to operate in virtual full-duplex mode. As our main result, it turns out that in a high signal-to-noise ratio (SNR) regime, the OND protocol asymptotically achieves the min-cut upper bound of DoF even in the presence of inter-relay interference and half-duplex assumption, provided the number of relays, , scales faster than , which is the minimum number of relays required to guarantee our achievability result. Numerical evaluation also indicates that the OND protocol has higher sum-rates than those of other relaying methods under realistic network conditions (e.g., finite and SNR) since the inter-relay interference is significantly reduced owing to the opportunistic gain. For comparison, the OND scheme without alternate relaying and the max-min SNR scheme are also shown as baseline schemes. Note that our protocol basically operates with local channel state information (CSI) at the transmitter and thus is suitable for distributed/decentralized networks.

Our main contributions are fourfold as follows:

• In the channel with interfering relays, we introduce a new achievability scheme, termed OND with virtual full-duplex operation.

• Under the channel model, we completely analyze the optimal DoF, the required relay scaling condition, and the decaying rate of the interference level, where the OND scheme is shown to approach the min-cut upper bound on the DoF.

• Our achievability result (i.e., the derived DoF and relay scaling law) is validated via numerical evaluation.

• We perform extensive computer simulations with other baseline schemes.

### I-C Organization

The rest of this paper is organized as follows. In Section II, we describe the system and channel models. In Section III, the proposed OND scheme is specified and its lower bound on the DoF is analyzed. Section IV shows an upper bound on the DoF. Numerical results of the proposed OND scheme are provided in Section V. Finally, we summarize the paper with some concluding remarks in Section VI.

### I-D Notations

Throughout this paper, , , and indicate the field of complex numbers, the statistical expectation, and the ceiling operation, respectively. Unless otherwise stated, all logarithms are assumed to be to the base 2.

Moreover, TABLE I summarizes the notations used throughout this paper. Some notations will be more precisely defined in the following sections, where we introduce our channel model and achievability results.

## Ii System and Channel Models

As one of two-hop cooperative scenarios, we consider the channel model with interfering relays, which fits into the case where each S–D pair is geographically far apart and/or experiences strong shadowing (thus requiring the response to a huge challenge for achieving the target spectral efficiency). In the channel model, it is thus assumed that each source transmits its own message to the corresponding destination only through one of relays, and thus there is no direct path between an S–D pair. Note that unlike the conventional channel, relay nodes are assumed to interfere with each other in our model. There are S–D pairs, where each receiver is the destination of exactly one source node and is interested only in traffic demands of the source. As in the typical cooperative relaying setup, relay nodes are located in the path between S–D pairs so as to help to reduce path-loss attenuations.

Suppose that each node is equipped with a single transmit antenna. Each relay node is assumed to operate in half-duplex mode and to fully decode, re-encode, and retransmit the source message i.e., decode-and-forward protocol is taken into account. We assume that each node (either a source or a relay) has an average transmit power constraint . Unlike the work in [10, 12], relays are assumed to interfere with each other.222If we can cancel the interfering signals among multiple relays, then the existing achievable scheme of the channel can also be applied here. To improve the spectral efficiency, the alternate relaying protocol in [28, 29, 30] is employed with a modification. With alternate relaying, each selected relay node toggles between the transmit and listen modes for alternate time slots of message transmission of the sources. If is sufficiently large, then it is possible to exploit the channel randomness for each hop and thus to obtain the opportunistic gain in multiuser environments. In this work, we do not assume the use of any sophisticated multiuser detection schemes at each receiver (either a relay or a destination node), thereby resulting in an easier implementation.

Now, let us turn to channel modeling. Let , , and denote the th source, the corresponding th destination, and the th relay node, respectively, where and . The terms denote the channel coefficients from to and from to , corresponding to the first and second hops, respectively. The term indicates the channel coefficient between two relays and . All the channels are assumed to be Rayleigh, having zero-mean and unit variance, and to be independent across different , , , and hop index . We assume the block-fading model, i.e., the channels are constant during one block (e.g., frame), consisting of one scheduling time slot and data transmission time slots, and changes to a new independent value for every block.

## Iii Achievability Results

In this section, we describe the OND protocol, operating in virtual full-duplex mode, in the channel with interfering relays. Then, its performance is analyzed in terms of achievable DoF along with a certain relay scaling condition. The decaying rate of the interference level is also analyzed. In addition, the OND protocol with no alternate relaying and its achievability result are shown for comparison.

### Iii-a OND in the K×N×K Channel With Interfering Relays

In this subsection, we introduce an OND protocol as the achievable scheme to guarantee the optimal DoF of the channel with inter-relay interference, where relay nodes among candidates are opportunistically selected for data forwarding in the sense of producing a sufficiently small amount of interference level. The proposed scheme is basically performed by utilizing the channel reciprocity of TDD systems.

Suppose that and denote the indices of two relays communicating with the th S–D pair for . In this case, without loss of generality, assuming that the number of data transmission time slots, , is an odd number, the specific steps of each node during one block are described as follows:

• Time slot 1: Sources transmit their first encoded symbols , where represents the th transmit symbol of the th source node.333For notational convenience, we use scalar notation instead of vector notation for each coding block from source nodes, but the size of each symbol is assumed to be sufficiently long to achieve the Shannon-theoretic channel capacity. A set of selected relay nodes, , operating in receive mode at each odd time slot, listens to (note that a relay selection strategy will be specified later). Other relay nodes and destinations remain idle.

• Time slot 2: The sources transmit their encoded symbols . The selected relays in the set forward their first re-encoded symbols to the corresponding destinations. If the relays in successfully decode the corresponding symbols, then is the same as . Another set of selected relay nodes, , operating in receive mode at each even time slot, listens to and decodes while being interfered with by . The destinations receive and decode from . The remaining relays keep idle.

• Time slot 3: The sources transmit their encoded symbols . The relays forward their re-encoded symbols to the corresponding destinations. Another relays in receive and decode while being interfered with by . The destinations receive and decode from . The remaining relays keep idle.

• The processes in time slots 2 and 3 are repeated to the th time slot.

• Time slot : The relays in forward their re-encoded symbols to the corresponding destinations. The sources and the other relays remain idle.

At each odd time slot (i.e., ), let us consider the received signal at each selected relay for the first hop and the received signal at each destination for the second hop, respectively.

For the first hop (Phase 1), the received signal at is given by

 y(1)π1(i)(l)= K∑k=1h(1)π1(i)kx(1)k(l)+K∑n=1h(r)π1(i)π2(n)x(2)π2(n)(l−1)+z(1)π1(i)(l), (1)

where and are the th transmit symbol of and the th transmit symbol of , respectively. As addressed earlier, if relay successfully decodes the received symbol, then it follows that . The received signal at is corrupted by the independent and identically distributed (i.i.d.) and circularly symmetric complex additive white Gaussian noise (AWGN) having zero-mean and variance . Note that the second term in the right-hand side (RHS) of (1) indicates the inter-relay interference, which occurs when the relays in the set , operating in receive mode, listen to the sources, the relays are interfered with by the other set , operating in transmit mode. Note that when , relays have no symbols to transmit, and the second term in the RHS of (1) becomes zero. Similarly when , sources do not transmit symbols, and the first term in the RHS of (1) becomes zero.

For the second hop (Phase 2), assuming that the selected relay nodes transmit their data packets simultaneously, the received signal at is given by

 y(2)k(l)=K∑n=1h(2)kπ2(n)x(2)π2(n)(l−1)+z(2)k(l), (2)

where is the i.i.d. AWGN having zero-mean and variance . We also note that when , there are no signals from relays.

Likewise, at each even time slot (i.e., ), the received signals at and (i.e., the first and second hops) are given by

 y(1)π2(i)(l)= K∑k=1h(1)π2(i)kx(1)k(l)+K∑n=1h(r)π2(i)π1(n)x(2)π1(n)(l−1)+z(1)π2(i)(l)

and

 y(2)k(l)=K∑n=1h(2)kπ1(n)x(2)π1(n)(l−1)+z(2)k(l),

respectively. The illustration of the aforementioned OND protocol is geographically shown in Fig. 1 (two terms and are specified later in the following relay selection steps).

Now, let us describe how to choose two types of relay sets, and , among relay nodes, where is sufficiently large (the minimum required to guarantee the DoF optimality will be analyzed in Section III-B).

#### Iii-A1 Step 1 (The First Relay Set Selection)

Let us first focus on selecting the set , operating in receive and transmit modes in odd and even time slots, respectively. For every scheduling period, it is possible for relay to obtain all the channel coefficients and by using a pilot signaling sent from all of the source and destination nodes due to the channel reciprocity before data transmission, where and (note that this is our local CSI assumption). When is assumed to serve the th S–D pair , it then examines both i) how much interference is received from the other sources and ii) how much interference is generated by itself to the other destinations, by computing the following scheduling metric :

 ~Li,k=K∑m=1m≠k(∣∣h(1)im∣∣2+∣∣h(2)mi∣∣2), (3)

where and . We remark that the first term in (3) denotes the sum of interference power received at for the first hop (i.e., Phase 1). On the other hand, the second term indicates the sum of interference power generating at , which can be interpreted as the leakage of interference to the receivers expect for the corresponding destination, for the second hop (i.e., Phase 2) under the same assumption.

Suppose that a short duration CTS (Clear to Send) message is transmitted by the destination who finds its desired relay node (or the master destination). Then according to the computed metrics in (3), a timer-based method can be used for the relay selection similarly as in [31].444The reception of a CTS message, which is transmitted from a certain destination, triggers the initial timing process at each relay. Therefore, no explicit timing synchronization protocol is required among the relays [31, 32]. Moreover, it is worth noting that the overhead of relay selection is a small fraction of one transmission block with small collision probability [31]. Since our relay selection procedure is performed sequentially over all the S-D pairs and the already selected relays for a certain S-D pair are not allowed to take part in the selection process for another S-D pair, the collision probability is thus at most times that of the single S-D pair case [31]. Note that the method based on the timer is considerably suitable in distributed systems in the sense that information exchange among all the relay nodes can be minimized. At the beginning of every scheduling period, the relay computes the set of scheduling metrics, , and then starts its own timer with initial values, which can be set to be proportional to the metrics.555To avoid a situation such that a malicious relay deliberately sets its timer to a smaller value so as to win the chance, prior to the relay selection process, a secret key may be shared among legitimate nodes including relays. If a malicious relay who did not share the key wants to participate in communication, then one can neglect his/her message (e.g., RTS (Request to Send) message). Thus, there exist metrics over the whole relay nodes, and we need to compare them so as to determine who will be selected. The timer of the relay with the least one among metrics will expire first, where and . The relay then transmits a short duration RTS message, signaling its presence, to the other relays, where each RTS message is composed of bits to indicate which S–D pair the relay wants to serve. Thereafter, the relay is first selected to forward the th S–D pair’s packet. All the other relays are in listen mode while waiting for their timer to be set to zero (i.e., to expire). At the stage of deciding who will send the second RTS message, it is assumed that the other relays are not allowed to communicate with the th S–D pair, and thus the associated metrics are discarded with respect to timer operation. If another relay has an opportunity to send the second RTS message of bits in order to declare its presence, then it is selected to communicate with the corresponding S–D pair. When such RTS messages, consisting of at most bits, are sent out in consecutive order, i.e., the set of relays, , is chosen, the timer-based algorithm for the first relay set selection terminates, yielding no RTS collision with high probability. We remark that when (i.e., the single S–D pair case), relay nodes are arbitrarily chosen as the first relay set since there is no interference in this step.

#### Iii-A2 Step 2 (The Second Relay Set Selection)

Now let us turn to choosing the set of relay nodes (among candidates), , operating in receive and transmit modes in even and odd time slots, respectively. Using RTS messages broadcasted from the relay nodes in the set , it is possible for relay node to compute the sum of inter-relay interference power generated from the relays in , denoted by . When is again assumed to serve the th S–D pair , it examines both i) how much interference is received from the undesired sources and the selected relays in the set for the first hop and ii) how much interference is generated by itself to the other destinations by computing the following metric , termed total interference level (TIL):

 LΠ2i,k =~Li,k+K∑k=1∣∣h(r)iπ1(k)∣∣2 =K∑m=1m≠k(∣∣h(1)im∣∣2+∣∣h(2)mi∣∣2)+K∑k=1∣∣h(r)iπ1(k)∣∣2, (4)

where and . We note that Steps 1 and 2 cannot be exchangeable due to the fact that the inter-relay interference term is measured after determining the first relay set . If the relay set selection order is switched, then the metric TIL in (4) will not be available.

According to the computed TIL , we also apply the timer-based method used in Step 1 for the second relay set selection. The relay computes the set of TILs, , and then starts its timer with initial values, proportional to the TILs. Thus, we need to compare TIL metrics over the relay nodes in the set in order to determine who will be selected as the second relay set. The rest of the relay set selection protocol (i.e., RTS message exchange among relay nodes) almost follows the same line as that of Step 1. The timer-based algorithm for the second relay set selection terminates when RTS messages are sent out in consecutive order. Then, relay nodes having a sufficiently small amount of TIL are selected as the second relay set .

###### Remark 1

Owing to the channel reciprocity of TDD systems, the sum of inter-relay interference power received at any relay , , also turns out to be sufficiently small when is large. That is, it is also guaranteed that selected relays in the set have a sufficiently small amount of TIL.

###### Remark 2

The overhead of each scheduling time slot (i.e., the total number of bits required for exchanging RTS messages among the relay nodes) can be made arbitrarily small, compared to one transmission block. From the fact that RTS messages, consisting of at most bits, are sent out in each relay set selection step, only bit transmission could suffice.

#### Iii-A3 Step 3 (Data Transmission)

The selected relays request data transmission to their desired source nodes. Each source () then starts to transmit data to the corresponding destination () via one of its two relay nodes alternately ( or ), which was specified earlier. If the TILs of the selected relays are arbitrarily small, then i) the associated undesired source–relay and relay–destination channel links and ii) the inter-relay channel links are all in deep fade. In Section III-B, we will show that it is possible to choose such relays with the help of the multiuser diversity gain.

At the receiver side, each relay or destination detects the signal sent from its desired transmitter, while simply treating interference as Gaussian noise. Thus, no multiuser detection is performed at each receiver, thereby resulting in an easier implementation.

### Iii-B Analysis of a Lower Bound on the DoF

In this subsection, using the scaling argument bridging between the number of relays, , and the received SNR (refer to [9, 18, 19, 20] for the details), we shall show 1) the lower bound on the DoF of the channel with interfering relays as increases and 2) the minimum required to guarantee the achievability result. The total number of DoF, denoted by , is defined as [2]

 DoFtotal=K∑k=1(limsnr→∞Tk(snr)logsnr),

where denotes the transmission rate of source . Using the OND framework in the channel with interfering relays where transmission slots per block are used, the achievable is lower-bounded by

 DoFtotal≥L−1LK∑k=12∑s=1⎛⎜ ⎜⎝limsnr→∞12log(1+min{sinr(1)πs(k),% sinr(2)k,πs(k)})logsnr⎞⎟ ⎟⎠, (5)

where denotes the received signal-to-interference-and-noise ratio (SINR) at the relay and denotes the received SINR at the destination when the relay transmits the desired signal ( and ). More specifically, the above SINRs can be formally expressed as666Note that at the first time slot for the relays , the third term in the denominator of (i.e., the inter-relay interference term) becomes zero.

 sinr(1)πs(i)=P∣∣h(1)πs(i)i∣∣2N0+P∑Kk=1k≠i∣∣h(1)πs(i)k∣∣2+P∑Kk=1∣∣h(r)πs(i),π~s(k)∣∣2 sinr(2)i,πs(i)=P∣∣h(2)iπs(i)∣∣21+P∑Kk=1k≠i∣∣h(2)iπs(k)∣∣2,

where the second term in the denominator of indicates the interference power at relay received from the sources while the third term indicates the inter-relay interference, and the second term in the denominator of indicates the interference power at the destination received from the active relays. Here, , i.e., if , and vice versa.

We focus on the first relay set ’s perspective to examine the received SINR values according to each time slot. Let us first denote for . For the first hop, at time slot (i.e., each odd time slot), , the received at is lower-bounded by

 sinr(1)π1(i) ≥snr∣∣h(1)π1(i)i∣∣21+snr∑Kk=1k≠i(∣∣h(1)π1(i)k∣∣2+∣∣h(2)kπ1(i)∣∣2)+snr∑Kk=1∣∣h(r)π1(i),π2(k)∣∣2 =snr∣∣h(1)π1(i)i∣∣21+snr(~Lπ1(i),i+∑Kk=1∣∣h(r)π1(i),π2(k)∣∣2) =snr∣∣h(1)π1(i)i∣∣21+snrLΠ1π1(i),i ≥snr∣∣h(1)π1(i)i∣∣21+snr∑Ki=1LΠ1π1(i),i, (6)

where indicates the scheduling metric in (3) when is assumed to serve the th S–D pair (, ). For the second hop, at time slot (i.e., each even time slot), , the received at is lower-bounded by

 sinr(2)i,π1(i) ≥snr∣∣h(2)iπ1(i)∣∣21+snr∑Ki=1∑Kk=1k≠i∣∣h(2)iπ1(k)∣∣2 ≥snr∣∣h(2)iπ1(i)∣∣21+snr∑Ki=1∑Kk=1k≠i(∣∣h(1)π1(i)k∣∣2+∣∣h(2)kπ1(i)∣∣2) =snr∣∣h(2)iπ1(i)∣∣21+snr∑Ki=1~Lπ1(i),i ≥snr∣∣h(2)iπ1(i)∣∣21+snr∑Ki=1LΠ1π1(i),i, (7)

where the second inequality holds due to the channel reciprocity. The term in the denominator of (6) and (7) needs to scale as , i.e., , so that both and scale as with increasing SNR, which eventually enables to achieve the DoF of per S–D pair from (5).777We use the following notation: i) means that there exist constant and such that for all . ii) if . iii) means that  [33]. Even if such a bounding technique in (6) and (7) leads to a loose lower bound on the SINR, it is sufficient to prove our achievability result in terms of DoF and relay scaling law.

Now, let us turn to the second relay set . Similarly as in (6), for the first hop, at time slot , , the received at is lower-bounded by

 sinr(1)π2(i)≥snr∣∣h(1)π2(i)i∣∣21+snr∑Ki=1LΠ2π2(i),i, (8)

where indicates the TIL in (4) when is assumed to serve the th S–D pair (, ). For the second hop, at time slot , , the received at can also be lower-bounded by

 sinr(2)i,π2(i)≥snr∣∣h(2)iπ2(i)∣∣21+snr∑Ki=1LΠ2π2(i),i. (9)

The next step is thus to characterize the three metrics , , and ( and ) and their cumulative density functions (CDFs) in the channel with interfering relays, which is used to analyze the lower bound on the DoF and the required relay scaling law in the model under consideration. Since it is obvious to show that the CDF of is identical to that of , we focus only on the characterization of . The scheduling metric follows the chi-square distribution with degrees of freedom since it represents the sum of i.i.d. chi-square random variables with 2 degrees of freedom. Similarly, the TIL follows the chi-square distribution with degrees of freedom. The CDFs of the two metrics and are given by

 F~L(ℓ)=γ(2K−2,ℓ/2)Γ(2K−2) (10) FL(ℓ)=γ(3K−2,ℓ/2)Γ(3K−2), (11)

respectively, where is the Gamma function and is the lower incomplete Gamma function [34, eqn. (8.310.1)]. We start from the following lemma.

###### Lemma 1

For any , the CDFs of the random variables and in (10) and (11) are lower-bounded by and , respectively, where

 C1=e−12−(2K−2)Γ(2K−1) (12) C2=e−12−(3K−2)Γ(3K−1), (13)

and is the Gamma function.

{proof}

The detailed proof of this argument is omitted here since it essentially follows the similar line to the proof of [9, Lemma 1] with a slight modification.

In the following theorem, we establish our first main result by deriving the lower bound on the total DoF in the channel with interfering relays.

###### Theorem 1

Suppose that the OND scheme with alternate relaying is used for the channel with interfering relays. Then, for data transmission time slots,

 DoFtotal≥(L−1)KL

is achievable if .

{proof}

From (5)–(9), the OND scheme achieves provided that the two values and are less than or equal to some constant , independent of SNR, for all S–D pairs. Then, a lower bound on the achievable is given by

 DoFtotal≥POND(L−1)KL,

which indicates that DoF is achievable for a fraction of the time for actual transmission, where

 POND=limsnr→∞Pr{snrK∑i=1LΠ1π1(i),i≤ϵ0 and snrK∑i=1LΠ2π2(i),i≤ϵ0}. (14)

We now examine the relay scaling condition such that converges to one with high probability. For the simplicity of the proof, suppose that the first and the second relay sets and are selected out of two mutually exclusive relaying candidate sets and , respectively, i.e., , , , , and . Then, we are interested in how and scale with SNR in order to guarantee that tends to one, where denotes the cardinality of for . From (14), we further have

 POND=limsnr→∞ (15)

Let with and be the candidate set associated with the second relay set and the th S-D pair and its cardinality, respectively. For a constant , we can bound the second term in (15) as follows:

 Pr{K∑i=1LΠ2π2(i),i≤ϵ0snr} ≥Pr{max1≤i≤KLΠ2π2(i),i≤ϵ0Ksnr} (a)≥1−Pr{∃i:LΠ2π2(i),i≥ϵ0Ksnr} (b)≥1−K∑i=1Pr{LΠ2π2(i),i≥ϵ0Ksnr% } (c)≥1−KPr{minj∈N2LΠ2j,i≥ϵ0K% snr} (d)≥1−K(1−FL(ϵ0Ksnr))|Bi| (e)≥1−K⎛⎝1−C2(ϵ0Ksnr)3K−2⎞⎠∣∣N2∣∣−K+1, (16)

where the inequality