A General Algorithm for Interference Alignment and Cancellation in Wireless Networks

Transcription

1 A General Algorithm for Interference Alignment an Cancellation in Wireless Networks Li Erran Li, Richar Alimi, Dawei Shen, Harish Viswanathan an Y. Richar Yang Bell Labs MIT Yale University Abstract Physical layer techniques have come a long way an can achieve close to Shannon capacity for single pointto-point transmissions. It is apparent that, to further improve network capacity significantly, we have to resort to concurrent transmissions. Multiple concurrent transmission techniques (e.g., zero forcing, interference alignment an istribute MIMO) are propose in which multiple seners jointly encoe signals to multiple receivers so that interference is aligne or cancele an each receiver is able to ecoe its esire information. In this paper, we formulate the interference alignment an cancellation problem in multi-hop mesh networks. We show that the problem is NP-har in general. We then propose a convex programming base algorithm to ientify interference alignment an cancellation opportunities. Our algorithm effectively utilizes knowlege of both local network topology an overhear packets at the sener sie as well as the receiver sie. We implement our system using GNU Raio to evaluate key practical implementation issues. I. INTRODUCTION Interference in traitional wireless networks has been consiere harmful, as supporte by both theoretical analysis (e.g., [6]) an experimental measurements (e.g., [8], [9]). The etrimental effects of interference are particularly severe as a traitional wireless network becomes larger. However, as point-to-point link throughput approaches Shannon capacity, it becomes increasingly important to allow simultaneous transmissions in orer to substantially improve wireless network capacity. As a result, techniques for achieving simultaneous transmissions an receptions have been a research topic of intense interest. In the communications community (e.g., [], []), novel techniques such as zero forcing [] an interference alignment [] are propose. In these techniques, multiple seners jointly encoe signals to multiple receivers such that interfering signals will cancel out, an each receiver is able to ecoe its esire information. In this paper, we refer to all of these cooperative sener-sie techniques as cooperative interference alignment techniques, or interference alignment for short. Receivers can also utilize overhear packets or even exchange receive packets through wireline links to cancel interference in orer to extract the esire packets. We refer to the receiver-sie technique as interference cancellation. Previous investigations on interference alignment an cancellation either target specific opportunities (e.g., [7], [4]) an thus miss beneficial opportunities or are mainly theoretical by focusing on asymptotic behaviors. Specifically, in [0], Niesen, Gupta an Shah show that, in arbitrary extene networks, optimal capacity scaling cannot be achieve using the traitional point-to-point link abstraction for α 3, where signal ecays with istance to the power of α; cooperative schemes are require to achieve optimal scaling. In [], Ozgur, Leveque an Tse propose a hierarchical cooperative transmission scheme. They show that, in ranom extene networks where the area is fixe an the ensity of noes increasing, the total capacity of the network scales linearly with the number of noes n; in ranom extene networks where the ensity of noes is fixe an the area increasing linearly with n, the capacity scales as n α for α 3 an n for α 3. In this paper, we seek to esign a general algorithm that can ientify the best interference alignment an cancellation opportunities in practical settings where a noe has only local information. In particular, the local information inclues only local topology (one or two hops), an the set of packets that each sener or receiver has. We make the following contributions: We ientify iverse, novel scenarios for using interference alignment an cancellation to improve network throughput. We formulate the general problem of optimal interference alignment an stuy its computational complexity. We show that it is computational challenging (NP-har). We present a promising, istribute algorithm for ientifying a wie range of opportunities for interference alignment an cancellation. The algorithm makes elegant use of channel state, egree of freeom, an opportunistically receive packets at both the sener sie an the receiver sie. To further progress towars making interference alignment practical, we implement our algorithm in GNU raio. We ientify two key issues. The first is time synchronization. It is a common assumption in previous stuies (e.g., [], [], []) that transmissions be synchronize. We investigate how ifficult it is to meet synchronization requirements in a istribute setting. The secon issue is channel estimation. Channel status can be particularly helpful in interference alignment. Can channel estimation be achieve for multiple packet urations? Specifically, for physical-layer implementation of interference alignment, we make the following contributions: Leveraging OFDM, we o not nee precise time synchronization as long as multiple transmissions are synchronize within an OFDM cyclic prefix. Using implementation in GNU Raio, we show that we can achieve this if the largest propagation elay ifference from seners to any receiver

2 Fig.. A MIMO transmission where both sener an receiver have antennas. is relatively small. Using GNU Raio, we stuy the accuracy of channel estimation. In commercial harware, channel estimates may be more stable; however we have not observe this using GNU Raio. This makes it ifficult to achieve interference alignment using opportunistically receive packets at the sener sie. Interference cancellation at the receiver sie is not affecte. II. RELATED WORK Although there is a large boy of literature on relate topics, there are fewer stuies on practical systems issues. Our investigation is base on previous interference alignment techniques, incluing interference cancellation [], zero forcing [] an interference alignment []. There has been some recent work on practically applying interference alignment an cancellation techniques (e.g., [7], [4], [4]). Katti et al. [7] have propose ANC (conceptually similar to [4]). ANC exploits interference cancellation opportunities in multi-hop wireless networks. In particular, the -hop 3-noe line topology an 5-noe X topology are evaluate using GNU Raio. It is shown that ANC can improve throughput up to 70%. They o not consier zero forcing or interference alignment techniques. Gollakota, Perli an Katabi [4] have esigne an interference cancellation an alignment scheme in a specific setting where synchronization is not require. They o not exploit zero-forcing techniques. In aition, their results are only for one-hop wireless networks. vector transpose). By rewriting the receive signal as y y H 0 x H 0 x () we can view the receive signal as the sum of two scale vectors. The ecoing of each packet, say x, can be viewe as projecting H 0 T x onto a vector that is orthogonal to the vector H 0 T carrying the interfering signal x. If the sener knows the channel H, the sener can multiply H with X an sen the resulting signal X. Again, ignoring noise, we can verify that receiver antenna will receive only x, no mixing of x ; similarly, receiver antenna will receive only x. This technique is often calle zero forcing. The channel inverse is a specific type of encoing vector. This vector view of receive signal enables the general technique of interference alignment. Rather than nulling all interference at the receivers like zero forcing, the seners can encoe the signals such that interfering signals are aligne in a irection that is ifferent from the esire signal. To ecoe, the receiver just projects the receive signal onto a vector that is orthogonal to the vector of the interfering signal. Although effective in many settings, centralize MIMO has many limitations. In particular, the number of antennas that can be place on a noe to allow inepenent channels is limite ue to the size limitation of communication evices. Thus, it is esirable to exten interference alignment to a istribute setting. IV. DESIGN CHALLENGES A general multi-hop wireless network using interference alignment is shown in Figure. The general network setting creates substantial challenges. Below, we enumerate two key challenges: () limite knowlege of packets at istribute seners an receivers; an () egree of freeom constraints. III. BACKGROUND: MIMO AS A SPECIAL CASE OF INTERFERENCE ALIGNMENT We use centralize MIMO to illustrate the basic iea of interference alignment. As shown in Figure (see also [4]), a MIMO sener an receiver, each with M antennas, can potentially achieve throughput M times that of using a single antenna, uner the same total transmit power constraint. Let s briefly illustrate how this is achieve when M. If we represent the signals corresponing to packets an as x, x respectively, ignoring noise, we have that the receive signals at receiver antennas an are y h x h x, an y h x h x respectively. Here h i j is a complex number whose magnitue an phase represent signal attenuation an elay from sener antenna i to receiver antenna j. The receiver estimates the channel H as shown in Figure. It can recover x an x by multiplying H, the inverse of H, with the receive signal vector Y (Y y y T where T represents Fig.. Interference alignment in istribute settings. Limite knowlege of packets at istribute seners an receivers: In a istribute setting, to achieve interference alignment using the preceing MIMO technique, either we nee to exchange packets such that all n seners have the same set of packets when computing interference alignment or we nee to exchange receive samples among receivers. In a no-infrastructure support environment, it is infeasible to exchange receive samples. Take a setting where each sener

3 has antenna on a 0 MHz 80. channel. Then the raw sample rate will be 40 Msamples/secon. If each sample is represente by 8 bits, it translates into 30 Mbps information for a receiver to sen. Although it is possible to exchange packets at the sener sie, this introuces substantial overhea. Without exchanging packets (or with limite exchange) an no exchange of receive signal in sample form, a istribute interference alignment algorithm is face with the challenge of effectively utilizing the following ientifie opportunities: The sener makes use of overhear packets for constructing interference nulling or alignment; The receiver makes use of overhear packets for interference cancellation; The algorithm exploits the channel structure between istribute seners an receivers. That is, ue to signal attenuation, sener i may just slightly raise the noise floor of receiver j. In this case, we can set h i j 0. Degree of freeom (DOF) constraints: The number of inepenent signals that can be prouce at a sener is typically referre to as the egree of freeom (DOF) of the sener. Generally, if the channels from a given sener s antennas to those of the receivers are inepenent, then the sener with M antennas is sai to have M egrees of freeom. The number of inepenent frequencies is also counte as egree of freeom. For example, 80.a/g has 48 use ata subcarriers. The number of inepenent subcarriers is the number of egree of freeom that these subcarriers have. It has been shown [] that, using interference alignment every sener-receiver pair can potential achieve half of its egree of freeom among parallel transmissions. However, the construction for the alignment scheme is by assuming that each sener has many egrees of freeom (e.g., more than 000 for a 4x4 parallel transmission). In reality, the number of DOF that a sener has is limite. Thus, it is a challenging problem on what the best interference alignment can o with limite egree of freeom. V. DIVERSE SCENARIOS The preceing section ientifies two key systematic esign challenges. In particular, limite knowlege of packets emans efficient utilization of available packets to construct interference alignment an cancellation. In this section, we show novel, practical scenarios beyon simple scenarios in [7], [4]. In the examples below, each sener has only one egree of freeom. For simplicity, we ignore noise an assume that channels are known to neighboring noes; we use a simple slotte moel to illustrate basic concepts; we assume that there is a triggering mechanism to start concurrent transmissions of multiple transmitters. In each scenario, if there is no ege between a sener an a receiver, then the sener s transmission causes minimal interference at the receiver an thus can be ignore. In our notation, packet i s signal is enote as x i. Helper noe with native packets: Traitionally, a noe transmits only when it has a packet to its intene receiver. Utilizing interference alignment, a noe can transmit even if it s u u 3 s u Fig. 3. A network with two seners an two receivers: a helper noe can help two seners. has no intene receivers. We refer to such a noe as a helper. Furthermore, a helper noe can help more than one sener. Figure 3 is a simple example to emonstrate the aforementione benefits. In this example, the existence of a link from one noe to another noe inicates that the transmissions from the first noe can be receive by the secon one. The traffic in the example is that s has with final estination through u, an s has with final estination through u. Without interference alignment, it takes 4 slots for the two packets to reach their estinations (e.g., s to u, s to u, u to, an u to in slots,,3,4 respectively). With interference alignment, noe u 3 in the mile can help u an u to transmit simultaneously. In slot, s sens to u ; in slot, s sens to u ; in slot 3, u u u 3 transmit at the same time. Let h i j be the channel from u i to j where i 3 an j. Then, u 3 transmits x 3 which is constructe from overhear packets an in the preceing rouns: h x 3 h h 3 x h 3 x. One can verify that for y j at j, where j, we have that y j h j x h j x h 3 j x 3 is a scale h version of x j. For example, y h 3 h h 3 x ; that is, the interference from the other packet is nullifie. Helper noes with mixe packets: A helper can help even with only mixe packets. As shown in Figure 4(a), packet s next hop is u an estination is, an packet s next hop an estination is. If there is no helper, it will take 3 time slots for both packets to arrive at their estinations: s to u, u to an s to in slots,,3 respectively. With helper noe u, it takes only two time slots: in the first slot, both s an s transmit (s s transmission oes not cause any interference at ); in the secon slot, both u an u transmit. u u have only a mixe packet (the combine signal of an ). Let h i j be the channel between s i an u j ; g i j be the channel between u i an j. We can see that u i will receive y i h i x h i x. If u sens h g y an u sens h g y, then will only receive h h h h g g x. So far the gains of our examples are no more than 50%. The example in Figure 4(b) shows that we can achieve 00% gains. Without helper noes v an v, we nee 4 time slots (s to u, s to u, u to, an u to in slots,,3,4 respectively). With helper noes, we nee only slots: in slot, both s an s transmit; in slot, u, u, v an v, each sens a scale mixe packet. This results in that receives a scale version of an receives a scale version of. Essentially, v nulls the interference component of in u s

4 s s u u (a) helper noe s u v v s u (b) helper noes Fig. 4. Networks with two seners an two receivers: helper noes have only mixe packets. mixe packet. Similarly, v nulls the interference component of in u s mixe packet. Note that, v oes not interfere at an v oes not interfere at. s u s u Fig. 5. A network with two seners an two receivers: interference alignment an cancellation. Interference alignment an cancellation: In Figure 5, in a traitional network, it will take four time slots for both an to receive their respective an from s an s. With interference alignment an cancellation, s an s can transmit their respective packets simultaneously in the first slot. Note that, u receives a mixe signal. In the secon slot, u an u transmit simultaneously without any particular encoing. can use receive in the first slot to cancel interference. To implement this scenario, one possibility is that u can be a coorinating noe to transmit a triggering message. The triggering message instructs s s to transmit simultaneously, an u u to sen immeiately thereafter. The triggering message also informs of its ecoing vector. s u u v Fig. 6. A network with two seners an two receivers: multi-hop interference alignment. Multi-hop interference alignment: It can be beneficial to make interference alignment ecisions two hops away from the receivers. Consier the example in Figure 6. Assume that noes u u first receive packets from S in two time slots. Now consier two ifferent transmission strategies. In the first strategy, u u encoe their packets to allow v v to ecoe, respectively. Then v an v have to transmit in separate time slots to relay the two packets to their estinations. As a result, the first strategy takes a total of 5 slots. To escribe the secon strategy, let the channel matrix between u u an v v be H, an between v v to be H. Let the encoing vector at u i be ω i. Let the receive signals at be y y respectively. If we let ω ω T H H, then y y T Γ x x T, where Γ is a v u s Fig. 7. A network with two seners an two receivers; local routing creates interference control opportunities. iagonal matrix. That is, u u encoe with ω ω respectively, an transmit simultaneously to v v in the thir slot. In the fourth slot, v v will amplify their receive signals with the same magnitue an transmit at the same time. The secon strategy will result in a total of 4 instea of 5 slots. The nee for local-routing: In Figure 7, there are two flows: goes from s to an goes from s to. Suppose that the original routing from s to is through s. Then it will take 3 slots to eliver the two packets: it takes one slot for to arrive at s ; since s cannot sen an receive at the same time, it takes s one slot to forwar an one slot to sen. However, if we reroute the first packet from s to u, we nee only two time slots. In the first slot, both s an s can sen. Both an will receive. Then u will receive a mixe packet an can just forwar the mixe packet. Noe can ecoe using the store packet. VI. PROBLEM FORMULATION AND COMPUTATIONAL s COMPLEXITY The preceing section gives concrete examples. In this section, we precisely efine the problem to be solve. A. Problem Definition We first state our esign assumptions. We assume that a noe computes multiple sener transmission opportunities for its neighbors an triggers the transmissions using signaling information piggybacke in either ACK packets or DATA packets. We assume that the noe can ientify such opportunities using overhear flow information, an channel estimations. We o not assume that hea-of-line packets are always transmitte in a concurrent transmission opportunity. A higherlevel mechanism will scheule transmissions to integrate a given fairness objective. We assume that all native packets are linearly inepenent. Consier a specific noe s in the wireless network. For ease of escription, we consier a single rate network (i.e., all transmissions use the same rate ρ). Noe s uses local information available at itself to compute interference alignment an cancellation opportunities. The local information consists of overhear packets, exchange packet ientifiers, channel information, a transmission graph G V E, an a interference graph G I V E I, where V consists of noes in a local neighborhoo of s (e.g., within two hops away from it in G). An ege e u v means that transmissions with rate ρ are feasible between u an v. If there is no ege between u v

5 in G I, then u s transmissions will cause negligible interference at v, an vice-versa. Note that our interference graph is noebase. For simplicity, we assume that all noes have the same egree of freeom DOF. We assume a set S of seners. We use s i to enote sener i. There is also a set R of receivers. We use j to enote receiver j. Denote the channel matrix between the sener an receivers as H. In the case of DOF, we implicitly assume that j s receiver is j. We assume that the set of seners an the set of receivers are isjoint. There may be more seners than receivers. In that case, some seners act as helper noes. They will transmit overhear packets. Denote the coing coefficient matrix as Φ. The receive signals at the receivers can be enote as Y HΦX Z, where X is the vector of packets, Y the vector of receive signals at the receivers, an Z the vector representing noise at the receivers. Note that, if DOF, we take a block view of HΦ where each non-iagonal element is a vector of imension DOF. A iagonal element for receiver j consists of λ j vectors (each of imension DOF), where λ j is the number of packets that receiver j wants to receive from the seners. Ieally, uring each time slot, we woul like to pack the maximum number of concurrent transmissions at rate ρ. In the best case, we fin a Φ such that the non-iagonal elements of HΦ are either zero or aligne in subspaces not containing the iagonal vectors, an all the elements at the iagonal support the given fixe rate. This makes sure that every receiver gets its sener s packet. However, this may not be always possible. If not possible, we want to fin a Φ with a submatrix G of HΦ of maximum size with the following property: non-iagonal elements are either zero or aligne, an iagonal elements are all non-zero. We refer to our problem as the generalize interference alignment an cancellation (GIAC) problem. B. Computational Complexity As we have shown in the preceing section, there are iverse scenarios where we can apply interference alignment. A natural question to ask is: can we efficiently ientify these opportunities? Even for the simple case of DOF=, we can show that the interference alignment an cancellation problem is NP-har. The proof is by a reuction from the MAX INDEPENDENT SET problem. Given a graph G V E, the maximum inepenent set asks for a set B V with the maximum carinality such that, for any two noes u v B, u v E. Theorem : Assume that DOF =, an that packets are linearly inepenent. The general interference alignment an cancellation (GIAC) problem is NP-har. Proof: Our reuction is from inepenent set. Given a G V E, we create an instance of GIAC problem as follows. We label the vertices as u u uv. For each u i V, we create a sener s i an receiver i. Only sener s i has packet i. Let h ii 0. For each ege u i u j, we a channels h i j h ji with non-zero values. We see that if there is an ege u i u j in the inepenent set problem, an s i transmits its packet i, then s j cannot concurrently transmit j, because the interference cause by s j at receiver i cannot be force to zero given that no other noes have j an all packets are inepenent. Thus, an inepenent set in G correspons to a set of seners in our GIAC problem where simultaneous transmissions can all be ecoe correctly at each intene receiver, an vice-versa. Note that, our proof has assume that arbitrary interference among noes are possible. We leave the complexity for specific interference moels to future work. VII. DISTRIBUTED ALGORITHMS Given the computational complexity of fining an optimal solution, we focus on esigning practical algorithms with soun heuristics. The algorithm we esign in this section can fin all of the opportunities presente in Section V as well as those presente in [7], [4]. A. Optimal Algorithm for a Special Case There is an optimal algorithm for interference alignment with complete channel matrix an no receiver sie information (a receiver has not receive any packets from any sener). Specifically, if the elements in the channel coefficient matrix are all non-zero an the matrix has full rank, then the problem can be solve optimally. DOF=: For simplicity, the algorithm where all noes have DOF of is shown in Figure 8. We assume that the set of seners an the set of receivers are isjoint. Let S enote the set of seners, an a ik whether sener s i has k. Lines -8 are simply fining the maximum set PKT of packets (receivers, since for DOF=, there is a one-to-one corresponence of packet an receiver) such that each packet k PKT has at least PKT seners. This ensures that there are enough inepenent equations to zero force k at the other unintene receivers. As an example, consier Figure 3 after s an s transmit. For the next roun, the seners are u, u an u 3 ; the receivers are an. Noe u has packet (to ), u has packet (to ), an u 3 has both packets. Thus, the packet availability matrix is A 0 0 Running lines -8 of the algorithm, we ientify that each packet has two seners (n = n = ). Thus, both packets will be selecte for transmission (the while loop is not execute). Lines 0-5 is an optimization to select the minimal number of seners. We use a greey algorithm to remove reunant seners one by one. We can also replace lines 0-5 by a greey k-set cover heuristic. Theorem : In the case of no receiver sie information (no receiver has any packets of any seners), the interference alignment algorithm is optimal if the elements in H are all non-zero an H has full rank. Proof: Sketch. The key insight is that n i represents the number of noes that can be transmitte simultaneous

6 S. Let n k i a ik k K. Orer n k such that n π n π n πk 3. Let PKT be the set of packets to be transmitte 4. Let t 5. while PKT n πt 6. PKT PKT πt 7. t t 8. enwhile 0.Let T be the subset of S whose packets in PKT. Let T S. For any sener s i S T 3. if remove s i from T, each PKT still has PKT number of seners 4. T T s i 5. enfor Fig. 8. An optimal algorithm for interference alignment with complete channel matrix, no sie information, an DOF = for all seners. Fig. 9. An example case with DOF. with i at sener i. It oes not matter which set of noes. We remove packets one by one if they limit the number of simultaneous transmissions until we reach the optimal. Note that the algorithm assumes that the iagonal elements of HΦ support rate ρ for all seners. In practice, there may be power limitations. We will return to this issue in our general algorithm. DOF : We can exten the preceing algorithm to the case of DOF. Figure 9 shows an example with four seners an three receivers, where DOF i i 3. In this example, seners s, s, s 3, s 4 have,, 3, 4 respectively. Each sener has a corresponing receiver,, 3,. Sener s overhear 3. If DOF is one for all noes, then we can sen only two packets using zero forcing. However, in the case of DOF for all, three packets ptk 3 can be sent through interference alignment [4], [5]. This is because each DOF provies an inepenent equation to align the two other interfering signals. In this example, with overhearing, we can actually sen four packets combining zero forcing an interference alignment. Specifically, s encoes 3 with vector v v 5 v 6 respectively; s s 3 encoes 3 with vector v v 3 respectively; s 4 encoes 4 with v 4. Equation () shows the encoing, where H i j enotes the by channel matrix from sener s i to receiver j. y y y 3 H v H 4 v 4 H v H v 5 H 3 v 3 H v 6 H v H 4 v 4 H v H v 5 H 3 v 3 H v 6 H 3 v H 43 v 4 H 3 v H 3 v 5 H 33 v 3 H 3 v 6 x x 4 x x 3 Specifically, in the encoing, we zero force the non-iagonal elements of the first row by H v H v 5 0 an H 3 v 3 H v 6 0. We then align the non-iagonal elements of the secon an thir row by H v H 4 v 4 H 3 v 3 H v 6 an H 3 v H 43 v 4 H 3 v H 3 v 5. Note that for our problem with complete channel matrix an receiver sie information (i.e., receivers may have overhear some seners packets), we prove that it is still NP-har. Theorem 3: In the case of receiver sie information (a receiver may have packets of other seners a prior, e.g., through overhearing), the general interference alignment an cancellation (GIAC) problem is NP-har. Proof: Sketch. The reuction is via the clique problem. Given a graph G V E. For each noe u i V, we create a sener s i with packet i, an we also create an intene receiver i. For each ege u i u j in E, we let i has j an j has i. Two seners s i s j can transmit simultaneously iff ege u i u j E. A clique of size m correspons to a set of m seners whose packets can be transmitte simultaneously. The key intuition why the receiver sie is ifferent from the sener sie is as follows. A packet i at the sener sie (say with s j ) provies one inepenent equation that can null interference of s i s transmission of i at any receiver; this is not the case at the receiver sie. A packet i at receiver j can be use to only null interference of i at itself. With receiver sie information, we nee to moify the interference alignment algorithm. For each receiver j an each packet it has, say k, we create a pseuo sener s k j. This pseuo sener has a unit channel coefficient with j an a zero channel coefficient with all of the other receivers. We efer the algorithm to the general one-hop case. B. Greey Algorithm for General One-Hop Case The preceing iscussion leas naturally to a greey algorithm for the general case. DOF=: Again, we first look at the simpler version when DOF is for all noes. In the preceing algorithm, we compute n k as the number of seners with k. The semantics of n k is the maximum number of receivers who can receive their own packets even if k is transmitte. Since the channel matrix is complete, n k is compute as we previously efine. In the general case when the channel matrix is incomplete, the seners of k may not create any interference at some receivers. Thus, we nee to moify the computation of n k. For each packet, we first fin the maximum matching between the ()

7 . Let PKT be the set of packets to be transmitte. A a pseuo sener for any packet PKT a receiver has 3. while CanNotXmitS(PKT, ρ) 4. n k maxnonintr PKT k k PKT 5. Let be the one with minimal n k k K 6. PKT PKT 7. one 8. Prune seners one by one while CanNotXmitS PKT ρ false Fig. 0. General algorithm for interference alignment with sie information in the one-hop case. seners who has k an the set of receivers (receiver k must be in the matching). Let the size be m k. We then fin the set of receivers which are not interfere by the set of seners in the matching. Let the size of this set be l k. Then n k m k l k. An an example, if sener s k s oes not create any interference at any receiver an no other seners have the packet, then m k an l k K. Line 4 in Figure 0 uses the function maxnonintr to compute n k accoring to this revise efinition. At the en of the general algorithm, we nee to test whether a set of packets can be transmitte simultaneously at a given rate. Let H be the channel matrix (incluing helper noes an pseuo seners). Let Φ be the coing coefficient matrix. If there is a feasible solution such that HΦ has only non-zero iagonal elements, then the set of packets can be transmitte simultaneously. This can be solve by convex programming an a graient metho. This is one in Function CanNotXmitS PKT ρ in Figure 0. We now present its formulation. Let W w w w K. minimize f W K w i ρ i K w i i where f W is solve by the following convex program: K maximize f W w i BWlog hii N i i H HΦ i j i j K : hi j 0 i M : K φ i j j P (4) Equation (3) can be solve by a simple graient approach where the largest rate among all sener-receiver pairs ecreases its weight whereas the smallest increases its corresponing amount. Figure illustrates how maxnonintr is calculate. In this example, there are three packets. The figure shows the matching step of (the re packet) with maximum matching M of size. Since 3 is not interfere by the transmission of (3) Fig.. An example illustrating the general algorithm: one-hop case., we have L. Thus, n M L 3. Similarly, n n 3 3. Thus, all three packets can be transmitte at the same time. DOF : To exten to the case of DOF greater than, all we nee to change is the following. In Equation (4), if n k is smaller than the carinality of the current set of packets PKT, we o not set hi j 0 for all i j. Instea, we replace PKT n k of them with interference alignment equations (IAE). If the total number of DOF is not enough, then we know line 3 will be true an we o not nee to explicitly solve the convex programming problem. If receivers are allowe to exchange ecoe packets, after we know that a receiver j can ecoe its packet k with enough equation in the IAE step, we can assume that k is aligne at an interfering receiver j without consuming DOF. Note that, we can also exten to the case where a sener wants to sen more than one packets. All we nee to take care of is that the interference alignment at its receiver will have less imension to operate. Extension to two-hop an rerouting case The one-hop case algorithm can be easily moifie for the two-hop an rerouting case. The main ifferences are () we have to compute twohop matching when we compute maxnonintr PKT k ; () the channel matrix H will be a prouct of the first hop an seconhop channel matrices H, H ; (3) we nee to eal with helper noes with mixe packets. We pick the minimal n i. If there is a helper noe u one hop away from receiver i that receives a mixe signal containing j PKT j i, then we increase n i by one. Once u is picke as a helper for a certain packet, it cannot be chosen for another packet in the case of DOF. We o this iteratively until we cannot increase n i. We o not go beyon more than two-hops as it can be challenging in practice to keep channel information up-to-ate in such cases. VIII. IMPLEMENTING ALIGNMENT IN PHYSICAL LAYER AND MAC The preceing section presents our general algorithm. To achieve the benefits of the algorithm, it is essential to evaluate the possibility of implementing istribute interference alignment an cancellation in real settings. Implementing interference alignment requires two important functionalities in the physical layer. First, multiple transmitters must be able to start transmitting signals simultaneously using istribute algorithms. Secon, transmitters must obtain the

8 channel state information so they can perform the necessary encoing for interference alignment. We outline the issues we face, our solutions, an areas that require further improvement. Coorination: Interference alignments have to be coorinate. Without coorination, interference oes not get zeroe or aligne, then it will be ifficult to ecoe. To coorinate transmission, we use an appropriate triggering noe that ientifies opportunities an sens a control message with an encoing vector for each sener, an ecoing vector for each receiver. Clock Synchronization: Interference alignment requires that signals from interfering seners align at the receivers. This in general requires synchronization at the sener sie. With OFDM, precise synchronization is not require. However, the signals from ifferent seners must arrive within the cyclic prefix of the esire signal at each receiver. In our implementation, the local time maintaine by each noe is a counter of the samples sent to the USRP (0-value samples are sent to the USRP when there is no packet). This allows a simple clock to be maintaine in userspace with the accuracy of the USRP master clock. Each noe maintains the time offset to each of its neighbors. The sening time is encoe in each packet, allowing receivers to upate their time offset to the sener. Let i j enote the time offset between sener u i an receiver j. When beginning interference alignment transmission, the triggering noe inclues a time t j at a reference noe at which the transmission is to begin. Next, each sener u i computes its local time at which it shoul start the transmission as t i i j t j. Noe i begins sening the first sample of the packet when its counter reaches t i. Carrier Frequency Correction: Practical oscillators cannot generate or sample signals at the precise carrier frequency. There is a frequency offset. Each sener nees to correct this offset when transmitting the signal. Otherwise, the vector representation of each sener s signal will rotate an angle proportional to the sener s frequency offset at the receiver. Because the frequency offset is ifferent at ifferent seners, this will estroy the interference nulling or alignment effect. Thus, frequency offset must be correcte to sufficient accuracy so that resiual error creates insignificant interference at each receiver. Channel Estimation: Seners nee to know the channel to interfering receivers in orer to encoe its signal such that zero forcing or interference alignment can be achieve. How can we obtain these channel estimates? In 80.a/g, there are 48 ata subcarriers. They are ivie into 4 groups. We nee at least 4 channel estimations, one for each group. Each channel estimate can be represente as two 3-bit numbers (4 bytes). So for a M M transmission, the total number of channel estimates that nee to be exchange can be 6M bytes. Even for one byte channel representation, the total bytes will be 4M. As we sai, we can cut the number of channels neee by ignoring certain channels. One can also make use of channel reciprocity. If the sener has the channel estimation for the reverse channel, it can get the estimation for the forwar channel. Techniques such as quantization (e.g., [3]) may also be useful to compress the transmitte estimations, but further analysis an experimentation is neee to ientify how much quantization is tolerable in practical settings. Since channels can change within 5 to a few hunre millisecons, the estimations nee to be exchange frequently. In our implementation, the complexity for channel estimation over the harware evices is obtaining channel phase offset estimations, as it is necessary to know the true channel phase offset to perform the encoing. Since estimating the channel s phase offset is sensitive to the frequency offset between transmitter an receiver, we implement frequency offset correction at the transmitter using a numerically-controlle oscillator (NCO) running at the frequency offset obtaine by the receiver. IX. EVALUATIONS Our implementation is base upon the GNU Raio OFDM implementation. Our experiment uses a 5-point FFT with 00 occupie subcarriers, each moulate using BPSK. The cyclic prefix is 8 samples, an the sampling rate is set to 500KHz. This setup prouces an overall bit rate of 56.5 Kbps. A. Time Synchronization The first functionality we require is time sychronization between noes. The basic requirement is that a set of noes must be able to start transmitting simultaneously when using interference alignment. Timing offset between TX an RX (samples) Time (s) Fig.. Timing offset between sener an receiver. To accomplish interference alignment, we must first ensure that timing offsets can be estimate accurately. Figure shows the rift in time offset as one noe transmits to another noe over a perio of 7 secons. In this experiment, the transmitter sens packet every 0.5 secons, an we report the time offset compute by the receiver. The rift is steay at 0.75 samples per secon (i.e., a rift of.5 microsecon per secon). The require upate frequency epens on the uration of the cyclic prefix. For example, 80.a/g has a 0.8 microsecon cyclic prefix. Note that relative noe placement between transmitters an receivers also affects this requirement, as propagation elays can become significant for large variations in istances (e.g., light travels about 40 meters in

9 0.8 microsecons). In particular, the requirement may be more strict epening on istance between noes an to what egree noe istances are known or estimate. B. Channel Estimation Next we evaluate channel estimation, in particular frequency offset. Total Freq. Offset (Hz) Fig Time (s) Measure frequency offset between sener an receiver. Using our implementation, we can observe how the measure frequency offset varies with time. Figure 3 shows the offset between sener an receiver for a particular -secon interval. In this experiment, we continuously sen packets from the sener to receiver, an recor the frequency offset compute at the receiver. We observe that the frequency offset can vary within a 5-0 Hz range very quickly. Resiual Freq. Offset (Hz) Time (s) Resiual frequency offset after transmitter-sie correction. Fig. 4. Next, to unerstan the possible effects on interference alignment transmissions, we enable transmitter-sie correction of the frequency offset using the last estimate fe back from the receiver. Figure 4 plots the results. We can observe that the resiual frequency offset compute by the receiver can still be high. Using transmitter-sie frequency offset correction with an inaccurate frequency offset estimate can cause the estimate channel phase to rotate with time. The NCO runs at the frequency offset estimate, to counter the frequency offset between the sener an receiver. The inaccurate frequency estimate causes the NCO to accumulate a phase error over time. To put the previous figure in perspective, we compute a simple example. With a frequency offset estimate error of only Hz, the phase error will be about π 4 after 60ms. The total time for setting up the interference alignment transmission in our implementation is larger than 60 ms. As we can observe from the figures above, the implementation is currently not able to achieve a frequency offset accuracy uner Hz. Thus, the challenge to operating this system using the harware evices is coping with frequency offset estimation errors to obtain the channel s true phase offset. X. CONCLUSION AND FUTURE WORK Interference alignment involving sener cooperation can provie ample throughput gains in wireless mesh networks. We explore the challenges involve. We present a istribute algorithm that can systematically compute the opportunities in a local neighborhoo of a noe. Given that interference alignment also requires time synchronization an accurate channel feeback, we implement in GNU Raio to stuy feasibility. We fin that time synchronization can be one accurately enough, if the propagation elay ifference from seners to any receiver is small compare with the uration of cyclic prefix when using OFDM. However, ue to elays in software processing, accurate channel feeback appears to require harware support. For our future work, we are investigating more powerful methos [], [3] to achieve interference alignment. We inten to incorporate them in our algorithm. We are exploring another software raio platform with more harware support. We woul like to fully implement our istribute algorithm an esign a full-flege triggering MAC protocol in this platform. We then woul like to evaluate our opportunistic interference alignment framework using a larger testbe. REFERENCES [] V. R. Caambe an S. A. Jafar. Interference alignment an the egrees of freeom for the k user interference channel. IEEE Transactions on Information Theory, 54(8): , 008. [] S. Changho an D. Tse. Interference alignment for cellular networks. In Proceeings of the 4n Annual Allerton Conference on Communication, Control, an Computing, 008. [3] S. W. Choi, S. A. Jafar, an S.-Y. Chung. 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