IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.XX, …szm0001/papers/TVT2402191_final.pdfIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.XX, NO.YY, MONTH YEAR 1 Interference Alignment

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.XX, NO.YY, MONTH YEAR 1

Interference Alignment Improves the Capacity ofOFDM Systems

Yi Xu, Student Member, IEEE, Shiwen Mao, Senior Member, IEEE, and Xin Su, Senior Member, IEEE

Abstract—Multi-user Orthogonal Frequency Division Multi-plexing (OFDM) and Multiple Output Multiple Output (MIMO)have been widely adopted to enhance the system throughput andcombat the detrimental effects of wireless channels. Interferencealignment has been proposed to exploit interference to enableconcurrent transmissions of multiple signals. In this paper, weinvestigate how to combine these techniques to further enhancethe system throughput. We first reveal the unique characteristicsand challenges brought about by using interference alignmentin diagonal channels. We then derive a performance bound forthe multi-user (MIMO) OFDM/interference alignment systemunder practical constraints, and show how to achieve this boundwith a decomposition approach. The superior performance of theproposed scheme is validated with simulations.

Index Terms—Interference alignment; Multiple Input andMultiple Output (MIMO); Orthogonal Frequency Division Mul-tiplexing (OFDM); Multi-User OFDM.

I. INTRODUCTION

The past decade has witnessed drastic increase of wire-less data traffic, largely due to the so-called “smartphonerevolution.” It is foreseeable that the capacity of existingand future wireless networks will soon be greatly stressed.Many advanced wireless communication technologies, suchas Orthogonal Frequency Division Multiplexing (OFDM) andMultiple Input Multiple Output (MIMO), are widely adoptedto enhance the system capacity, while a huge amount ofwireless access networks/base stations (BS) are deployed everyyear to accommodate the compelling need for larger capacity.Given the increasing wireless data volume and the more andmore crowded BS deployment, interference is apparently themajor factor that limits wireless network performance.

Traditionally, interference is considered harmful and oftentreated as background noise. As the performance of point-to-point transmission techniques is approaching Shannon ca-pacity, there is now considerable interest on exploiting in-terference for further capacity gains. It is shown that wheninterference is large, it can be decoded and canceled from

Manuscript received July 30, 2014; revised Dec. 6, 2014; accepted Feb. 4,2015. This work was supported in part by the US National Science Foundationunder Grants CNS-0953513 and CNS-1247955. This work was presented inpart at IEEE GLOBECOM 2012, Anaheim, CA, December 2012 [1].

Y. Xu and S. Mao are with the Department of Electrical and Computer Engi-neering, Auburn University, Auburn, AL 36849-5201, USA. X. Su is with theResearch Institute of Information Technology, Tsinghua University, Beijing,P. R. China, 100084. Email: [email protected], [email protected],[email protected].

Copyright c©2015 IEEE. Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained fromthe IEEE by sending a request to [email protected].

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier XXXX/YYYYYY

the mixed signal (as in interference cancellation), while wheninterference is comparable, interference alignment can beadopted to enable concurrent transmissions. Although inter-ference is harmful in many cases, it could be beneficial forenhancing system throughput as long as the interference canbe aligned. This kind of interference can be termed beneficialinterference.

Interference alignment was first proposed in [2], and thefeasibility condition was investigated in [3]. Since interfer-ence alignment is expected to greatly enhance the systemthroughput, many existing works focus on applying interfer-ence alignment to practical wireless networks. For instance,interference alignment with limited feedback or incompletechannel state information (CSI) were investigated in [4]–[6]. In a more general scenario, multi-cell opportunistic in-terference alignment was examined in [7]. Combined withcognitive radios, in [8], a cognitive interference alignmentscheme was presented to suppress both cross-tier and co-tier interferences in two-tier networks. Moreover, the authorsin [9], [10] investigated the behaviors of primary users andsecondary users under a Stackelberg game theory framework,where distributed interference alignment was adopted to enablespectrum leasing in the cognitive radio network. There are alsosome existing studies that aim to adopt interference alignmentin more advanced systems, such as heterogeneous network(HetNet) [11] and multi-hop mesh networks [12] The authorsin [13] also considered combining interference alignment andinterference cancellation. In [14], the authors proposed touse multimode MIMO antennas instead of the typical omni-directional antennas with interference alignment, while in [15],the impact of antenna spatial correlation on the performanceof interference alignment systems was investigated.

Although many applications of interference alignment havebeen investigated, few works have studied how to applyinterference alignment to OFDM systems due to the difficultyof analyzing throughput with structured channel (e.g., diagonalchannel). Since in a large network, there were many usersbut limited dimensionality, the authors in [16] proposed theconcept of “best-effort” interference alignment, and adoptedan iterative algorithm for optimization. However, how to useinterference alignment to enhance the throughput of a practicalOFDM system was not fully considered. Another attempt canbe found in [17], where the precoder design was studied.However, it only considered the case when the cyclic prefixwas insufficient. General precoding design specifically forOFDM system was not discussed. Shi et al. in [18] alsoconsidered the problem of interference alignment in multi-carrier interference networks. But the number of subcarriersconsidered was small, typically 3 subcarriers. So it’s not clear

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TVT.2015.2402191

Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].


if the approach can be extended to the general case of alarge number of subcarriers. In a more recent work [19],the authors proposed two schemes to adopt interferencealignment in multi-cell MIMO OFDM systems. In the firstscheme, interference alignment was used to remove the inter-cell interference, while zero-forcing precoding was used tosuppress the intra-cell interference. In the second scheme,interference alignment was also used for inter-cell interferenceremoval, while the OFDMA access scheme was applied forintra-cell interference cancellation. However, the fundamentalperformance bound of multi-user MIMO OFDM systems withinterference alignment has not been discussed. Some otherworks tried to understand the performance bounds of OFDMsystem with interference alignment, such as [20] [21]. In [20],the authors derived the necessary and sufficient conditions forthe three-user OFDM system with interference alignment inthe time domain. However, these conditions cannot be appliedto system with more users or under other conditions. Ayachet al. in [21] studied the feasibility of MIMO-OFDM systemwith interference alignment over measured channels.

In this paper, we show how interference alignment worksin OFDM system under practical constraints and are mainlyconcerned about the theoretical bound when interference align-ment is incorporated in the OFDM system. Specifically, weconsider the problem of incorporating interference alignmentin multi-user (MIMO) OFDM systems. We first examinethe fundamental characteristics and practical constraints onadopting interference alignment in a multi-user OFDM system.We show that, for a K user N subcarrier OFDM system,KN/2 concurrent transmissions that is achievable for genericstructureless channels [2], cannot be achieved for a practicalmulti-user OFDM network with diagonal channels and alimited number of subcarriers. We then investigate effectiveschemes to exploit interference in multi-user OFDM systems.With an integer programming problem formulation, we derivethe maximum efficiency of the Multi-user OFDM/interferencealignment system. We also show how to achieve the maximumefficiency with a decomposition approach, and derive theclosed-form precoding and decoding matrices. Finally, weextend the above analysis to the multiple antenna scenarios.All the proposed schemes are evaluated with simulations andtheir superior performance is validated.

Notation: in this paper, a capital bold symbol like H denotesa matrix, a lower case symbol with an arrow on top like ~vdenotes a vector, and a lower case letter like v denotes ascalar. [·]T means transpose and [·]−1 means inversion. Hij

and hij are the channel gain matrix and channel gain fromthe i-th transmitter to the j-th receiver, respectively. Vi is theprecoding matrix for transmitter i; ~vji is the j-th column ofVi. Ui denotes the interference cancellation matrix for the i-threceiver, while ~uji is the j-th column of Ui. Let h, v, u denotethe entries of H, V, and U, respectively.

Note that with these notations, the entries of Hij takesslightly different ordering from conventional ones. For in-stance, if transmitter 1 and receiver 2 are both equipped with

Tx1 Tx2 TxK

Rx1 Rx2 RxK

1

11h

1

21h

1

1Kh

Fig. 1. Multi-user OFDM using interference alignment.

M antennas, the channel gain is:

H12 =

h11 h21 · · · hM1

h12 h22 · · · hM2

......

. . ....

h1M h2M · · · hMM

. (1)

The rest of this paper is organized as follows. Section IIdescribes the background and preliminaries. Section III investi-gates how to adopt interference alignment in multi-user OFDMsystem. Section IV extends the analysis to the multiple an-tenna scenario. Simulation results are presented in Section V.Section VI concludes the paper.

II. BACKGROUND AND PRELIMINARIES

A. Orthogonal Frequency Division Multiplexing

While higher data rates can be achieved by reducing symbolduration, severe inter-symbol-interferences (ISI) will be causedover time dispersive channels. OFDM is an effective approachto allow transmissions at a high data rate and combat thedestructive effect of channel. By dividing the channel intonarrow bands, in which the signal experiences flat fading,OFDM can effectively mitigate ISI and maintain high datarate transmissions. Interested reader are referred to [23] andthe references therein for details.

B. Multiple Input and Multiple Output

With the single antenna transmission technique being welldeveloped, it is natural to extend to multiple antenna sys-tems. The MIMO transmission techniques have been evolv-ing rapidly since last decades. Generally speaking, multi-ple antennas or an antenna array can be used to attainthe diversity gain, multiplexing gain, or antenna gain, andthereby reduce the system error rate, enhance the systemthroughput, or strengthen the signal to interference and noiseratio (SINR) [24], [25]. Given M1 transmitting antennas andM2 receiving antennas, the maximum multiplexing gain isknown to be min{M1,M2}. Throughout this paper, we assumethat channel state information is perfectly known at eachtransmitter and receiver as in prior works [2].




C. Interference Alignment

It is shown in [2] that in a K user wireless network, with(n + 1)q + nq symbol extensions, totally K/2 normalizeddegrees of freedom (DoF) can be achieved using interferencealignment, where q = (K−1)(K−2)−1 and n ∈ N. In singleantenna systems, the normalized DoF is 1. With interferencealignment, the system throughput is enhanced by a factor ofK/2 for K ≥ 2. Note that there is no interference if there isonly one user occupying the time or frequency resource.

Observation 1: The system throughput could be improvedif alignable interference is introduced among users.

This observation is useful for OFDM systems, where thechannel gain matrix is diagonal. Since the gain of interferencealignment is proportional to K, we should have more userstransmit at the same time slot or frequency band if thetransmitted vectors can be aligned. That is why we call thiskind of interference beneficial interference in this paper.

III. MULTI-USER OFDM WITH INTERFERENCEALIGNMENT

In this section, we investigate the problem of interfer-ence alignment in multi-user OFDM systems. The systemmodel is illustrated in Fig. 1. We first examine fundamentalcharacteristics and practical constraints, and then demonstratehow to exploit interference in multi-user OFDM systems. Wederive the maximum throughput when interference alignmentis adopted, as well as closed-form precoding and decodingmatrices to achieve the maximum throughput.

A. Subcarriers versus Antennas

In traditional interference alignment, deploying multipletransmitting antennas allows us to precode data packets andalign them at the receiver. Deploying multiple receiving anten-nas provides multidimensional signal space, so that interfer-ence can be aligned into a sub-signal space that is orthogonalto the desired signal. Therefore, deploying multiple antennascan provide the needed freedom in the signal space.

In OFDM systems, we observe that subcarriers can functionin similar ways as antennas in MIMO/interference alignmentsystems, since subcarriers could also provide a multidimen-sional signal space. To some extent, subcarriers can be re-garded as a counterpart of antennas. So we could compressthe interference at each receiver in no more than half of thesubcarriers, and leave the other half subcarriers free frominterference.

However, note that there is a distinguishing difference be-tween the two systems: there is no cross-talk among differentsubcarriers in OFDM.

B. Precoding in OFDM

The main idea of interference alignment is to compress theinterference space to no more than half of the total receivedsignal space at each receiver, leaving the remaining part of thespace for desired signals [2]. This goal is achieved throughprecoding at every transmitter and zero forcing interferencecancellation at every receiver.

In OFDM systems, data is transmitted over multiple carriersbetween transmitters and receivers, as shown in Fig. 1. Since inOFDM systems, subcarriers can also provide multidimensionalsignal space for the transmitter and receiver as multiple an-tennas, we could precode over multiple subcarriers to achieveinterference alignment for OFDM system as multiple antennasfor MIMO system. Suppose there are N subcarriers. Ignoringnoise, if there is no precoding, the received signal for eachreceiver ~y is an N × 1 vector given by:

~y = H~x, (2)

where ~x is the desired signal in the form of an N × 1vector, and H is the N × N channel gain matrix betweenthe transmitter and receiver. Since different subcarriers havedifferent frequencies, the channel gain matrix is diagonal ifthere is no severe frequency shift. It can be seen from laterdiscussions that this property makes interference alignment inOFDM system quite different from the general channel case.

Going one step further, we can precode the data beforetransmission. If d packets are to be transmitted in an Nsubcarrier OFDM system, an N×d precoding matrix V couldbe used. The system equation is rewritten as follows.

~y = HV~x. (3)

If we let d = N and V = IN , where IN is an N ×N identitymatrix, (3) is reduced to (2).

In general, we could control what to be transmitted on thesubcarriers by adjusting the precoding matrix accordingly. Fora single user single antenna OFDM system with N subcarriers,the maximum number of packets can be transmitted is N . Notethat, here N is normalized by the QAM (Quadrature Am-plitude Modulation) modulation level. However, inspired bythe idea of interference alignment, we show that a throughputhigher than N can be achieved in the following subsections.

C. Interference Alignment in a K-User OFDM System

As discussed, we consider the problem of interferencealignment in multiuser OFDM systems. Basically, we aim toanswer the following questions.

(i) What are the practical constraints for adopting interfer-ence alignment in such systems?

(ii) What is the maximum throughput that can be achieved?(iii) How to achieve the maximum throughput (i.e., deriving

closed-form precoding and decoding matrices)?1) Dependence of Precoding and Decoding Vectors in Di-

agonal Channels: In this section, we show the difference onapplying interference alignment between a diagonal channeland a general channel, as well as the challenges to adoptinterference alignment in the former case.

It was shown in [3] that given M1 transmitting antennasand M2 receiving antennas in a K user interference channel,the DoF for each user, denoted by d, must satisfy

d ≤ (M1 +M2)/(K + 1). (4)

For example, given two transmitting and receiving antennasin a three-user interference channel, (4) indicates that eachuser could transmit one packet simultaneously. With a generic




structureless channel, the throughput Kd = 3 can be achievedas follows.

At each receiver, we align the signals from the other twousers. Recall the channel gain matrices as defined in (1) andlet the user i signal be ~vi, i = 1, 2, 3. It follows that

H21~v2 = H31~v3, H12~v1 = H32~v3, H13~v1 = H23~v2. (5)

Solving (5), we have

~v1 = eig(H−112 H32H−1

31 H21H−123 H13) (6)

~v2 = H−123 H13~v1 (7)

~v3 = H−132 H12~v1, (8)

where eig(A) stands for the eigenvector of matrix A.This scheme works well for generic structureless channels,

but not for the case of diagonal channels. For instance, if 2subcarriers (instead of two antennas) are used in OFDM, allthe channel gain matrices in (6), (7) and (8) are diagonal.Since the product of diagonal matrices is still diagonal, wehave from (6) that

~v1 =

(10

)or(

01

).

If ~v1 = [1, 0]T , we derive ~v2 = [c1, 0]T from (7) and ~v3 =[c2, 0]T from (8), where c1 and c2 are scalars. To cancel theinterference at receiver 1, the cancellation vector ~u1 must be~u1 = [0, c]T , where c is also a scalar. However, the desiredpacket is also canceled since ~u1 is orthogonal to ~v1. Therefore,we cannot simultaneously transmit 3 packets in this system.

The reason behind is that for a diagonal channel, itseigenvectors have only one nonzero entry. If we align inter-ferences at receiver r by letting Hjr~vj = · · · = Hir~vi, forj 6= · · · 6= i 6= r, the precoding vectors are dependent toeach other. Consequently, when interference is canceled at areceiver, the desired packet will also be canceled.

2) Interference Alignment with Multi-user OFDM–Performance Bound: It is shown in [2] that in a K usersystem with (n + 1)q + nq symbol extensions, totally K/2normalized DoF can be achieved using interference alignment,where q = (K − 1)(K − 2) − 1 and n ∈ N. In light of thisresult, one may think that KN/2 concurrent transmissionsis achievable in a K-User, N subcarrier OFDM system.However, we will show that this is unachievable for large Kin practical systems in the following.

It is worth noting that an assumption made in [2] isthat the symbol extensions can be infinitely large. This as-sumption may not hold true in practical systems. Given afinite bandwidth, the number of subcarriers is the bandwidthdivided by the subcarrier spacing. Typically, the value ofsubcarrier spacing is 10− 20 KHz. Then even for a 100 MHzbandwidth, we can have at most 104 subcarriers. For instance,in 802.16m and LTE, the maximum number of IFFT is 2,048,and maximum number of effective subcarriers is 1,200.

Therefore, the problem is to maximize system throughputgiven a finite number of subcarriers, denoted by Nmax. It isshown in [2] that with (n + 1)q + nq symbol extensions, thetotal normalized DoF is [(n+1)q+(K−1)nq]/[(n+1)q+nq].

So we aim to maximize (n + 1)q + (K − 1)nq and have thefollowing formulation.

max{n,K}

(n+ 1)q + (K − 1)nq (9)

s.t. q = (K − 1)(K − 2)− 1 (10)(n+ 1)q + nq ≤ Nmax, n ∈ N (11)K ≥ 3,K ∈ N. (12)

The physical meaning of problem (9) is that we try tomaximize the unnormalized DoF given finite number of sub-carriers. Note that all the variables are integers. Constraint (11)indicates that for practical OFDM systems, the number ofsubcarriers N = (n + 1)q + nq is upper bounded by Nmax.Although this integer programming problem is NP-hard ingeneral, by careful inspection, we can find the solution underpractical constraints.

In particular, we find the feasible region is very small forpractical Nmax values. Also the objective value is monotonewith respect to the two variables n and K. In problem (9),assuming K = 5, we have q = 11 from (10). For eachvalue of n, we can derive the number of subcarriers needed,Nmax, from (11) for the problem to be feasible, as wellas the throughput of the system (i.e., the objective value of(9)). The corresponding degree of freedom, d, is the ratio ofthe throughput and the number of subcarriers required. Thesenumbers are presented in Table I.

Table I shows that if there are K = 5 users, 2, 049 and179, 195 subcarriers are needed when n = 1 and n = 2,respectively. As discussed, a practical system usually do nothave more than 104 subcarriers. So n can only be 1 in thiscase, with efficiency dmax = 1.002. Therefore, interferencealignment is not useful in this case, since we can simply allowonly one user to transmit over one time-slot or a particularfrequency band to get d = 1 (i.e., single user OFDM).

If there are K = 6 transmitters, we have q = 19. Evenif n = 1, the number of subcarriers needed is 524, 289,which is not feasible for practical systems. Since the numberof subcarriers (n+ 1)(K−1)(K−2)−1 + n(K−1)(K−2)−1 growsexponentially with (K2−3K+1), it can be readily concludedthat K cannot be more than 4 for interference alignment tobe beneficial in multi-user OFDM systems.

Since the objective value of (9) is an monotone increasingfunction of K, the maximum feasible value K = 4 is ofparticular interest. We have q = 5 when K = 4. Table Ialso shows that under this condition, the maximum efficiencyfor practical system is dmax = 1.38 for the practical casewith at most 2, 000 subcarriers. When K = 3, we haveq = 1. The objective function (9) becomes 3n + 1, and theconstraint (11) becomes 2n + 1 ≤ Nmax. If the maximumnumber of subcarriers is Nmax = 2, 001, the system achievesits maximum efficiency dmax = 1.4998.

The above analysis can be summarized as follows.

Lemma 1. For a practical multi-user OFDM system with lessthan 2, 002 subcarriers, the maximum efficiency is dmax =1.4998, which is achieved when there are K = 3 users usingN = 2, 001 subcarriers.




TABLE ISYSTEM EFFICIENCY

When K = 5 and q = 11

n No. of subcarriers No. of packets Normalized DoF d

1 2,049 2,052 1.0022 179,195 185,339 1.03



1 33 35 1.062 275 339 1.233 1,267 1,753 1.384 4,149 6,197 1.49



1 3 4 1.3332 5 7 1.403 7 10 1.4294 9 13 1.444

100 201 301 1.4981000 2001 3001 1.4998

However, in the later discussions, we will show that thisconjecture does not hold true.

3) Interference Alignment with Multi-user OFDM–Realization: It is shown in [2] how to design the precodingmatrices to transmit 3n + 1 packets over 2n + 1 symbolextensions in a three-user interference channel (i.e., for a three-user system, we have q = 1 and N = (n+1)q+nq = 2n+1).We will derive the precoding/decoding procedure forinterference alignment with multi-user OFDM and prove itsefficacy in this section.

The precoding matrices proposed in [2] for the case of threeusers are as follows.

V1 = A, V2 = H−123 H13C. V3 = H−1

32 H12B, (13)

where A = [~w T~w T2 ~w · · · Tn ~w]B = [T~w T2 ~w · · · Tn ~w]C = [~w T~w T2 ~w · · · Tn−1 ~w]T = H21H−1

12 H32H−123 H13H−1

31

~w = [1 1 · · · 1]T .

(14)

Thus, the received signal at receiver 1 is:

~y1 = H11V1~x1 + H21V2~x2 + H31V3~x3. (15)

In the general case, since the data streams are independentof each other, the received mixed signal spans 3n+ 1 dimen-sions of the space. In interference alignment with multi-userOFDM, the received signal spans only 2n+ 1 dimensions ofspace. Solving these 2n + 1 equations will yield the desiredpackets. However, the challenge is, if 2n + 1 is too large,we may not be able to solve these equations efficiently (ascan be seen from the later discussions). This problem canbe addressed with a decomposition approach as given in thefollowing theorem.

Theorem 1. For an N subcarrier OFDM system, we candivide the subcarriers into bN/(2n+1)c groups, where n ∈ N,

and precode and decode the groups separately to achieve theinterference alignment gain.

Proof: Recall that the channel gain matrix in OFDM isdiagonal. Generally, if every user tries to transmit d packetsover the N subcarriers, we have

HV =

h1 0 · · · 00 h2 · · · 0...

.... . .

...0 0 · · · hN

v11 · · · v1d

v21 · · · v2d

.... . .

...vN1 · · · vNd

.

The precoding vectors must satisfy the conditions given in(13)-(14). Let the precoding matrix assume the following form.

V =

V1 0 · · · 0

0 V2 · · · 0...

.... . .

...0 0 · · · Vg

, (16)

where g = N/(2n+ 1) is the number of groups and Vi is theprecoding matrix for group i with dimensions (2n+1)×(n+1)or (2n+ 1)× n (i.e., user 1 sends (n+ 1) packets, and eachof the other users sends n packets over (2n+ 1) subcarriers.)Without loss of generality, we assume N is dividable by 2n+1.Rewriting H in the form of multiple diagonal sub-matriceswith the same dimensions, we have

HV =

H1V1 0 · · · 0

0 H2V2 · · · 0...

.... . .

...0 0 · · · HgVg

. (17)

For instance, when N = 6 and n = 1, we have fortransmitter 1

HV =

h1v11 h1v12 0 0h2v21 h2v22 0 0h3v31 h3v32 0 0

0 0 h4v41 h4v42

0 0 h5v51 h5v52

0 0 h6v61 h6v62

. (18)

If there are 3 users, we can let H21V2 = H31V3 at receiver 1to get

h(1)21 v

(1)2 0 · · · 0

h(2)21 v

(2)2 0 · · · 0

h(3)21 v

(3)2 0 · · · 0

0 h(4)21 v

(4)2 · · · 0

0 h(5)21 v

(5)2 · · · 0

0 h(6)21 v

(6)2 · · · 0

......

. . ....

0 0 · · · h(N−2)21 v

(N−2)2

0 0 · · · h(N−1)21 v

(N−1)2

0 0 · · · h(N)21 v

(N)2




=

h(1)31 v

(1)3 0 · · · 0

h(2)31 v

(2)3 0 · · · 0

h(3)31 v

(3)3 0 · · · 0

0 h(4)31 v

(4)3 · · · 0

0 h(5)31 v

(5)3 · · · 0

0 h(6)31 v

(6)3 · · · 0

......

. . ....

0 0 · · · h(N−2)31 v

(N−2)3

0 0 · · · h(N−1)31 v

(N−1)3

0 0 · · · h(N)31 v

(N)3

,

which indicates:

h(i)21 v

(i)2

h(i+1)21 v

(i+1)2

h(i+2)21 v

(i+2)2

=

h(i)31 v

(i)3

h(i+1)31 v

(i+1)3

h(i+2)31 v

(i+2)3

, i = 1, 4, · · · , N − 2.

(19)

Since the above conditions can also be obtained by sepa-rately encoding the N/(2n+1) groups of subcarriers, we coulddecompose the problem into a number of subproblems, one foreach group, and precode and decode the groups separately.

It remains to show how to decode the packets for thisscheme. Without loss of generality, we also assume K = 3.If this scheme is adopted, each time we sequentially take out2n+ 1 subcarriers. The received signal at receiver 1 is

~y1 = H11V1~x1 + H21V2~x2 + H31V3~x3

= H11V1~x1 + H21H−123 H13C~x2 + H31H−1

32 H12B~x3

= H11V1~x1 + H21H−123 H13C~x2 + H31H−1

32 H12TC~x3

= H11V1~x1 + H21H−123 H13C(~x2 + ~x3)

= (H11V1 H21V2) ·(~x1 ~x2 + ~x3

)T. (20)

Taking the inverse of matrix (H11V1 H21V2) and discard thepackets from transmitters 2 and 3, we can recover the desiredpackets ~x1. Note that we exploit the commutative property ofdiagonal matrices in (20).

At receiver 2, the received signal is

~y2 = H12V1~x1 + H22V2~x2 + H32V3~x3

= H12(~w B)~x1 + H22V2~x2 + H12B~x3

= H12 ~wx(1)1 + H22V2~x2 + H12B

x

(2)1 + x

(1)3

...x

(n+1)1 + x

(n)3

= (H22V2 H12 ~w H12B)·(

~x2, x(1)1 , x

(2)1 + x

(1)3 , · · · , x(n+1)

1 + x(n)3

)T. (21)

Taking the inverse of matrix (H22V2 H12 ~w H12B), we get ~x2.

At receiver 3, the received signal is

~y3 = H13V1~x1 + H23V2~x2 + H33V3~x3

= H13(C Tn ~w)~x1 + H13C~x2 + H33V3~x3

= H13C

x

(1)1 + x

(1)2

...x

(n)1 + x

(n)2

+ H13Tn ~wx(n+1)1 + H33V3~x3

= (H33V3 H13C H13Tn ~w)·(~x3, x

(1)1 + x

(1)2 , · · · , x(n)

1 + x(n)2 , x

(n+1)1

)T. (22)

Taking the inverse of matrix (H33V3 H13C H13Tn ~w), wecan decode ~x3. After decoding each group separately, we thencombine the decoded data. The theorem is thus proved.

Note that the proof of Theorem 1 also leads to an algorithmto achieve interference alignment gains for any large N ∈ N.

4) Practical Issue of Large Channel Variance: Here weexamine another practical problem of adopting interferencealignment for multi-user OFDM.

A necessary condition to achieve interference alignment inOFDM is that the channel gain is drawn from a continuousdistribution. As a result, if the variance of the channel islarge, some of the channel gains can be very small in certainconditions, while some other channel gains can be verylarge. When precoding over all the subcarriers, after takingthe inverse of the channel gain matrix, some entry of theprecoding matrix could be 104 times (or even more) largerthan some other ones. The result is that the power of onesubcarrier could be 108 times (or even more) larger than that ofanother subcarrier. Given certain power constraints, the errorperformance of the system will suffer from great degradation,which makes interference alignment less useful.

In our proposed scheme, if the channel variance islarge, there is also a certain chance that some entriesof T can be much larger than the others, since T =H21H−1

12 H32H−123 H13H−1

31 = H21H32H13H−112 H−1

23 H−131 . If we

precode and decode over large n, since the last column of V1,V2 and V3 are all obtained by multiplying Tn, the situationcould be further exacerbated. The consequences are as follows.

(i) Since some of the entries can be extremely small, thedecoding matrices can be close to singular. Thus thedesired signal cannot be decoded.

(ii) Even if the decoding matrices is invertible, due to thetransmitter power constraint, the system error perfor-mance could be rather poor.

In fact, even if n = 1, there is still a chance that some matricesare not invertible. These are the reasons why we cannotprecode and decode for large N . This issue also demonstratethe importance of the proposed decomposition theorem (seeTheorem 1).

Take V1 for instance. The constraint is the power on onesubcarrier cannot be 10a (e.g., a = 3) times larger than thepower on another subcarrier. If the constraint is violated, thesystem is considered to be in the outage state. Let

T = diag{t1, t2, · · · , t2n+1}, (23)




where ti = h(i)21h

(i)32h

(i)13 /(h

(i)12h

(i)23h

(i)31 ), i = 1, 2, . . . , (2n+ 1).

t1, t2, . . . , t2n+1 can be regarded as i.i.d (independent identi-cally distributed) random variables. Let t denote the commondistribution of t1, t2, . . . , t2n+1. Define t(1), t(2), . . . , t(2n+1)

be the order statistics of t1, t2, . . . , t2n+1 with t(1) = mini ti,t(2n+1) = maxi ti.

Let γ = t(2n+1)/t(1). From (13) and (14), we have γ2n ≤10a, thus

γ ≤ 10a/(2n), (24)

which means t(2n+1) cannot be 10a/(2n) times larger than t(1).On the other hand, since γmax = 10a/(2n), we have

1−(

Pr

{t ≥

t(2n+1)

10a2n

})2n+1

≤ Pr

{t(1) ≤

t(2n+1)

γ

}≤ 1.

(25)It can be seen that Pr

{t ≥ t(2n+1)

10a2n

}is a decreasing function

of n. With the power of 2n + 1, Pr{t(1) ≤ t(2n+1)/γ

}will

quickly converge to 1. That means, with large n, P (t(2n+1) ≥γt(1)) = 1. Therefore, with large n the constraint (24) willnot be satisfied.

Next, we show how large n could be for given con-straint (24). The joint probability density function (PDF ) oft(1) and t(2n+1) is found as follows.

ft(1)t(2n+1)(x, y) =

∂2Ft(1)t(2n+1)(x, y)

∂x∂y, (26)

where Ft(1)t(2n+1)(x, y) is the joint cumulative distribution

function (CDF) of t(1) and t(2n+1). By the definition of partialderivative, we have:

ft(1)t(2n+1)(x, y)

=∂

∂y{ lim

∆x→0[Ft(1)t(2n+1)

(x+ ∆x, y)−

Ft(1)t(2n+1)(x, y)]/∆x}

= lim∆x→0,∆y→0

[Ft(1)t(2n+1)(x+ ∆x, y + ∆y)−

Ft(1)t(2n+1)(x, y + ∆y)− Ft(1)t(2n+1)

(x+ ∆x, y)+

Ft(1)t(2n+1)(x, y)]/(∆x∆y)

= lim∆x→0,∆y→0

[Pr{x ≤ t(1) ≤ x+ ∆x, t(2n+1) ≤ y + ∆y}−

Pr{x ≤ t(1) ≤ x+ ∆x, t(2n+1) ≤ y}]/(∆x∆y)

= lim∆x→0,∆y→0

Pr{x ≤ t(1) ≤ x+ ∆x,

y ≤ t(2n+1) ≤ y + ∆y)/(∆x∆y}. (27)

To calculate the probability of the last equality, for any x <y, we can divide the x axis into five disjoint intervals as: I1 =(−∞, x), I2 = (x, x+∆x), I3 = (x+∆x, y), I4 = (y, y+∆y)and I5 = (y+∆y,∞). For each ti, the probability it falls intoeach interval can be calculated as follows.

p1 = Pr{ti ∈ I1} = Ft(x)p2 = Pr{ti ∈ I2} = Ft(x+ ∆x)− Ft(x)p3 = Pr{ti ∈ I3} = Ft(y)− Ft(x+ ∆x)p4 = Pr{ti ∈ I4} = Ft(y + ∆y)− Ft(y)p5 = Pr{ti ∈ I5} = 1− Ft(y + ∆y).

(28)

To make (x ≤ t(1) ≤ x + ∆x, y ≤ t(2n+1) ≤ y + ∆y)happen, the statistics {t1, t2, . . . , t2n+1} must have exactly

1 sample falling into interval I2, 1 falling into interval I4,(2n− 1) falling into interval I3, and 0 elsewhere, which is amultinomial problem. So we have

Pr{x ≤ t(1) ≤ x+ ∆x, y ≤ t(2n+1) ≤ y + ∆y} =(2n+ 1

0, 1, (2n− 1), 1, 0

)p0

1p12p

(2n−1)3 p1

4p05. (29)

It follows that

ft(1)t(2n+1)(x, y)

= lim∆x→0,∆y→0

{(2n+ 1)!

(2n− 1)!

Ft(x+ ∆x)− Ft(x)

∆x×

Ft(y + ∆y)− Ft(y)

∆y× [Ft(y)− Ft(x+ ∆x)]2n−1

}= (2n+ 1)(2n)ft(x)ft(y)[Ft(y)− Ft(x)]2n−1. (30)

Since ti = h(i)21h

(i)32h

(i)13 /(h

(i)12h

(i)23h

(i)31 ), i = 1, 2, . . . , 2n+ 1,

and each h(i) is a random variable, the distribution of ti isdifficult to be explicitly found. Here we continue our analysisby approximating ti as a Uniform distributed or Rayleighdistributed random variable.

If ti is approximated as a Uniform distributed randomvariable and ti ∈ (0, 1), we have:

Pr

{t(1) ≤

t(2n+1)

γ

}(31)

=

∫ 1

0

∫ yγ

0

(2n+ 1)(2n)(y − x)2n−1dxdy. (32)

=

[1−

(1− 1

γ

)2n]≥ 1− (1− 10−

a2n )2n, (33)

where the last inequality is a direct result of (24). Tak-ing derivative of (33), it can be found that Poutage =

Pr{t(1) ≤

t(2n+1)

γ

}is an increasing function of n. For a = 3,

if n = 1, Poutage = 0.0622; if n = 2, Poutage = 0.5431;and if n = 3, Poutage = 0.8978. For a system with manysubcarriers, it indicates that we can only precode over n = 1.

If ti is approximated as a Rayleigh distributed randomvariable with PDF f(x | σ) = x

σ2 exp(− x2

2σ2 ), x ≥ 0, then

ft(1)t(2n+1)(x, y) = (2n+ 1)(2n)

xy

σ4exp

(−x

2 + y2

2σ2

)×(

exp

(− x2

2σ2

)− exp

(− y2

2σ2

))2n−1

.

There’s no closed-form solution of Pr{t(1) ≤

t(2n+1)

γ

}in this

case. The numerical results are shown in Fig. 2. It can be seenthat the conclusion still holds, i.e., we can only precode overn = 1.

Recall that Lemma 1 shows dmax = 1.4998 when K = 3and n = 1000. Here we can see that this maximum DoF cannotbe achieved under practical settings. So we have the followingtheorem.

Theorem 2. For a practical multi-user OFDM system withless than 4149 subcarriers, the maximum DoF is dmax = 1.33,which is achieved when there are three transmitter/receiverpairs precoding over 3 subcarriers each time.




1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

Pro

bab

ilit

y o

f O

uta

ge

Unifrom

Rayleigh

Fig. 2. Probability of System Outage.

IV. MULTI-USER MIMO OFDM WITH INTERFERENCEALIGNMENT

In previous sections, we have considered applying inter-ference alignment to OFDM systems. Since MIMO transmis-sion technique can also be adopted to enhance the systemthroughput, we consider incorporating interference alignmentto MIMO-OFDM systems in this section.

Suppose we have M antennas at both the transmitter andreceiver sides, and N subcarriers in total. The signals receivedat the i-th receiver on subcarrier n can be represented as:

~yi(n) = Hii(n)Vi(n)~xi(n) +∑j 6=i

Hji(n)Vj(n)~xj(n), (34)

where Hij(n), Vi(n), and ~xi(n) are the channel matrix fromtransmitter i to receiver j, precoding matrix at transmitteri, and data at transmitter i, respectively; and all of themare at subcarrier n. From (34), we can see that, the signalsreceived can be represented as a matrix, with each columnbeing the signals received from each subcarrier, i.e., Yi =[~yi(1) ~yi(2) . . . ~yi(n)]. Or we could vectorize this matrix sothat we get the following simpler form.

~yi = HiiVi~xi +∑j 6=i

HjiVj~xj . (35)

Since each antenna pair could operate on any subcarrier andthere is no crosstalk between subcarriers, the wireless channelHij between transmitter i and receiver j is of the form asshown in (36).

Theorem 3. For a MIMO-OFDM system with N subcarriersand M antennas at each transmitter and receiver side, we candivide the subcarriers into bN/(2n+1)c groups, where n ∈ N,and precode and decode the groups separately to achieve theinterference alignment gain.

Proof: In Theorem 1, we have actually established thatfor a system of diagonal channels, we could separately precodeand decode each group of subcarriers. Now consider the casewhen all the devices are equipped with multiple antennas.We can still divide the subcarriers into different groups, thenprecode and decode them separately, since we are able todistinguish the signals from different antennas and different

subcarriers. In other words, upon receiving a signal, thereceiver has the knowledge of from which antenna and whichsubcarrier it gets the signal. So by properly adjusting the orderof the data transmitted, the channel is essentially of the formin (37). We can readily identify that (37) is actually in theblock diagonal form with the i-th block corresponding to thechannels associated with the i-th subcarrier. Within each block,we have standard MIMO channels. Letting V , with dimensionMN × d, assume the form of (16), by similar argumentsas in Theorem 1, we could precode and decode the groupsseparately to achieve the interference alignment gain.

Lemma 2. All the channel matrices and matrix T are invert-ible.

Proof: As shown in (38), the inverse of a block matrixcan be found by calculating the inverse of each block. Sincefor each block, we have a standard MIMO channel matrixand each of its entry is drawn from a continuous randomdistribution, each block is invertible with probability 1. Soeach channel matrix is invertible. Since the product of invert-ible matrices is still invertible, according to (14), matrix T isinvertible.B1 0 0

0 B2 00 0 B3

−1

=

B−11 0 00 B−1

2 00 0 B−1

3

. (38)

Theorem 4. For a MIMO-OFDM system with N subcarriersand M antennas at each transmitter and receiver side, themaximum gain is 4

3M .

Proof: According to Theorem 3, we could precode anddecode over groups of subcarrier. Also, according to ourprevious results, we can only precode and decode over 3subcarriers. So subcarrier-wise, the normalized DoF is 4/3.

We next show that 43M is the maximum achievable DoF.

Firstly, we notice that by dividing the subcarriers into groupsof 3, taking H11 for instance, it is transformed from (39)to (40). With the establishment of Lemma 2, following theproof of Theorem 1, and replacing the scalars with blocks, wereadily have the maximum gain of 4

3M .

H11 =

h11

11 0 0 h4111 0 0

0 h2211 0 0 h52

11 00 0 h33

11 0 0 h6311

h1411 0 0 h44

11 0 00 h25

11 0 0 h5511 0

0 0 h3611 0 0 h66

11

. (39)

H11 =

h11

11 h2111 0 0 0 0

h1211 h22

11 0 0 0 00 0 h33

11 h4411 0 0

0 0 h3411 h44

11 0 00 0 0 0 h55

11 h6511

0 0 0 0 h5611 h66

11

. (40)




Hij =

h1,1ij 0 0 · · · hN+1,1

ij 0 · · · h(M−1)N+1,1ij 0 · · ·

0 h2,2ij 0 · · · 0 hN+2,2

ij · · · 0 h(M−1)N+2,2ij · · ·

......

. . ....

.... . .

......

. . ....

. (36)

Hij =

h1,1ij · · · hM,1

ij 0 · · · 0 · · · 0...

. . .... 0 · · · 0 · · · 0

h1,Mij · · · hM,M

ij 0 · · · 0 · · · 0...

......

. . . · · ·...

......

· · · · · · · · · · · ·. . . · · · · · · · · ·

0 · · · · · · 0 · · · hM(N−1)+1,M(N−1)+1ij · · · h

MN,M(N−1)+1ij

......

...... · · ·

.... . .

...0 · · · · · · 0 · · · h

M(N−1)+1,MNij · · · hMN,MN

ij

. (37)

We next show how to achieve this gain. We design V1, V2,and V3 as follows.

V1 = A, V2 = H−123 H13C, V3 = H−1

32 H12B, (41)

whereA = [~w T~w T2 ~w · · · T(n+1)M−1 ~w]

B = [TM ~w TM+1 ~w · · · T(n+1)M−1 ~w]

C = [TM−1 ~w TM ~w · · · T(n+1)M−2 ~w]T = H−1

12 H32H−131 H21H−1

23 H13

~w = [1 1 · · · 1]T .

(42)

It can be observed that

A = [~w T~w · · · TM−1 ~w B] (43)= [~w T~w · · · TM−2 ~w C T(n+1)M−1 ~w]. (44)

At receiver 1, the received signals can be written as

~y1 = H11V1~x1 + H21V2~x2 + H31V3~x3

= H11V1~x1 + H21H−123 H13C~x2 + H31H−1

32 H12B~x3

= H11V1~x1 + H21H−123 H13C~x2 + H31H−1

32 H12TC~x3

= H11V1~x1 + H21H−123 H13C(~x2 + ~x3)

= (H11V1 H21V2) ·(

~x1

~x2 + ~x3

). (45)

For signals at receiver 2, we have

~y2 = H12V1~x1 + H22V2~x2 + H32V3~x3

= H12(~w T~w · · · TM−1 ~w B)~x1 + H22V2~x2 + H12B~x3

= (H22V2 H12(~w T~w · · · TM−1 ~w) H12B)×

~x2

x(1)1...

x(M)1

x(M+1)1 + x

(1)3

...x

((n+1)M)1 + x

(nM)3

. (46)

And similarly for signals at receiver 3, we have

~y3 = H13V1~x1 + H23V2~x2 + H33V3~x3

= H13(~w T~w · · · TM−2 ~w C T(n+1)M−1 ~w)~x1+

H13C~x2 + H33V3~x3

=

H33V3

H13CH13(~w T~w · · · TM−2 ~w)

H13T((n+1)M−1) ~w

T

×

~x3

x(M)1 + x

(1)2

...x

((n+1)M−1)1 + x

(nM)2

x(1)1...

x(M−1)1

x((n+1)M)1

. (47)

From (45)–(47), we can see that the desired signals are allfree from interferences.

We can also calculate the probability of system outagewhen multiple antennas are deployed. So we need to find theprobability of Pr

{t(1) ≤

t((2n+1)M)

γ

}. With similar arguments,

the joint PDF of t(1) and t((2n+1)M) can be found as

ft(1)t((2n+1)M)(x, y)

= lim∆x→0,∆y→0

P (x ≤ t(1) ≤ x+ ∆x,

y ≤ t((2n+1)M) ≤ y + ∆y)/(∆x∆y)

=

((2n+ 1)M

0, 1, (2n+ 1)M − 2, 1, 0

)p0

1p12p

(2n+1)M−23 p1

4p05. (48)

If ti is approximated as a Uniform distributed variable inthe range of (0, 1), the probability Pr

{t(1) ≤

t((2n+1)M)

γ

}can




02

46

810

0

5

100

0.2

0.4

0.6

0.8

1

nM

Pro

ba

bili

ty o

f O

uta

ge

Fig. 3. Probability of System Outage with multiple antennas for Uniformdistribution.

be found as follows.

Pr

{t(1) ≤

t((2n+1)M)

γ

}= (2nM +M − 1)× (49)

(2nM +M)

∫ 1

0

∫ yγ

0

(y − x)2nM+M−2dxdy (50)

= 1−(

1− 1

γ

)(2n+1)M−1

(51)

≥ 1−(

1− 10−a

2(n+1)M−2

)(2n+1)M−1

.(52)

If ti is approximated as a Rayleigh distributed vari-able, there is no closed-form solution for probabilityPr{t(1) ≤

t((2n+1)M)

γ

}. The joint PDF of t(1) and t((2n+1)M)

can be derived as

ft(1)t((2n+1)M)(x, y) = ((2n+ 1)M)((2n+ 1)M − 1)

xy

σ4×

exp

(−x

2 +y2

2σ2

)[exp

(− x

2

2σ2

)−exp

(− y2

2σ2

)](2n+1)M−2

.

Figs. 3 and 4 illustrate the probabilities of system outagefor Uniform and Rayleigh distributions, respectively. Here wealso set a = 3. So the power on one subcarrier cannot be 103

times larger than that on any other subcarrier. We can see thatwhen n = 1 and M = 2, the system outage probabilities are0.8505 and 0.2758 for Uniform and Rayleigh distributions,respectively. For n = 1 and M = 3, the outage probabilitiesare even higher, i.e., 0.9962 and 0.7937, respectively. For n =2 and M = 2, the outage probabilities are unacceptably ashigh as 0.9981 and 0.9971, respectively. From the simulationresults to be presented in the next section, we will see that ahigh outage probability is undesirable.

V. SIMULATION STUDY

Simulations are conducted to evaluate the performance ofthe proposed schemes and verify the benefits brought aboutby incorporating interference alignment in multi-user OFDMsystems. We consider the case of 3 users. The number ofsubcarriers is 255. Each transmitter precodes over (2n+ 1)Msubcarriers. Block fading channels are used in the simulations,where channel gains are piece-wise constants for the duration

02

46

810

0

5

100

0.2

0.4

0.6

0.8

1

nM

Pro

ba

bili

ty o

f O

uta

ge

Fig. 4. Probability of System Outage with multiple antennas for Rayleighdistribution.

20 30 40 50 60 70 80 90 100182

184

186

188

190

192

194

196

198

200

202

Eb/N

0 (dB)

Un

no

rma

lize

d D

oF

MU OFDM with IA, n=5

Fig. 5. System throughput when n = 5.

20 30 40 50 60 70 80 90 100100

200

300

400

500

600

700

Eb/N

0 (dB)

Unn

orm

aliz

ed D

oF

Single User OFDMMU OFDM with IA, n=3


MU MIMO OFDM with IA, n=2, M=2




Fig. 6. System throughput comparison when the channel variance is large.

of each time slot drawn from a certain distribution. BPSK isused as the modulation scheme. So we transmit 1 bit on eachsubcarrier and we measure how many bits are successfullydecoded at the receivers. In this way, we are essentiallycalculating the number of interference-free channels in thesystem (we call it unnormalized DoF hereafter).

Fig. 5 depicts the system throughput when n = 5. Wecan see that the system performance is generally unstable.




20 30 40 50 60 70 80 90 100100

200

300

400

500

600

700

Eb/N

0 (dB)

Unnorm

aliz

ed D

oF

MU OFDM with IA, n=3MU OFDM with IA, n=2MU OFDM with IA, n=1

Single User OFDM




Fig. 7. System throughput comparison when the channel variance is small.

Comparing Fig. 5 with Figs. 6 and 7, we find that since n istoo large, the system performance is degraded. This confirmsour result that we could not precode over a large amount ofsubcarriers.

Fig. 6 and Fig. 7 illustrate the performances of differentschemes when the channel is drawn from an uniform distribu-tion on [0, 1] and [0.9, 1], respectively. Comparing these twofigures, we can see that when the channel variance is small,higher system throughput can be achieved. This conforms toour discussions about the precoding matrix in Section III-C4.It can also be observed that the trends and comparativerelationships are similar in Fig. 5 and Fig. 6.

We can see from Fig. 7 that when n = 1, multiuser OFDMwith interference alignment can achieve an unnormalized DoFof 339.98. Compared to the highest throughput of single userOFDM of 255, the DoF has been improved by a factor ofapproximately 1.33 by incorporating interference alignment.When n = 2, we can see from both figures that the throughputof multiuser OFDM with interference alignment has degradedwhen the SNR is in the range [0, 78] dB. That verifies ourtheorem that under certain power constraint, we can onlyprecode over 3 subcarriers. Same conclusions also hold forn = 3 of multiuser OFDM with interference alignment, whichexhibits poorer performance in the SNR range of [20, 100] dB.

For the case of multiuser MIMO OFDM with interferencealignment, when n = 1 with small channel variance, thehighest unnormalized DoF is 622.7, which is 2.44 times ofthe unnormalized DoF of the single user OFDM system. Thereason why it is slightly less than 2.66 is also due to the bigdifferences among the elements of the precoding matrices. Forn = 2 and M = 2, we can see that the performance is worsethan that of n = 1 and M = 2. When the devices are equippedwith 3 antennas, we let n = 1 and precode over 3 subcarriers.The highest unnormalized DoFs are 671.2 and 683.208 forlarge and small channel variance cases, respectively, whichare 2.63 and 2.68 times of that of the single user OFDMsystem. However, the maximum gain is suppose to be 4 timesthe single user OFDM system. The performance degradation isalso due to big difference among the elements of the precodingmatrices.

VI. CONCLUSIONS

In this paper, we investigated the problem of how to exploit-ing interference in OFDM systems. We provided an analysisand developed effective schemes on incorporating interferencealignment with multi-user (MIMO) OFDM to enhance sys-tem throughput. With an integer programming formulation,we derived the maximum efficiency for multi-user (MIMO)OFDM/interference alignment systems, and showed how toachieve the maximum efficiency under practical constraints.The performance of the proposed schemes were validated withsimulations. The proposed decomposition algorithm and themain results in this paper may serve as a guidance for practicalOFDM system design.

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Yi Xu (S’11) received the M.S. degrees in Elec-tronic Engineering from Tsinghua University, Bei-jing, China in 2010 and the B.S. degree in ElectronicInformation Engineering from University of Elec-tronic Science and Technology of China, Chengdu,China in 2007. Since 2011, he has been pursuingthe Ph.D. degree in the Department of Electrical andComputer Engineering, Auburn University, Auburn,AL, USA. His research interests include Optimiza-tion, Game Theory, massive MIMO, OFDM, IDMAand Cognitive Radio Networks.

Shiwen Mao (S’99-M’04-SM’09) received Ph.D.in electrical and computer engineering from Poly-technic University, Brooklyn, NY. Currently, he isthe McWane Associate Professor in the Departmentof Electrical and Computer Engineering, AuburnUniversity, Auburn, AL, USA. His research interestsinclude wireless networks and multimedia commu-nications, with current focus on cognitive radio,small cells, mmWave networks, free space opticalnetworks, and smart grid. He is a DistinguishedLecturer of the IEEE Vehicular Technology Society

in the Class of 2014. He is on the Editorial Board of IEEE Transactions onWireless Communications, IEEE Internet of Things Journal, IEEE Commu-nications Surveys and Tutorials, among others. He is the Vice Chair–Letters& Member Communications of the IEEE ComSoc Multimedia Communi-cations Technical Committee. He received the 2013 IEEE ComSoc MMTCOutstanding Leadership Award and the NSF CAREER Award in 2010. Heis a co-recipient of The IEEE ICC 2013 Best Paper Award and The 2004IEEE Communications Society Leonard G. Abraham Prize in the Field ofCommunications Systems.

Xin Su (SM’14) received the M.S. and Ph.D.degrees in Electronic Engineering from Universityof Electronic Science and Technology of China in1996 and 1999, respectively. Currently he is a fullprofessor and vice director of the Wireless andMobile Communications Technology R&D Centerof Tsinghua University. He is also the chairmanof IMT-Advanced technology work group in MIIT(Ministry of Industry and Information Technologyof People’s Republic of China) and vice chairman ofthe Innovative Wireless Technology Work Group of

CCSA (China Communications Standards Association). His research interestsinclude broadband wireless access, wireless and mobile network architecture,self organizing networks, software defined radio, and cooperative commu-nications. Dr. Su has published over 80 papers in core journals and topconferences, applied for more than 40 patents and obtained more than 20pieces of patent right.



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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.XX, …szm0001/papers/TVT2402191_final.pdfIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.XX, NO.YY, MONTH YEAR 1 Interference Alignment