Semi-Blind Interference Alignment Based on OFDM over Frequency Selective X Channels Manato Takai∗ Koji Ishibashi† , Won-Yong Shin‡ , Hyo Seok Yi§ , and Tadahiro Wada¶ ∗ Graduate

School of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, Shizuoka, 432-8561, Japan † Advanced Wireless Communication Research Center (AWCC), The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan ‡ College of International Studies, Dankook University, 152 Jukjeon-rom, Shuji-gu, Yongin-si, Gyeonggi-do 448-701, Korea § School of Engineering and Applied Science, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA ¶ Dept. of Electrical and Electronic Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, Shizuoka, 432-8561, Japan Abstract—In this paper, we focus on a X channel; a system with two transmitters and two receivers each equipped with a single antenna, where independent messages need to be conveyed over channels from each transmitter to each receiver. A key concept of that arises in the context of the X channel is interference alignment (IA) that refers to an overlap of signal spaces occupied by undesired interference at each receiver while keeping desired signal spaces distinct. However, all the channel responses and the precise time synchronization are required at every transmitter and receiver to realize the IA, which are seriously demanding in practice. In this paper, we propose semi-blind IA (S-BIA) scheme exploiting frequency selectivity of wireless channels using orthogonal frequency division multiplexing (OFDM), which only requires channel responses corresponding to each transmitter. Numerical results confirm that proposed S-BIA can achieve the same degrees of freedom as conventional IA even in the presence of time synchronization errors. Index Terms—Semi-blind interference alignment (S-BIA), orthogonal frequency division multiplexing (OFDM), degrees of freedom (DoF), X channel

I. I NTRODUCTION Recent dramatic evolution of wireless communications technology has caused severe shortage of wireless resources while emerging communications technologies such as machine-tomachine (M2M) communications assume communications with zillions of users, which require more and more wireless resources. Interference alignment (IA) has been recently proposed and gained much attention of the researchers since it can significantly enhance the bandwidth efficiency over multiple access channels [1]. IA enables to align the signal spaces occupied by undesired interference at each receiver with a given signal space while keeping desired signal spaces distinct. However, all the channel responses and the precise time synchronization are required at every transmitter and receiver, which are practically demanding. In order to overcome this practical difficulty, blind interference alignment (BIA) has been proposed [2]. The key to the BIA scheme is the ability of the receivers to switch between reconfigurable antenna modes to create short term channel

fluctuation patterns that are exploited by the transmitter. This scheme does not require the global knowledge of channel state information while it still requires several receive antennas and precise time synchronization at both transmitters and receivers. In this paper, we propose new semi-blind interference alignment (S-BIA), which can realize signal alignment without reconfigurable antennas, global channel knowledge, and precise time synchronization, while the transmitters should have their own channel knowledge to align the signal spaces of interference. In addition, S-BIA with orthogonal frequency division multiplexing (OFDM) systems is investigated as a practical scheme of S-BIA. Although the idea of S-BIA has been already proposed in [3], which is based on users’ multipath intensity profile (MIP). However, since this scheme only works under specific MIP conditions, the scheme is called “opportunistic IA” by the author of [3]. Meanwhile, our approach can always form the alignment space as long as number of delayed paths is greater than the number of transmitters. This paper is organized as follows: In Section 2, we briefly describe a system model, 2-user X channel, assumed throughout the paper. We propose the new IA scheme, S-BIA in Section 3. S-BIA over OFDM system is theoretically analyzed in terms of the achievable capacity in Section 4. In Section 5, some numerical results are provided in order to evaluate the performance of S-BIA. Finally, in Section 6, we conclude this paper. II. S YSTEM M ODEL Figure 1 illustrates a system model of 2-user X channel considered throughout the paper. This channel is comprised of two transmitters (Tx1 and Tx2) and two receivers (Rx1 and Rx2), and each transmitter has independent messages for every receiver where every terminal is equipped with only one antenna. We assume that the messages (a1 , a2 ) are transmitted by the Tx1, and the messages (b1 , b2 ) by the Tx2. Then, two messages (a1 , b1 ) are intended to be received by the Rx1 and

(k)

h11

Tx1

(a1 , a2 )

Rx1

(a1 , b1 )

frequency slots, which can be expressed as,  (1)      x1 α 0  (2)   1  a1 +  1  a2 .  x1  = (3) 0 β x1

(1)

(k)

h21

Also, the transmit signals at the Tx2 can be written by  (1)      x γ 0  2(2)    x2  = 1  b1 +  1  b2 , (3) 0 δ x

(k)

2

h12 Tx2

(b1 , b2 )

Fig. 1.

(2)

Rx2 (k)

h22

(a2 , b2 )

(k)

where xi represents the transmitted signals from the ith transmitter at the frequency slot k. Here, α, β, γ and δ are called as alignment coefficients, which can be respectively expressed as

System model of 2-user X Channel.

the remaining messages (a2 , b2 ) to be received by the Rx2, respectively. For instance, at the Rx1, the messages (a1 , b1 ) are the desired signals while the other messages (a2 , b2 ) become interference. Therefore, each transmitter should adequately encode messages in order to avoid interferences at the receivers. (k) Let hji represents the channel response between the ith transmitter and the jth receiver at the kth channel resource such as time or frequency. All the channel responses are assumed to follow zero-mean and unit variance Gaussian distribution, CN (0, 1), and are mutually independent and (k) identically distributed (i.i.d.). Also, nj indicates the zeromean additive white Gaussian noise (AWGN) with variance σ 2 at the jth receiver, i.e., CN (0, σ 2 ).

III. S EMI -B LIND I NTERFERENCE A LIGNMENT Conventional IA requires the global knowledge of all the channel coefficients in order to align the signal spaces of interference. Wang et al. [2] have proposed BIA which can align interference signals without knowledge of channel responses at both transmitters and receivers. However, this scheme essentially requires several antennas modes at each receiver and requires precise time synchronization among them. In this section, in order to facilitate these difficulties, we propose S-BIA. In our proposed scheme, each transmitter should know the channel coefficients related to itself. Moreover, different from the conventional IA, each transmitter align the interference signals by using the observed channel coefficients, which mitigate the computational complexity at the transmitter side. Note that our proposed S-BIA can be performed over the frequency selective channel with OFDM, which does not require even the precise transmitting time synchronization among the transmitters. Similar to the BIA, the Tx1 transmits signals over three

(2)

(2)

α=

h21

, (1) h21

β=

h11

, (3) h11

(2)

γ=

h22

, (1) h22

(2)

δ=

h12

(3)

.

(3)

h12

As observed from (3), all the alignment coefficients do not include the channel coefficients of the other transmitters. If the channel links between the transmitters and the receivers are assumed to be symmetric and vary slowly compared to the symbol duration, transmitters can easily estimate the channel coefficients from the received signals. Note that instantaneous power of these coefficients may be high compared to the average transmit power since every alignment coefficient shown in (3) contains an inverse of channel coefficient. This concern, however, would be negligible when we consider the use of OFDM in combination with S-BIA, which will be discussed in Section VI. (k) Let yj represent the received signals for the jth receiver at the kth channel resource such as a frequency slot of frequency division multiple access (FDMA). Hence, we can write the received signal at the Rx1 as    (1)  (1) (1) ] y1 h11 α h12 γ [ a1   (2)  (2)    y1  =  h(2) h12 11 b1 (3) 0 0 y1   (1)   0 0 [ ] n a2  (2)  1  (2)  h12  +  n(2) +  h11  .(4) 1 b2 (3) (3) (3) h11 β h12 δ n1 Similarly, the received signals at the Rx2 are given by  (1)    (1) (1) ] y2 h21 α h22 γ [ a1  (2)   (2)   y2  =  h(2)  h22 21 b1 (3) 0 0 y2    (1)  0 0 [ ] n a2  (2)  2  (2)  h22  +  h21 +  n(2)  .(5) 2 b2 (3) (3) (3) h21 β h22 δ n2 (3)

By subtracting y1

(2)

from y1 , the Rx1 can remove the

interference. We may get  (2) ] [ (1) h21 (1) y1  h11 (1) = h21 (2) (3) y1 − y1 (2) h11

(2) (1) h h12 22 (1) h22 (2) h12

[

+

  

[

(1)

a1 b1

] ]

n1 (2) (3) n1 − n1

(6)

Similarly, the Rx2 can remove the interference by subtracting (1) (2) y2 from y2 .   (2) (2) [ ] [ ] h21 h22 (2) (1) y2 − y2 a2  (2) (2)  = h h   (3) 11 (3) (3) b2 h21 (3) h22 12 y2 (3) h11 h12 [ ] (2) (1) n2 − n2 + (7) (3) n2 Note that the channel matrices of (6) and (7) have full rank, i.e., 2, and constitute equivalent 2×2 multi-input multioutput (MIMO) channels. Hence, each receiver can retrieve the original information sequences by using maximum likelihood detection (MLD). IV. D EGREES - OF -F REEDOM (D O F) A NALYSIS In this section, we first derive the capacity of the proposed approach and calculate its degrees of freedom (DoF) gain in order to show that our approach can achieve the identical DoF gain to the conventional BIA. A. Capacity Analysis It is well-known that the capacity of 2-user X channels at each receiver (i.e., Rx1 or Rx2) is given by [2] [ ( )] 3 ˜ ˜† 1 H , (8) C = E log det I + ρH 3 8 ˜ is the given channel matrix, where I is an identity matrix, H † ˜ ˜ and ρ indicates H is the transposed conjugate matrix of H, the average received signal-to-noise ratio (SNR) [2]. Since there are two receivers in the network, the sum capacity of the system is simply given by [ ( )] 2 3 ˜ ˜† C = E log det I + ρHH (9) 3 8 1) Capacity Analysis of BIA: We first analyze the capacity of the conventional BIA. As described in [2], the conventional BIA over 2-user interference channel can be considered as the 2×2 MIMO channel. However, because of the subtraction at the receiver, we need to normalize the power of channel responses. Then, the corresponding channel matrix can be rewritten as   (1) (1) h11 h12 (2)  ˜ BIA =  (10) H  h(2) h12  √11 √ 2 2 We can easily calculate the sum capacity by substituting the channel matrix in (9) with (10).

Fig. 2. Sum capacity of three different IA schemes; BIA, S-BIA, and TDMA.

2) Capacity Analysis of S-BIA: We further analyze the capacity of the proposed S-BIA. S-BIA can be equivalently considered as the MIMO channel. Although the channel matrix (2) (3) can be obtained from (6), the subtraction y1 −y1 makes the resulting noise power at the Rx1 double. Therefore, we again need to normalize the power of the channel response and the channel matrix can be rewritten as   (2) (2) (1) h21 (1) h22  h11 (1) h12 (1)  h21 h22  ˜ S-BIA =  H (11)   (2)  h(2)  h12 11 √ √ 2 2 As described in section III, S-BIA increases the average transmission power due to the alignment coefficients. Considering the normalization of average transmit power and (9), we can get the sum capacity of S-BIA as [ ( )] 2 3 ˜ ˜† C = E log det I + ρHH , (12) 3 P [ ] where P = E |α|2 + |β|2 + |γ|2 + |δ|2 + 4 represents a power limiting factor. B. Numerical Examples Figure 2 shows the sum capacity of BIA, S-BIA, and a well-known time division multiple access (TDMA) calculated by computer simulation, where the sum capacity of TDMA is simply given by C = log(1 + ρ/2). The DoF gain is represented by the slope of the capacity curve. From the figure, we clearly observe that the BIA and S-BIA achieve higher capacity than TDMA due to the DoF gain. Moreover, the capacity of S-BIA scheme has the identical slope of BIA. These results confirm that S-BIA achieves the same DoF gain to that of BIA, i.e., DoF= 4/3 in this case. The capacity of S-BIA, however, is slightly less than that of BIA. To analyze this capacity loss of S-BIA, we further analyze the statistical property of alignment coefficients by calculating its cumulative distribution function (CDF).

1

correlation value

number of subcarrier

"data1.dat"

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.5

0

-0.5

-1

number of subcarrier Fig. 3. CDFs of the received amplitude of resulting channel responses composed of channel coefficients and alignment coefficients.

Figure 3 exhibits the CDFs of channel responses with alignment coefficients. The distributions of channel responses obviously depend on the alignment coefficients and the average power of the resulting channel responses of S-BIA becomes less than that of BIA. Therefore, as shown in Fig. 2, the capacity of S-BIA slightly decreased due to the loss of the average received power. V. P RACTICAL S-BIA BASED ON OFDM OVER F REQUENCY S ELECTIVE C HANNELS Although we showed that our S-BIA can achieve the identical DoF gain to that of BIA, we so far assumed that the communications would be performed over the three orthogonal channels via FDMA. However, FDMA does not efficiently use frequency resources. In this section, we further consider SBIA in combination with the OFDM system over frequency selective fading channels to form the aligned spaces. A. Formation of Aligned Spaces Using Subcarriers Under the multipath fading environments, the channel response of subcarriers varies with respect to the number of independent paths and the delay spread [6]. Consequently, the number of independent subcarriers of each OFDM symbol basically depends on the number of independent paths of the multipath channel [7]. Figure 4 shows a cross correlation matrix calculated by computer simulations where we assumed that the number of subcarriers is 16, the length of guard interval (GI) is 10, and the wireless channel is assumed to be a frequency selective channel with three equal power taps. Also, the delayed signals are received at 4 and 8 samples after the symbol arrival timing of the direct path. For instance, we can find the high cross correlation value between the 0th and 4th subcarriers, 0th and 8th ones, and 4th and 8th ones. In contrast, between the 0th and 1st ones, 0th and 2nd ones, and 1st and 2nd ones, the cross correlation values become relatively low. To realize S-BIA scheme, three

Fig. 4. Cross correlation between subcarriers in one OFDM symbol over frequency selective channel with three equal power taps where the number of subcarriers is 16,the length of guard interval (GI) is 10, and the delayed signals are received at 4 and 8 samples after the symbol arrival timing of the direct path.

independent channel coefficients are necessary and thus the latter combination of subcarriers can be utilized for the S-BIA scheme. B. Effect of Clock Synchronization Errors Between Transmitters We further investigate the effect of clock synchronization errors between transmitters. Here, the wireless channel is assumed to be frequency selective quasi-static (block) Rayleigh fading channel composed of several independent ˜ i,m represent the fading coefficient of delayed paths. Let h the mth multiple propagation path at the Txi. Without loss of generality, the effect of GI is ignored and the number of multipaths are assumed to be two. The tth received OFDM signal at the Rx1 through the frequency selective channel can be expressed as (t)

y1 =

2 N −1 ∑ ∑

(k) ˜ j2π x1 h 1,m e

m=0 k=0 2 N −1 ∑ ∑

+

k(t−`1,m ) K

(k) ˜ j2π x2 h 2,m e

k(t−`2,m ) K

(13)

m=0 k=0

where k = 0, 1, . . . , N − 1 denotes the subcarrier position, N (k) is the number of subcarriers, xi is the transmitted symbol from the ith transmitter over the kth subcarrier, and `i,m is the number of propagation delay samples that corresponds to the mth delayed path and the ith transmitter. If all the channel links have the same delay profile, `2,m = L + `1,m would be the effective propagation delay samples including the effect of channel and time difference L between two transmitters where L ≤ 0. If `2 is shorter than the GI length, the demodulated signal

at the f th subcarrier is written by [ 2 N −1 N −1 ∑ ∑ ∑ (k) k(t−` ) (f ) ˜ 1,m ej2π N1,m = x1 h y1 t=0

=

≡

m=0 k=0 ] 2 −1 ∑ N∑ k(t−`2,m ) tf (k) ˜ j2π N x2 h2,m e + e−j2π N m=0 k=0 2 ∑ ˜ 1,m e−j2π Nf `1,m x(f ) h 1 m=0 2 ∑ ˜ 2,m e−j2π Nf `2,m x(f ) + h 2 m=0 (f ) (f ) (f ) (f ) h11 x1 + h12 x2 . (14)

Thus, if `2 is shorter than the GI length, the receiver can decode the symbol without inter-subcarrier interference (ISI). Therefore, the proposed S-BIA with OFDM should be robust against the clock synchronization error between transmitters.

B. BER of Practical S-BIA with OFDM

VI. N UMERICAL R ESULTS In this section, we show some numerical results of S-BIA over OFDM system via computer simulations. We show the of bit error rate (BER) performances of S-BIA with the three orthogonal channels and one with the OFDM to confirm our statement. Moreover, we also investigate the feasibility of our proposed approach in terms of time synchronization errors and peak-to-average power ratio. A. Preliminaries In order to evaluate the feasibility of our proposed approach with OFDM. We here define the PAPR of OFDM. The bandlimited baseband signal of OFDM s(t) is given by √ s(t) = J LPF[IFFTJN [x(0) , x(1) , · · · , x(N −1) , 0, 0, · · · , 0]], | {z } (J−1)N

Fig. 5. BER comparison between S-BIA with OFDM and ideal S-BIA using three orthogonal channel resources.

(15)

00 s

where LPF represents an ideal low pass filter, IFFTJN is JN -points inverse fast Fourier transform (FFT), and J is an oversampling factor. Then, the PAPR of the OFDM signal in (15) can be defined as 2 maxt s(t) [ ] . P AP R = (16) 2 E s(t) The complementary cumulative distribution function (CCDF) is the most commonly used measure of PAPR, which evaluates the probability that the PAPR of transmitted OFDM symbols exceeds a given threshold level P AR0 . In ordinary OFDM signals cases, the CCDF for OFDM signals with large number of subcarriers N (≥ 64) has been already theoretically analyzed [8], which is given by √π √ −P AR0 . (17) Pr [P AP R > P AR0 ] = 1 − e− 3 N P AR0 e

Figure 5 shows BER performance of S-BIA with the OFDM system. We assume that number of subcarrier is 64, length of GI is 32, three multipath waves arrive at the receiver with equal power, the number of delay sample is 16 and 32, and the others assumption is same as the previous subsection. From the figure, BER of S-BIA with OFDM is the same as the result of ideal environment with three orthogonal channel resources. Thus, if the wireless channels between transmitters and receivers are rich multipath scattering environments, the S-BIA scheme would be feasible by choosing adequate subcarrier sets. C. Effect of Time Synchronization Errors Figure 6 shows BER performance of the proposed S-BIA with OFDM in the presence of time synchronization error between transmitters, where the length of GI is assumed to be 40 and SNR is 30 dB. Other assumptions are the same as the previous section. As observed from the figure, S-BIA with time synchronization errors provides the same BER performance with that without synchronization errors depicted in Fig. 5 at 30 dB as long as the maximum delay is less than or equal to 8 samples. From the assumption, the number of the maximum delay samples is set at 32. Thus, we can allow the time difference of L ≤ 8 since the GI length is 40. From the above statements, our scheme is significantly robust against time synchronization errors between transmitters. D. PAPR of S-BIA with OFDM Finally, the PAPR of OFDM symbol in combination with S-BIA is investigated. Figure 7 shows the CCDF performance of S-BIA with OFDM system and theoretical CCDF given by (17) where the number of subcarriers is 64, oversampling factor is 4, and each subcarrier is modulated by QPSK. As observed from the figure, there is a gap between OFDM with S-BIA and conventional OFDM because of the alignment

showed that this proposed approach is significantly robust against the clock synchronization errors between transmitters. ACKNOWLEDGMENT The authors wish to thank Kaiji Mukumoto for his valuable and insightful comments. This work was supported in part by JSPS KAKENHI Grant Number 24760295. R EFERENCES

Fig. 6. BER performance in the presence of time synchronization errors between transmitters.

Fig. 7. CCDF performance of OFDM in combination with S-BIA. The theoretical CCDF curve of OFDM also shown as a benchmark.

coefficients, which is about 0.5 dB. This gap is sufficiently small and thus we can conclude that if the S-BIA is performed in combination with OFDM, the occurrence of high instantaneous power can be ignored. However, if we consider the PAPR reduction techniques for OFDM signaling, nonlinear distortion due to the reduction of peak power may severely affect the performance of S-BIA since S-BIA relies on the orthogonality between subcarriers. We do not consider this effect since this performance degradation is beyond the scope of the paper. VII. C ONCLUSIONS In this paper, we have proposed the new IA scheme named S-BIA and shown that S-BIA can achieve comparable DoF gain as BIA scheme where it only requires channel responses corresponding to each transmitter. Moreover, we proposed the practical S-BIA in combination with OFDM system and

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