Practical Semi-Blind Interference Alignment Exploiting Frequency Selectivity of Fading 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, 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 practically demanding. 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 show that it can achieve the same degrees of freedom as conventional IA.

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 requires 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. 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 blind interference alignment 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. The achievable 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 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. This paper is organized as follows: In Section 2, we briefly describe a system model, 2×2 X channel, assumed in this paper. Conventional BIA scheme is expressed in Section 3. After that, we proposed new IA scheme, S-BIA in Section 4. In Section 5, we give some numerical result and compare S-BIA and BIA. Finally, in Section 5, we conclude this paper. II. S YSTEM M ODEL Figure 1 illustrates a system model of 2×2 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 the remaining messages (a2 , b2 ) are for 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.

Fig. 3.

Fig. 1.

System model of 2 × 2 X Channel

Blind interference alignment (BIA) over 2×2 X Channel

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

Fig. 2.

Reconfigurable antenna for BIA scheme.

(k)

Let hij represents the channel response between the ith transmitter and the jth receiver at time (or frequency) slot k. All the channel responses are assumed to follow zero-mean and unit variance Gaussian distribution, CN (0, 1), and are mutually independent and identically distributed (i.i.d.). Also, (k) nj indicates the zero-mean additive white Gaussian noise (AWGN) with variance σ 2 at the jth receiver, i.e., CN (0, σ 2 ).

III. 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. Every receiver, however, should equip a reconfigurable antenna which is capable of switching several independent dumb antenna modes as conceptually illustrated in Fig. 2, where these antenna modes have mutually independent channel coefficients. Then, the receivers can switch these modes such as antenna selection according to predefined rules and create the staggered independent spaces as shown in Fig. 3. In the BIA, the Tx1 transmits three signals over three time slots. The transmit signals at the Tx1 can be given by 

     (1) x1 1 0  (2)    x1  = 1  a1 +  1  a2 . (3) 0 1 x1

The receivers switch antenna modes as Fig. 3 according to (k) predefined rules. In this paper, yj represents received signals for the jth receiver, at time slot k. Thus, the received signals by the receiver Rx1 are written by  (1)    ] y1 h11 (1) h21 (1) [ a1  (2)    h11 (2) h21 (2)  y1  = b1 (3) 0 0 y1  (1)    [ ] n 0 0 a2  1  +  h11 (2) h21 (2)  +  n(2) , 1 b2 (3) h11 (2) h21 (2) n1 (3) where hij (t) represents the channel response between the ith transmitter and the jth receiver at antenna mode t and follows CN (0, 1). At the receiver Rx2, we can then write  (1)    ] y2 h11 (1) h21 (1) [ a1  (2)   y2  =  h11 (1) h21 (1)  b 1 (3) 0 0 y2  (1)    [ ] n 0 0 a2  2  +  h11 (1) h21 (1)  +  n(2) . 2 b2 (3) h11 (2) h21 (2) n2 (4) From (3) and (4), it can be seen that interference is aligned into one dimension while desired signals appear through a full rank matrix. Therefore, BIA consists of four independent streams on three transmitted signals and obtains only desired signals. However, this scheme essentially requires several dumb antennas modes at each receiver and requires precise time synchronization among them. IV. S EMI -B LIND I NTERFERENCE A LIGNMENT (S-BIA)

(1)

In the previous section, we described the practical difficulties of both IA and BIA. In this section, in order to

facilitate these difficulties, we propose semi-blind interference alignment (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 time slots, which can be expressed as,  (1)      x1 α 0  (2)   1  a1 +  1  a2 . (5)  x1  = (3) 0 β x1 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

(6)

2

(k) xi

where represents the transmitted signals from the ith transmitter at the time slot k. Here, α, β, γ and δ are called as beamforming factor, which can be respectively expressed as (2)

h12

(2)

h11

(2)

h22

(2)

h21

, β = (3) , γ = (1) , δ = (3) . (7) (1) h12 h11 h22 h21 As observed from (7), all the coefficients does not include the 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 extremely high compared to average transmit power since every beamforming factor shown in (7) contains an inverse of channel coefficients. This concern, however, diminishes when we consider the use of OFDM in combination with S-BIA. We can write the received signal at the Rx1 as   (1)   (1) (1) ] y1 h11 α h21 γ [  (2)   (2)  a1 (2)  y1  =  h11 h21  b 1 (3) 0 0 y1    (1)  0 0 [ ] n a2  (2)  1  (2)  h21  +  h11 +  n(2)  .(8) 1 b2 (3) (3) (3) h11 β h21 δ n1 α=

Similarly, the received signals at the Rx2 are given by  (1)    (1) (1) ] y2 h12 α h22 γ [  (2)   (2)  a1 (2)  y2  =  h12 h22  b 1 (3) 0 0 y2    (1)  0 0 [ ] n2 a2  (2)  (2)  (2)  h22  +  h12 +  n2  .(9) b 2 (3) (3) (3) h12 β h22 δ n2

(3)

(2)

By subtracting y1 from y1 , the Rx1 interference. We may get  (2) (2) ] [ (1) h22 (1) h12 (1) h h y1  21 (1) =  11 h(1) h22 (2) (3) 12 y1 − y1 (2) (2) h11 h21 [ +

can remove the   

[

a1 b1

(1)

] ]

n1 (2) (3) n1 + n1

(10)

Similarly, the Rx2 can remove the interference by subtracting (1) (2) y2 from y2 .   (2) (2) [ ] [ ] h12 h22 (2) (1) y2 − y2 a2  (2)  (2) =  (3) h11  b (3) h (3) h12 (3) h22 21 2 y2 (3) h11 h [ 21 ] (2) (1) n2 + n2 + (11) (3) n2 Note that the channel matrices of (10) and (11) have full rank, i.e., 2, and constitute an equivalent 2×2 multi-input multioutput (MIMO) channel. Each receiver can retrieve the original information sequences by using maximum likelihood detection (MLD). V. P RACTICAL S-BIA BASED ON OFDM OVER F REQUENCY S ELECTIVE C HANNELS In the previous section, we assumed that the communications would be performed over three time slots and each receiver would exploit the three independent channel coefficients. However, in practice, if the channel responses vary in time, it is difficult for the transmitters and receivers to share the channel knowledge because of the difficulty of channel estimation. However, if the fading is sufficiently slow, it would be almost infeasible to obtain three independent channel coefficients. 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 [5]. Consequently, the number of independent subcarriers of each OFDM symbol basically depends on the number of independent paths of the multipath channel [6]. Figure 4 shows a 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, 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 3rd ones, the cross correlation

If `002 is shorter than the GI length, decoded symbols at frequency g is written by [ 2 N −1 N −1 ∑ ∑ ∑ (n) n(k−` ) (g) ˜ 1s ej2π N i = x1 h d1

number of subcarrier

correlation value

1

"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. 4.

≡

Cross Correlation matrix of subcarriers

values become relatively low. To realize S-BIA scheme, three independent channel coefficients are necessary and thus the latter combination of subcarriers can be utilized for 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 having several statistically independent mul˜ is is a zero-mean and tipath delays. The fading coefficient h circularly symmetric complex Gaussian random variable with unit variance of the sth multipath delay at Txi. The OFDM signal z1 ( Nk TS )(k = 1, 2, · · · , N ) which is transmitted to Rx1 from Tx1 can be expressed as ( ) ∑ 2 N −1 ∑ n k (n) ˜ (k−`1s ) j2π N z1 TS = , (12) x1 h 1s e N s=0 n=0 where N is number of samples, k is the time index, n is the frequency index, TS is the OFDM symbol length, and xi is the transmitted symbol from the ith transmitter. `1s is the number of delay samples which corresponds to the sth multipath delay. Here, the number of multipath delays is set as two. k Similarly, the transmitted OFDM signal z2 ( N TS ) from Tx2 to Rx1 is ) ∑ ( 2 N −1 ∑ n k (n) ˜ −j2π N (k−`02s ) TS = x2 h z2 , (13) 2s e N s=0 n=0 where `02s = J + `2s is the number of delay samples including the time difference J between two transmitters, where J ≤ 0. Thus, the received signals by Rx1 can be expressed as z

= z1 + z2 2 N −1 ∑ ∑ n(k−` ) ˜ 1s ej2π N i = dn1 h s=0 n=0 2 N −1 ∑ ∑

+

s=0 n=0

˜ 2s ej2π dn2 h

n(k−`00 i ) N

.

(14)

k=0 s=0 n=0 ] 2 N −1 ∑ ∑ n(k−`00 ) kg t (n) ˜ e−j2π N + x2 h2s ej2π N s=0 n=0 2 2 ∑ ∑ ˜ 1s e−j2π Nn τi x(g) + ˜ 2s e−j2π Nn τt00 x(g) h h 1 2 s=0 s=0 (g) (g) (g) (g) h11 x1 + h21 x2 . (15)

Thus, if `002 is shorter than the GI length, the receivers can decode symbols without inter-subcarrier interference (ISI). Therefore, the proposed S-BIA with OFDM is significantly robust against the clock synchronization errors between transmitters. VI. N UMERICAL R ESULTS In this section, we show several numerical results of S-BIA via computer simulations. By comparing capacity of S-BIA with conventional schemes, we show that S-BIA can obtain the identical degrees of freedom (DoF) gain to BIA. We further compare S-BIA with BIA in terms of bit error rate (BER) performance. We further investigate the BER performance of S-BIA over the OFDM channel. A. Capacity Analysis In order to show the DoF gain of S-BIA, we here analyze the capacity of the proposed scheme. It is well-known that the capacity of 2×2 X channels for each Rx1 and Rx2 is given by [2] [ ( )] 1 3 ˜ ˜† C = E log det I + ρHH , (16) 3 8 ˜ is the given channel matrix, where I is an identity matrix, H ˜ † is the transposed conjugate matrix of H, ˜ and ρ indicates 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 easily given by [ ( )] 2 3 ˜ ˜† C = E log det I + ρHH (17) 3 8 1) Capacity Analysis of BIA: We first analyze the capacity of conventional BIA. From (3), The conventional BIA 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   h11 (1) h21 (1) ˜ BIA =  h11 (2) h21 (2)  (18) H √ √ 2 2 Then, we can calculate the sum capacity by substituting the channel matrix in (17) with (18).

Fig. 5.

Sum capacity of BIA, S-BIA,TDMA

Fig. 6.

CDF of Channel response amplitude

2) Capacity of S-BIA: We further analyze the capacity of the proposed S-BIA. Similar to the analysis of BIA, SBIA can be equivalently considered as the MIMO channel. Although the channel matrix can be obtained from (10), the (3) (2) 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 written as   (2) (2) (1) h12 (1) h22  h11 (1) h21 (1)    h12 h22  ˜ HS-BIA =  (19) (2)  h(2)  h21 11 √ √ 2 2

B. Impact of Beamforming Factor

As described in section IV, S-BIA increases the average transmission power due to the beamforming factors. Considering the normalization of average transmit power and (17), we can get the sum capacity of S-BIA as [ ( )] 2 3 ˜ ˜† C = E log det I + ρH H , (20) 3 P

C. BER Comparison between BIA and S-BIA

[ ] where P = E |α|2 + |β|2 + |γ|2 + |δ|2 + 4 represents a power limiting factor. 3) Numerical Examples: Figure 5 shows the sum capacity of BIA, S-BIA and well-known time-division multiplexing access (TDMA) by computer simulation, where the sum capacity of TDMA is C = log(1 + ρ/2). From this figure, we clearly observe that the BIA and SBIA achieve higher capacity than TDMA due to the DoF gain. Moreover, the capacity of S-BIA scheme has the same slope of BIA. Thus, S-BIA achieves the same DoF gain to that of BIA. The capacity of S-BIA, however, is slightly less than that of BIA because of the beamforming factor, which will be investigated in detail in the following subsection.

In previous subsection, we have described that the beamforming factors would affect the capacity and cause the loss of the resulting capacity. In this subsection, we further analyze their statistical property by calculating cumulative distribution function (CDF). Figure 6 exhibits the CDFs of channel responses with beamforming factors. As observed from the figure, the CDF of S-BIA is less than that of BIA scheme. The distributions of channel responses obviously depend on the beamforming factors and the averages of the channel response amplitude of S-BIA is reduced compared to BIA. Thus, as shown in Fig.5, the capacity of S-BIA slightly decrease due to loss of the average received power.

Figure 7 shows the BER performances of BIA and S-BIA. All the curves are obtained by computer simulations where we assume the multi-path Rayleigh fading, all transmitters have equal power, and all channels have equal average SNRs. Moreover, uncoded quadrature phase shift keying (QPSK) is assumed and MLD is utilized to retrieve the information. Because of the symmetric property of X channel, we only plot results of the message a1 in Fig. 7. From Figure 7, we see that BIA is the best of all scheme, and it is drastically reduce error rate than S-BIA. The reason for this is the impact of beamforming factor to channel response. D. BER of Practical S-BIA with OFDM In Figure 8, we compare BER of S-BIA over the OFDM system and previous subsection result. We assume that number of subcarrier is 64, GI length is 33, three multipath waves arrive at receiver with equal power , and the number of delay sample is 16 and 32, and the others introduction is same as previous subsection. Consider the Figure 8, BER of S-BIA with OFDM is same as previous result which is

Fig. 7.

BER Comparison between BIA and S-BIA.

Fig. 9.

BER with time synchronization errors.

scheme is significantly robust against time synchronization errors between transmitters. VII. C ONCLUSION In this paper, we have proposed the new IA scheme named S-BIA. We have shown that S-BIA can achieve comparable DoF gain as BIA scheme and only requires channel responses corresponding to each transmitter. Moreover, we proposed the practical S-BIA in combination with OFDM system and showed that this proposed approach is significantly robust against the clock synchronization errors between transmitters. ACKNOWLEDGMENT The authors would like to acknowledge many helpful discussions with Dr. Kaiji Mukumoto who have guided this work. This work was supported in part by MEXT/JSPS KAKENHI Grant Number (24760295). Fig. 8.

BER comparison between S-BIA with OFDM and ideal S-BIA.

under ideal environment. Thus, if under multipath environment and each subcarrier in a set is independent, S-BIA over the OFDM can remove interference by each transmitter assign these subcarriers. E. Effect of Time Synchronization Errors Figure 9 shows the BER in the presence of time synchronization errors 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 Fig. 9, BER can be achieved the same performance to that in Fig. 8 at 30 dB as long as the maximum delay is less than 9 samples. From the assumption, the number of the maximum delay samples is set as 32. Thus, we can allow the time difference of J ≤ 8 since the GI length is 40. From the above statements, our

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