As an inevitable phenomenon in wireless networks, interference has been always an important concern in designing communication networks. To combat this problem, one of the most promising techniques, interference alignment, is proposed. While most research about interference alignment aims at deriving or realizing the maximum achievable degrees of freedom (DoF), the bit error rate (BER) performance is of equal importance. To improve the BER performance, this paper presents an Alamouti coded blind interference alignment (BIA) scheme, which combines interference alignment with space-time coding. First, the Alamouti coded BIA scheme for a two-user MIMO X channel is proposed. In this scheme, 4/3 DoF is obtained with no channel state information at transmitters. To decrease the decoding complexity, then we explore the partial interference cancellation (PIC) group decoding and compare the impact of different grouping schemes. Simulation results show that the proposed scheme achieves better BER performance than the original one. It is also shown that the BER performance can be further improved by a special grouping scheme.
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Interference management is one of the biggest challenges in wireless networks, in which multiple transmissions occur concurrently over a common communication medium. Interference is usually handled in practice by either interference avoidance where users coordinate their transmissions by orthogonalizing their signals in time or frequency, or by treating-interference-as-noise in which users increase their transmission power and treat each other's interference as noise. Interference decoding, although it is less practical, is another approach to deal with interference when interference is strong enough to be decoded along with the desired signal.
When using these conventional interference management approaches, the sum-capacity of the network will be limited by interference, regardless of the number of users, or the approach itself cannot be generalized for more than two user cases. Interference alignment is a surprising approach with which the sum-capacity of the time-varying interference networks using limited resources can be increased linearly with the number of users. It was first introduced by Maddah-Ali et al i, ii in the context of MIMO X channels. The scheme combines successive decoding and dirty paper coding and achieves ⌊4/3M⌋ DoF on the two-user X channel when all nodes are equipped with M antennas. The first explicit interference alignment scheme was presented in iii which showed that dirty paper coding and successive decoding are not required to achieve the maximum degrees of freedom on the two-user MIMO X channel. Interference alignment schemes have been explored for a variety of networks including X networks ii, iv compound broadcast channel v, vi interference networks vii, viii, cellular networksix, multi-hop (relay) networks x, xi, and bidirectional relay networks xii.
However, most studies about interference alignment assume that full channel state information (CSI) is available to all transmitters and receivers. In practice, the state of the channel can only be measured at the receivers, so it is often difficult for the transmitters to acquire the CSI precisely. Therefore, most exiting schemes are not feasible.
To make interference alignment feasible, several schemes have been proposed recently. Reference xiii proved that, even with no CSIT, the transmitters can still align interference only based on the knowledge of the temporal correlation structures of the channel variations associated with different users. In general, if the channel coherence does not display any special properties, the supersymbol architecture based on staggered block fading model cannot be created, so the problem of blind interference alignment remains open. In xiv, reconfigurable antennas are employed to manipulate the channel itself to create the opportunities that facilitate blind interference alignment.
In this paper, a new Alamouti coded blind interference alignment scheme is proposed, which combines interference alignment and Alamouti code xv. In this scheme, each transmitter sends four data symbols over two antennas, instead of sending two data symbols as shown in xiv, so two supersymbols are needed each time to decode the signal. In the first supersymbol we send eight data symbols, while in the second supersymbol we send the modified data symbols according to Alamouti scheme. In this scheme, we obtain an equivalent channel matrix with orthogonal structure, while the DoF remains unchanged. Then we explore the low complexity PIC group decoding proposed in xvi, and compare the impact of different grouping schemes.
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The rest of this paper is organized as follows. In section II, the system model is described. Section III contains the proposed Alamouti coded interference alignment scheme and the PIC group decoding scheme. Finally, the simulation results and the conclusion are given in section IV and section V, respectively.
Consider a K-user MIMO X channel, where each transmitter has antennas while each receiver is equipped with one reconfigurable antenna. X channel is a generalized model for Broadcast Channel (BC), Multiple-Access Channel (MAC) and interference channel. Every transmitter has an independent message for every receiver. A reconfigurable antenna is an antenna that can change its characteristics by dynamically changing its geometry xvii. Each distinct geometrical configuration corresponds to a different mode of operation. So it can switch among several preset modes, each of which has a channel that is linearly independent of the channel of other modes. In this paper, we focus on the two-user MIMO X channel with reconfigurable receive antennas, as is shown in Fig. 1.
Figure 1 Illustration of a two-userMIMO X channel with reconfigurable receive antennas
The received signal at receiver can be described as
Where, is the user index, is the time slot index, is the transmitted signal vector of transmitter and is the complex additive white Gaussian noise (AWGN) at the receiver. We assume that all noise terms are independent and identically distributed (i.i.d.) zero-mean complex Gaussian with unit variance. Furthermore, is the channel vector from transmitter to receiver , and denotes the preset mode of the receiver's reconfigurable antenna. The transmitters are required to satisfy the same power constraint . We assume that the channel vectors experience i.i.d. Rayleigh fading across time and space and are independent of receiver noise. That is, the elements of are i.i.d. standard complex Gaussian random variables across time and space. Besides, the channel coherence time should be long enough so that the channels stay constant across two supersymbols. The supersymbol will be defined later.
Blind Interference Alignment with PIC Group Decoding
In this section we first present the BIA scheme for the two-user MIMO X channel, and then explore the Alamouti coded BIA scheme and PIC group decoding.
Blind Interference Alignment
For the two-user MIMO X channel, our goal is to achieve 4/3 DoF. This can be done by sending four data symbols over three time slots. That is, every transmitter sends one data symbols to each receiver over three time slots. Because each data symbol carries one DoF, we can achieve 4/3 DoF. Although the transmission procedure is similar to the BIA scheme in xiv, there are still some differences. The channel model used in xiv is MISO broadcast channel, but we employ MIMO X channel, which is more general than the former.
First, the transmitted signal can be written as equation (2) when .
Where and are two independent data symbols intended from transmitter to receiver . At receiver , we seek to obtain the desired data symbols , while eliminate the data symbols intended to other receivers.
The channel vector defined above can be written as equation (3) when .
We only consider receiver 1 in the following discussion, because the processing at receiver 2 is similar. In the first time slot, the two transmitters send data symbols intended to both receivers. That is, transmitter 1 sends and , and transmitter 2 sends and . Besides, the receiver 1 is configured to mode 1, i.e. . So the received signal at receiver 1 is
Receiver 1 intends to decode and , and the data symbol and are interference. So we rewrite equation (4) to separate the intended signal from interference, as shown in (5).
In the second time slot, the two transmitters only send data symbols intended to receiver 1, and the receiver 1 is configured to mode 2. So the second received signal at receiver 1 is
In the third time slot, the two transmitters only send data symbols intended to receiver 2, and the receiver 1 is configured to mode 1 again. So the third received signal at receiver 1 is
Combining equation (5), (6) and (7) into a single equation, as is shown in (8).
From equation (8), we note that receiver 1 can eliminate the interference by simply subtracting the third received signal from the first one, as is shown in equation (9). This operation does not require any CSIT and produces interference-free signals, although the noise is doubled over the first signal. Finally, we can decode and from equation (9).
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As stated above, every transmission needs three time slots, so we name every three time slots as a supersymbol.
Alamouti Coded Blind Interference Alignment
To improve the BER performance of the BIA scheme, we seek to combine interference alignment with space-time coding. We can employ multiple supersymbols to construct space-time code. That is, we transmit the same data symbols over multiple supersymbols according to space-time code. However, to attain the same data rate, we have to send more data symbols in every supersymbol.
In the particular scenario of two-user 2-1 MIMO X channel, to attain 4/3 DoF, we can send eight data symbols twice over six time slots (two supersymbols). That is, in the first supersymbol we send eight data symbols, while in the second supersymbol we send the modified eight data symbols according to Alamouti scheme. So transmitter needs to send four data symbols, in which and are intended to receiver 1, while and are intended to receiver 2. This can be done by sending a superposition of two data symbols on each antenna, as is shown in equation (10).
According to equation (1) and (3), the received signal of receiver 1 is expressed as
For receiver 1, , ,and are intended signal and the other four symbols are interference, so we rewrite equation (11) to separate them.
Equation (12) presents the received signal in one time slot, which will be used to construct supersymbols in the following derivation.
We construct a supersymbol in the same way as the previous subsection did. That is, the two transmitters send data symbols intended to both receivers in time slot 1, send data symbols intended to receiver 1 in time slot 2, and send data symbols intended to receiver 2 in time slot 3, while the receive antenna is configured to mode 1, mode 2 and mode 1, respectively. Therefore, the received signal at receiver 1 in the first supersymbol is
To eliminate the interference, we subtract the third received signal from the first one, as is shown in (14)
We note that there are two equations including four unknown variables, so obviously we need another supersymbol to decode the received signals.
In the second supersymbol, we retransmit the eight data symbols in (10) according to the Alamouti scheme xv, as is shown in (15).
Using the same scheme discussed above, the received signal at receiver 1 in the second supersymbol is
We want to decode and ,, so we can remove the conjugate transpose of and to obtain
After subtracting the third received signal from the first one, the received signals without interference is
Thus, the other two equations including the four unknown variables are obtained. We combine equation (14) and (18), as is shown in (14).
Finally, we get four equations containing four unknown variables, so we can decode the desired signal. Besides, by employing Alamouti Code, the first two columns and the second two columns of the channel coefficients are orthogonal respectively. This property can be exploited in the PIC group decoding, which will be discussed in the following subsection.
These two schemes have similar equivalent channel matrix. The difference is that the channel coefficients are totally i.i.d. in the original scheme, but partly orthogonal in the proposed scheme. Although we can obtain optimal performance if we use ML decoding at receivers, the complexity is too high to implement, and the orthogonal property makes no advantages. So we study the low complexity PIC group decoding and the influence of different grouping schemes on the performance.
the channel coefficients are totally i.i.d. in the original scheme
PIC Group Decoding and Grouping schemes
Although the performance can be optimal if ML decoding is employed at receivers, the complexity is quite high. Hence we decide to employ low complexity decoding schemes, such as PIC group decoding. However, we will see later that the original BIA scheme is not suitable for PIC group decoding, because the channel coefficients are totally i.i.d. in equation (20).
PIC group decoding algorithm is an intermediate decoding algorithm between the ML decoding algorithm and the ZF decoding algorithm. The PIC group decoding provides a framework to adjust the complexity-performance tradeoff by choosing the sizes of the information symbol groups.
First, define as the information symbol vector that contains two symbols of . For example,
We can similarly define, which contains two column vectors of , whereare the column vectors of the equivalent channel matrix in (19). With these notations, equation (19) can be written as
Suppose that we want to decode the first symbol group. In the ZF decoding algorithm, to decode the symbol, the interferences from the other symbols are completely eliminated by a linear filter. The same idea can be applied here. We want to find a matrix such that by multiplying by to the left (linear filtering), all the interferences from the second symbol groups can be eliminated. Such a matrix has been defined in xvi
Due to, after projecting the received signal is
From (18), we can see that by passing the received signal vector through the linear filter, the interference from the second symbol group is completely canceled and the output only contains the component of the first symbol group.
Then we can decode using ML algorithm. The complexity of the ML decoding of the dimension-reduced system in (18) is obviously lower than that of the original system in (14). The PIC group decoding algorithm can be viewed as a decomposition of the original high-dimensional decoding problem with high complexity into low-dimensional decoding problem with relatively low decoding complexity.
Besides, different symbol grouping schemes result in different performance. We can normally group the first and the second column of the equivalent channel matrix, and denote in this as [1 2; 3 4] grouping. However, when grouping the first and the third column, denoted as [1 3; 2 4] grouping, the performance should be better. This is because when cancelling the interference, the orthogonal columns have little impact on each other.
In this section, we investigate the performance of our proposed Alamouti coded interference alignment scheme.
To compare the performance of the original BIA scheme and the Alamouti coded BIA scheme, they should have the same number of antennas. For a two-user MIMO X channel, there are a total of four transmit antennas and two receive antennas. So we choose the two-user MISO broadcast channel (BC), which also has four transmit antennas and two receive antennas xiv.
In the original BIA scheme for two-user MISO BC, the received signals at user 1is
We note that the noise is doubled over the first three symbols, which will obviously degrade the BER performance. The noise in the proposed scheme is doubled over two symbols, while the noise is doubled over three symbols in equation (4). Less noise The result shows that the proposed scheme works better than the original one.
In order to compare the performance of the original scheme and our proposed scheme properly, i.e., compare them with the same antenna number, we employ the two-user MISO broadcast channel in xiv as our counterpart, which also has four transmit antennas and two reconfigurable receive antennas. The DoF of this channel setting is 8/5, but the DoF of our proposed scheme is 4/3, so we employ 32-QAM modulation in the original scheme, and employ 64-QAM modulation in the proposed scheme, to obtain the same sum rate.
We now consider BER performance of our proposed Alamouti coded interference alignment scheme. As shown in Fig. 2, different PIC grouping schemes have little impact on the BER performance of the original scheme. This is because the equivalent channel coefficients of the original scheme are totally i.i.d. However, in the proposed scheme, different PIC grouping schemes result in different BER performance. We note that the [1 3; 2 4] grouping has better performance, due to the reasons explained above.
Figure 2 The BER performance of the original BIA scheme in xiv and the proposed scheme.
We propose a new Alamouti coded blind interference alignment for multi-user MIMO X channel, which combines interference alignment with space-time coding. In this scheme, we send more data symbols in every supersymbol and send the same data symbols several times, enabling us to construct Alamouti Code among supersymbols. With the Alamouti structure, the columns of the equivalent channel matrix are orthogonal. Low complexity decoding algorithms, such as PIC group decoding, can exploit this orthogonal property very well. Such property not only decreases the complexity, but also improves the BER performance. As the numerical results show, the proposed scheme has a better BER performance than the original one.
Several issues that will be studied in the future to evaluate the potential benefits of our proposed BIA scheme include the generalization of this scheme to K-user MIMO X channels and other more realistic channel models. Besides, we will seek to employ more complex space-time code in our scheme and to obtain better performance possibly some diversity gain.