Signal Detection with Interference Constellation Alignment Rui Wang and Yinggang Du Huawei Technologies, Co., Ltd. Email: [email protected] [email protected] Abstract—Although the mechanism of interference alignment and cancellation over signal scale, e.g., lattice alignment, has been extensively studied recently, there is still technical barrier towards its implementation in practical systems, where the mapping from information bits to transmission symbols is realized by channel coding and modulation. This is because the lattice code must be linear in the existing interference cancellation schemes; however, the concatenation of channel encoding and modulation may not be treated as a linear code in general. In this paper, we propose a novel scheme to detect the superposed interferences with aligned QAM constellations. Specifically, multiple interference transmitters align their QAM constellations at the signal receiver; the receiver first demodulates the aligned interference symbols with a novel constellation, then decodes the aligned interference message with conventional channel decoding mechanism and finally, performs interference cancellation and signal detection. Our proposed scheme can work even when the modulation and coding scheme is not linear. It’s shown by analysis and simulations that the proposed scheme could significantly improve the performance of the signal receiver compared with the existing receiving algorithms.

which claims that any linear combination of codewords is also a codeword, holds for these lattice codes, the receiver can decode the aligned interference according to the interference codebook. However, the application of lattice interference alignment at practical wireless systems is not trivial. In most of the wireless systems, the mapping from information bits to transmission symbols is separated into two main steps: channel encoding and modulation1 . Although most of existing channel coding schemes are linear, the combined modulation and coding scheme (MCS) may not be linear. For example, the convolutional code is a linear code with alphabet {0, 1}, however, the modulation and coding scheme with convolutional code and 16QAM is no longer a linear code with a alphabet of discrete complex numbers. Hence, the signal scale alignment schemes in the existing literatures can not be directly applied onto the practical wireless systems and the following questions remain unsolved: •

I. I NTRODUCTION Interference is believed as the major bottleneck limiting the throughput of future wireless systems [1]. Recent research works reveal that interferences from multiple sources can be aligned so that “clean space” can be squeezed out to allocate extra desired signals. Generally speaking, there are two alignment methodologies: namely vector space alignment and signal scale alignment. In vector space alignment [2]–[4], interferences are scheduled into some specific vector dimensions of the receivers, so that the other vector dimensions can be used to receive desired signals. This mechanism can be implemented in the practical multi-antenna wireless systems or single-antenna system with time/frequency extension by appropriate precoding design. In this work, we shall show that the interference alignment and cancellation in signal scale can also be implemented in the existing wireless systems, where the transmission signal is generated by practical channel encoding and QAM modulation, rather than linear lattice codes. A. Related Works and Motivation There have been a number of literates focusing on the signal scale interference alignment [5]–[7]. For example, in [6] a layered lattice coding scheme for a particular class of threeuser Gaussian interference channels is introduced and more than one degree of freedom can be achieved. In [7] signal scale alignment is used to develop the approximate capacity of many-to-one and one-to-many Gaussian interference channels. In all these works, interference transmitters are assumed to use the same linear lattice code. Since the property of linearity,

•

Which modulation and coding schemes can/cannot be treated as linear codes? How to design the signal scale alignment scheme for the existing wireless systems with practical channel coding and QAM modulation?

B. Contribution We would like to shed some lights on the above questions in this paper. We first show that the MCSs with linear channel codes and QPSK can be treated as linear codes. However, we find that this conclusion cannot be extended to the MCSs with higher modulation orders. Hence, the interference alignment and detection mechanism in the existing literatures can not be applied directly in general. To address this issue, in this paper we propose a uniform framework to detect the aligned interferences with the same MCS, which may not be linear. Specifically, the interference detection procedure can be divided into two main steps: demodulation and decoding. First, we propose a novel constellation to demodulate the aligned interferences where some constellation points may be mapped to soft bits (As a special case, all the constellation points are mapped to hard bits when QPSK is used). Then, the aligned interferences can be further detected by soft decoding and the desired signals can be obtained after interference cancellation. Finally, it’s shown by theoretical analysis and simulations that the proposed scheme could significantly improve the bit error rate (BER) performance of the signal receiver compared with the existing receiving algorithms. 1 There

are also some other steps, i.e., appending parity bits and scrambling.

signal receiver, H0 ∼ CN (0, 1) be the channel between the signal transmitter and the signal receiver. Denoting Y = [Y 1 , Y 2 , ..., Y β ] as the sequence of received symbols at the signal receiver, the channel model is given by Y n = H0 X0n +

K X

Hi Xin + Z n , n = 1, 2, ..., β,

(1)

i=1

where Z n ∼ CN (0, 1) (n = 1, 2, ..., β) is the complex Gaussian noise. To satisfy the peak power constraint, we have Fig. 1.

An illustration of example scenarios

The remaining of this paper is organized as follows. In Section II, the system model is elaborated. In Section III, the proposed alignment and cancellation scheme is first proposed for QPSK-modulated interference and then, generalized to interference with M-QAM modulation. In Section IV, we compare the performance of the proposed scheme with the existing schemes by both theoretical analysis and simulations. Finally, the conclusion is drawn in Section V, where the impacts of imperfect CSI on our proposed scheme is also discussed. II. S YSTEM M ODEL A. Channel Model In this paper, we consider a wireless system with K interference transmitters, one signal transmitter and one signal receiver. To simplify our elaboration, it’s assumed that each node has single antenna, however, our work can easily be extended to the scenario with multiple antennas. The interference transmitters deliver information to their own receivers respectively (which are outside the scope of our interests in this paper), raising strong interference at the signal receiver. To facilitate the signal receiving, the interference transmitters are assumed to cooperate and align their constellations at the signal receiver. Hence, the signal receiver can detect and cancel the interference before signal detection. This system configuration matches a number of application scenarios. For example in Fig. 1 (a), in Heterogeneous Networks (HetNet) defined by 3GPP [8], if one picocell locates at the edge of two macrocells, the uplink signals of the macrocell users at cell edge can be aligned at the pico base station (BS), so that it can detect the uplink signal of the pico users. Moreover, in Fig. 1 (b), if one picocell is deployed close to one macro BS, the multiple antennas of the macro BS can align their signals at the pico user, so that it can detect the downlink signal from the pico BS. Let Xi = [Xi1 , Xi2 , ..., Xiβ ] be a sequence of transmission symbols within one code block at the i-th interference transmitter, [X01 , X02 , ..., X0β ] be the sequence of transmission symbols at the signal transmitter, Hi ∼ CN (0, 1)2 be the channel between the i-th interference transmitter and the

Xin (Xin )H ≤ Pi n = 1, 2, ..., β; i = 0, 1, 2, ..., K,

where AH is the conjugate of A, P0 and Pi (i = 1, 2, ..., K) are the maximum transmission power of the signal transmitter and the i-th interference transmitter respectively. As elaborated in (1), the signal receiver should detect the desired signal in the presence of interferences. There are two existing approaches in signal detection: • Baseline 1: The signal receiver detects the desired signal by treating all the interferences as noise. • Baseline 2: The signal receiver constructs an demodulation constellation which is a superposition of the signal constellation and the interferences’ constellations, and detect the desired signal according to this constellation [9]. The Baseline 1 is only suitable for the regime with small interference power. The Baseline 2 utilizes the modulation structure of the interferences, leading to better performance than Baseline 13 . In the remaining of this paper, we shall show that a better performance can be achieved by aligning and decoding the superposed interference signals. Specifically, we can align the constellations of the interferences, and decode the superposed interference message as if the interference were transmitted by one single source with a particular channel coding scheme and modulation constellation. Since the detection on interference is protected by channel coding, the accuracy of interference cancellation can be improved, leading to better performance on signal detection. To elaborate the signal detection algorithm, we first introduce the model of the interference transmitters in the following section. B. Interference Transmitter Model Although the baseband signal processing at transmitters may vary a lot in different wireless systems, it can generally be divided into the four steps, namely appending (CRC) parity bits, channel encoding, scrambling and modulation4 . In this paper, we have the following assumptions on the baseband signal processing at all interference transmitters: 1. Each interference transmitter has the same size of code block (Thus, each interference transmitter collects same number of information bits for channel encoding), and the transmission opportunity of each code block at each interference transmitter is aligned. 3 The

2 CN (0, 1)

variance 1.

denotes complex Gaussian distribution with zero mean and

(2)

performance of Baseline 1 and 2 is compared in [9] may also be interleaving and bit puncturing at the transmitter. They are treated as components of channel encoding procedure in this paper. 4 There

2. The CRC generator polynomial at each interference transmitter is the same. 3. All interference transmitters use the same modulation scheme and linear channel coding scheme. Remark 1 (Mild Assumptions): The above assumptions is analogy to the assumption in lattice alignment [6], [7] that all interference transmitters use the same lattice code. In fact, these assumptions are mild. For example, the first and second assumptions are naturally satisfied in LTE system, and the third assumption can be satisfied by some cooperation between interference transmitters. Let the information bits delivered by the i-th interference transmitter be ai = [a1i , a2i , ..., aα i ], where α is the size of code block, the baseband signal processing at the i-th interference transmitter (i = 1, 2, ..., K) is modelled in the following. • Appending Parity Bits: γ parity bits are generated and appended with the information bits according to mapping γ 1 1 2 α [a1i , a2i , ..., aα i , pi , ..., pi ] = Fcrc ([ai , ai , ..., ai ]),

where [p1i , ..., pγi ] are the parity bits. Fcrc is the same at all interference transmitters. Moreover, we assume that the same number of zeros are padded before calculating the parity bits at all interference transmitters. Hence, it can be verified that, ∀i, j [a1i ⊕ a1j , ..., pγi •

⊕ pγj ]

=

α Fcrc ([a1i ⊕ a1j , ..., aα i ⊕ aj ]),

(3)

where ⊕ denotes the exclusive OR (XOR) operation. Channel Coding: The coded bit sequence [b1i , b2i , ..., bmβ i ] is generated according to one linear channel coding scheme γ 1 2 α 1 [b1i , b2i , ..., bmβ i ] = Fenc ([ai , ai , ..., ai , pi , ..., pi ]),

where m = log2 M and M is the modulation level used by the interference transmitters. Since the same linear channel code is used at all interference transmitters, we have ∀i, j,

•

γ γ 1 1 [b1i ⊕ b1j , ..., bmβ ⊕ bmβ i j ] = Fenc ([ai ⊕ aj , ..., pi ⊕ pj ]). (4) Scrambling: Let [s1i , s2i , ..., smβ ] be the scrambling sei quence at the i-th interference transmitter, [c1i , c2i , ..., cmβ i ] be the bit sequence after scrambling, we have cki = bki ⊕ski (k = 1, 2, ..., mβ). Thus, mβ i 1 2 [c1i , c2i , ..., cmβ i ] = Fscr ([bi , bi , ..., bi ]).

As a remark notice that the scrambling sequences of different interference transmitters can be different, which usually depend on the user ID in cellular systems. For the elaboration convenience, we define Fi,j scr as the scrambling mapping using scrambling sequence [s1i ⊕ ⊕ smβ s1j , ..., smβ i j ]. Hence, ∀i, j mβ i,j 1 1 [c1i ⊕ c1j , ..., cmβ ⊕ cmβ ⊕ bmβ j ] = Fscr ([bi ⊕ bj , ..., bi j ]). i (5)

•

M-QAM Modulation: Every m = log2 M bits in [c1i , c2i , ..., cmβ i ] is mapped to one M-QAM symbol, thus [Ui1 , Ui2 , ..., Uiβ ] = Fmod ([c1i , c2i , ..., cmβ i ]).

•

Constellation Rotation: To align the interference constellation at the receiver, the modulated symbols are rotated and scaled by one complex number Ti . Thus, we have Xin = Ti Uin n = 1, 2, ..., β, where Ti is given by min

Ti =

n=1,2,...,K

√ |Hn | Pn e−∠Hi .

|Hi |

C. Linearity of the Interference Transmitters For the elaboration convenience, we define the mapping from the information bits to the transmission symbols as Fiinf = Fmod · Fiscr · Fenc · Fcrc . Thus, Xi = Ti ∗ Fiinf (ai ). Hence, we have the following lemma on the linearity of interference transmitter with QPSK modulation. Lemma 1 (Linear Interference Transmitter): When QPSK is used, the interference transmitters Fiinf (i = 1, 2, ..., K) are linear in superposition at the signal receiver with appropriate modulo and shift. Specifically, Hi ∗ Ti ∗ (Fmod · F1,...,K · Fenc · Fcrc )(a1 ⊕ ... ⊕ aK ) scr X  K = Hi Ti Fiinf (ai ) mod Λ − S, (6) i=1

where S = H√i T2 i + i H√i T2 i , the modulo operation is applied on each element, and Λ denotes the square Voronoi cell in the complex plane specified by

−

H√ i Ti 2

− i H√i T2 i , − H√i T2 i +



i Ti √i Ti , 3H √i Ti + i 3H √i Ti , 3H √i Ti − i H√ i 3H . 2 2 2 2 2

Proof: With the modulo and shift operation, the superposition of interferences is equivalent to applying element-wise XOR on the bit sequences after channel coding. Hence, the conclusion is straightforward due to the linearity in (3,4,5). The conclusion in Lemma 1 can not be extended to the situation with higher modulation levels. Fig. 2 illustrates an example with two 16QAM-modulated interferences, where the highlighted constellation point X may be obtained by superposing the constellation points A and B or the points C and D. The two combinations ask for different bit mapping on X. Hence, the proof of Lemma 1 cannot be applied and the linearity of the interference transmitters does not hold any more. To address this issue, we shall propose in the next section a general framework to decode the aligned interferences with arbitrary QAM modulation, where novel interference demodulation constellation is introduced. Remark 2 (Differentiation from Linear Lattice Code): In the existing literatures on lattice alignment, it’s assumed that all the interference transmitters use the same linear lattice code Cla . Hence, the aligned interference at the signal receiver is still one codeword in Cla , and the interference

Fig. 2.

An illustration of aligning two 16QAM constellations.

detection is naturally followed. Our proposed scheme is different from the existing literatures since we consider the practical channel coding and modulation schemes instead of linear lattice codes. Specifically, aligning two sequences of symbols generated by the same coding and modulation scheme, denoted as Cmcs , may not always lead to another sequence in Cmcs . Hence, decoding the aligned interference is not trivial in our problem. III. I NTERFERENCE A LIGNMENT & C ANCELLATION In this section, we shall first propose an interference alignment and signal detection algorithm for QPSK-modulated interferences and then, extend the algorithm to align the interferences with general M-QAM. A. Interference with QPSK Modulation In our proposed algorithm, there are two detection procedures at the receiver: (i) to detect the aligned interferences and; (ii) to detect desired signal according to known interference. Due to the linearity in (3,4,5), the superposed interference is decodable in the first procedure if [c11 ⊕ ... ⊕ c1K , ..., cmβ ⊕ 1 mβ ... ⊕ cK ] can be estimated from the received symbol sequence Y = [Y 1 , Y 2 , ..., Y β ]. In the second procedure, the receiver can partially cancel the interference from the received signal sequence, and the desired signal can be detected in the presence of residual interference. In the following, we first introduce two demodulation constellations for the above two procedures and then, summarize the overall receiving algorithm. Definition 1 (Demodulation Constellation for Interference): A constellation demodulating the aligned interference with QPSK modulation is constructed by the following two steps: • The set of points on this constellation, denoted as Calign , is constructed by  K X H0 Calign = C0 + Ci C1 , ..., CK ∈ CQP SK ; H1 T1  i=1 C0 ∈ Csig , where CQP SK and Csig are the sets of points on the QPSK constellation and signal constellation, respectively.

Fig. 3.

•

∀C =

An illustration of the constellation in Definition 1

H0 H 1 T1 C 0

+

K P

Ci , where C1 , ..., CK ∈ CQP SK

i=1

and C0 ∈ Csig , the mapping from constellation points to demodulated bits is given by falign (C) = fQP SK (C1 ) ⊕ ... ⊕ fQP SK (CK ),

(7)

where ⊕ denotes the element-wise XOR, and fQP SK : CQP SK → {0, 1}2 is the bit mapping on the QPSK constellation points. Remark 3 (Example on Interference Demodulation): Fig. 3 illustrates how to construct the constellation according to Definition 1 when K = 2: the two interference constellations are first superposed to form a medium constellation (whose bit mapping is the XOR of the original bit mappings); the signal constellation is then superposed with the medium constellation (The bit mapping follows that of the medium constellation). It’s clear that demodulation according to this constellation can obtain an unbiased estimation of the XOR of the coded interference bits. Remark 4 (Uniqueness on Bit Mapping): Some constellation points on Calign can be constructed by various combinations of QPSK points. For example in Fig. 3, when the two interference symbols are (01, 10) or (00, 11), the superposed interference symbols are the same. Despite of various possible superposition combinations, it can be easily proved that the bit mapping of each constellation point in Calign is unique. Definition 2 (Demodulation Constellation for Signal): A constellation demodulating the desired signal given two coded bits of aligned interference [b1 , b2 ] is constructed by the following two steps: • The set of points on the signal constellation is given by  K X H0 C0 + Ci C1 , ..., CK ∈ CQP SK ; Crx = H1 T1 i=1 fQP SK (C1 ) ⊕ ... ⊕ fQP SK (CK ) = [b1 , b2 ];  C0 ∈ Csig . •

PK ∀C = HH1 T0 1 C0 + i=1 Ci , where C1 , ..., CK ∈ CQP SK and C0 ∈ Csig , the bit mapping is given by frx (C) = fsig (C0 ),

(8)

received symbol Y n (n = 1, 2, ..., β). The remaining processing on the demodulated information bits is the same as the conventional receiver. B. Interference with General QAM

Fig. 4. An illustration of the constellation in Definition 2 when the coded bits of the aligned interference are 00

where fsig is the bit mapping of original signal constellation. Remark 5 (Example on Signal Demodulation): Fig. 4 illustrates how to construct the signal demodulation constellation given two coded interference bits 00. As a remark notice that when the coded interference bits are 00, 01 or 10, the interference cannot by completely cancelled before demodulating the desired signal; whereas, when the coded interference bits are 11, the interference can be completely cancelled. As shown in this figure, even though the interference is only partially cancelled, the distance between constellation points is generally larger than that in Fig. 3, and therefore, the demodulation performance is improved. This performance gain comes from the interference detection procedure. As a result, the receiving algorithm is elaborated below: Algorithm 1 (Receiving Algorithm for QPSK Interference): n

1. Demodulate the normalized received symbols { HY1 T1 |n = 1, 2, ..., β} according to the constellation in Definition 1, yielding [ˆ c1 , cˆ2 , ..., cˆ2β ] which is an unbiased estimation 2β 1 of [c1 ⊕ ... ⊕ c1K , ..., c2β 1 ⊕ ... ⊕ cK ]). 1 2 2β 2. Descramble [ˆ c , cˆ , ..., cˆ ] by sequence [s11 ⊕ ... ⊕ 2β 1 ˆ1 ˆ2 ˆ2β sK , ..., s1 ⊕ ... ⊕ s2β K ], yielding [b , b , ..., b ]. 3. Decode [ˆb1 , ˆb2 , ..., ˆb2β ] by the channel decoding algorithm ˆ2 , ..., a ˆα , pˆ1 , ...ˆ pγ ]. corresponding to Fenc , yielding [ˆ a1 , a 4. Use the last γ parity bits, [ˆ p1 , ..., pˆγ ], to correct potential error in transmission, yielding [˜ a1 , a ˜2 , ..., a ˜α ]. As a remark notice that if the channel coding scheme of the interference transmitter is sufficiently strong and the code rate is sufficiently low, [˜ a1 , a ˜2 , ..., a ˜α ] = [a11 ⊕ ... ⊕ 1 α α aK , ..., a1 ⊕ ... ⊕ aK ] holds with high probability. 5. Reconstruct the coded bits of aligned interference from [˜ a1 , a ˜2 , ..., a ˜α ] by the mapping F1,...,K · Fenc · Fcrc , scr yielding [˜ c1 , c˜2 , ..., c˜2β ]. As a remark notice that if α [˜ a1 , a ˜2 , ..., a ˜α ] = [a11 ⊕ ... ⊕ a1K , ..., aα 1 ⊕ ... ⊕ aK ], then 2β 2β [˜ c1 , c˜2 , ..., c˜2β ] = [c11 ⊕ ... ⊕ c1K , ..., c1 ⊕ ... ⊕ cK ]. 6. Use every two bits in [˜ c1 , c˜2 , ..., c˜2β ] to assist the demodulation of desired signal according to Definition 2. Thus, the information bits c˜2n−1 c˜2n is used to demodulate the

As illustrated in Fig. 2, when general M-QAM is used at the interference transmitters, there may not exist a unique bit mapping on the aligned constellation such that [c11 ⊕ ... ⊕ c1K , ..., cmβ ⊕ ... ⊕ cmβ 1 K ] can be estimated in an unbiased manner. In this section, we address this issue by introducing soft bits on the interference demodulation constellation. Specifically, the demodulation constellation for the aligned interference and the desired signal are introduced below. Definition 3 (Demodulation Constellation for Interference): A constellation demodulating the aligned interference with M-QAM constellation is constructed by the following two steps: M • The set of points on this constellation, denoted as Calign , is constructed by   K X H0 M Ci C1 , ..., CK ∈ CM QAM ; C0 ∈ Csig , C0 + Calign = H1 T1 i=1

•

where CM QAM is the set of points on the M-QAM constellation. M ∀C ∈ Calign = , define the set S(C)  K H P Ci = C, C1 , ..., CK ∈ (C1 , C2 , ..., CK ) H1 T0 1 C0 + i=1  CM QAM , C0 ∈ Csig . Hence, the mapping from constellation points to demodulated bits is given by P fM QAM (C1 ) + ... + fM QAM (CK ) M falign (C) =

S(C)

|S(C)|

,

where + denotes the element-wise summation, |S(C)| is the cardinality of S(C), and fM QAM : CM QAM → {0, 1}log2 M is the bit mapping on the M-QAM constellation points. Remark 6 (Example on Bit Mapping): Consider an example with K = 2 and M = 16. The aligned interference constellation is illustrated in Fig. 2. The highlighted point X in the aligned constellation can be obtained by six different combinations: {(0111, 1111), (0011, 1011), (0010, 1001), (0110, 1101), (1110, 0101), (1010, 0001)}. Applying elementwise XOR on each combination, we have {1000, 1000, 1011, 1011, 1011, 1011}. Hence, the bit mapping of this constellation point is (1, 0, 2/3, 2/3). Note that the last two bits of this constellation point are soft. The value 2/3 means that there is a larger probability that they are mapped to 1. Definition 4 (Demodulation Constellation for Signal): A constellation demodulating the desired signal given m = log2 M coded bits [b1 , ..., bm ] of the aligned interference is constructed by the following two steps:

•

The set of points on the constellation is given by

A. Analysis

In the analysis, we assume there are two interference transmitters in the system using QPSK modulation. The = transmission symbol of the signal transmitter is given by √ m −∠H0 P0 U0m (m = 1, 2, ..., β), where U0m is one fM QAM (C1 ) ⊕ ... ⊕ fM QAM (CK ) = [b1 , ..., bm ];X0 = e  QPSK symbol. Moreover, to simplify the analysis, we assume C0 ∈ Csig . interference constellations are also aligned in Baseline 2. Hence, we have the following lemma. PK Lemma 2 (Performance Comparison): Suppose the four H0 • ∀C = H T C0 + i=1 Ci , where C1 , ..., CK ∈ CM QAM 1 1 transmission symbols of each interference transmitter appear and C0 ∈ Csig , the bit mapping is given by with equal probability. For a given channel realization between M the transmitters and receiver {H0 , H1 , ..., HK }, we have frx (C) = fsig (C0 ). • The symbol error rate of the Baseline 1 is 2   v Hence, the receiving algorithm is given below. 1u u 2|H0 |2 P0 Algorithm 2 (Receiving Algorithm for M-QAM Interference): , (9) PBL1 = 1 − 1 − Q u K P 2t 1+ Pi n i=1 1. Demodulate the normalized received symbols { HY1 T1 |n = where Q(·) is the Q-function defined in [10]. 1, 2, ..., β} according to the constellation in Definition 3, • With the constellation alignment, the symbol error rate yielding [ˆ c1 , cˆ2 , ..., cˆmβ ] where some bits may be soft. of the Baseline 2 is 2. Descramble [ˆ c1 , cˆ2 , ..., cˆmβ ] according to  1 2 mβ [ˆb , ˆb , ..., ˆb ] = [|s11 ⊕ ... ⊕ s1K − cˆ1 |, ..., |smβ ⊕ 1 1 mβ mβ PBL2 = E{H1 ,...,HK } 1 − (4 − 4Qa − 3Qb + 3Qb+2a ... ⊕ sK − cˆ |]. Note that there may be soft bits 16 in [ˆ c1 , cˆ2 , ..., cˆmβ ], the minus is used in descrambling +3Q2b+a − 3Q2b+3a − Q3b+2a + Q3b+4a  instead of exclusive OR. 1 2 mβ 2 3. Decode [ˆb , ˆb , ..., ˆb ] by the channel decoding +Q4b+3a − Q4b+5a ) , (10) algorithm corresponding to Fenc , yielding √ √ | P0 [ˆ a1 , a ˆ2 , ..., a ˆα , pˆ1 , ...ˆ pγ ]. Note that soft decoding √0 where Qx = Q(|H1 T1 |x), a = |H and b = 22 − a. 2H1 T1 may be needed as there may be soft bits. • If the receiver can successfully decoded the aligned inter4. Use the last γ parity bits, [ˆ p1 , ..., pˆγ ], to correct potenference, the symbol error rate of the proposed interference tial error in transmission, yielding [˜ a1 , a ˜2 , ..., a ˜α ]. As a alignment and cancellation scheme is remark notice that if the channel coding scheme of the  1 interference transmitter is sufficiently strong and the code PIC = E{H1 ,...,HK } 1 − (4 − 4Qa − Q2b+a + Q2b+3a 1 2 α 1 16 rate is sufficiently low, [˜ a ,a ˜ , ..., a ˜ ] = [a1 ⊕ ... ⊕  α ] holds with high probability. ⊕ ... ⊕ a a1K , ..., aα 2 1 K +Q4b+3a − Q4b+5a ) (11) 5. Reconstruct the coded bits of aligned interference from 1 2 α 1,...,K · Fenc · Fcrc , [˜ a ,a ˜ , ..., a ˜ ] by the mapping Fscr Proof: The proof of this lemma can be established by yielding [˜ c1 , c˜2 , ..., c˜mβ ]. analysing the region of successful demodulation. 6. Use every m = log2 M bits in [˜ c1 , c˜2 , ..., c˜mβ ] to assist To compare the three symbol error rate in the above lemma, the demodulation of desired signal according to Defi- we let P = ... = P 1 K = ηP0 , where η > 1 (we nition 4. The remaining processing on the demodulated are interested in the strong interference scenario). It can be information bits is the same as the conventional receiver. observed that P and P decrease with the growing P , M Crx



K X H0 C0 + Ci C1 , ..., CK ∈ CM QAM ; H1 T1 i=1

BL2

IC

0

Notice that with interference detection, there are less constellation points in Definition 4 than that in Definition 3. Hence, the quality of signal detection can be improved as the inter-symbol distance on the constellation is enlarged. Moreover, note that when QPSK is used at the interference transmitters, Algorithm 2 will reduce to Algorithm 1. Thus, Algorithm 1 is a special case of Algorithm 2.

however, PBL1 is almost unchanged when increasing P0 . Hence, both Baseline 2 and the proposed scheme have much better SER performance than Baseline 1. Moreover, it can be proved that PIA > PBL2 is always true for Gaussian noise. This demonstrates the benefits on interference decoding and cancellation.

IV. P ERFORMANCE C OMPARISON

Fig. 5 and 6 compare the proposed scheme with Baseline 1 and 2 by simulations. In both figures, there are two interference transmitters in the system, and the convolutional code is used. In Fig. 5, all the interference and signal transmitters use QPSK. The code rates of the interference and signal are 1/2 and 3/4 respectively. In Fig. 6, interference transmitters use 16QAM

In this section, we shall compare the performance of the proposed interference alignment and cancellation scheme with the two baselines elaborated in Section II-A by both analysis and simulations.

B. Simulations

0

0

10

BER

BER

10

−1

10

Baseline 1, INR=10dB Baseline 2, INR=10dB Proposed, INR=10dB Baseline 1, INR=20dB Baseline 2, INR=20dB Proposed, INR=20dB

−2

10

0

5

10 Signal SNR (dB)

−1

10

Baseline 1, INR=20dB Baseline 2, INR=20dB Proposed, INR=20dB Baseline 1, INR=30dB Baseline 2, INR=30dB Proposed, INR=30dB

−2

15

20

10

0

5

10

15 Signal SNR (dB)

20

25

Fig. 5. Comparison of BER performance, where the MCS of interference transmitter is QPSK + CC 1/2 and the MCS of the signal transmitter is QPSK + CC 3/4.

Fig. 6. Comparison of BER performance, where the MCS of interference transmitter is 16QAM + CC 1/2 and the MCS of the signal transmitter is QPSK + CC 1/2

and the signal transmitter uses QPSK. The code rates of all the transmitters are 1/2. In the simulation, it’s assumed that P1 = P2 and the curves for different interference powers (Interference-to-noise ratio, INR) are drawn. It can be observed that the Baseline 2 performs better than the Baseline 1. This is because structure information of the interference is utilized in the Baseline 2. Moreover, the proposed scheme has significant performance gain over the two baselines. The gain is because (1) with interference constellation alignment, some constellation points of the interference transmitters are overlapped, hence, its distortion to the desired signal is reduced; (2) with the interference detection, interference can be partially cancelled before signal detection.

nel estimation and constellation alignment, which models a more common situation in practical systems. Specifically, the receiver can still use the aligned constellation in interference and signal detection, and the alignment error can be treated as an equivalent noise. Moreover, the aligned constellation should be designed by considering the distribution of alignment error.

V. C ONCLUSION & D ISCUSSION In this paper, a general framework for interference constellation alignment and signal detection is proposed. We show that when QPSK is used, the interference transmitters are linear in the sense that the aligned interference can be treated as if it were generated by single transmitter with QPSK modulation. However, when higher modulation levels are used, this linear property does not hold. To tackle this issue, we propose a general algorithm to construct a demodulation constellation to detect the aligned interference, where soft bit mapping may be used. With the detected interference, the quality of signal detection can be improved. Finally, we compare the performance of the proposed scheme with the existing receiving algorithms, where significant performance gain can be observed. While this paper assumes perfect channel estimation and constellation alignment at the signal receiver, the proposed scheme can be applied in the scenario with imperfect chan-

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