Constellation Shaping and LDPC Coding in a ... - IEEE Xplore

Constellation Shaping and LDPC Coding in a Bidirectional Full Duplex Communication ∗

A. Sadeghi∗ , F. Lahouti+∗ , M. Zorzi# WMC Lab, School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Iran + Electrical Engineering Department, California Institute of Technology, USA # Department of Information Engineering, University of Padova, Italy [email protected], [email protected], [email protected]

Abstract—We investigate the ability of the physical layer approaches of constellation shaping and forward error correction to enhance the performance of full duplex (FD) wireless radio systems. We use low density parity check and optimized shaping codes to transmit non-equiprobable constellation points from an amplitude phase shift keying modulation with imperfect power amplifier and in the presence of FD residual self interference. A simple approach to the design of optimized shaping codes for a given signal to noise ratio and residual self interference power is presented. Simulation results show that a FD transceiver with bit interleaved coded modulation, optimized shaping and iterative demodulation and decoding noticeably improves the performance in comparison to the same system without constellation shaping.

Keywords: Full Duplex Wireless, Constellation Shaping, LDPC coding, Nonlinear Self Interference Channel. I. I NTRODUCTION The wireless revolution has caused a drastically increasing demand on the scarce wireless spectrum resources. Currently, almost all of the wireless bidirectional systems rely on half duplex (HD) transmission strategies, using either time (TDD) or frequency (FDD) division duplexing. The full duplex (FD) radio technology can enhance the spectrum efficiency by enabling simultaneous transmission and reception in the same frequency band at the same time. This new emerging technology has the potential to double the physical layer capacity [1]. A FD physical layer can also be utilized for more efficient communications at the higher layers. FD operation can enable a wireless radio to detect collisions while transmitting in a contention based network, or to receive instantaneous feedback over the same channel from the other party. Consequently, redesigning higher layers of the OSI model such as Medium Access Control (MAC), while considering this new capability at the physical layer, can improve the throughput even more than 2.5X in comparison with conventional HD networks [1]– [3]. The main obstacle to enabling a wireless node to operate in FD mode is the so called self interference (SI). It is the high ratio of SI’s power to the desired signal’s power that makes FD challenging. This power difference is simply due to the fact that the SI signal has to travel a much shorter distance compared to the desired signal. Thus, the desired signal soaks in the SI; accordingly, an important characteristic of a FD radio is its ability to suppress its own SI from the coupled received signal. In principle, since all the transmitted data is known at the transceiver, the SI waveform can be regenerated at the receiver and mostly eliminated. However, because of

the many imperfections in the transmitting or receiving chains such as phase noise, nonlinearities, I/Q imbalances, receiver quantization noise and constraints on the accuracy of the SI channel estimation, the FD transceivers are incapable of fully canceling the SI signal [4], [5]. Suppressing SI in a FD radio via a combination of active and passive suppression mechanisms has been investigated in several works [6]–[16]. Passive suppression is a wirelesspropagation-domain mechanism to attenuate the SI signal. Antenna separation used in [8], [10], directional antennas [17], cross polarization [18], transmit beamforming [19], [20] or use of a circulator in the TX/RX antenna [7] are the most common passive suppression mechanisms. Different analog and digital active cancellation techniques are proposed to suppress SI. Analog cancellation refers to canceling SI before it enters the receiver front end. Unless analog cancellation is used, the receiver front end will saturate due to the high power of SI and the limited dynamic range of analog to digital converters (ADCs) [10]. The authors in [8] have used a transformer called Balun in the analog domain in order to create a perfectly inverted copy of the transmitted signal. In [9] the authors propose an extra transmit chain in order to generate an analog cancellation signal. In [7], the analog canceling signal is generated by passing the original analog signal through a number of parallel wire lines with different but fixed delays and variable attenuations. Following passive and active analog SI cancellation, an active digital canceler aims to clean out any remaining SI in the digital baseband domain. The performance of digital SI cancelers is typically constrained by the limited dynamic range of analog to digital converters. Thus, sufficient suppression of self interference by passive and analog mechanisms is needed before a digital baseband technique can be utilized. The residual self interference (RSI) which remains after all active and passive cancellation steps is similar to noise, uncorrelated from the original transmitted signal, and consequently cannot be estimated and eliminated easily [7], [9], [21]. The degraded performance in a FD wireless radio is mainly due to this RSI which prohibits a perfect FD communication. The more RSI at the transceiver the lower the SINR and consequently the worse the performance. In this paper, we mitigate the negative effects of RSI in the performance of a FD communication system by constellation shaping and forward error correction coding. Constellation shaping is a technique to transmit lower power constellation points with higher probability compared to higher power constellation points. We consider an experimentally driven channel model for RSI, which takes into account the imperfection

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Fig. 1: A two user bidirectional full-duplex communication.

(nonlinearity) of the transmit power amplifier. In addition to the conventional shaping gain due to the resulting non-uniform modulation constellations, shaping particularly helps in the FD setting, as the more frequent use of symbols with lower power leads to reduced distortion imposed to the system and hence improves the performance. To the best of the authors’ knowledge this is the first paper to investigate the effect of constellation shaping and coding in a FD communication system and its ability to enhance the system performance. The rest of the paper is organized as follows. The RSI signal and channel model for linear and nonlinear components are presented in Section II. Section III presents the transmitter and receiver structure of the bit interleaved coded modulation with optimized shaping and iterative demodulation and decoding. It also presents the shaping code design. Simulation results are presented in Section IV and Section V concludes the paper. II. F ULL -D UPLEX T RANSCEIVER M ODELING A block diagram representing a bidirectional FD communication is depicted in Fig. 1, where two identical FD radios communicate with each other in the same frequency band at the same time. In this figure hab represents the channel between the transmit antenna of node a and the receive antenna of node b; and hSI,a shows the SI channel impulse response from the transmitter antenna to the receiver antenna at node a. The FD transceiver block diagram shown in Fig. 1 is based on [9]. This diagram is presented from the perspective of node a, but everything will be the same at node b. A. Full-Duplex Transceiver Signal Modeling In the first stage, the transmit chain will upsample and pulse shape the desired transmitting symbols from the modulator, xa [n]. The digital pulse shaped signal will be converted to the analog domain through a digital to analog converter (DAC) and finally the transmit radios will upconvert the analog baseband signal to a narrowband radio frequency signal, xRF,a (t). Because of several nonlinearities in the transmitter chain, and specifically power amplifier induced nonlinearities, the output RF signal contains higher order terms of the desired signal. These harmonics can be modeled by Taylor Series expansion [7], [22]; consequently, the RF transmitted signal is:  αm [xp (t)] m (1) xRF,a (t) = m odd

where xp (t) represents the ideal passband analog signal corresponding to xa [n]. In (1), m = 1 corresponds to the linear component (ideal passband signal, α1 = 1), and αm , m > 1 represent Taylor Series expansion coefficients to generally

model the power amplifier nonlinearities. After antenna separation each of these nonlinear components will go through a specific SI channel to be received at the receiver chain. The received SI signal at the receiver antenna can be modeled as [7]  αm [xp (t)] m ∗ hm (2) ySI,a (t) = SI,a (t) m odd th where hm order SI,a (t) indicates the SI channel that the m nonlinear component experiences from the transmitter antenna to the receiver antenna in node a; these should be estimated in order to effectively cancel the SI components. Due to the narrowband characteristic of xp (t) the convolution in (2) may be approximated by a product.

B. Residual-Self-Interference Channel Model In order to understand the effect of the RSI signal on the performance of a FD wireless radio we need to have a RSI channel model able to precisely represent the relationship between the output signal of the transmitter chain of a FD wireless radio and the corresponding input to its demodulator. The self interference channel is the channel between two adjacent TX and RX antennas in a FD transceiver. Consequently, this channel can be modeled by a Ricean distribution with large K−factor. After each step of active cancellation the strong LOS component of this channel will be suppressed so the RSI channel can be modeled as a Ricean distribution with smaller K−factor [9]. Based on the results reported in [9] the RSI of a narrowband transmitted symbol x can be represented as  (3) yRSI = x · h ΩI /αRSI  θRSI    hRSI

where x is the complex transmitted symbol, yRSI is its RSI after analog and digital cancellation, and θRSI and hRSI are the phase and amplitude of the RSI channel. In (3), hRSI is described by three variables: (i) h which is a Ricean distributed random variable with parameters Ω = 1 and −10 dB < K < +10 dB. (ii) ΩI represents the level of passive suppression in the FD node. This parameter depends on the distance between the two transmitting and receiving antennas in the FD transceiver (d in Fig. 1) and its typical value is in the order of −30 dB. (iii) αRSI reflects the effect of different cancellation mechanisms on the RSI signal, and is a function of the received (transmitted) symbol power. The dependency of αRSI on the received symbol power is due to the nonlinearity of the RSI channel. Based on the experiments

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Fig. 2: System model for the transceiver chains. The upper part shows the transmission chain, the lower the receiver chain.

in [9], the authors have proposed a linear relation to describe this dependency αRSI = λPRI + γ (4) where λ = 0.12 dB/dBm, γ = 35.49 dB, and PRI is the received symbol power in dBm. We have PRI (dBm) = PT (dBm) + ΩI

(5)

in which PT is the transmitted symbol power. Based on (2) and (3) the RSI signal from a FD transmitter to itself can be modeled in baseband as follows, 2

1 3 + α3 xa |xa | h3RSI,a  θRSI,a + ... yRSI = xa h1RSI,a  θRSI,a (6)

The first term in (6) corresponds to the linear RSI component, the second term corresponds to the third order nonlinearity m is the phase of the corresponding and so on. In (6), θRSI,a RSI channel. Since the receiver does not make any attempt to estimate the phase of the RSI harmonics, in (6) we assume that m ∼ Uniform [0, 2π]. they are uniformly distributed, i.e., θRSI When nodes a and b set up a two-way communication the received baseband signal at node a after active and passive suppression will be given by  2 1 3 + β3 xb |xb | h3ba  θba + ... + ya = xb h1ba  θba     

transmitted signal from node b

xa h1RSI,a 

2 1 3 θRSI,a + α3 xa |xa | h3RSI,a  θRSI,a + ... +νa   RSI

(7) m where θba represents the phase of the corresponding channel between the transmit antenna of node b and the receive antenna of node a, and βl represents the l-th Taylor series expansion coefficient of the power amplifier nonlinearities at node b. The second bracket shows the baseband model of the RSI signal presented in (6) and νa is the receiver noise at node a, which is a combination of the receiver AWGN noise and the SI transmitted noise [7]. We assume that the desired symbol channel is known at the receiver (full CSI scenario), and

consequently the received signal entering the demodulator can be modeled as 1 3 3 h1RSI,a  θRSI,a 2 hba  θba + β x |x | y a = xb + xa 3 b b 1 1 + h1ba  θba h1ba  θba 3 3  2 hRSI,a θRSI,a + ν (8) α3 xa |xa | 1 h1ba  θba In (8), only the third order nonlinearities are considered and ν  indicates the noise; also the components are ordered in decreasing power order, i.e., the desired signal power (xb ’s power) is the dominant term at the receiver, then is node a’s own linear component of RSI and so on. For ease of exposition the time index is dropped. III. LDPC C ODING AND C ONSTELLATION S HAPING A. Transmitter Chain A block diagram of the transmitter prior to the modulator is represented in Fig. 2 based on [23]. The data bits, b, are encoded by a rate Rc = Kc /Nc binary LDPC encoder. Interleaver Π1 interleaves the codeword u. The interleaved codeword, v, separates into two disjoint bit streams d and q. A shaping encoder encodes the bit stream d. The shaping encoder produces more zeros than ones. The non-equiprobable bits at the output of the shaping encoder will be used to shape the APSK constellation points. The bit stream generated by the shaping encoder, c, goes through interleaver Π2 to give S1 and bit stream q goes through the padder to generate bit stream S2 . The bits in stream S2 are equiprobable; however, the bits in stream S1 are produced with a specific and unequal probability. Shaping bits, S1 , partition the constellation into several non-equiprobable sub-constellations. While the unshaped bits, S2 , select a symbol from the selected sub-constellation with uniform probability. Consider a constellation χ with M signal points {x1 , x2 , ..., xM } where M = 2m . Consider g as the number of shaping bits per symbol (g < m), thus the overall constellation χ is partitioned into 2g sub-constellations, each containing 2m−g symbols. The APSK modulator transmits m consecutive bits per transmission, g shaping bits from S1 and m−g bits from S2 , to select a single constellation point. Thus, the ratio between the number of bits in S1 to S2 must be equal to g/(m − g); the padder in Fig. 2 is responsible for adding an adequate number of zeros at the end of stream S2 in order to keep this ratio.

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B. Receiver Chain In our BICM-ID receiver, the demodulator, the shaping decoder, and the LDPC decoder exchange their LLRs in a turbo iterative manner to enhance the system performance. The APSK demodulator uses the following to derive the Log Likelihood Ratios (LLRs) [23] ⎛ ⎞ 2 m 



|y−x | exp ⎝− + fn (x ) La (zn )⎠  Le (zk ) = ln

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(9) Based on (9) the demodulator calculates the k th bit extrinsic LLR of a received symbol, by using both the received symbol, y, and the apriori LLRs provided by LDPC and shaping decoders, La (zk ). In (9), fk (x) is a function that represents the k th bit of the constellation point x; χqk is a set, containing all constellation points whose k th bit position is labeled with q ∈ {0, 1}. We assume that the RSI entering the demodulator is treated as a complex Gaussian noise with variance σ 2 . Consequently, the overall variance of the noise and interference in (8) is N0 = σ 2 + N0 . The shaping decoder is a MAP decoder. In each time unit it accepts two apriori LLRs, La (c) with length ns and La (d) with length ks , and generates two extrinsic LLRs, Le (c) and Le (d) [23] ⎛ ⎞ n k s s  



 exp ⎝ fn d La (cn ) + d l La (dl )⎠ Le (dk ) = ln

0 d ∈Dk

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ns

n=1

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⎞ ks  

fn d La (cn ) + d l La (dl )⎠ l=1 l=k

(10) Dkq

indicates the set of messages, whose k th bit position is q ∈ {0, 1}; fn (c ) returns the nth bit of the message associated with codeword c . The same procedure is done in the shaping decoder to generate Le (c). The LDPC decoder runs the iterative sum product algorithm (SPA). A variable and a check node exchange their LLRs through an edge interleaver (de-interleaver), to generate a posteriori LLRs.

C. Designing the Shaping Code The shaping code is an (ns , ks ) block code, which generates more zeros than ones. The shaping code rate is Rs = ks /ns and the codebook is described by 2ks ns -tuples. We construct our shaping encoder in two steps. At the first step we use mutual information to find the optimized P0∗ , and then in the second step we design our shaping code for a specific rate (ns , ks ) producing the desired P0∗ . To obtain the optimized value of the shaping probability P0 , we examine how it affects the mutual information between the transmitted signal at node b and the received signal at node a, I(Xb , Ya ). Using the following relation, Fig. 3 depicts this mutual information for given SNR and RSI power (σ 2 ),   P ( ya | xb ) I (Xb , Ya ) = E log P (ya ) ⎛ ⎛ ⎞⎞  = E ⎝log P ( ya | xb ) − log ⎝ P (ya |x )P (x )⎠⎠ x ∈χ

(11) In (11), P (x ) represents the probability of transmitting the constellation point x . Since we assumed a Gaussian distribution for the interference in (8), taking statistical expectation in (11) can be done numerically by Gaussian Hermit Quadratures. It can be inferred from this figure that P0∗ at SNR = 15 dB lies in [0.65, 0.80]. Moreover, by increasing σ 2 , the power of RSI, P0∗ also increases. Our investigations for other values of SNR revealed the same trend. Thus in general, P0∗ is an increasing 1 . Consequently, the values of function of both σ 2 and Es /N 0 both these parameters need to be known in order to find the optimal P0∗ . The (ns , ks ) shaping codebook (encoder) can be represented by a mapping matrix, consisting of 2ks rows and ns columns. Each row determines how a specific ks -bit input sequence maps to a codeword. Note that this does not necessarily result in a linear code, and encompasses nonlinear codes as well. To generate a desired P0∗ at the output of the shaping encoder, we need to have (1 − P0∗ ) × 2ks × ns ones within this mapping matrix. As the number of ones is small in general, to get an integer number, one may use the closest larger integer to this value. The shaping decoder is used as a constituent decoder in the iterative receiver. As such in shaping code design (positioning the ones within the matrix), we aim at maximizing the Hamming distance between any two codewords. Our codebook is designed in a recursive manner as

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Algorithm 1: Generating Mapping Matrix input : ns , ks , P0∗ . output: Mapping Matrix, M . 1 2 3 4 5 6 7 8 9

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M ← [0]  2ks ×ns  N1 ← (1 − P0 ) × 2ks × ns k←0 if N1 ≥ 2ks − 1 then S←φ while k < N1 do S ← φ  while |S S  | < 2ks × ns do (i, j) ←Randomly Choose an entry of Matrix M  if (i, j) ∈ / S S  then M (i, j) ← 1 k ← k +1  S  = S {(i, :) , (:, j)} S = S {(i, j)} return: M else return: There is no possible Mapping

IV. S IMULATION R ESULTS We simulated the bit error performance of a bidirectional FD wireless communication system. We assumed that the two

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described in Algorithm 1. In this Heuristic algorithm, M is the mapping matrix, S is a set containing the indices of all entries of M selected to have a 1, N1 is the number of 1s that should be replaced within M , and |S| represents the cardinality of S. In each iteration of the proposed algorithm, we randomly choose an entry of M , row and column, if it is a suitable choice we add a 1 in that entry and add the index of that entry to S. In each attempt, we try to add a 1 bit in an entry which not only does not have a 1 bit but also maximizes the Hamming distance between any two rows of M . The algorithm runs until all 1s are placed within this matrix. In the next section, we will show by simulation that our proposed scheme has good performance. The development of an analytical approach for the study of our scheme is currently being pursued. As mentioned, P0∗ is a function of both SNR and σ 2 . Thus, designing the shaping encoder needs knowledge about both these parameters. For SN R = 10 dB and σ 2 = −20 dB, the optimal shaping probability is found to be P0∗ = 0.75. Consequently, the number of 1 bits that have to be positioned within the mapping matrix of a, for instance,  (ns = 3, ks = 2) shaping encoder is (1 − P0∗ ) × 2ks × ns = 0.25 × 22 × 3 = 3. Accordingly, we only need to find the places of these three 1 bits within the mentioned matrix based on the described method. It is noteworthy that in general shaping codes are selected with small sizes for a manageable decoding complexity; fortunately, as we shall see in the results significant shaping gain can be achieved even by shaping codes of small dimension [24]. There are two points to be noted for the design: (i) we find places of 1s since the number of 1 bits is much less than that of 0 bits; (ii) for a given  s , ks ), the  shaping encoder, (n number of ones must satisfy (1 − P0∗ ) × 2ks × ns ≥ 2ks −1.

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Fig. 4: Effect of shaping and K factor of the RSI channel on the BER performance of a FD bidirectional communication, ΩI = − 30 dB.

nodes a and b use identical encoding and decoding processes, and the same structure of LDPC and shaping encoders. In addition, we assumed α3 = β3 = 0.1 and αm , βm  1 for m > 3 [21]. A (1000,500) random LDPC code (free of cycles of length 4), a (3, 2) shaping encoder, as described in Section III-C, and a 32-APSK modulation with bit mapping based on the DVB-S2 standard, and only one bit for shaping, g = 1, are used. In addition, we assume that the path loss between the two nodes is -60 dB (based on [25], this path loss in a 2.4 GHz corresponds to a distance of 8 meters). The parameters of the FD channel model are taken from [9].We investigate the BER at node a, while the same results are expected at node b. Fig. 4 shows the BER performance of a FD bidirectional communication system with optimized and uniform shaping for two RSI channel K factors. As evident, the system with optimized shaping code provides a 2 dB improvement over a system without shaping (uniform probability for shaping). Indeed with optimized shaping, lower power symbols are transmitted more frequently and since the equivalent RSI channel is nonlinear, lower power symbols induce less distortion at the receiver. Thus, in addition to the conventional shaping gain of a non-uniform constellation, we observe a distinct advantage in a FD communication system. As evident in Fig. 4, changing the K parameter over a wide range does not have a significant effect on the system performance. Fig. 5 compares the performance of the system in ΩI = −30, −29 dB for the system with optimized and uniform shaping. As evident, the higher ΩI , the worse the performance and the more the gain from shaping, and the performance is relatively sensitive to the variations in ΩI . It is noteworthy that in general one may design and use a shaping code optimized for any specific pair of SNR and σ 2 for improved performance. However, to keep the complexity small, here we simply considered a fixed (3, 2) shaping code with P0∗ = 0.75 (as described in Section III-C) as it provides good performance over a wide range of parameters. We are currently investigating this trade-off in greater detail. V. C ONCLUSIONS In this paper, we investigated the impact of constellation shaping in mitigating the negative effects of residual self interference in full duplex wireless communications. Our study reveals that due to the nonlinear effects in imperfect power amplifiers and RSI channels, a shaped constellation would

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