Physical-Layer Network Coding Using GF(q) Forward Error Correction Codes

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1 Physical-Layer Network Coding Using GF(q) Forward Error Correction Codes Weimin Liu, Rui Yang, and Philip Pietraski InterDigital Communications, LLC. King of Prussia, PA, and Melville, NY, USA Abstract Consider a physical-layer network coding (PNC) method in a two-way relaying scheme in which two wireless nodes exchange information through a relay in two time slots. The two nodes transmit to the relay simultaneously, and the relay receives the superposition of the signals from the two nodes and extracts the summation of the bits before broadcasting back to the two nodes such that data can be exchanged. This paper deals with the core problem of decoding the superposition of the signals from the two nodes. This coding and decoding method uses linear forward error correction (FEC) codes defined on Galois field of order q, GF(q), where q is prime, in order to accommodate higher-order modulation over high-snr channels. Application of GF(2) and GF(3) linear trellis codes together with BPSK, QPSK, 3PAM, and 9QAM is demonstrated. Keywords-physical-layer; network coding; FEC; forward-error correction; Galois field; GF(q); higher-order modulation; superposition; PAM; QAM cancellation and possibly channel decoding. In this approach, performance is limited due to the noise at the relay which is also amplified and forwarded to the receiving nodes along with the signal. Time slot 1 m 1 x 1 x 2 Node 1 R Time slot 2 r(m 1, m 2 ) m 2 I. INTRODUCTION Wireless communication systems often need to exchange information between two end nodes in the network with the help of a relay node. This is commonly achieved either in a frequency-domain or time-domain duplexing fashion to avoid interference. Here we focus on a relaying scheme in which the information exchange between two end nodes can be completed in two time slots as illustrated in Figure 1. In the first slot (the multiple access phase), the two end nodes, Node 1 and transmit messages m 1 and m 2, which are respectively modulated to x 1 and x 2, to the relay node R simultaneously. The relay receives the superposition of the modulated signals, x 1 + x 2, from these two nodes, and then performs decoding of some function of the messages, r(m 1, m 2 ). In the second slot (the broadcast phase), the relay broadcasts r(m 1, m 2 ) back to the two nodes. The function r(m 1, m 2 ) must be chosen to contain enough information so that Nodes 1 and 2 can extract the message, m 2 and m 1, respectively, from their counter node thereby accomplishing information exchange. This paper addresses the critical question how r(m 1, m 2 ) can be chosen, especially when higher-order modulation beyond BPSK or QPSK and channel coding are required. One straightforward approach to relaying is amplify-andforward in which the relay does not decode the signals, x 1 and x 2, from the two nodes at all, but rather the relay simply broadcasts the signal it receives, together with the noise, to the two nodes subject to certain power constraint. Each of the two nodes can decode the data intended for it by performing echo Node 1 R Figure 1. Illustration of the two-way relaying scheme. In time slot 1, nodes 1and 2 send m 1 and m 2 to the relay R simultaneously. In time slot 2, the relay broadcasts r(m 1, m 2), some function of m 1 and m 2, to nodes 1 and 2, allowing them to decode m 2 and m 1, respectively. For better performance, physical-layer network coding at the relay is a promising approach in which the relay performs some level of decoding of the messages from the two nodes, with the most common choice being the exclusive-or of the message bits m 1 m 2, and encodes the compound message before sending it back to the two nodes. Since Node 1 knows m 1, it can decode m 2 from m 1 m 2. Similarly can decode m 1 from m 1 m 2. Note that there is no need for the relay to decode m 1 and m 2 explicitly. Some of the recent work on this subject includes that of Zhang, et al, [1], which demonstrated the gain of physical-layer network coding over uncoded BPSK and QPSK transmission. Later on, the same authors extended the approach to 4PAM (pulse amplitude modulation) and 16QAM (quadrature amplitude modulation). Kioke-Akino, et al, [2] [3], studied the denoising map for the relay receiver for physical-layer network coding over QPSK modulation and showed that an unconventional quinary (base-5) constellation would improve denoising performance.

2 The approaches in the previous work are not easily applicable to higher-order modulation or forward-error correction (FEC) coding, which are required for practical communication systems. Figure 2. illustrates the failure of GF(2) physical-layer network coding over 4PAM modulation. The summation of x 1 and x 2 results in ambiguities at values of 2 and +2 in terms of m 1 m 2 over GF(2) the decoder cannot determine whether they are 00 or 11. The 4PAM constellation in this illustration uses Gray code, as is common in practical applications, but ambiguities exist for any binary code. x 1 +x =00 x 1 x = = = = = = = = = Figure 2. Illustration of the failure of GF(2) physical-layer network coding over Gray-coded 4PAM modulation. The summation of x 1 and x 2 results in ambiguity in terms of m 1 m 2 over GF(2) at values of 2 and +2. Zhang, et al, [1], demonstrated physical-layer network coding over QPSK modulation. However, the most commonly used channel codes are designed over GF(2), and the method cannot be extended to transmission using such FEC codes for the reason illustrated in Figure 2. In order to combine channel coding and physical-layer network coding, Zhang and Liew [4] employed Repeat Accumulate (RA) code, which is a turbolike channel code, together with BPSK modulation, utilizing a property of RA codes that the accumulation is modulo-2. In this paper, we present a different approach to higherorder modulation and channel coding using physical-layer network coding in which the algebraic basis for the modulation and channel code is extended beyond 2 (which is conventional) to all prime numbers. The number of points in a constellation is no longer a power of 2 but rather a prime number. The channel code is no longer binary but rather defined on GF(q), where q is prime. The channel code does not have to be turbolike; it can be any linear forward error correction code. The critical problem in a physical-layer network coding scheme, as applied to two-way relays in which two nodes exchange information exclusively through a relay, can be stated as the following. Node 1 wishes to transmit bits m 1 to, and wishes to transmit bits m 2 to Node 1. The relay scheme involves two steps: 1. In the first time slot (MAC, or multiple access phase), Nodes 1 and 2 encode and modulate bit streams m 1 and m 2, respectively, and transmit the modulated signals, x 1 and x 2, to the relay at the same time. The relay receives the superposition of the two signals, x 1 + x 2, and decodes r(m 1, m 2 ), a network coding function of m 1 and m 2 with presumably fewer bits than m 1 and m 2 combined. Generally, decoding r(m 1, m 2 ) should be easier (less SNR needed) than decoding both bit streams m 1 and m 2 explicitly. 2. In the second time slot (broadcast phase), the relay encodes r(m 1, m 2 ) and broadcasts the encoded and modulated signal x R to both Node 1 and. Node 1 receives x R, decodes bits r(m 1, m 2 ), and then derives m 2 from r(m 1, m 2 ) and m 1 using a network decoding function g(r(m 1, m 2 ), m 1 ). Note that Node 1 has perfect knowledge of m 1. Similarly, derives m 1 from r(m 1, m 2 ) and m 2 using network decoding function g(r(m 1, m 2 ), m 2 ). The physical-layer network coding problem becomes finding forward error correction (FEC) coding and modulation scheme P: m 1 x 1, and m 2 x 2, such that: 1. A network coding function r(m 1, m 2 ) exists; 2. A network decoding function g( ) exists such that m 2 = g(r(m 1, m 2 ), m 1 ), and m 1 = g(r(m 1, m 2 ), m 2 ); and 3. P shall be scalable to accommodate higher-order modulations beyond BPSK and QPSK. In this paper, we move away from the adherence to binary exclusive-or, or GF(2) addition, of the messages in the definition of r(m 1, m 2 ). Instead, we consider higher-order Galois fields for channels with higher SNR (signal to noise ratio). We propose a physical-layer network coding approach in which the messages m 1 and m 2 are base-q, and the transmission by Nodes 1 and 2 employs qpam or q 2 QAM, and the FEC code is over GF(q), Galois field 1 of order q, where q is prime and is chosen based on the SNR of the channel. Instead of binary m 1 m 2, the relay shall decode m 1 m 2 over GF(q). Note that we have simplified the channel between the relay and Nodes 1 and 2 to be AWGN. For practical wireless applications, the scheme must work over arbitrary channels, which we will address a sequel of this paper. II. PHYSICAL-LAYER NETWORK CODING USING LINEAR FEC CODE OVER GF(Q) We propose a physical-layer network coding scheme P = {(C q, M q ) q all primes} that can be summarized below: For a given signal-to-noise ratio (SNR) or single-tointerference-and-noise ratio (SINR), choose a prime number q such that there are q points in a PAM (pulse amplitude modulation) or q 2 points in a QAM (quadrature and amplitude modulation). In the case of complex modulation, construct a square QAM from two orthogonal qpams consisting of combined q 2 points. 1 A Galois field (GF) is a field with finite number of elements. The order, or the number of elements, of Galois fields takes the form q n, denoted as GF(q n ), where q is a prime number, and n is a positive integer. An example of Galois field is GF(2), binary numbers 0 and 1, over which the addition is exclusive-or, and the multiplication is logical-and. Only when n = 1, i.e., over GF(q), the addition is defined as real addition modulo-q. Over GF(q n ), addition is defined as digit-wise modulo-q (not modulo-q n ), which is not suitable for the PNC approach in this paper.

3 The value of q can be chosen in a similar manner as the modulation and coding scheme (MCS) in a communication system. The value of q should be chosen so that qpam or q 2 QAM is best suited for the SNR or SINR of the channel in order to achieve the desired performance goal and spectral efficiency. We denote the qpam or q 2 QAM modulation as M q. The table below lists the constellation power of qpam and q 2 QAM relative to that of BPSK. q Power (dbr) qpam q 2 QAM Linear FEC codes C q are constructed over GF(q), Galois field of prime order q. Messages m 1 and m 2 are also expressed in GF(q). Particularly, m 1 and m 2 are taken from integers {0, 1, 2,, q 1}. They are each encoded with the linear FEC code C q, b 1 = C q (m 1 ), b 2 = C q (m 2 ), modulated, and then transmitted using qpam or q 2 QAM: x 1 = M q (C q (m 1 )), and x 2 = M q (C q (m 2 )). The network coding function r(m 1, m 2 ) is chosen to be the addition over GF(q), denoted as to distinguish it from addition over reals. The addition over GF(q) is modulo-q real addition of the possible messages. Namely, r(m 1, m 2 ) = m 1 m 2 = m 1 +m 2 mod q. Since m 1 GF(q), and m 2 GF(q), by closure property r(m 1, m 2 ) GF(q). Message r(m 1, m 2 ) is decoded and broadcasted by the relay. Decoding is possible due to the linearity of the FEC code, C q (m 1 ) C q (m 2 ) = C q (m 1 m 2 ). Note that without considering noise the relay receives We can construct y = x 1 +x 2 = M q (C q (m 1 ))+M q (C q (m 2 )). ŷ = x 1 +x 2 mod(q) = M q (C q (m 1 )) M q (C q (m 2 )) subtraction operation over GF(q), which is the network decoding function g( ). Then we can see the sufficiency of r( ): g(r(m 1, m 2 ), m 2 ) = m 1 m 2 m 2 = m 1, and g(r(m 1, m 2 ), m 1 ) = m 1 m 2 m 1 = m 2, which means Nodes 1 and 2 are able to decode m 2 and m 1, respectively and uniquely, if the relay broadcasts r(m 1, m 2 ). Transmission of r(m 1, m 2 ) by the relay in the broadcast phase can use any capacity-achieving modulation and channel coding. GF(q) linear FEC code can be one such choice, but its use is not required for the proposed physical-layer network coding scheme. The order q is limited to prime numbers due to the existence of finite fields whose addition is defined as real addition modulo-q instead of modulo-2. The most common field used in convolutional and block codes is GF(2). The rest of q (for q>2) are a significant departure from conventional linear codes. Another practical code is Reed-Solomon, which is defined on GF(2 n ). Codes defined on GF(2) are suitable for transmission over BPSK (one-dimensional constellation) or QPSK (two-dimensional constellation) in our physical-layer network coding scheme. However, codes defined on GF(2 n ), for n>1, cannot be applied to constellation of size 2 n directly for physical-layer network coding, as the addition defined on GF(2 n ) is bit-wise modulo-2, not modulo-2 n. Similarly, by limiting q to prime numbers we only consider qpam or q 2 QAM modulations. Again, except for BPSK and QPSK, the rest are constellations of unconventional size. III. SYSTEM MODEL AND APPLICATIONS OF NETWORK CODING The system model for the two-way relaying scheme using linear GF(q) forward error correction coding and modulation is illustrated in Figures 4 and 4. The input and output data are represented in base-q. Binary data in conventional systems can be converted to base-q representation, and vice versa, using modulus conversion. The MAC phase (time slot 1) of the relaying scheme is shown in Figure 4, which illustrates how m 1 m 2 is decoded by the relay, by taking advantage of the linearity of the code over GF(q). Note that the relay does not need to decipher either m 1 or m 2 individually and only decodes m 1 m 2. = M q (C q (m 1 ) C q (m 2 )) = M q (C q (m 1 m 2 )). The properties of the addition operator over GF(q) implies the existence of its additive inverse operator, the

4 m1 Linear FEC over GF(q) b1 qpam or q 2 QAM Modulation x =0 0 1=1 1 0=1 1 1= Node 1 (a) (b) m2 x1+x2+n Modulo q Linear FEC over GF(q) b2 qpam or q 2 QAM Demod Relay qpam or q 2 QAM Modulation b1 b2 x2 Decode Linear GF(q) FEC m1 m2 Figure 3. Illustration of the MAC phase (time slot 1) in a two-way relaying scheme. Nodes 1 and 2 send m 1 and m 2 to the relay simultaneously, and the relay decodes m 1 m 2. In the broadcast phase (time slot 2), as illustrated in Figure 4, the relay broadcasts the decoded m 1 m 2 back towards Nodes 1 and 2. Nodes 1 and 2 then decode m 2 and m 1, respectively. m 1 m 2 x 12 Demod Channel Coding Relay Channel Decoding Node 1 m 12 x 12 x 12 Demod Channel m Decoding 12 GF(q) Modulation m 1 GF(q) m 2 m 2 Figure 4. The broadcast phase (time slot 2) of the two-way relaying scheme. The relay broadcasts the decoded m 1 m 2, and Nodes 1 and 2 decode m 2 and m 1, respectively. We first review how qpam is mapped and how modulo-q addition is done by first examining an uncoded BPSK system and modulo-2 over GF(q) where q=2. Figure 5 shows the BPSK constellation and the superposition of two BPSK signals. In BPSK, a bit m 1 = 0 is mapped to x 1 = 1, and a bit m 1 = 1 is mapped to x 1 = +1. Assuming the channels have a unity gain, two BPSK signals summed together over the air, x 1 + x 2, will be received by the relay. There are 4 possible combinations resulting in 3 possible amplitudes of x 1 + x 2 : 2, 0, and +2. m 1 Figure 5. (a) BPSK constellation, and (b) superposition of two BPSK constellations. Here denotes addition over GF(2), namely addition modulo-2, or exclusive-or. The relay does not need to decode m 1 and m 2. It decodes m 1 m 2, where denotes addition over GF(2) here (modulo-2 addition). As we can see from Figure 5. (b), there is no ambiguity as far as m 1 m 2 is concerned, and m 1 m 2 can be decoded perfectly given sufficiently low noise. For example, x 1 + x 2 > 1 implies m 1 m 2 =0, otherwise m 1 m 2 = 1. The above scheme works for both uncoded and coded systems. In a system employing forward-error correction coding (FEC) over GF(2), both the input and output are represented over the GF(2) alphabet (0 and 1). The FEC must be a linear code for the relaying scheme to work. In other words, if b 1 and b 2 are codewords, b 1 b 2 must also be a codeword. QPSK is simply two independent dimensions of BPSK. Systems and relays transmitting QPSK employing FEC over GF(2) will work the same way as described above. The following table shows the definition of addition and multiplication over GF(3). It is easy to verify that they satisfy the definition of a finite field, and the operators are simply real addition and multiplication modulo * Similarly one can construct a definition of addition and multiplication over GF(5). Again it is easy to verify they define a finite field and are equivalent to real addition and multiplication modulo-5. Note that each row and column in the addition table is simply a permutation of the elements in GF(5), which guarantees the existence of, the inverse of addition * Figure 6. illustrates a ternary one-dimensional 3PAM constellation and the superposition of two 3PAM constellations. It is shown that there is no ambiguity in terms of m 1 m 2. For coded systems, a linear FEC code over GF(3) is needed.

5 =0 0 1=1 0 2=2 1 1=2 1 2=0 2 2=1 Presented here are the performance simulation of linear GF(q) convolutional code and qpam modulation for q=2 and q= (a) Figure 6. (a) Ternary constellation, and (b) superposition of two ternary constellations. Here denotes addition over GF(3). Similarly one can extend the above superposition scheme to larger constellations. However, as discussed earlier, the cardinality q (the number of points in the constellation) must be prime. Only over prime q is the addition defined over GF(q) as real addition modulo-q. There is no restriction on the type of FEC used for the encoding, as long as the code is linear over GF(q). The code can be a block code, a convolutional code, or a concatenated (turbo) code. Most wireless applications use a two-dimensional constellation, such as QAM, in which data are coded into inphase (I) and quadrature (Q) components. It is straightforward to consider a q 2 QAM as two orthogonal qpams. Figure 7. (a) illustrates how a 9QAM constellation encoding two ternary digits can be decomposed into two orthogonal 3PAMs encoding one ternary digit each. The GF(3) code for each dimension is not necessarily the same as the other. Figure 7. (b) illustrates the summation of two 9QAMs. Again the I and Q phases can be treated separately, and there is no ambiguity in terms of m 1 m 2 over GF(3). (b) A. Binary Convolutional Code and BPSK The convolutional code C 2 used in the simulation is a rate-⅓ constraint length-7 trellis code (polynomials 133, 171, and 165 octal, as used in 3GPP Long Term Evolution (LTE) for control and broadcast channels (PDCCH and PBCH). The BPSK constellation M 2 has symbol power of 1. Figure 8 shows the comparison bit error rates (BER) among (1) uncoded BPSK M 2, (2) decoding of m from M 2 (C 2 (m)) using a conventional Viterbi decoder, and (3) decoding of m 1 m 2 from M 2 (C 2 (m 1 ))+M 2 (C 2 (m 2 )) using a conventional Viterbi decoder. The assumptions are: All transmitters transmit at power = 1, which is the power of M 2 = { 1, +1}. In the network coding case, two signals arrive at the relay simultaneously, and the signal power is 2; All channels have a unity gain; The noise power is as indicated in the abscissa with reference level 0 dbrn = 1, i.e., for uncoded BPSK and trellis coded M 2 (C 2 (m)), a noise power of 3dBrn would correspond to SNR of 3dB; Soft decoding is used in the Viterbi decoder. Q Q I I (a) (b) 10 Figure 7. (a) 9QAM constellation encoding two ternary digits, and (b) superposition of two 9QAMs showing the encoded m 1 m 2 over GF(3). Similarly, one can construct 5PAM/25QAM, 7PAM/49QAM, 11PAM/121QAM, etc. in the same manner demonstrated here. For a given channel SNR or SINR in an application, one can choose the appropriate q for channel coding and modulation. IV. SIMULATION The performance of the receiver in the relay is considered since this is the bottleneck of the two-way relaying scheme employing physical-layer network coding under the assumptions of equal transmit power and AWGN channel. Figure 8. Bit error rate comparison among (1) uncoded BPSK, single link, (2) rate-1/3 constraint length-7 trellis code (polynomials 133, 171, and 165 octal), single link, (3) network coding using the same GF(2) trellis code. Compared to the convolutional code C 2 itself, decoding m 1 m 2 from M 2 (C 2 (m 1 ))+M 2 (C 2 (m 2 )) loses about 1.2 db in performance asymptotically. This loss is attributed to the 3- point constellation in Figure 5. (b), which is less reliable than the BPSK constellation in Figure 5. (a). This is a small performance gap compared to decoding m 1 and m 2 individually, which would result in high BER for any noise level due to 0 db signal-to-interference ratio. This performance gap is the greatest for BPSK and diminishes for increasingly higher order constellations.

6 B. Ternary Convolutional Code and 3PAM We have designed a GF(3) convolutional code C 3 and a GF(3) Viterbi decoder for performance simulation. C 3 is a rate- ⅓ constraint length-5 trellis code (polynomials 21022, 20122, and 11122, free distance = 18 Euclidean), and M 3 is the 3PAM modulation shown in Figure 6. (a). Figure 9. shows the comparison of symbol error rates (SER) among (1) uncoded 3PAM M 3, (2) decoding of m from M 3 (C 3 (m)) using the GF(3) Viterbi decoder, and (3) decoding of m 1 m 2 from M 3 (C 3 (m 1 ))+M 3 (C 3 (m 2 )) using the GF(3) Viterbi decoder. The assumptions are: All transmitters transmit at power = 2/3, which is the power of M 3 = { 1, 0, +1}. In the network coding case, two signals arrive at the relay simultaneously, and the signal power is 3dB higher at 4/3; All channels have a unity gain; The noise power is as indicated in the abscissa with reference level 0 dbrn = 1, i.e., for uncoded 3PAM M 3 and trellis coded M 3 (C 3 (m)), a noise power of 0dBrn corresponds to SNR of 10log 10 (2/3) = db; Soft decoding is used in the Viterbi decoder. Compared to the convolutional code C 3 itself, decoding m 1 m 2 from M 3 (C 3 (m 1 ))+M 3 (C 3 (m 2 )) loses about 1 db in performance asymptotically. Again this loss is attributed to the 5-point constellation in Figure 6. (b), which is less reliable than the 3PAM constellation in Figure 6. (a). Compared to BPSK, the performance gap is smaller for 3PAM. For large constellations, the performance gap will diminish. V. SUMMARY AND FUTURE WORK In this paper we present a physical-layer network coding scheme in which linear forward error correction coding defined on Galois field of order q, where q is prime, and qpam or q 2 QAM modulation are employed. In this scheme, two nodes transmit their messages to the relay simultaneously, and the relay is able to decode the GF(q) sum of the messages from the two nodes. The validity of this method has been verified with simulation in the GF(2) and GF(3) cases using convolutional codes. This paper only deals with the ideal scenario in which the channels from the two nodes to the relay are identical. In practical applications, this assumption may not be valid due to imperfect or absent channel state information feedback or power constraints at the transmitters. We have studied the general case of arbitrary channel gains and phases and will present the result in a sequel. Future study on the physical-layer network coding approach presented in this paper may include practical impact of using odd-sized constellations, extension into MIMO channels, using more powerful forward error correction such as turbo codes, and ways to make general GF(q) encoder and decoder computationally efficient for all prime q. REFERENCES Figure 9. Symbol error rate comparison among (1) uncoded 3PAM over a single link, (2) rate-1/3 constraint length-5 GF(3) trellis code (polynomials 21022, 20122, and 11122), single link, and (3) network coding using the same GF(3) trellis code. [1] S. Zhang, S.C. Liew, P.P. Lam, Hot topic: physical-layer network coding, MobiCom 06, Sept , 2006, Los Angeles, Calif. [2] T. Koike-Akino, P. Popovski, and V. Tarokh, Denoising maps and constellations for wireless network coding in two-way relaying systems, Proceedings IEEE GLOBECOM, [3] T. Koike-Akino, P. Popovski, and V. Tarokh, Optimized constellation for two-way wireless relaying with physical network coding, IEEE J. Selected Areas in Communications, Vol. 27, No. 5, pp , June [4] S. Zhang and S.C. Liew, and P.P. Lam, Channel coding and decoding in a relay system operated with physical-layer network coding, IEEE J. Selected Areas in Comm., Vol. 27, No.5, June 2009.