Algebraic Number Precoded OFDM Transmission for Asynchronous Cooperative Multirelay Networks

This paper proposes a space-time block coding (STBC) transmission scheme for asynchronous cooperative systems. By combination of rotated complex constellations and Hadamard transform, these constructed codes are capable of achieving full cooperative diversity with the analysis of the pairwise error probability (PEP). Due to the asynchronous characteristic of cooperative systems, orthogonal frequency division multiplexing (OFDM) technique with cyclic prefix (CP) is adopted for combating timing delays from relay nodes.The total transmit power across the entire network is fixed and appropriate power allocation can be implemented to optimize the network performance. The relay nodes do not require decoding and demodulation operation, resulting in a low complexity. Besides, there is no delay for forwarding the OFDM symbols to the destination node. At the destination node the received signals have the corresponding STBC structure on each subcarrier. In order to reduce the decoding complexity, the sphere decoder is implemented for fast data decoding. Bit error rate (BER) performance demonstrates the effectiveness of the proposed scheme.


Introduction
Recently, cooperative communication networks are known to have significant potential in increasing network capacity and transmission reliability.In recent years, cooperative networks have attracted substantial interest from the wireless networking and communications research [1][2][3][4][5][6][7][8][9][10][11][12][13].The basic idea is that intermediate relay nodes act as a virtual distributed antenna array to assist the source node in transmitting its information to the destination node.The amplify-and-forward (AF) and decode-and-forward (DF) are the most famous schemes for cooperative systems [14].
Space-time block code (STBC) [15,16] is an effective approach to achieve spatial diversity for cooperative transmissions.Since the relay nodes are in different locations and have different oscillators, there may exist timing errors.Therefore, it is difficult to design proper space-time coding schemes.The transmission schemes, which are based on orthogonal frequency division multiplexing (OFDM), are exploited for combating the loss of timing phase [2][3][4][5].In [4], a simple Alamouti scheme is proposed to achieve cooperative diversity, where only a few simple operations such as time-reversion and complex conjugation are required at the relay nodes; the fast ML Alamouti decoding is used at the destination node.However, this scheme is only useful for the case of two relay nodes.In [6,7], the clustered orthogonal space-time block code (OSTBC) schemes for four or more relay nodes are proposed by clustering the relay nodes.It is shown that limited performance improvement is achieved while the number of relay nodes increases.
In [17][18][19][20][21], a new multidimensional modulation scheme is proposed to increase the modulation diversity.It is feasible to achieve full modulation diversity by applying optimum rotation to signal constellation.The algebraic number theory is introduced in [22] to construct a multidimensional rotation matrix.By properly selecting the algebraic number field, a generator matrix that guarantees full modulation diversity and maximizes the minimum product distance for precoded information symbols can be obtained.
In this paper we study the diagonal algebraic space-time (DAST) block coding for asynchronous cooperative network.We consider the OFDM technique with enough cyclic prefix (CP) [23,24] to combat the timing errors.The information symbols are precoded by a matrix multiplication at the source node.Throughout the proper operation at the relay node, the received signals hold the DAST block code structure on each subcarrier; hence it is convenient to decode data by using ML decoding or fast sphere decoder.It is assumed that the relay nodes do not have to know any information about the channels but the destination node knows all channel information through training.Therefore the relay nodes do not need to decode and demodulate signals received from the source node.Only a few simple signal processings are needed at the relay nodes.
This paper is organized as follows.In Section 2, the relay network model is described and a new transmission scheme is proposed.In Section 3, the algebraic number theory is introduced and the optimal rotation constellation scheme is presented.The PEP is analyzed with the optimal power allocation in Section 4. Section 5 contains the simulation results.Finally, the conclusions are given in Section 6.
Notation.For a vector or matrix ,   ,   , and ‖‖ indicate the transpose, Hermitian, and Frobenius norm of , respectively. ∘  denotes the Hadamard product of  and , that is, the componentwise product.* indicates the conjugate operation.⊛ denotes the circular convolution.diag{ 1 , . . .,   } is a diagonal matrix with   being its th diagonal entry. indicates the expectation and  indicates the probability.Gaussian integer is a complex number whose real and imaginary parts are both integers.

Relay Network Model and Cooperative Protocol
Consider a relay network with one source node, one destination node, and  ( = 2  ,  = 1, 2, . ..) relay nodes, as shown in Figure 1.There is only one antenna at every node.insertion, the OFDM symbols of length  + ℓ cp are transmitted to the destination node through the relay nodes.We assume that the knowledge of timing errors can be obtained at the destination node.It is emphasized that the CP length must be larger than the maximum time delay  max .Besides, we make the assumption that the channels between any two nodes are quasi-static flat  fading.The fading coefficient from the source node to the th relay node is denoted by ℎ  , and the fading coefficient from th relay node to the destination node is denoted by   .These coefficients are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance.
Denote by  1 , . . .,   the  consecutive OFDM symbols, where   consists of IFFT(  ) and the corresponding CP for  = 1, . . ., .The th subcarrier output of th OFDM symbols in frequency domain can be represented by The 1/ √  multiplier in terms of IFFT guarantees the power of signal symbols invariant after IFFT operation.We assume the channel coefficients remain unchanged during the transmission of  OFDM symbols.At the th relay node in the th OFDM symbol duration the received signals are where  1 is the average transmit power at the source node and  , is the corresponding additive white Gaussian noise (AWGN) with zero mean and unit variance at the th relay node, in the th OFDM symbol duration.Note that due to the additive white Gaussian noise, the average power of received signal is  1 + 1.
The relay nodes would simply process and transmit the received noisy signals.Only unary positive and negative operations are needed.For instance, in the case of  = 4 the processed signal matrix is given by We can find that it holds the structure similar to the Hadamard matrix.Apart from the orthogonal STBC scheme, which is required to switch the OFDM symbols and has to wait to start process and transmit until the next several OFDM symbols arrive at the relay nodes [4,5], in our proposed scheme the relay nodes can process and broadcast the received signals immediately without waiting the other symbols arriving.As a result, the proposed scheme would not induce the transmission delay.
Let  , =  ,  , be the transmit signal from the th relay node, in the th OFDM symbol duration, where  , belonging to {+1, −1} is the entry of Hadamard matrix  of  dimensions, and scalar  = √ 2 / ( 1 + 1) guarantees the average transmit power is  2 for one transmission at every relay node.At the destination node, the received signal in the th OFDM symbol duration can be written as where  is the corresponding AWGN at the destination node and   is an  point vector whose (  + 1)th element is one and the others are zero.Time delays in the time domain are expressed by circular convolution with   .After CP removal and -point FFT transformation, the received signals can be rewritten as where / ]  meaning the phase change in frequency domain corresponding to the sample time delay   in time domain,  , = FFT( , ), and   = FFT(  ).For every subcarrier , 1 ≤  ≤ , we have where we have defined is an  ×  matrix with the entries { ,, }, 1 ≤  ≤ , 1 ≤  ≤ .The Hadamard transform is useful for reducing the high peak-to-average power ratio over different transmit antennas resulting in power amplification.

Rotated Constellations Using Algebraic Number Theory
In order to achieve full modulation diversity over the Rayleigh fading channel and Gaussian fading channel, the rotation of a multidimensional signal symbol vector is discussed in this section.The minimum product distance of the constellation considered is defined as where  and   belong to an -dimensional constellation (QAM or PAM).The algebraic number theory was employed to construct a proper precoding rotation matrix , which maximizes the minimum product distances in certain dimensions.The diversity order is the minimum Hamming distance between any two coordinate vectors of constellation points.It is emphasized that a rotation matrix constructed by algebraic number theory is a Vandermonde matrix, which can simplify the encoding and reduce the computational complexity in a similar way of fast Fourier transformation.Furthermore, this algebraic construction of rotations is useful for reduction of peak-to-mean envelope power ratio (PMEPR).
It can be considered that the algebraic constellation has full spatial diversity if the associated minimum product distance  min is strictly positive; equivalently, the components of vectors  =  and   =   (with  ̸ =   ) in the rotated constellation are all different.To construct a precoding matrix  of dimension  ( = 2  ,  = 1, 2, . ..) with full modulation diversity, we apply the canonical embedding to some totally complex cyclotomic number fields.The reader can refer to [25,26] for more comprehensive details about cyclotomic number fields and canonical embedding.
The algebraic norm for real integer in number fields at first appears in solving problems such as the integer solutions of finding all   +   =   , for  = 2, which is stated in the Fermat theorem.It is emphasized that () is an integer and () = 0 if and only if  = 0; hence () ≥ 1 for  ̸ = 0.The algebraic norm should be compared with the canonical embedding.With the application of the canonical embedding   to each element of basis [1, ,  2 , . . .,  −1 ] of , the generator matrix is given by where   is the corresponding column of  for  = 1, 2, . . ., .Due to the characteristic of Vandermonde matrix and   −  V ̸ = 0 for 1 ≤  ̸ = V ≤ , the matrix  has full rank.This means that matrix  is eligible to be a generator complex matrix for multidimensional rotated constellation.It is important to select the roots   ,  = 1, 2, . . .,  for generating full modulation diversity rotations and maximizing the minimum product distance.With proper   the matrix  becomes an orthogonal matrix; that is,    = .According to the property of algebraic norm, the minimum product distance is a nonzero integer.It is revealed that  min is related to the special properties of algebraic number field when the full modulation diversity is obtained.In fact, the diversity product is 1 for the optimal cyclotomic rotation matrix no matter what the space-time code size and constellation size are, which have been proved in [26].
We consider any two columns   and  V of , , V = 1, 2, . . ., ; without loss of generality, we assume V < ; then the complex inner product of   and  V is The problem transforms to the  − 1 power symmetric functions with  complex roots, with the constraints of the properties of the minimal polynomial   ().Due to the fact that the complex roots   are on the unit circle, we hold    *  = 1 and  is an orthogonal matrix, so that we obtain We assume the minimal polynomial where Notice that it is an elementary symmetric polynomial.According to Newton's identities, we have Since ∑  =1 (  )  = 0 for  = 1, 2, . . .,  − 1, we have   = 0 for  = 1, 2, . . .,  − 1.Notice the coefficients of the minimal polynomial   () are a Gaussian integer and   1 =   = , where  = √ −1, which yields Using the above results it is easy to achieve the complex roots   = exp((4 − 3)/2) for  = 1, . . ., .

PEP Analysis and Optimal Power Allocation
The optimum power allocation between the source node and the relay nodes is discussed in [10] to minimize the pairwise error probability (PEP), which is well known to be an important measure for performance analysis.In this section we focus on the high power regime and the upper bound of PEP and provide the theoretical bases for reducing the upper bound of PEP with the precoding matrix.At first, for simplicity's sake we rewrite (6) as follows: where  = diag((   ∘ ℎ ∘ )  ) and    = (  ∘ )(   ∘ ) +   .The expression (15) implies that equivalently the rotated constellation symbols are transmitted over the diagonal channel matrix in space and time.Since |    | = 1 for any , it is obvious to see that    is an independent Gaussian random vector, so that we can obtain (   ) = 0 and Var(   ) = (1 +  2 ∑  =1 |  | 2 )  .Assuming ideal channel state information (CSI), the maximum-likelihood (ML) decoding is implemented.The PEP of mistaking ŝ by ŝ  is given by where  and   are corresponding code matrix of ŝ and ŝ  , respectively.When  goes to infinity, by the law of large numbers, it shows that It is also implied that the fading has little effect when  is large.We assume the total transmit power in the whole system is  per symbol transmission; then we have  =  1 +  2 and It is not hard to see that the probability can be minimized when  1 = /2 and  2 = /2; that is, it is optimal to allocate half of the total power to the transmit node and the other half to the relay nodes.It is significant that every relay node only consumes a little amount of power to contribute to the transmission.
To obtain the upper bound of PEP we have to compute the expectation over ℎ, .Note that  =   ℎ, where   = diag( . By using the tight upper bound of Gaussian tail function, we have the following approximate inequality: Due to the fact that with 2 degrees of freedom and the corresponding probability density function is (  ) =  −  , we obtain Mathematical Problems in Engineering 5 where is the exponential integral function.This upper bound is sufficient to derive the optimization criteria.On the high power regime of log  ≫ 1, we have thus Therefore, the achieved diversity gain is (1−loglog/ log ), which is linear in the number of relay nodes.If  increases greatly (log  ≫ loglog), full diversity of  is obtained, the same as the multiple-input multiple-output (MIMO) system with  transmit antennas and one receive antenna.

Simulation Results
In this section, we present some simulated performance of DAST codes for wireless asynchronous cooperative networks.The MATLAB 8.0 sumulation tool was used for the simulation (on a Core i7-3770 3.4 GHz PC).The different values of the number of relays  and total transmit power  are considered.We assume that the length of OFDM subcarriers is  = 64 and the length of cyclic prefix ℓ cp = 16 with the total bandwidth of 10 MHz, and the OFDM symbol duration is   = 6.4 s.To satisfy the conditions that the time delay must be less than the CP length,   is randomly chosen from 0 to 15 with the uniform distribution, and  1 is assumed to be 0 for the first relay node.The information symbol is modulated by normalized 4-QAM.We fix the total transmit power , which is measured in decibel.Using the optimal power allocation strategy, the transmit power of the source node is /2 and the relay nodes share the other power.The average SNR at the destination node can be calculated to be  2 /4( + 1).When  ≫ 1, the SNR becomes /4.At the destination node, the ML decoding of the DAST block code can be implemented by the sphere decoder to obtain almost the same performance at a moderate complexity.The performances of the unrotated constellation for the relay network and the DAST block code for the multiple-input multiple-output (MIMO) system are given.For the sake of comparison, the performances of Alamouti code scheme [4] are also shown.
In Figure 2, we show the decoding performance of the network system equipped with two relay nodes.Assume all the systems have the same total transmit power and all the nodes in the relay network systems have the same power constraint.As expected, we can see the performance for unrotated constellation is poor.For  = 2, since the Alamouti scheme is the unique complex orthogonal design at a transmission rate of 1, it seems hard for the DAST scheme to outperform it.However, there are some drawbacks for Alamouti scheme such that the relay nodes have to stack every two OFDM symbols and then exchange the transmit order of the two OFDM symbols, which leads to one OFDM symbol time slot delay.The advantage of DAST scheme will be immediate and significant when more relay nodes are employed.Figures 3(a) and 3(b) show the decoding performance of the four-relay network systems and the MIMO system with 4 transmit antennas and 1 receive antenna. Figure 3(a) shows the BER performances with respect to the total transmit power.Figure 3(b) shows the BER performances with respect to the receive SNR.In the MIMO system the receive SNR is assumed to be .We observe that the slopes of the BER curves of the DAST scheme approache the slopes of the BER of the DAST MIMO systems when the total transmit power  or the receive SNR increases, which indicates that the new scheme can achieve diversity degree of 4. We can see that the distance between the curves of the MIMO system and DAST relay system is large because the transmit antennas can fully cooperate in MIMO system.However, at the same receive SNR, the gap of the two curves is diminishing, which can be seen clearly in Figure 3(b).Since OSTBC design cannot achieve full transmission rate for more than two antennas, it is not applicable for the system with more than 2 relay nodes.Therefore, the cluster Alamouti code method [5] with rate 1/2 is used for the network with 4 relay nodes.16QAM is considered to maintain the same transmission rate.We can see that DAST scheme has a gain of about 3 dB at 10 −6 in Figure 3(b).It should be noted that this gain will be enhanced when increasing the size of the constellation.In addition, the cluster Alamouti code scheme has one OFDM symbol time delay to perform signal processing at the relay nodes.
Finally, the example for  = 8 is given to show the BER performance of the DAST scheme and the MIMO system with respect to the total transmit power in Figure 4.It also achieves a diversity order similar to that of DAST MIMO systems.When more relay nodes are utilized, the curves descend much faster at the high SNR regime.This implies the better BER performance is achieved.

Conclusions
In this paper, we propose the use of DAST block codes for asynchronous cooperative relay networks.OFDM technique is implemented at the source node, and only +/− operations are required at the relay nodes, without decoding and transmission delay.By using this method the received signal symbols at the destination node hold the STBC structure after removing CP and IFFT operation.This structure is useful for decoding.It can be considered efficient to adopt fast decoding algorithms such as sphere decoder to maintain the ML decoding performance in a polynomial time.It is not required with the transmitted signal symbols and channel information at the relay nodes.We analyze the PEP and observe that the proposed scheme is capable of achieving full spatial diversity for high total transmit power.Simulation results on the DAST scheme are demonstrated, which verifies our results.
For possible future works, it would be interesting to investigate further of imperfect CSI.In the existing works, the channels are assumed to be perfectly estimated.However, the channel estimation errors are inevitable in practice and affect the BER performance.Therefore, the investigation of impact of imperfect CSI on the proposed schemes would be our further work.Moreover, the symbol time offset and carrier frequency offset in OFDM technique can also be explored.
We denote the average signal energy   = 1 by a normalizd QAM constellation.OFDM technique is used to combat the time errors.It is assumed that the number of subcarriers is  and the transmission delay of the signals from th relay node at the destination node is   , which is a multiple of   with   denoting the information symbol duration.At the source node the information bits are modulated into complex symbols by normalized QAM constellation.The consecutive  symbols are denoted by  , ,  = 1, 2, . .., ,  = 1, 2, . .., .Then we let   = [ ,1 ,  ,2 , . ..,  , ] represent the th block consisting of  symbols.Let  be an  ×  precoding matrix.We obtain the precoded symbols x = [ 1, , . . .,  , ]  = ŝ  , where ŝ = [ 1, , . . .,  , ]  .Then the new  consecutive OFDM block is denoted by   = [ ,1 ,  ,2 , . . .,  , ].After -point IFFT operation and CP of length ℓ cp