WITH the introduction of space-time codes (STC) it has

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE Pragmatic Space-Time Trellis Codes: GTF-Based Design for Block Fading Channels Velio Tralli, Senior Member, IEEE, Andrea Conti, Senior Member, IEEE, and Marco Chiani, Fellow, IEEE Abstract The pragmatic approach for the construction of space-time codes over block fading channels leads to low complexity encoders/decoders, making them suitable for various applications including cooperative communications. We propose a design methodology for pragmatic space-time codes, which is based on an extension of the concept of generalized transfer function. Our search algorithm provides code generators for arbitrary number of states, antennas, and fading rates. Numerical results show that the performance of the pragmatic space-time codes obtained by our design methodology is generally better than that of previously known space-time trellis codes in block fading channels. Index Terms Block fading channels (BFC), generalized transfer function, performance evaluation, pragmatic space-time codes (STC). I. INTRODUCTION WITH the introduction of space-time codes (STC) it has been shown how multiple transmitting antennas can be used to improve system performance without sacrificing spectral efficiency with the use of proper trellis codes [1] [6]. The information-theoretic capacity of multiple-input multiple-output (MIMO) systems in block fading channels (BFC) [7] [9] has been studied in [10] assuming the availability of channel state information (CSI) at both transmitter and receiver. One of the key conclusions is that antenna diversity can be a substitute for temporal diversity from the capacity point of view, but this does not necessarily hold when (CSI) is only available at the receiver [11]. In this scenario, the code design objective is to exploit both spatial and temporal diversity. The design of STC over quasi-static flat fading (i.e., fading level constant over a frame and independent frame by frame) has been addressed in [3], where some handcrafted trellis codes for the case of two transmitting antennas have been proposed. Extensions of this work have eventually appeared in the literature to design good codes for different scenarios. STC with improved coding gain have been presented in [12] [14]. In [15] it has been highlighted that the diversity achievable by STC for Manuscript received December 10, 2009; revised July 30, 2010 and November 29, 2010; accepted February 04, Date of publication February 22, 2011; date of current version May 18, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Gerald Matz. This work was supported in part by the FP7 European project OPTIMIX under Grant Agreement V. Tralli and A. Conti are with ENDIF, University of Ferrara, Ferrara, Italy and WiLab, Universiy of Bologna, Bologna, Italy ( velio.tralli@unife.it; a.conti@ieee.org). M. Chiani is with DEIS/WiLab, University of Bologna, Bologna, Italy ( marco.chiani@unibo.it). Digital Object Identifier /TSP binary phase shift keying (BPSK) and quartenary phase shift keying (QPSK) modulations can also be investigated through a binary design criteria, instead of looking for the Euclidean distances among complex transmitted sequences. This approach has been extended to MIMO BFC in [11] where the notion of universal STC is introduced to exploit diversity over an arbitrary number of fading blocks per codeword. The binary design criterion proposed in [11] and [15] is based on the pairwise error probability (PEP). It is quite well known that, despite their simplicity, code design methodologies based on optimization of the worst case PEP could be not appropriate over the BFC, since the worst case pairwise error event is not necessarily the only dominant error event [9], [16] [18]. For example, [17] proposed a distance spectrum analysis for comparing the relative merit of STC based on trellis code modulation in quasi-static fading channel. The determination of STC with maximum diversity gain and large coding gain remains a difficult task, especially as the number of transmitting antennas and states increases. Furthermore, the design of STC for fast fading channels is still an open problem. A criterion to achieve maximum diversity is given in [3], where, however, coding gain optimization is not addressed and the STC require ad hoc encoders and decoders. An alternative architecture for BFC has been analyzed in [19], while bit-interleaving has been investigated in [20] [23]. The pragmatic space-time codes (P-STC) 1 are of interest since they simplify the structure of the encoder and decoder, and allow for a feasible method to search for good codes in BFC. The pragmatic approach consists in the use of common convolutional encoders and Viterbi decoders over multiple transmitting and receiving antennas [16]. This approach has been also applied in the context of cooperative communications [25] where two or more transmitters use different convolutional encoders with the same trellis to encode the same bit sequence. In this paper we show that P-STC are able to achieve maximum diversity in BFC, and better performance than other existing schemes, with no need for specific encoder or decoder other than those used for convolutional codes (CC). The Viterbi decoder for P-STC requires only a simple modification of the branch metrics. The contributions of the paper are the following: a method to evaluate bounds on the performance of STC over BFC; an algorithm for searching the CC generators for P-STC over BFC; 1 This coding scheme was proposed in [16] by extending to multiple antennas the approach [24]. A similar structure is also considered in [11] and denoted as algebraic STC X/$ IEEE

2 2810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 code generators for various combinations of number of states, number of antennas, and fading rates, based on a new search methodology. The numerical results show that our P-STC perform better than the best known STC, confirming the validity of the new methodology based on the generalized transfer function (GTF). With respect to [11] and [15] this work provides, for arbitrary number of fading blocks per codeword, codes optimized with respect to the frame error probability (FEP) rather than on the worst case PEP. The paper is organized as follows: in Section II the channel model and system architecture are described. In Section III the PEP and FEP are analyzed by means of the GTF, and geometrical uniformity is discussed. In Section IV the algorithm for searching optimum codes over BFC is proposed. Numerical results are reported in Section V, and conclusions are given in Section VI. Through the paper the superscripts,, and denote conjugation and transposition, transposition only, and conjugation only, respectively. II. SYSTEM ARCHITECTURE AND CHANNEL MODEL We consider a space-time coding scheme with transmitting and receiving antennas. With the notation we indicate a supersymbol, that is a vector of unitary norm symbols simultaneously transmitted at discrete time on the antennas. A codeword (hereafter also called frame) is a sequence of supersymbols generated by the encoder. We focus here on symbol interleaving (bit interleaving is addressed in [22]). Then, each codeword is interleaved (we refer to intra-codeword interleaving) to obtain the sequence, where is a permutation of the integers and is the interleaving function. With we will indicate the inverse of the interleaving function (de-interleaving). The supersymbol is transmitted at time (i.e., ) through the channel described by the matrix with and, where is the channel gain between transmitting antenna and receiving antenna at time. In the BFC model [7] [9] these channel matrices do not change for consecutive transmissions, hence there are possible distinct channel matrix instances per codeword. 2 By denoting with the set of channel blocks, we have for and. When the length of a fading block is equal to one,wehave the ideally interleaved fading channel (i.e., independent fading levels from symbol to symbol), while for we have the quasi-static fading channel (fading level constant over a codeword). By varying we can describe channels with different correlation degrees or velocities [7] [9]. 2 For simplicity, we assume that N is an integer multiple of B. (1) At the receiving side, the sequence of received signal vectors is and after de-interleaving we have, where the received vector at time is with components Here, represents the signal-space representation of the signal received by antenna at time, the noise terms are independent, identically distributed (i.i.d.) complex Gaussian random variables (RVs), with mean zero and variance per dimension. The RVs are assumed i.i.d. (we consider spatially uncorrelated channels) complex Gaussian with mean zero and variance 1/2 per dimension, hence are Rayleigh distributed RVs with unitary power. Due to the normalization on fading gains the transmitted energy per symbol is equal to the average received symbol energy (per transmitting antenna). The total energy transmitted per supersymbol is and the energy transmitted per information bit is where is the number of bits per modulation symbol and is the code-rate. Thus, the spectral efficiency is given by [bps/hz]. For the discussions in the following sections it is worthwhile to recall that, over a Rayleigh fading channel, a system achieves a diversity if the error probability for large signal-to-noise ratio (SNR) is where is a constant depending on the asymptotic coding gain [26] [29]. The pragmatic approach consists in using common convolutional codes as space-time codes with the architecture presented in Fig. 1 [16]. Here, information bits are encoded by a convolutional encoder with rate. The output bits are divided into streams, one for each transmitting antenna, of BPSK or QPSK symbols that are obtained from a natural (Gray) mapping of bits, thus having for BPSK and, with for QPSK. The extension to other formats such as M-ary quadrature amplitude modulation (M-QAM) is also possible. We indicate the STC obtained with this scheme as -P-STC, where is the encoder constraint length and the associated trellis has states. The maximum likehood (ML) decoder for this pragmatic scheme is the usual Viterbi decoder for the same trellis of the convolutional encoder, with only a modification in the branch metrics. More precisely, for transmitting antennas and perfect CSI at the decoder the branch metric is where is the branch label. Thus, the advantages of P-STC with respect to STC are that the encoder is a common convolutional encoder, that the (Viterbi) decoder is the same of the original convolutional code, except for the metrics, and that, as we will show later, P-STC allow easy analysis and optimization even on BFC. (2) (3)

3 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2811 Fig. 1. Equivalent low-pass scheme for the proposed pragmatic space-time codes. III. ERROR PROBABILITY ANALYSIS OVER BFC In this section we analyze the performance of the general class of STC over BFC, with the aim to provide the tools to design good codes. A. Pairwise Error Probability The pairwise error probability is defined as the probability that, given the transmitted codeword, and having the two possible choices and with, the decoder chooses the codeword. The PEP conditional to the set of fading levels is given by where is the complementary Gaussian error function. The squared Euclidean distance at the channel output, conditional to, is (4) and is the set of indexes where the channel matrix is equal to. The set depends on the interleaving strategy adopted. Note that in our scheme (Fig. 1) the interleaving is done horizontally for each transmitting antenna and the set is independent on (i.e., the interleaving rule is the same for all antennas). The matrix defined in (7) gives, for the two codewords and, the distance calculated over the supersymbols transmitted on the same fading block. The matrix is also Hermitian nonnegative definite. Therefore, it can be written as, where is a unitary matrix and is a real diagonal matrix whose diagonal elements are the eigenvalues of counting multiplicity. Note that the eigenvalues are a function of. Thus, we can express through the eigenvalues of as (5) where is the vector of fading coefficients for the receiving antenna, and the matrix with elements is Hermitian and nonnegative definite. The vectors are the supersymbols at time in the sequences and, respectively. In BFC, for each frame and each receiving antenna the fading channel is described by different vectors, where is the th row of. By grouping these vectors, we can rewrite (5) as where for the fading block (6) where. This equation generalizes to BFC a similar expression reported in [3]. Since represents a unitary transformation, has independent complex Gaussian elements with zero mean and variance 1/2 per dimension. Moreover, the vectors and are independent for all due to the BFC model. Thus, using (8) in (4) and averaging over the fading, the asymptotic behavior of the PEP for large SNR is where, the integer is the number of nonzero eigenvalues of, and (8) (9) (10) (7) In the following we will denote as pairwise transmit diversity.

4 2812 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 Then, the PEP of codewords and shows a diversity that is the product between the transmit diversity and the receive diversity. Equation (9) generalizes the PEP for the quasi-static channel given in [3] to the BFC, accounting, through (7), for the number of fading levels per codeword and the interleaving rule. The analysis is valid for STC in general and applied next to P-STC. B. Derivation of Frame Error Probability and Bounds Given the transmitted codeword, the union bound on the frame error probability is the FEP in (13) can be approximated for large SNR by where (15) (11) where is upper bounded for large SNR by (9). We note that in (9) the dominant terms are those with minimum (remember that the parameters and depend on codewords and ). So, by retaining dominant terms only in (9), the FEP in (11), conditioned to the transmit codeword, is approximated for large SNR by where and (12) is the set of codeword sequences with diversity. About the asymptotic FEP in (12), we remark that: (i) the asymptotic behavior shows that the diversity gain increases linearly with the number of receiving antenna; (ii) the transmit diversity order plays the same role of the code free distance in additive white Gaussian noise (AWGN) channels. Clearly, the unconditional FEP (averaged over all possible transmitted codewords) coincides with when dealing with codes for which does not depend on the transmitted codeword (see also the discussion in Section III-F). However, in general, the FEP is upper bounded by (13) where is the probability of transmitting the codeword. By using (12) and observing that the dominant terms are those with minimum transmit diversity (14) We observe that the asymptotic FEP of STC over BFC given in (15) depends on both the achievable diversity and on the performance factor (16) Note also that and the weights for each pair and do not depend on the number of receiving antennas. Therefore, when a code reaches the maximum diversity in a system with one receiving antenna, the same code reaches the maximum diversity when used with multiple receiving antennas. However, due to the presence of the exponent in each term of the sum in (16), the best code for a given number of antennas (i.e., the code having the smallest performance factor) is not necessarily the best for a different number of receiving antennas. Thus, a search for optimum codes in terms of both diversity and performance factor must in principle be pursued for each. To summarize, the derivation of the asymptotic FEP for STC requires the computation of the rank and of the product of the nonzero eigenvalues for the matrices defined in (7). Similarly to [30], we also observe that: the union bound becomes tighter when the set of sequences is restricted to those diverging only once from the path of codeword in the trellis diagram; the first event error probability at time is obtained when the set of sequences is restricted to those diverging only at time from the path of codeword in the trellis diagram. In the particular case of periodical interleaving over the BFC we can use the first event error probability at to obtain a simpler but looser union bound as 3 (17) where is the set of codewords restricted to the first event errors at used to evaluate the asymptotic performance (15); we can easily obtain a FEP approximation by truncating the summation in (15) to the most significant terms (this aspect will be considered in Section III-F). 3 This is also known as first event error probability analysis.

5 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2813 C. Review of the GTF Concept for CC in BFC Here, we briefly review the concept of GTF applied to the performance evaluation of convolutional codes in BFC [18]. Let us consider a system where a block of information bits is encoded by means of a rate CC, interleaved, and finally transmitted with binary antipodal modulation over a BFC. For the sake of simplicity we also assume that the fading is constant over at least a trellis transition ( -tuple based interleaving). The convolutional encoder with constraint length starts encoding from the zero state and terminates the trellis by inserting a tail of zeros at the encoder input. If is the length of the binary input block including the zero-tail, is the codeword length. The encoder is described by a trellis diagram of length with states, where for each state and for each -tuple of input bits at the discrete time, there is a transition to the designated state at time carrying a label which is the -tuple of encoded bits. Each codeword obtained by encoding the input sequence is represented by a path in the trellis diagram. The error probability for any arbitrary transmitted sequence, conditional to the set of fading levels, is given by (18) where is the Hamming distance between the subset of symbols of the codewords and that are transmitted over the -th block of the BFC. To evaluate the GTF we first introduce the error trellis diagram to describe all possible error sequences 4 of length, for a given, where and are the input sequences corresponding to output codewords and. This trellis has the same set of states (CC are group trellis codes [31]), but each state, which is represented by the bits stored in the encoder, is given by the bitwise sum of the states related to sequences and. The label of the state transition corresponding to is simply. When is the all-zero sequence, as usually considered, the error trellis is exactly the trellis of the code. Then for the BFC we introduce variables, one for each distinct fading block. The GTF of the error trellis diagram is the multivariate scalar function (19) where is the number of sequences characterized by the same set of Hamming distances from. The relation between the GTF and the conditional FEP is 4 With 8 we denote the element-wise binary sum. (20) To obtain the GTF, we first replace the labels in the error trellis diagram, for each generic transition from to, with (21) where is a scalar and is the Hamming distance between and. Then, we specialize the error trellis diagram for a system with BFC by replacing the set of scalars with the set of variables related to the fading blocks. The mapping between and takes into account the interleaving function adopted. As a last step, we evaluate for each node of the trellis, that is for each state a weighting polynomial, given by the sum over all transitions reaching of the products of the transition label and the weighting polynomial of the node from which the transition departs at time. Finally, if we set to 1 the weight of the initial state of the trellis, denoted by (the zero-state at time 0), we obtain the GTF as where is the final zero-state and the contribution of the all-zero error sequence (the polynomial 1) is subtracted. Some algorithms to evaluate it are discussed in [18]. D. The Space-Time Generalized Transfer Function for P-STC in BFC The evaluation of the FEP bound for P-STC can be carried out by generalizing the methodology described in Section III-C. This leads to the definition of the space-time generalized transfer function (ST-GTF) for BFC, whose difference with respect to CC is that it accounts for the space-time fading channel. To define the ST-GTF let us first introduce the error sequences and discuss their role in the evaluation of the FEP. P-STC are built using common binary convolutional codes, therefore they are group-trellis codes [31]. Let us consider the input bit sequences and, both of length, generating the output codewords and, respectively, and define as the input error sequence for the transmitted codeword and decoded codeword. Then, we can rewrite the FEP bound, conditional to the set of fading levels,as (22) where is the encoding function giving and, is the a-priori probability of the input bit sequence, and is given by (4). As before, the upper bound is preserved by restricting the set of error sequences to those diverging only once from the all-zero sequence. The bound is also simplified by considering the error sequences diverging from the all-zero sequence at. Following the methodology reported in Section III-C, we first associate a variable to each fading block. Then, since

6 2814 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 in (7) accounts for the codeword distance over the fading block, we define the ST-GTF for BFC as a polynomial in the variables, where the exponents are the matrices 5, as follows: sition with a new matrix label whose elements are given by 7 where with is a variable associated to the trellis transitions at time and is the matrix (23) The meaning of the operation will be illustrated in the next paragraphs. In (23) each pairwise error event is characterized by a set of matrices, and enumerates (including the weight ) the error sequences producing. Note that the most important terms in (15) are those related to the matrices having minimum diversity, that is the minimum value of. So, it will be useful to derive these terms from (23), and in particular the weight, given by the product of the nonzero eigenvalues of. Let us now introduce the method proposed to derive the ST-GTF by using, for the sake of clarity, the four steps (a) (d) outlined below. This method generalizes to P-STC the procedure summarized in the Section III-C. Step a) Construction of the error trellis diagram of the P-STC, starting from the trellis diagram of length describing the P-STC. This trellis can be used for both the set of input sequences and the set of error sequences (in both cases it is the same trellis diagram of the convolutional code but with different meanings of input and output sequences). Let us denote with and the binary vectors representing the generic state at time when the trellis is referred to the input sequences and to the error sequences, respectively. States and belong to, where is the number of states, being the constraint length of the CC. Let us build the error trellis diagram according to [32, Ch. 12] by labeling each trellis edge connecting to with a matrix. Each matrix has elements, with, for each state transition on the trellis of the input sequence, given by 6 where is the output supersymbol labeling the transition from state to state and is the output supersymbol labeling the transition from state to state (this is a supersymbol of the sequence ). The elements of matrix tell us, for each error block of the error sequence, which are the admissible pairs of coded supersymbols compatible with that input error. Step b) Construction of a modified error trellis diagram by replacing in the error trellis diagram the label of the generic tran- 5 To simplify the notation the dependency of F on c;g will be omitted in the following derivations. 6 The element E is void if no transition from S to S exists in the trellis. The elements are not matrices themselves, but symbolic elements intended to replace the single terms or, with scalar, in the polynomial structure of the GTF of the CC (see [18] and Section III-C). The following rules are defined for sum and product of powers with matrices at the exponent: (24) where are generic nonnegative definite matrices, is the zero matrix, and are scalars. Hence, these symbolic elements which include matrices can be used in sums and products in the same way as the terms of the GTF of CC. This trellis diagram, named error trellis with error matrices, depends on the sequence of scalar variables related to the multiple fading channel levels seen by each supersymbol of the codeword. Due to BFC, for each realization there are different fading levels per frame. Step c) Construction of the error trellis diagram with error matrices specialized for the BFC. This is done by mapping the sequence of variables to a sequence of scalar variables, each one related to one of the fading blocks of the BFC. As an example, in case of quasi-static channel only one variable is needed. In the opposite case of perfect symbol interleaving, the number of variables would be taken equal to the frame length. However, since they correspond to i.i.d. RVs, only one variable may be used for evaluating the average FEP. The mapping can be done through the position (25) where is the inverse of the interleaving function defined in Section II. Note that the effect of the interleaving is taken into account at this step of the procedure. Step d) Evaluation of the transfer function for the error trellis diagram by generalizing the method in Section III-C. With respect to Section III-C, error matrices with special elements appear, instead of scalars, as labels of the error trellis diagram. 7 The entry E is 0 if no transition from S to S exists in the trellis. In case of zero-tailed terminated codes the term 2 must be removed from labels at t = N 0 +1;N 0 +2;...;N 0 1.

7 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2815 Hence, products and sums apply to matrices, taking into account the rules (24). First, following Section III-C, we evaluate for each node of the error trellis with error matrices (thus for each state at time )a weighting vector. The novelty here is that we have a vector of multivariate functions whose elements are of the kind (see also the example reported in Appendix). The weighting vector can be evaluated as the sum over all the transitions reaching of the vectors obtained by multiplying the transpose of each transition label, which is a matrix, by the weighting vector of the node at time from which the transition departs. Next, if we set to the weighting vector of the initial state of the trellis, denoted by, i.e. the zero-state at the time 0, and if we set for each at time 0, we obtain 8 the ST-GTF as (26) where is the final state of the trellis (the zero-state at time ) and the contribution of the correct sequence (the polynomial 1) is subtracted. This expression holds for zero-tailed codes, and the term accounts for the effect of termination. This structure of ST-GTF becomes the one reported in (23), since the successive application of the first rule (24) in the GTF evaluation reproduces for all pairs the sum in (7). The algorithms discussed in [18] can be used to efficiently evaluate it. In case of periodic interleaving and infinite codeword size the GTF can be evaluated with standard algorithms on the state diagram. An example on how these steps can be applied to a simple P-STC is given in Appendix. E. Application of ST-GTF to the Evaluation of FEP By following the approach usually adopted for convolutional codes (see Section III-C), the ST-GTF in (26) can be directly used to bound the FEP as in (13). Through the well-known bound for we have (27) where the terms in the argument of the exponentials come from (4) and (8) with and. The linear operator maps symbolic terms to a product of variables as (28) where are the nonzero eigenvalues of and. As alternatives, the tighter exponential bound, or the approximation can also be used [33]. Finally, the performance factor (16) is given by 8 It is assumed that the encoding starts from the all-zero state at time 0. (29) where is the set of error matrices giving, i.e. The value of can be derived from the ST-GTF looking at the terms of the sum characterized by the sequence of error matrices with minimum transmit diversity. This is illustrated in the example of Appendix. Important remarks on the evaluation of FEP through the ST-GTF are the following. First, the ST-GTF depends on both encoder and interleaver structures, which have been suitably considered to build the error state diagram for BFC. Second, if we are interested in evaluating the FEP conditioned to a selected reference codeword (usually given by all-zero input sequence) then we need to define a ST-GTF referred to, which can be easily obtained by using scalar (not matrix) labels in both the error trellis diagram and the modified error trellis diagram. In this case, the generic transition of the error trellis has to be labeled with where is the matrix given by and is the sequence of encoder states leading to the output sequence. At each state the associated polynomial is a scalar (not a vector), and the ST-GTF is simply obtained as. Third, if the goal is to find the error probability according to (17) or to a tighter bound (obtained by limiting the set of decoded sequences to those corresponding to paths in the trellis diagram diverging only once from the path of codeword ), then we can define a modified error trellis diagram by splitting the all-zero state at each time,, into two states: the diverging state, having only transitions departing to all the other states, and the converging state, having only the transition departing to and all the transitions arriving from. By defining the time- ST-GTF of this diagram as, when the initial settings are,,, and for, we obtain the transfer function which can be used in place of to refine the FEP bound, and the time-0 ST-GTF which can be used to evaluate (17) through. F. Discussion on Geometrical Uniformity Note that the FEP given in (11) is in general a function of the reference codeword. The conditions under which there is no dependence on the transmitted codeword are related to the concept of geometrical uniformity, that has been introduced in [34] with respect to Euclidean distance. Geometrically uniform codes are codes with the same distance profile for all pairs of codewords. In AWGN channels, the

8 2816 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 geometrical uniformity guarantees that the performance is independent of the particular transmitted codeword. Thus, the FEP can be evaluated by assuming the transmission of a particular codeword, that can be the all-zero codeword. Clearly, this condition greatly simplifies the code design. However, the application of the concept of geometrical uniformity to STC requires a careful investigation, as highlighted in [35]. Indeed, it can be noticed that the PEP depends on the Euclidean distance between the coded signals after the MIMO channel and hence on the eigenvalues of matrices like those defined in (7), that can change with the reference codeword. For this reason, in general the design of STC should consider all possible transmitted codewords. For P-STC we can easily verify the following properties. (a) The P-STC before the channel (the set of codewords ) are geometrically uniform with respect to the Euclidean distance. In fact, for the P-STC with Gray mapping the Euclidean distance between the symbols of two generic codewords is for BPSK, and for QPSK, where denotes Hamming distance and the superscripts P, Q refer to the in-phase and the in-quadrature components, respectively. Since we are using CC, the Hamming distance spectrum is independent of the reference codeword, and therefore the same is true for the Euclidean distance spectrum. (b) In a system with BFC and ideal symbol interleaving (i.e., ), the PEP in (4) depends on the Hamming distance between the two codewords, but not on the specific reference codeword chosen. In fact, the PEP depends on the statistical distribution of the distance after the channel, defined in (5). However, in Rayleigh fading channels each is a complex zero-mean Gaussian distributed RV with variance 1/2 per dimension; then the generic term is still zero-mean complex Gaussian with variance, which is proportional to the Hamming distance previously discussed. The resulting overall variable is still zero-mean complex Gaussian, with a variance that depends only on the Hamming distance between the codewords and. Since for ideal interleaving the RVs are independent also in, we can conclude that the distribution of the RV defined in (5) depends only on the Hamming distance between the codewords. 9 Thus, since for P-STC the Hamming distance spectrum is invariant with the reference codeword, the same applies to the FEP bound (11). 9 It can be also shown that in (6) the matrix F (c; g)=a (c; g) has only one nonzero eigenvalue given by = jc 0 g j directly related to the Hamming distance of supersymbols C and G. (c) In other cases and especially for quasi-static fading channels, although the code is geometrically uniform before the channel, we could expect that the PEP depends in general on the reference codeword. In fact, it depends, through (8), on the eigenvalues of the matrix defined in (7). However, in many cases we have numerically verified that the conditional FEP does not change significantly with the selected reference codeword. This happens in particular when: the number of fading blocks in the BFC is large enough with respect to the length of the error sequences (in this case the behavior of the ideally interleaved code is approached); the memory of the code is small, and consequently the error sequences are short. In this case for a large class of codes the distance in (8) has a distribution, over the set of all possible codewords, which is mainly driven by the sum of the eigenvalues, that is, the trace of. Again, this is related to terms and therefore to the Hamming distance between the codewords. Thus, the performance is mainly determined by the Hamming distance spectrum that, in P-STC, is invariant with the reference codeword. This will be verified numerically in Section V. In addition, it is also worth noting that for P-STC with antennas and BPSK modulation the FEP evaluated with the all-zero sequence as a reference codeword is always the worst-case error probability. This can be proved by looking at the structure of matrix and its eigenvalues (not included here for conciseness). In general, we will not rely on the geometrically uniformity assumption (that holds before the channel but not after), and we then analyze and design P-STC averaging over all possible transmitted sequences. However, we will also show that fixing a particular reference codeword often gives similar results. IV. SEARCH FOR THE OPTIMUM P-STC ON BFC In this section, we address the efficient search of optimum generators for P-STC in BFC. Our search criterion is based on the asymptotic FEP in (15). The optimum code with fixed parameters, among the set of noncatastrophic codes, is that which (i) maximizes the achieved diversity ; (ii) minimizes the performance factor ; where the values of and can be extracted from the ST-GTF (23) of the code. An exhaustive search algorithm should evaluate the ST-GTF for each code of the set. Another search criterion for STC has been investigated in [12], [14] where a method based on the evaluation of the worst PEP was proposed. Although the worst PEP carries information about the achieved diversity,, it is incomplete with respect to the coding gain. Our method based on the union bound is still approximate with respect to coding gain, but it includes more information than the other methods, leading to better codes. When applying our search criterion we must consider, as shown in [36], that the union bound for the average error probability is loose and in some cases (long codes and small diversity) can be far from the actual value. This problem can be

9 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2817 partially overcame by truncating the sum to the most significant terms, leading to an approximation, but giving good results in reproducing the actual performance ranking of the codes among those achieving the same diversity. This will be numerically verified in Section V. Since cannot be larger than both and the free distance of the CC used to build the P-STC, then the free distance of a good code for a given BFC should be at least or larger to capture the maximum diversity per receiving antenna offered by the channel (i.e., ). On the other hand, there is a fundamental limit on the achievable diversity for BFC related to the Singleton bound [9]. In fact, if we define the reference BFC for the system as the ideal equivalent BFC with fading blocks that would describe the space-time fading channel if the transmitters determine independent channels, then the achievable diversity, which cannot be larger than the diversity achievable on the reference BFC, is upper bounded by (30) As an example, to achieve full diversity with a P-STC on a quasi-static channel the value cannot be larger than 1, thus the code rate of the CC can not be larger than, or the value of cannot be smaller than (see also [3]). Different methodologies can be used to compute the ST-GTF of the error trellis diagram. By following a classical approach, the error state diagram could be derived by splitting the all-zero error state from the related trellis and in principle classical techniques could be used to evaluate, but this approach is limited to long codewords with periodical interleaving. To overcome these limits, the ST-GTF has to be computed on the error trellis. A simple way is to proceed along the error trellis with an iterative algorithm which evaluates for each state the weighting vector starting from and ending in with initial conditions and for. Since branches connect the states of the trellis at each step, there are products of multivariate functions, that can be reduced to to account for zeros in the matrix labels, and further reduced to if labels are scalars. With the aim of using the ST-GTF to compare different codes in a systematic search for best codes, the previous algorithm still maintains a large complexity due to the growth of the number of terms in the node weights when increases, and only conventional simplification rules are available to reduce the complexity, which do not allow a significant improvement in the computational efficiency. The last issue was addressed in [18] for CC over BFC, where an algorithm and some simplification rules were given to largely reduce the computation complexity. We propose here to derive and apply similar rules aimed at discarding the parts inside weighting vectors which do not contribute to the most significant terms of the ST-GTF, namely, those having small diversity order and product-degree, enabling the evaluation of and. To formulate this method let us start from the error state diagram modified by splitting the all-zero states at each time (see the remark at the end of Section III-E and also the example in the Appendix) and, for each state at time, let us consider the weighting vector which can be evaluated for the state by using the initial settings, and for. Let us also denote with the set of weighting vectors obtained for all at time in the modified error state diagram (zero states split at time ). The sets of weighting vectors can be computed along the error trellis, by using the same iterative algorithm presented in [18, Appendix] for scalar weighting polynomials. The algorithm starts from and ends at with the required initial setting for the weighting vectors of the zero-states along the trellis. By using the weighting vectors at the end of the trellis, we finally obtain and. The evaluation of the ST-GTF through the iterative computation of for each leads to a much more efficient computation of the truncated asymptotic bound, because it allows the application of suitable simplification rules. To derive them we start from the two following properties of nonnegative definite Hermitian matrices [37]: Property 1: the rank of the sum of two nonnegative definite Hermitian matrices is greater than, or equal to, the rank of each matrix. Property 2: the product of the nonzero eigenvalues of the sum of two nonnegative definite Hermitian matrices is greater than, or equal to, the product of the nonzero eigenvalues of each matrix. These properties can be used to derive simplification rules useful to eliminate terms 10 in the multivariate functions of weighting vectors which do not affect the final value of and. More precisely, Property 1 allows to eliminate from each element of the polynomial terms with rank of the exponent strictly greater than the minimum rank of the exponent of the polynomial terms in. Property 2 allows to eliminate from each element of vector the polynomial terms with the product of the nonzero eigenvalues of the exponent much greater, e.g. 100 times, than the product of the nonzero eigenvalues of the polynomial terms with minimum rank in. V. NUMERICAL RESULTS This section presents the results of the proposed search algorithm used to design P-STC in BFC, and the simulated FER versus the SNR per receiving antenna element,. Let us now consider the search for optimum generators in the sense defined in Section IV. In Table I we report, for quasi-static fading channel and QPSK, the characteristic parameters and performance of the best generators for the (4,2,2) 2-P- STC, compared with codes proposed in [3]. It is possible to show that for these parameters the code given in [3] is amenable of a P-STC representation with generators. Note that all codes achieve the available diversity, but with different performance factors. It is remarkable that the 10 With a slight abuse in terminology we denote them as polynomial terms.

10 2818 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 TABLE I COMPARISON OF RATE 2/4 P-STC WITH QPSK, n = m =2, =2ON BFC WITH L =1, =15dBAND N =130.THE PERFORMANCE FACTOR IS TRUNCATED TO =9.THE FIRST TWO CODES ARE THE BEST PRODUCED BY THE SEARCH (THERE ARE 12 FIRST-CLASS CODES WITH ALMOST THE SAME BEHAVIOR, AND 48 SECOND-CLASS CODES). THE THIRD CODE IS THE CODE PROPOSED IN [3] TABLE II OPTIMUM RATE 1/2 P-STC WITH BPSK, n =2, m =1ON BFC WITH L =1.PARAMETERS WITH SUPERSCRIPT REFER TO THE CODES OBTAINED WITH BEST CONVOLUTIONAL CODES FOR AWGN CHANNEL. THE PERFORMANCE FACTOR IS TRUNCATED TO =2d 0 1. ERROR PERFORMANCE REFERS TO = 20 db AND N = 130 ratio between performance factors is approximately the same as the ratio of the simulated FERs. Note also that the performance factor evaluated by fixing a reference codeword provides a slightly different ranking of generators, giving as best code the generator. This code is not the best according to (that would give ), but is very close to it in terms of performance factor, and the best in terms of FER. As will be clarified in the following, we checked that there are 11 codes out of behaving as the best and 47 behaving as the second best, meaning that there is not a single best code but a class of codes that perform similarly. This fact suggested us to carefully investigate the performance differences among generators through exhaustive simulations that we performed for all possible (4,2,2) 2-P-STC with in terms of FER in quasi-static fading channel, with. In Fig. 2 we report the FER for all 4-states P-STC obtained through 2/4 convolutional encoders (i.e., generators ordered in abscissa). A remarkable outcome is that noncatastrophic codes can be divided in few classes, with almost the same performance for codes in the same class. 11 Note that within the class of codes providing the best performance, there is the one obtained through our searching methodology, that gives a FER of about. With our code searching algorithm we saved about two order of magnitude in time with respect to the exhaustive search by simulation (requiring several days of computation even for this small code); this highlight the importance of algorithmic methods. Then, we investigate the impact of the number of states on the performance of (2,1, ) 2-P-STC with BPSK in quasi-static fading channel. In Table II we report the best codes obtained through the search algorithm for 2, 4, 8, and 16 states, for which we indicate the achieved diversity, the performance factor, and the FER. We also report the performance factor for AWGN optimal generators [32] with the same number of states. Note that all codes achieve the maximum diversity and that increasing the number of states does not produce relevant performance 11 Thus, also for P-STC it is verified a phenomenon similar to what already discussed in [18] for CC in BFC. Fig. 2. Exhaustive search for MIMO(2,2) P-STC in terms of FER: QPSK, quasi-static fading channel (L =1), =9dB. improvements. Moreover, on the quasi-static channel the error probability bound tends to become looser, especially when the free distance of the convolutional code increases with respect to the achieved diversity. The behavior is different in BFC with time diversity available. This case is illustrated in Table III for, where it is shown that increasing the number of states results in a larger diversity. Note also that the optimum P-STC are able to achieve a diversity equal to that achieved by using the optimal generators for the AWGN channel. Both are not able to reach the diversity available on the reference BFC with fading levels per codeword, that is, CC are more suitable to collect time diversity than spatial diversity. We report in Tables IV, V, and VI the optimum generators obtained through the search algorithm for, 3, 4, respectively, with BPSK and QPSK modulations, and for different number of states. The corresponding performance factors are also reported. It is worth noting that, although the codes are geometrically uniform at channel input but not at channel output, in

11 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2819 TABLE III OPTIMUM RATE 1/2 P-STC WITH BPSK, n =2, m =1ON BFC WITH L =8.PARAMETERS WITH SUPERSCRIPT REFER TO THE CODES OBTAINED WITH BEST CONVOLUTIONAL CODES FOR AWGN CHANNEL. THE PERFORMANCE FACTOR IS TRUNCATED TO =2d 0 1. THE ~ IS THE DIVERSITY ACHIEVABLE ON THE REFERENCE BFC WITH nl FADING BLOCKS [18]; THE VALUE OF THE SINGLETON BOUND IS 9 TABLE IV OPTIMUM P-STC FOR A SYSTEM WITH n =2, m =1ON BFC WITH L =1.SYMBOL 3 INDICATES THAT THE SEARCH BASED ON F TO THE SAME CODE AS THE FULL SEARCH (c ;m) LEADS TABLE V OPTIMUM P-STC FOR A SYSTEM WITH n =3, m =1ON BFC WITH L =1.SYMBOL 3 INDICATES THAT THE SEARCH BASED ON F SAME CODE AS THE FULL SEARCH; N:E: = not evaluated (c ;m) LEADS TO THE TABLE VI OPTIMUM P-STC FOR A SYSTEM WITH n =4, m =1ON BFC WITH L =1.SYMBOL INDICATES THAT THE SEARCH BASED ON F (c ;m) LEADS TO THE SAME CODE AS THE FULL SEARCH; N:E: = not evaluated most of the cases the code search based on the ST-GTF with a fixed reference codeword leads to the same code as the search over all possible transmitted codewords, or to a code with similar performance. Finally, in Fig. 3 we compare our 2 bps/hz (4,2,2) 2-P-STC for in quasi-static fading channel with others similar STC schemes proposed in [3], [12], and [14]. It is worth noting that our scheme outperform the previously known schemes in terms of coding gain. The gain at with respect to the STC in [3] is about 1.4 db. VI. CONCLUSION In this paper, we extended to space-time codes the concept of generalized transfer function, which was introduced for convolutional codes in block fading channels. This results in the possibility to rank different P-STC with an efficient algorithm based on the asymptotic error probability. Then, a search methodology to obtain optimum generators over BFC has been proposed. It has been shown that the search leads to codes which achieve better performance compared to previously known STC, for several combinations of spectral efficiency, number of antennas, and fading rates. Therefore, P-STC have been shown to represent a valuable choice both in terms of implementation complexity and performance. APPENDIX EXAMPLE OF ST-GTF FOR P-STC An example of computation of the ST-GTF for P-STC is reported here. We consider a (2,1,2) 2-PSTC with receiving antenna over a quasi-static BFC and obtained from a rate 1/2 CC with generators and BPSK modulation format. These generators are the best for two-states CC in AWGN, and give free distance 3. When used to build a P-STC, this choice of generators produces the second best code in quasistatic fading channels, achieving diversity 2, that is the maximum available diversity. This code has been chosen as simple enough to illustrate the evaluation of time-0 ST-GTF,,

12 2820 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 Fig. 3. Comparison of 2 bps/hz (4,2,2) 2-P-STC with m =2in quasi-static fading channel. Our best code (CCT) has generators (06; 13; 11; 16). Other space-time codes are: TSC [3], BBH [12], and YB [14]. and to allow the use of standard algorithms on the modified error state diagram for periodic interleaving. The two-states trellis of the CC is depicted in Fig. 4(a), whereas the trellis of the P-STC is in Fig. 4(b). The associated possible output symbols and are in the set with,,, and. Thus, the matrix is in the set with Fig. 4. Trellis diagram for the example in the Appendix. The states represented by empty circles are those with weighting vector [0; 0]. Error trellis diagrams can be derived according to the procedure outlined in Section III-D. In Fig. 4(c), (d), and (e) the error trellis, the error trellis diagram with error matrices for BFC ( here) and the error trellis diagram modified by splitting the zero-state, respectively, are illustrated. To understand how to obtain the labels we can refer to Table VII. The error matrices result The corresponding ST-GTF can be evaluated after having derived the weighting vectors and on the error trellis diagram at each time as follows: When at time is computed, we can evaluate. As an example, for a code of length we would have. The time-0 ST-GTF can be easily derived with the same procedure, but applied to the error trellis of Fig. 4(e). As an example, for a code of length we would derive. In the case of quasi-static fading channel or in the case of periodical interleaving, the time-0 ST-GTF can also be derived by using standard algorithms on the error state diagram modified by splitting the zero-state. In this case the results holds for codes of infinite length. For this example, the modified error state diagram is drawn in Fig. 5(c). Thus, by following the same steps applied to the error trellis diagram, the error state diagram can be modified for BFC introducing the error matrices, as in Fig. 5(b). The corresponding ST-GTF can be evaluated as follows:

13 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2821 TABLE VII CONSTRUCTION OF THE LABELS APPEARING IN THE ERROR TRELLIS DIAGRAMS FOR THE EXAMPLE IN APPENDIX (32) Therefore, to obtain the ST-GTF of the code we need to compute the eigenvalues of matrices of the form or, with integer, leading to (33) Fig. 5. Trellis and state diagrams modified by splitting zero-state for the P-STC investigated in Appendix. which becomes giving (31) With respect to the ST-GTF evaluated on the error trellis diagram, the term accounting for zero-tailing is not needed (the state diagram is for sequences of infinite length) and is replaced by. It can also be expanded in where and are the two eigenvalues of or (in this example, ). Since the product of the eigenvalues is simply,we also obtain and,. Note that the approximate expression of derived on error state diagram is not convergent for the code in the example, whereas converges. It is also interesting to note that since both and have four equal elements, and and can be interchanged in without altering the eigenvalues, the ST-GTF does not depend on the particular reference sequence. Therefore, this code is geometrically uniform even at the output of the channel. As further check, we can evaluate for the all-zero reference codeword by exploiting the error trellis or the error state diagram with scalar labels; to this aim it is sufficient to replace, and, thus obtaining (34) which is equal to in (31), if we replace with, and leads to the same eigenvalues. To evaluate the ST-GTF of the considered P-STC in a BFC with and periodical interleaving, that is, we can modify the error state diagram (with scalar labels) by replacing the state with two states and, thus obtaining transitions with output, with output, with output,

14 2822 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011 with output and with output [see Fig. 5(d)]. Then, the ST-GTF results which gives and, after expansion in which only (35) (36) (37) has two terms with rank 1. Therefore implying and. ACKNOWLEDGMENT The authors wish to thank the editor and anonymous reviewers for valuable suggestions and comments. REFERENCES [1] A. Wittneben, A new bandwidth efficient transmit antenna modulation diversity scheme for linear digital modulation, in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, Jun. 1993, pp [2] J.-C. Guey, M. P. Fitz, M. R. Bell, and W.-Y. Kuo, Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels, in Proc. 46th Ann. Int. Veh. Technol. Conf., Atlanta, GA, Sep. 1996, pp [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inf. 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15 TRALLI et al.: PRAGMATIC SPACE-TIME TRELLIS CODES 2823 Velio Tralli (S 93 M 94 SM 05) received the Dr.Ing. degree in electronic engineering (with honors) and the Ph.D. degree in electronic engineering and computer science from the University of Bologna, Bologna, Italy, in 1989 and 1993, respectively. From 1994 to 1999, he was a Researcher of CNR at CSITE, University of Bologna, and an Assistant Professor with the University of Ferrara, Ferrara, Italy. Since 2001, he has been an Associate Professor with the University of Ferrara, teaching electrical communications and digital transmission. He conducts research within the telecommunication group at ENDIF Department, Ferrara University, and within the WILAB group in Bologna. His research interests are mainly concerned with wireless communication, digital transmission and coding, including radio resource management, cross-layer design and scheduling, wireless multihop networks, OFDM and multiantenna SDMA systems. He participated by playing a significant role to several national and European research project regarding short-range communications systems for vehicles, high-speed wireless LAN, wireless sensor networks, systems for immersive communication, as well as industrial project regarding 3G wireless networks, and smart antenna systems. Dr. Tralli serves as a reviewer for IEEE TRANSACTIONS/JOURNALS and Conferences, and as a TPC member for several international conferences. He has published many papers in refereed journals, including IEEE TRANSACTIONS, and international conferences. He served as a Co-Chair for the Wireless Communication Symposium of the 2006 IEEE International Conference on Communication (ICC 2006). Andrea Conti (S 99 M 01 SM 11) received the Laurea degree (summa cum laude) in telecommunications engineering and the Ph.D. degree in electronic engineering and computer science from the University of Bologna, Italy, in 1997 and 2001, respectively. In 2005, he joined the University of Ferrara, Italy, where he is an Assistant Professor. He was Researcher with Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT) from 1999 to 2002 and with the Istituto di Elettronica e di Ingegneria dell Informazione e delle Telecomunicazioni, Consiglio Nazionale delle Ricerche (IEIIT/CNR) from 2002 to 2005 at the Research Unit of Bologna. During summer 2001, he was with the Wireless Systems Research Department, AT&T Research Laboratories. Since 2003, he has been a frequent visitor to the Wireless Communication and Network Sciences Laboratory, Massachusetts Institute of Technology (MIT), Cambridge, where he presently holds the Research Affiliate appointment. He is a coauthor of Wireless Sensor and Actuator Networks: Enabling Technologies, Information Processing and Protocol Design (New York: Elsevier, 2008). His research interests involve theory and experimentation of wireless systems and networks including network localization, adaptive diversity communications, cooperative relaying techniques, interference management, and sensor networks. Dr. Conti is currently an Editor for the IEEE COMMUNICATIONS LETTERS. He served as an Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 2003 to 2009 and Guest Editor for the EURASIP Journal on Advances in Signal Processing (Special Issue on Wireless Cooperative Networks) in He is the Technical Program Chair for a number of IEEE conferences. He served as reviewer for IEEE and IET journals and as Technical Program Committee Member for numerous IEEE conferences. He was elected Vice Chair and Secretary of the IEEE Communications Society s Radio Communications Technical Committee for the terms and , respectively. Marco Chiani (M 94 SM 02 F 11) was born in Rimini, Italy, in April He received the Dr. Ing. degree (magna cum laude) in electronic engineering and the Ph.D. degree in electronic and computer science from the University of Bologna, Italy, in 1989 and 1993, respectively. He is a Full Professor in Telecommunications and the current Director of the Industrial Research Center on ICT at the University of Bologna. During summer 2001, he was a Visiting Scientist at AT&T Research Laboratories, Middletown, NJ. He is a frequent visitor at the Massachusetts Institute of Technology (MIT), Cambridge, where he presently holds a Research Affiliate appointment. His research interests include wireless communication systems, MIMO systems, wireless multimedia, lowdensity parity-check codes (LDPCC), and UWB. He is leading the research unit of the University of Bologna on cognitive radio and UWB (European project EUWB), on Joint Source and Channel Coding for wireless video (European projects Phoenix-FP6 and Optimix-FP7), and is a consultant to the European Space Agency (ESA-ESOC) for the design and evaluation of error correcting codes based on LDPCC for space CCSDS applications. Dr. Chiani has chaired, organized sessions, and served on the Technical Program Committees at several IEEE International Conferences. In January 2006, he received the ICNEWS award for Fundamental Contributions to the Theory and Practice of Wireless Communications. He was the recipient of the 2008 IEEE ComSoc Radio Communications Committee Outstanding Service Award. He is the past chair ( ) of the Radio Communications Committee of the IEEE Communication Society and past Editor of Wireless Communication ( ) for the IEEE TRANSACTIONS ON COMMUNICATIONS.