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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 4, APRIL 2003

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Trellis-Coded Differential Unitary Space–Time Modulation Over Flat Fading Channels Meixia Tao, Student Member, IEEE, and Roger S. Cheng, Member, IEEE

Abstract—Coding and modulation for multiple-antenna systems have gained much attention in wireless communications. This paper investigates a noncoherent trellis-coded scheme based on differential unitary space–time modulation when neither the transmitter nor the receiver know the channel. In a time-varying flat Rayleigh fading environment, we derive differentially noncoherent decision metrics and obtain performance measures for systems with either an ideal interleaver or no interleaver. We demonstrate that with an ideal interleaver, the system performance is dominated by the minimum Hamming distance of the trellis code, while without an interleaver, the performance is dominated by the minimum free squared determinant distance (a novel generalization of the Euclidean distance) of the code. For both cases, code construction is described for Ungerboeck-type codes. Several examples that are based on diagonal cyclic group constellations and offer a good tradeoff between the coding advantage and trellis complexity are provided. Simulation results show that, by applying the soft-decision Viterbi decoder, the proposed scheme can achieve very good performance even with few receive antennas. Extensions to trellis-coded differential space–time block codes are also discussed. Index Terms—Antenna diversity, differential detection, multiple-antenna systems, space–time code, trellis-coded modulation, wireless communications.

I. INTRODUCTION

R

ECENT theoretical studies [1], [2] show that employing multiple antennas at the transmitter side and/or the receiver side promises high spectral efficiency in wireless communications. Many transmission schemes, such as layered space–time codes [3]–[5] and coordinated space–time codes [6]–[9], have been developed to exploit this advantage. Most of them require perfect channel state information (CSI) at the receiver, which is reasonable only when the channel varies slowly enough that pilot signals can be used to estimate the CSI. However, when the channel varies rapidly, precisely tracking the channel variation may be difficult. Moreover, while a large number of transmit antennas increase the training period, which significantly reduces the system efficiency, a large number of receive antennas increase the complexity of doing the channel Paper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received November 19, 2001; revised March 27, 2002. This work was supported in part by the Hong Kong Research Grant Council and the CMC at the Hong Kong University of Science and Technology (HKUST). This paper was presented in part at the IEEE Wireless Communications and Networking Conference, Orlando, FL, USA, March 17–21, 2002. The authors are with the Center for Wireless Information Technology, Electrical and Electronic Engineering Department, Hong Kong University of Science & Technology, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.810844

estimation. Hence, for rapid fading channels, it is desirable to develop techniques that do not require channel estimation at the receiver. Motivated by this, [10], [11], and [12] studied the channel capacity of multiple-antenna systems when CSI is unavailable at either the transmitter or the receiver. To exploit the noncoherent capacity, Hughes [13] and Hochwald et al. [14] independently introduced a differential unitary space–time modulation (DUSTM) scheme. Their scheme can be viewed as an extension of the traditional differential phase-shift keying (DPSK) scheme used in single-antenna systems. The main difference is that the signal constellation is no longer the set of scalar symbols with unit amplitude, but the set of unitary matrices, called the unitary space–time modulation. The constellations proposed in [13] and [14] form groups under matrix multiplication, thus simplifying the differential transmission process. Constellations based on space–time block codes (STBC) [16]–[19] have an orthogonal structure and, therefore, allow simple differential detection. Other group and nongroup constellations are discussed in [15]. From the perspective of information theory, it is necessary to apply channel coding in front of DUSTM to further approach the channel capacity. In this paper, we will present a combined (not concatenated) trellis coding and unitary space–time modulation scheme with differential transmission, namely, trellis-coded differential unitary space–time modulation (TC-DUSTM). In a time-varying flat Rayleigh fading environment, we demonstrate that if a matrix-wise interleaver with sufficient length is used before differential transmission, the error probability is dominated by the minimum Hamming distance of the trellis code, which contributes linearly to the order of diversity. If the system is delay sensitive and the interleaver cannot be used, the error probability is dominated by the minimum free squared determinant distance (a novel generalization of the Euclidean distance to be defined later), which measures a lower bound of the coding gain. These results are similar to the well-known results of trellis-coded modulation (TCM) in single-antenna systems [21]. Thus, the trellis-code design is relatively easy in our proposed scheme, even though the system has multiple antennas. The Ungerboeck-type trellis code construction is described, as well as a general set-partitioning method for unitary space–time modulation. Several code examples based on the diagonal cyclic group constellations [14] are also provided. The proposed design methods are generally suitable for any signal constellation. However, for the special class STBC [16]–[19], we show that the encoding and decoding processes can be simplified through orthogonality. The rest of this paper is organized as follows. In Section II, we introduce some background information on channel model

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and DUSTM. In Section III, we first present the system model of TC-DUSTM, and derive the differentially noncoherent decision metrics. Then we propose the trellis-code design criteria. Section IV describes code construction, as well as some code examples. Some simulation results are shown in Section V. Extensions to the trellis-coded differential STBC are discussed in Section VI. Finally, Section VII concludes the paper. Throughout this paper, matrices (vectors) are denoted using and stand for the boldface upper (lower) case letters. zero matrix and the identity matrix, respectively, where the dimensions may be dropped if there is no confusion. Superscripts “ ” and “ ” denote transpose and condenotes the Krojugate transpose operations, respectively. is denecker product. The covariance of a random matrix fined as the covariance of the vector that is formed by stacking . If a random matrix is comthe columns of and covariance plex Gaussian distributed with mean , we simply say is distributed. Particularly, for a complex Gaussian distributed random variable with distributed. zero-mean and unit-variance, we say is II. BACKGROUND A. Channel Model We consider a point-to-point wireless communication link transmit antennas and receive antennas. It equipped with is assumed that the channel is time-varying flat Rayleigh fading and the channel coefficients for different transmit–receive antenna pairs are statistically independent. We further assume that the channel coefficients remain unchanged within channel coherent time intervals and are not known to either the transmitter or the receiver. denote the baseband transmitted signal on the th Let transmit antenna at time slot and satisfy the constraint

i.e., the average energy summation of the transmitted signals antennas during time intervals is . The baseband over all on the th receive antenna at time slot is received signal given by (1) is the complex channel coefficient from the th where transmit antenna to the th receive antenna, which is distributed; is the additive white complex Gaussian noise on disthe th receive antenna at time slot , which is tributed; and is the average transmitted signal energy.1 Hence, the average signal-to-noise ratio (SNR) per receive antenna can . be computed as 1In this paper, both  and  are assumed to be fixed parameters. As demonstrated, they do not affect our maximum-likelihood decisions, and hence, can be ignored in the decision metrics. In an independent work [20], relevant but different channel assumptions are considered and explicit log-likelihood functions that are valuable in certain situations are derived.

Considering time intervals together, we can rewrite both the received signals and transmitted signals in matrix form and remodel (1) as (2) received signal matrix with the th ; is the transmitted signal matrix with ; is the channel matrix with ; and is the noise matrix with . It has been shown in [11] that without the knowledge of and conditioned on , has independent and identically distributed (i.i.d.) columns, and that the covariance matrix of each column is . Thus, is distributed with conditional probability density function (pdf) given by

where entry

is the

(3) where “tr” and “det” stand for trace and determinant, respectively. One of the most interesting observations from (3) is that will not change if we right multiply by any arbitrary unitary matrix . Therefore, and are indistinguishable for noncoherent detection. B. DUSTM DUSTM was proposed in [13] and [14] as an extension of DPSK in multiple-antenna systems. We will briefly review it in this subsection. Since the signals are transmitted block by block, each contime intervals, it is convenient to use to taining denote the block index; within the th block, the time index is . Let denote denoted as unitary signal matrix transmitted over transmit the antennas during the th block. The differential transmission is . Then, initiated by sending the identity matrix, i.e., with differential encoding, we have at block

where , with , is the data matrix at time block and is selected from a unitary space–time modulated constellation of size , i.e., . The transmitted generally does not belong to the constellation signal matrix unless the constellation itself forms a group under matrix multiplication. At the receiver, the noncoherent demodulation is based on and . We group them two consecutive observations, matrix as . Similarly, we into a . It is assumed that the channel is constant let within two consecutive time blocks. By applying (3) and using can be simplified to the unitary property of , the pdf of

(4)

TAO AND CHENG: TC-DUSTM OVER FLAT FADING CHANNELS

Fig. 1.

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Transmission system diagram of TC-DUSTM.

which is independent of . Hence, after ignoring irrelevant based on the log-likelihood terms, the decision metric of can be written as function

or or

(5)

and denote the squared Frobenius norm and where are not present the real part, respectively. Notice that and in the decision metric (5). That implies the receiver does not need to estimate and . Later on, similar observations will be made in the proposed TC-DUSTM. It can also be seen from (5) that the implementation of the decision metric can be either in quadratic form, correlation form, or minimum distance form. These are all equivalent. At high SNR, the system performance is determined by the so-called diversity product of the signal constellation [14]

Full transmit-antenna diversity can be achieved as long as , and a larger results in better performance. As will be demonstrated in Section III, it is not physically straightforward to link with the design of TC-DUSTM. Hence, we reformulate as a new relevant performance parameter, called the determinant and distance, denoted as . Between any pair of elements in the constellation, we have2

two parts: trellis encoder and USTM mapper. A sequence of information binary bits is passed through a trellis encoder to generate an encoded bit stream . This trellis encoder is a rate code that receives input bits and generates coded bits during each trellis transition. Then the encoded bit stream is divided into groups of bits. Each group is mapped at every selected from an time block into an element unitary space–time modulated constellation, based upon a certain set-partition method. Hence, each transition branch in the trellis corresponds to one coded signal matrix. This is the combined TC-USTM. Following the encoder, the coded matrix seis reordered by a matrix-wise interleaver as quence . The purpose of the matrix-wise interleaver is to break the correlation of the fading that affects adjacent coded signal matrices. The reordered coded matrix sequence from the interand transmitted by leaver is then differentially encoded as transmit antennas. The channel is flat Rayleigh fading and time intervals so the channel coherent time is greater than that the channel coefficients within two consecutive time blocks can be assumed constant. With respect to the interleaver design, we consider two extreme cases. 1) Ideal interleaver: we assume the system can allow infinite delay so that the length of the interleaver can be made sufficiently large. The channel seen by different coded signal matrices can be assumed to be statistically independent. 2) No interleaver: we assume the system is delay sensitive and that interleavers cannot be used. In the following, we will show the decision metrics and design criteria for these two cases, respectively. A. Ideal Interleaver

(6) Indeed, maximizing the minimum mizing Moreover, when

is equivalent to maxi-

Consequently, the determinant distance can be viewed as a novel generalization of the Euclidean distance for signals in matrix form. III. PERFORMANCE MEASURE In this section, we consider TC-DUSTM with noncoherent decoding. The simplified transmission system diagram under investigation is depicted in Fig. 1. For convenience, we break the trellis-coded unitary space–time modulation (TC-USTM) into

V V0

2Based on matrix equations, the squared determinant distance between and is also the geometric mean of the eigenvalues of ( ) ( ).

V

V

V 0V

At the receiver, the received signal matrix sequence is partitioned into groups of two blocks in such a way that two adjacent groups are overlapped by one time block. The overlapping is necessary, since the last block in the previous group must be used as a reference in the current group for the differential decoding to be applied to each trellis branch. Note that due to overlapping, the noise terms in different groups are not independent. The partitioned sequence is then deinterleaved group by group. The th deinterleaved group and the associated transmitted signal matrices are denoted by the same and . Therefore, notations as those in Section II-B, i.e., under the assumption of an ideal interleaver/deinterleaver, both the channel coefficients and the noise terms contained in difare made independent. Hence, given the coded maferent , ’s are independent trix sequence of each other and the joint pdf of is simply given by

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where the second equality holds because of (4). Hence, the decision metric of selecting the maximum-likelihood (ML) path through the trellis is

(7) is the corresponding branch metric. It can where be written as any one of the forms in (5). Thus, the ML decoding can be efficiently implemented using the soft-decision Viterbi algorithm. Let us assume that the transmitted coded matrix sequence and the erroneously decoded sequence is . Then the is pair-wise error probability (PEP) is given by

(12) it is also observed that the coding advantage is the th root of the product of the squared determinant distance

From the two observations, we reach the following design criteria. Design criteria for the case of ideal interleaver: in flat • in order to achieve the diversity of order Rayleigh fading channels with an ideal matrix-wise interleaver, the Hamming distance between any two distinct coded matrix sequences must be at least ; • the minimum product of the squared determinant distance between any pair of coded matrix sequences having the , minimum Hamming distance, denoted as must be maximized. B. No Interleaver

(8) Applying the Chernoff bound on (8) gives

(9)

In the case without an interleaver, we assume the channel is time blocks.3 constant within a certain period of time, say Without the knowledge of CSI at the receiver, the pdf of the blocks conditioned received matrix sequence over the on the coded matrix sequence is [11]

is the Chernoff bound parameter to be optimized. where It is observed that the th product term in (9) is exactly the Chernoff bound of

which corresponds to the PEP of an uncoded system . As the explicit expression of the Chernoff bound of is given by [13]

(10) we obtain the Chernoff bound of the PEP (8) as

(11) Assuming the SNR ther upper bounded by

where is the differentially transmitted matrix sequence. Then we can obtain the ML decision metric as (13) It is seen that (13) cannot be simplified to the sum of each defined in (5). A direct implebranch metric mentation of such ML sequence decoding becomes unfeasible. We are seeking a suboptimal decoder that is much less complicated and can be easily implemented in practice. For each trellis branch, we take into consideration two consecutive re, and compute the total deciceived blocks sion metric as the sum of the branch metrics at all time blocks, i.e.

is high enough, the PEP can be fur-

(12) and is the where is the set of all that size of , termed as the Hamming distance between and . From (12), it is observed that the achieved diversity order is the and the Hamming distance . product of antenna diversity Hence, when the Hamming distance increases by one, the order . This indicates the substantial adof diversity increases by vantage of using multiple antennas over a single antenna. From

(14) Therefore, the suboptimal decoder is to select the path through the trellis that has the largest total decision metric, and can be implemented using the soft-decision Viterbi algorithm in an efficient manner. Compared with the ML decision metric (13), the suboptimal decision metric (14) is a simplified version of (13) are ignored. in which all cross terms between different

K

3If is large enough, i.e., slow fading, coherent schemes with channel estimation may be preferred. Nevertheless, even in slow fading channels, the differential scheme is still a reasonable choice in practical systems when the receiver does not want to estimate the channel.

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Nevertheless, the performance gap may be narrowed by using an approach similar to that of multiple-symbol differential detection in [23] for DPSK in single-antenna systems. Using (14), we are unable to derive the PEP due to the memory channel and nonwhite Gaussian noise among different . However, because we know that DPSK modulation can be viewed as PSK modulation but with a 3-dB higher noise power, it is reasonable to expect that the trellis code structure designed for TC-PSK systems should work well in TC-DPSK systems, but with around 3-dB performance degradation. As DUSTM is an extension of DPSK, we will also expect that a good trellis code designed for TC-USTM with coherent decoding is also a good code in TC-DUSTM with noncoherent decoding but with some performance degradation. The simulation results in Section V will help to verify this statement. Thus, in the following, we will derive the design criterion of coherent TC-USTM when CSI is available at the receiver and then apply it to the noncoherent case. In TC-USTM without differential encoding, the transmitted is the same as the coded matrix signal matrix sequence is sequence . Thus, the coherent ML decision metric of clearly given by (15) Based on (15), the Chernoff bound of the PEP of transmitting but detecting is given by [6]

Assuming bounded by

(16) is sufficiently high, (16) is further upper

(17) From (17), it can be seen that the TC-USTM achieves a full and a coding advantage of antenna diversity of order (18) However, (18) is not helpful enough for designing good trellis codes. Hence, a simpler but slightly looser criterion will be derived instead, as follows. Recall Minkowski’s determinant inequality [22], which states that

for positive definite matrices and , and the equality for . Applying it in (18) yields holds if and only if a lower bound of (19) which is defined as the minimum free squared determinant . Hence, an alternative design criterion for distance

TC-USTM is to maximize between any two distinct coded matrix sequences. Now, let us come back to TC-DUSTM when CSI is unavailable at the receiver. We arrive at the following criterion. Design criterion for the case of no interleaver: • in order to maximize the coding advantage, the minimum free squared determinant distance between any pair of distinct coded matrix sequences must be made as large as possible. Note that we design trellis codes that work well in the coherent case and, by merely adding an inner differential encoder at the transmitter, expect them to work well too in the noncoherent case. IV. CODE CONSTRUCTION The design criteria of TC-DUSTM derived in the previous section for both the ideal interleaver case and the no interleaver case are quite close to the well-known design criteria of TCM in Rayleigh fading channels and AWGN channels, respectively [21]. Thus, we propose relatively easy code construction methods in this section. We focus our attention on the design trellis codes. The coded of Ungerboeck-type rate matrix is selected from a unitary space–time modulated con. stellation of size The first step before designing any code is to do set partition based on its determinant distance profile for the employed constellation. In the following, we shall describe a general set-partitioning method with an idea from conventional set partitions for PSK or QAM-type constellations. The set partition starts from the minimum determinant distance. Consider the whole set as the zeroth level. It is split into two subsets at the first level with an interdistance equal to the minimum determinant distance. At the second level, each subset from the first level is partitioned into two smaller subsets with an interdistance equal to the intradistance of the parent subset in the first level. This procedure continues for the succeeding levels and stops at a desired level. The subsets after partition should exhibit similar distance properties and provide a maximum of symmetry. Of all the existing signal constellations, the diagonal cyclic group constellations proposed in [14] are the most systematic ones and can be designed for any number of transmit antennas with any spectral efficiency. Thus, we take this class of constellations as an example and briefly review it here. Other constellations can be easily extended. , in the considThe th element , with ered diagonal cyclic group constellation is generated by , with

and taken from the integer set and chosen to maximize the minimum determinant distance. Hence, the constellation is fully specified by the parameters and the size . We denote such a constellation for simplicity. Later on, we also by

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TABLE I DETERMINANT DISTANCE PROFILE

G = ([1 3]; 8)

Fig. 2. Set partition of

OF

G = ([1 3]; 8).

TABLE II DETERMINANT DISTANCE PROFILE OF

G = ([1 7]; 32)

According to the above design rules, we provide three code examples based on the diagonal cyclic group constellations with two transmit antennas. The code parameters are listed in Table III, where denotes the spectral efficiency in terms of information bits per second per Hz. The trellis diagrams are illustrated in Fig. 4. All three codes can achieve the minimum Hamming distance of 2. As the employed constellations in our examples have a diagonal structure and each diagonal element is chosen from the -PSK type symbols, a layered coding strategy, i.e., the coding is done on each diagonal element individually and independently, is also possible. However, in order to ensure the same diversity advantage at the same spectral efficiency as that of the proposed joint coding strategy, a higher decoding complexity is required. To demonstrate this point, we measure the decoding complexity using the average number of branch metrics that need to be calculated through the soft-decision Viterbi decoder for each information (info) bit. For the joint coding strategy with the minimum Hamming distance , the decoding complexity is (20) For the layered coding strategy, the minimum Hamming dis. tance for each component trellis code has to be Hence, the total decoding complexity is

simply use integer , , to represent in the constellation. The group properties of diagonal cyclic group constellations characterize that the determinant distance between any two elements as and depends only on the absolute value of and, thus, it allows efficient set partitions. Two examples are provided to demonstrate this procedure. Example 1: Set partition of The determinant distance profile of is listed in Table I. The resulting set partition is shown in Fig. 2. Example 2: Set partition of The determinant distance profile of this example is given in Table II. The set partition is illustrated in Fig. 3. From the above two examples, it can be observed that all the subsets in the same partition level have the same size and an identical intradistance. Such partition is called a fair partition. For other constellations, fair partition may not be guaranteed. Using the two design criteria derived in Section III, we propose two corresponding sets of design rules in the following subsections. A. Ideal Interleaver Set the target minimum Hamming distance to be . We have the following design rules. • All elements in the constellation should be equally probable. . Parallel • The minimum number of trellis states is transitions are not allowed. branches departing from a common state or con• The verging to a same state must be assigned with elements from one subset at the first level of set partitioning.

(21) times larger than (20). Moreover, for the joint which is coding strategy, the coding advantage within each matrix block is already maximized, as the matrix constellation itself is optimal in the sense of maximizing the minimum determinant distance. Hence, compared with the layered coding strategy, it is relatively easier to maximize the overall coding advantage. In component codes are fact, in the layered coding strategy, the independent and since they do not interfere with each other due to the diagonal transmission, the system performance should be identical to that of the system using one component code and transmitting with only one antenna continuously. Therefore, there is no point in using the layered coding strategy. B. No Interleaver Following Ungerboeck’s rules we have the following criteria for assigning elements to the trellis branches. • All elements should be used equally often. • Parallel transitions should be assigned with elements from the subset with the greatest intrasubset distance. transitions that diverge from a common state or • The remerge into a same state must be assigned with elements from one subset at the first level of set partitioning. When comparing this design rule set for the no interleaver case with the previous one for the ideal interleaver case, it is found that they differ only in the second rule. In particular, parallel transitions can be used in the no interleaver case, but are not allowed in the ideal interleaver case. This is because in the case

TAO AND CHENG: TC-DUSTM OVER FLAT FADING CHANNELS

Fig. 3. Set partition of

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G = ([1 7]; 32). TABLE III PARAMETERS OF CODES DESIGNED FOR IDEAL INTERLEAVER CASE

with an ideal interleaver, the minimum Hamming distance of a trellis code quantifies the order of time diversity and should be made as large as possible. If parallel transitions exist, the minimum Hamming distance becomes one. However, in the case with no interleaver, the dominating parameter is the minimum free squared determinant distance. Parallel transitions can help to increase this value, especially when there are only a few trellis states. Two code examples based on the same signal constellations as before are given in this subsection. The trellis diagrams are shown in Fig. 5. Both of the codes have parallel transitions . For code (a), the code rate is 2/3 with the denoted by b/s/Hz. It is not difficult to see that spectral efficiency the error path within the parallel transitions achieves the min. In this imum free squared determinant distance case, the equality in (19) holds. Hence, the actual coding advantage is also four. With a coherent receiver, its asymptotic with coding gain over uncoded constellation at the same spectral efficiency is expected to be dB. For code (b), the code rate b/s/Hz. It is examined that the parameters is 4/5 with and are equal to 1.5999 and 1.7631, respectively, where the former is slightly smaller than the latter. The asymptotic with coding gain over uncoded constellation at the same spectral efficiency is 4.78 dB when the receiver knows CSI. V. SIMULATION RESULTS In this section, we present the simulation results of the code examples provided in the previous section. Due to limited space, only the results of the codes with spectral efficiency b/s/Hz are reported. In all our figures, notation ( , ) is used transmit antennas and receive to denote a system with antennas.

In Fig. 6 we provide the bit-error rate (BER) performance of the two codes with b/s/Hz designed for the ideal interleaver case shown in Table III and Fig. 4. The channel is modeled as flat Rayleigh fading with an ideal interleaver/deinterleaver. The channel coefficients are kept constant within each time block and are not known to either the transmitter or the receiver. The derived ML differential decoder (7) is employed. For comparison, the BER performance of uncoded constellation at the same spectral efficiency is also provided. Gray mapping is used from the binary information bits to the elements in the constellation. As can be seen in this figure, a significant improvement is observed. In particular, at a BER of and with one receive antenna, the four-state code achieves around 5-dB gains compared with the uncoded case, while the eight-state code is seen to provide more than 6.5-dB improvement over the uncoded case. When the number of receive antennas is increased to two, the improvement is not so large, but can still be achieved by these a gain of 3 4 dB at a BER of two codes. b/s/Hz designed for the no interFor the code with leaver case shown in Fig. 5(a), we first demonstrate its performance with direct transmission and coherent receiver, and then show its performance with differential transmission and suboptimal noncoherent receiver (14). This is done to confirm our statement in Section III-B that a good TC-USTM also works well in the noncoherent case if an inner differential encoder is employed. The channel is assumed to be constant during a frame of 100 transmission time intervals, and an additional two time intervals are required to send an initial signal matrix in the differential transmission case. As the errors caused by fading tend to occur in bursts, frame-error rate (FER) rather than BER is simulated. Fig. 7 illustrates the results of direct transmissions with a coherent receiver for this rate 2/3 four-state code. The results of the corresponding uncoded transmission at the same spectral ef-

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VI. EXTENSIONS TO TRELLIS-CODED DIFFERENTIAL STBC All the code examples provided in this paper are based on the diagonal cyclic group constellations. Nevertheless, the proposed design methods are suitable for any USTM constellations. One special class is the STBC. Due to its orthogonal structure, the trellis-code design correspondingly has particular rules. They are discussed in this section. Let the coded matrix during the th transmission block consist of independent PSK-type symbols , with unit energy. For any pair of distinct , and , we have [8] coded matrices at block ,

(22) , , are the symbols contained in where . We first consider the case without an interleaver. The coding advantage in (18) and the minimum free squared determinant in (19) are equal because of the orthogonality distance shown in (22). We rewrite (18) as

(23)

Fig. 4.

Trellis diagrams of the codes designed for ideal interleaver case.

ficiency are given for comparison. It can be observed in Fig. 7 that there is no diversity improvement but a coding gain of 3 dB over a wide range of FER is achieved by this code, as expected, compared with the uncoded case at both one and two receive antennas. The results of differential transmission with the noncoherent receiver are illustrated in Fig. 8. The simulated coding gain over the uncoded system is about 2.4 dB at both one and two receive antennas. A gain of 0.6 dB is lost, compared with the gain in direct transmission and coherent receiver. A comparison with Fig. 7 shows that the differentially noncoherent results have a performance loss of 3.5 dB. In Fig. 8, this code is also compared with the traditional TC-DPSK designed for single transmit antenna. For a fair comparison, both the spectral efficiency and the trellis state b/s/Hz is number are set the same. The TC-DPSK for constructed using the rate 1/2 four-state convolutional code with Gray-mapped QPSK. The antenna diversity enhancement is obvious by the use of TC-DUSTM. This yields a substantial improvement in FER, especially when there is only one receive antenna.

and are the symbol sequences where and , respeccontained in the coded matrix sequences differs from . Using tively, and is the set of all that (23), the design criterion becomes maximizing the minimum free squared Euclidean distance between any distinct symbol sequence pairs. Therefore, using the conventional set-partition method, we simply map the binary coded bits to the symbols directly, rather than the matrix . Next, we consider the case when an ideal interleaver/deinterleaver is used. We rewrite the PEP in (12) as

We further propose a layered structure for this system. The information bits are divided into substreams (or layers), each of which is then encoded using an individual component trellis at the th code with outputs mapped to symbols time block for the th layer. All the component codes are set to be identical for simplicity and have the minimum Hamming order distance of . Then, the whole system achieves of combined antenna and time diversity. At the receiver, since every coded data symbol is decoupled without the loss of any information, all the component codes can be decoded individually. To better understand the benefit of having a layered structure, we compare its decoding complexity with that of the general joint coding structure. The decoding complexity is measured in the same way as introduced in Section IV-A, i.e., the average number of branch metric calculations for each info

TAO AND CHENG: TC-DUSTM OVER FLAT FADING CHANNELS

Fig. 5.

595

Trellis diagrams of the codes designed for no interleaver case. (a) Rate 2/3 four-state with

G

G = ([1 3]; 8). (b) Rate 4/5 16-state with G = ([1 7]; 32).

G

Fig. 6. BER of rate 2/3 four-state and eight-state = ([1 3]; 8) TC-DUSTM compared with uncoded = ([1 1]; 4) DUSTM, ML differential decoder and ideal interleaver, = 1 b/s/Hz.

Fig. 7. FER of rate 2/3 four-state = ([1 3]; 8) TC-USTM compared with uncoded = ([1 1]; 4) USTM, ML coherent decoder and no interleaver, frame length = 100, = 1 b/s/Hz.

bit through the soft-decision Viterbi decoder. Let the spectral be b/s/Hz, and each component code has efficiency trellis states to ensure the minimum the minimum Hamming distance is . Then the complexity of the layered structure is

Comparing (24) with (25), it can be observed that the proposed layered structure significantly reduces the decoding complexity. ( in this case), b/s/Hz As an example, let , the layered strategy has a complexity of and branch metric calculations per info bit, while the joint coding branch metric calculations per strategy has that of info bit.

R

G

(24)

G

R

VII. CONCLUSION However, at the same spectral efficiency and diversity order, the complexity of the joint coding structure has to be (25)

In this paper, we proposed TC-DUSTM. No channel estimation is required in this scheme. Therefore, it reduces the receiver complexity and relaxes the design requirement on the pilot signals. We derived decision metrics for differential decoding and proposed design criteria for the trellis codes. We also described a

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G=

Fig. 8. FER of rate 2/3 four-state ([1 3]; 8) TC-DUSTM compared with uncoded = ([1 1]; 4) DUSTM and rate 1/2 four-state TC-DPSK, suboptimal differential decoder and no interleaver, frame length = 102, = 1 b/s/Hz.

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general set-partition method for unitary space–time modulated constellations based on their determinant distance profile, and code constructions for Ungerboeck-type trellis codes. Several code examples, based on diagonal cyclic group constellations, were provided. Simulation results exhibited excellent performance with computationally efficient noncoherent receivers. We also considered the extensions to trellis-coded differential STBCs. Therein, the decoupling property retained by STBCs allows efficient and simple encoding and decoding structures. REFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [3] G. J. Foschini, “Layered space–time architecture for wireless communication is a fading environment when using multiple antennas,” Bell Labs. Tech. J., vol. 1, no. 2, pp. 41–59, Autumn 1996. [4] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Yalenzuela, “V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel,” in Proc. 1998 URSI Int. Symp. Signals, Systems, Electronics, New York, NY, 1998, pp. 295–300. [5] V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Combined array processing and space–time coding,” IEEE Trans. Inform. Theory, vol. 45, pp. 1121–1128, May 1999. [6] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data-rate wireless communication: performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [7] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [8] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999. [9] G. Ganesan and P. Stoica, “Space–time block codes: a maximum SNR approach,” IEEE Trans. Inform. Theory, vol. 47, pp. 1650–1656, May 2001. [10] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 45, pp. 139–157, Jan. 1999. [11] B. M. Hochwald and T. L. Marzetta, “Unitary space–time modulation for multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 46, pp. 543–564, Mar. 2000.

[12] L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inform. Theory, vol. 48, pp. 359–383, Feb. 2002. [13] B. L. Hughes, “Differential space–time modulation,” IEEE Trans. Inform. Theory, vol. 46, pp. 2567–2578, Nov. 2000. [14] B. M. Hochwald and W. Sweldens, “Differential unitary space–time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [15] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, “Representation theory for high-rate multiple-antenna code design,” IEEE Trans. Inform. Theory, vol. 47, pp. 2335–2367, Sept. 2001. [16] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, pp. 1169–1174, July 2000. [17] H. Jafarkhani and H. Tarokh, “Multiple transmit antenna differential detection from generalized orthogonal designs,” IEEE Trans. Inform. Theory, vol. 47, pp. 2626–2631, Sept. 2001. [18] G. Ganesan and P. Stoica, “Differential modulation using space–time block codes,” IEEE Signal Processing Lett., vol. 9, pp. 57–60, Feb. 2002. [19] M. Tao and R. S. Cheng, “Differential space–time block codes,” in Proc. IEEE GLOBECOM 2001, San Antonio, TX, Nov. 25–29, 2001, pp. 1098–1102. [20] E. G. Larsson, P. Stoica, and J. Li, “On maximum-likelihood detection and decoding for space–time coding systems,” IEEE Trans. Signal Processing, vol. 50, pp. 937–944, Apr. 2002. [21] E. Biglieri, D. Divsalar, P. J. McLane, and M. K. Simon, Introduction to Trellis-Coded Modulation With Applications. New York: Macmillan, 1991. [22] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U. K.: Cambridge Univ. Press, 1985. [23] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990.

Meixia Tao (S’00) was born in Jiangxi, China, in 1978. She received the B.S. degree in electronic engineering from Fudan University, Shanghai, China, in 1999. She is currently working toward the Ph.D. degree in the Department of Electrical and Electronic Engineering at the Hong Kong University of Science and Technology, Kowloon, Hong Kong. Her research interests are in the area of communications, including multiple-antenna systems, space–time coding, and OFDM techniques. Miss Tao has been a student member of the IEEE Communications Society since 2000.

Roger S. Cheng (S’85–M’91) received the B.S. degree from Drexel University, Philadelphia, PA, in 1987, and both the M.A. and Ph.D. degrees from Princeton University, Princeton, NJ, in 1988 and 1991, respectively, all in electrical engineering. From 1991 to 1995, he was an Assistant Professor in the Electrical and Computer Engineering Department of the University of Colorado at Boulder. In June 1995, he joined the Faculty of Hong Kong University of Science and Technology (HKUST), Kowloon, Hong Kong, where he is currently an Associate Professor in the Department of Electrical and Electronic Engineering. He also held visiting positions with Qualcomm, San Diego, CA, in the summer of 1995, and with Institute for Telecommunication Sciences, NTIA, Boulder, CO, in the summers of 1993 and 1994. He was the Director of the Center for Wireless Information Technology at HKUST from 1999 to 2001 and the Associate Director for the Consumer Media Center at HKUST. His current research interests include wireless communications, OFDM, space–time processing, CDMA, digital implementation of communication systems, wireless multimedia communications, information theory, and coding. Dr. Cheng is the recipient of the Meitec Junior Fellowship Award from the Meitec Corporation in Japan, the George Van Ness Lothrop Fellowship from the School of Engineering and Applied Science of Princeton University, and the Research Initiation Award from the National Science Foundation. He has served as Editor in the area of Wireless Communications for the IEEE TRANSACTIONS ON COMMUNICATIONS, Guest Editor of the special issue on Multimedia Network Radios in the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and Membership Chair for of the IEEE Information Theory Society.