IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
783
Space–Time Code Design With Continuous Phase Modulation Xiaoxia Zhang and Michael P. Fitz
Abstract—This paper addresses the space–time code design for Rayleigh-fading channels using continuous phase modulation (CPM). General code construction is desirable due to the nonlinearity and inherent memory in CPM signals which make hand design or computer search computationally impractical. Several sufficient conditions for full rank are identified for CPM signaling schemes and for linear representations of CPM modulation. Simulation results verify the resulting performance. Index Terms—Continuous phase modulation (CPM), linear decomposition, rank criterion, Rayleigh fading, space–time code.
I. INTRODUCTION
O
NE OF THE major challenges in wireless communication is to overcome multipath fading. The severe attenuation that can result from channel fading makes it impossible for the receiver to reliably determine the transmitted information unless some kind of diversity resource is used. The studies by Telatar in [1] and Foschini and Gans in [2] have shown the enormous capacity and performance promised by multiple antenna systems in fading channels. Transmit diversity—space–time coding, has turned out to be a very effective diversity strategy [3]–[7]. The design criteria for space–time coding in quasi-static fading were originally formulated to optimize the worst case pair-wise error probability (PWEP). PWEP is a function of the rank and the product of nonzero eigenvalues of a “signal matrix” defined by the pairwise error event [4], [5]. In quasi-static fading, this signal matrix is a function of the codeword difference matrix of the error event. Further work has shown that the “Product Distance” spectrum of a space–time code is a more complete characterization of the performance with a small number of receive antennas ( 4) [8] and the Euclidean distance spectrum is a more complete characterization of the performance with a large number of receive antennas ( 4) [9]–[11]. All design criteria proposed for space–time coded systems have been for linear modulations. In [4], Tarokh et al. further present some simple design rules that guarantee full diversity for linear modulations with two transmit antennas. Full diversity is the condition that all error events associated with the code have a rank equal to the number of antennas. These design rules are extended to any levels of diversity by constraining the codeword difference matrices to
Manuscript received May 1, 2002; revised November 1, 2002. This work was supported in part by the National Science Foundation under Grant NCR9706372. X. Zhang is with the Standards Group, Qualcomm Inc., San Diego, CA 92121 USA (e-mail:
[email protected]). M. P. Fitz is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/JSAC.2003.810343
be upper or lower triangular [12], [13] which is referred as zeroes symmetry. Coding with linear modulation to achieve full diversity implies a restriction on the achieved rate and the code complexity. However, these design rules have been shown to be sufficient but not necessary to guarantee full diversity. In fact, there are space–time trellis codes which achieve full diversity that violate these rules while having a better coding gain [12], [14]. Based on the rank and determinant criteria many researchers have proposed space–time trellis codes for linear modulation which have resulted from a computer search [12], [14], [15]. The main obstacle in deriving general space–time trellis code design rules is that the diversity gain and coding gain criteria are defined over the complex “signal matrix,” whereas the traditional code designs are carried out in finite fields or rings. Recently, research on systematic design procedures for space–time trellis codes in quasi-static fading has been investigated. In [16], binary rank criteria that ensure full diversity for binary binary phase-shift keying (BPSK) codes and quaternary phase-shift keying (QPSK) codes are presented. In [17], the rank criteria are generalized for higher order quadrature amplitude modulation (QAM) constellations. Since the criteria are on the generator matrix pertaining to the unmodulated code words over the finite fields or rings, the search for space–time trellis codes can be greatly simplified. For both power and bandwidth limited communication links such as mobile satellite communication, continuous phase modulation (CPM) has been an attractive scheme for digital transmission [18], [19]. The constant envelope of CPM signals makes it possible to use low-cost, nonlinear power amplifiers. On the other hand, CPM maintains phase continuity through an introduced memory to achieve bandwidth efficiency. The cost of this memory is to produce a more complex optimum demodulator. Combining CPM with space–time coding (ST-CPM) can provide better performance in wireless links. ST-CPM code design is more difficult than linear modulation due to both the nonlinearity of the modulation and the more complex resulting performance metrics. In this paper, we first derive the design criterion of ST-CPM for quasi-static fading and similarly as in linear modulation we define the signal matrix associated with the PWEP. We can see that the computation of the signal matrix both involves complex integration and depends on the particular CPM scheme. Due to the memory inherent in CPM signal, the whole system is much more complicated than its linear modulation counterpart. Computer search for codes in CPM system is not easy and is computationally challenging. It is desirable that some general code constructions be proposed to guarantee full diversity. Fortunately, through the linear decomposition of
0733-8716/03$17.00 © 2003 IEEE
784
Fig. 1.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Block diagram of ST-CPM.
CPM signals [20]–[22], we are able to identify rank criteria for certain CPM schemes which include GMSK, used in GSM standard, as a special example. It should also be pointed out that the space–time block codes proposed in [6], [7], [23] can not be generalized to CPM signals easily as both orthogonality and phase continuity of the CPM waveforms need be achieved. This paper is organized as follows. Section II gives the system model. Performance criteria are presented in Section III. In Sections IV–V, the general rank criteria are derived. Simulation results are provided in Sections VI and VII concludes.
symbol time, is the symbol energy, and is the AWGN at the th receive antenna with single-sided power spectral den. For quasi-static fading the channels are modeled sity being as independent complex Gaussian random variables of variance 0.5 per dimension. Denote1 the original information sequence , the coded space–time M-ary signals with a matrix and the space–time modulated pulse amplitude modulation (PAM) with a matrix . In parallel to the prior work with linear modulation, the code symbol matrix has columns which correspond to the rows corresponding the time intervals, transmit antenna and i.e.,
II. SYSTEM MODEL In this paper, we consider a mobile communication system receive antennas. In equipped with transmit antennas and informathe transmitter block diagram shown in Fig. 1, the are input into the space–time encoder to get tion bits streams of data, . Each stream of data is used as the input to a CPM modulator. The modulated sigtransmit antennas. nals are simultaneously transmitted from Consequently, the signal received at each receive antenna is a noisy superposition of the transmitted signals corrupted by a corresponding Rayleigh fading and an independent zero-mean complex additive white Gaussian noise (AWGN). A simple space–time CPM (ST-CPM) model is considered. First, we assume quasi-static fading. Though utility might exist in using different CPM formats in a ST-CPM system, this paper will limit itself to considering schemes that use the same CPM format at each transmit antenna (with a single modulation index). With these assumptions the signal received by antenna , can be given by [19]
.. .
.. .
.. . (4)
is the th modulation symbol of the th antenna where is the frame size in terms of symbol time. Similarly, the and codeword matrix is defined as
.. .
.. .
.. .
(5)
and . CPM is a nonlinear modulation that is defined by a phase , where trellis. With rational modulation indices, say and are relatively prime integers, at each symbol time is constrained to lie on a trellis, i.e.,
Note that
(1) (6) (2)
where
mod
and (7)
(3) ’s denote the channel gain between transmit antenna and is the phase receive antenna , is the modulation index, smoothing response function with memory length , is the
The state at time
for each transmit antenna is defined as (8)
A represents a scalar term, A~ represents a vector and A a matrix.
1
ZHANG AND FITZ: SPACE–TIME CODE DESIGN WITH CONTINUOUS PHASE MODULATION
Assume the space–time coding is feedforward2 with constraint length , then the overall transmitter can be represented by a being super trellis with the state at time (9)
where (17) and
III. DESIGN CRITERION Optimum demodulation of ST-CPM is a maximum-likelihood (ML) sequence demodulator (MLSD). denote3 When as the signals received from all antennas. For the given model the ML metric of codeword in the presence of perfect channel state matrix is information
785
is just the signal matrix defined as
.. .
.. .
.. . (18)
with (19) is the continuous time difference between the transmitted signal and the decoded signal on the th antenna. The quadratic form of PWEP is then
(10)
(20)
The union bound utilizing the PWEP is the typical method to analyze the performance of a MLSD. The PWEP is the probability of a code symbol matrix having a larger metric than the and is denoted as transmitted code symbol matrix . The PWEP is mathematically expressed as
The tools of quadratic forms of complex Gaussians allow a closed form analysis of the PWEP. Following the similar method in [24], it can be shown that the PWEP can be upper bounded by:
(21)
(11) is transmitted
Noting that given
(12)
(13) Expanding the magnitude-square reduces each of the conditional ML metrics to a quadratic form of complex Gaussian random variables. More specifically, denote (14) (15)
.. . 2Feedback
.. .
.. .
.. .
are the nonzero eigenvalues of the signal matrix where , is the number of its nonzero eigenvalues. Equation (21) shows that the performance and design of ST-CPM have a direct analogy to the performance and design of space–time coded linear modulation [4], [24]. Consequently, all the literature and the derived insights (rank criterion, determinant criterion, etc.) for space–time linear modulations are directly applicable with only the consideration of a different “signal” matrix (18). IV. DESIGN RULES The characteristics of for CPM produce a more difficult design problem than encountered in linear modulation. The marequires a time integration which is a function of the partrix ticular CPM format to evaluate each entry of the matrix. Completing a computer search would be prohibitively expensive for all but the simplest situations. Here we concentrate on the diversity level of ST-CPM as this is usually the most dominant characteristic. In order to arrive at an alternate way to check full diversity, we propose the following proposition. Proposition 1: A necessary and sufficient condition for to be full rank is to make
(16)
coding can be accommodated in a similar fashion. 3In this paper, capital letters denote random variables while the lowercase denote realizations. We will also use subscripts to distinguish different codewords.
(22) complex field unless for all . Proof: Given in Appendix A.
786
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Proposition 1 states that full spatial diversity is guaranteed if and only if the waveform differences from all transmit antennas are linearly independent over the complex field. It also indicates that the initial phase offset for each transmitter will not change the performance, hence for simplicity, we assume all phase trajectories start at a common phase. Dealing with (22) is easier since it does not contain any integration. than computing Based on Proposition 1 the following conclusions can be drawn: 1) Zeroes symmetry (with delay diversity being a special case) always works with CPM transmission as it does with linear modulation. 2) Full diversity can be achieved trivially if different CPM schemes are used over each antenna. 3) When the number of transmit antennas is less than the of the data, full diversity can be obtained alphabet size with memoryless repetition coding if different mapping rules are used over each antenna. This technique does not work with linear modulation. This result has been shown in [25] and the reason is that with linear modulation full rank means linear independence of the codeword difference over each antenna, while with CPM, full rank means linear independence of the waveform difference over each antenna. From now on, we will only consider the case that diversity is obtained through space–time trellis coding as it allows for more choices of code design without bandwidth expansion. A feedforward trellis code is characterized with a generator matrix . The output codeword is given as .. . .. .
.. .
.. .
.. .
where is the corresponding M-ary symbols. The physical which is the tilted phase modulo 2 tilted phase (denoted as ) is given as
(25) where
(26) means modulo . Here, with a slight abuse of notation, and is written as , where and are relatively prime. The introduced physical tilted phase is now time-invariant and it is completely specified by where (27) is a power of , i.e., and is an integer, only depends on the least significant digit (LSD) of the input, that is
When
(23)
vector of all zeroes. The operations where represents a denoted by (23) are over a finite field that is a characteristic of the trellis code. Since the nonlinearity of CPM signals makes it hard to arrive at some general code construction, linear decomposition is used. In [20], Rimoldi has shown that any CPM system can be decomposed into a continuous phase encoder and a memoryless modulator in such a way that the former is linear and time invariant (modulo some integer ) and the latter is also time invariant. With this linear decomposition, we can identify some exact general code construction for certain CPM schemes by applying the rank criteria available in linear modulation given in [16]. The exact performance of a given space–time code will be determined by its product distance spectrum [8], but that is not pursued in this paper.
(28) (29) is linear on the As , the continuous phase encoder (CPE) then takes the form in Fig. 2 and the memoryless modulator is a mapping transforming to the time function . B. Full Response 2 -ary CPM With Theorem 1: Let be a linear space–time code with . Suppose that the binary projection of every nonzero is a matrix of full rank over the binary 2 -ary codeword , field . Then, for any full response 2 -ary CPM with the space–time code achieves full spatial diversity . Proof: It can be observed that a sufficient condition to make (22) hold is to make the following matrix:
A. Decomposition of CPM .. .
Reference [20] introduces the tilted phase function4 as (24)
4Here,
we consider the single antenna case to simplify the discussion.
.. .
.. .
(30)
in the memoryfull rank, which corresponds to the , the th less mapping in the decomposition. For with input channel CPE is a differential encoder in and the physical tilted phase at is a being
ZHANG AND FITZ: SPACE–TIME CODE DESIGN WITH CONTINUOUS PHASE MODULATION
787
is a matrix of full rank over the binary field . Then for binary , the space–time code achieves 2RC and 2REC with full spatial diversity . Proof: Things get more complicated when the partial response is concerned. With binary input and memory length is given as being two, the physical tilted phase at time
(34) where (35)
= 1.
Fig. 2. Block diagram of continuous phase encoder (CPE), L
BPSK mapping from . Since there is a frequency offset between the phase and the tilted phase as is shown in (31)–(32), at the bottom of the page, where the just the BPSK mapping from matrix with
.. .
.. .
(33)
.. .
rank . The differential encoder is a and rank which does not change nonsingular transformation in rank property, applying the BPSK binary rank criterion in linear modulation we can arrive at Theorem 1. C. Full Response 4 -ary CPM With
and
Theorem 2: Let be a linear space–time code over with . Suppose that the projection of every satisfies QPSK binary rank criterion, nonzero code word and , the then for any 4 -ary CPM with space–time code achieves full spatial diversity . and Proof: For full response 4 -ary CPM with , the approach in the proof of Theorem 1 applies similarly, the only difference is that the differential encoder is now and the memoryless mapping at time is defined on . Applying the QPSK binary rank a QPSK mapping from criterion in linear modulation we can arrive at Theorem 2. D. Binary 2RC and Binary 2REC With Theorem 3: Let be a linear space–time code with . Suppose that every nonzero binary code word
.. .
.. .
.. .
and the memoryFor both 2RC and 2REC, less mapping can be regarded as a QPSK mapping from ). is a constant which only rotates the ( constellation but does not affect the rank property. As long as the original binary code satisfies BPSK rank criterion, we can apply [16, Th. 32] which states: if and are binary space–time codes satisfying the BPSK binary rank criterion, is an then the -valued space–time code space–time code that satisfies the QPSK binary rank criterion and thus, for QPSK modulation, achieves full spatial diversity . Consequently, the overall ST-CPM achieves full diversity. V. DESIGN CRITERION FOR LINEAR MODULATION APPROXIMATION TO CPM The theorems in the previous section are very useful, but unfortunately, most CPM schemes considered in these theorems are very limited and are not very often encountered in real applications due to the relatively small memory length. In this section, we use the results in [21] and [22] to approximate the CPM signals as a sum of linearly modulated waveforms, then we apply the rank criteria derived for linear modulation. As the linear approximation tends to capture the main energy in the signal, it will be highly likely that the rank criteria derived for a linear approximation still apply to the original CPM waveform. A. Binary CPM With Theorem 4: Let be a linear space–time code with . Suppose that every nonzero binary code word is a matrix of full rank over the binary field . Then for any , the Laurent approximation to the binary CPM with CPM space–time modulation achieves full spatial diversity .
.. .
.. .
.. .
.. .
(31)
(32)
788
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Proof: In [21], Laurent derives the PAM decomposition of the binary CPM signals. Laurent further gives the approximate representation as
achieves full diversity when used in a GMSK CPM scheme like that used in the GSM standard. and
B. 4-ary CPM With (36) (37) is the modified data stream, is the principle where is the linear modulation approximapulse shape and , tion in . Note that in the case of full response, i.e., (36) is an exact expression. is Using the PAM decomposition in [21], making equivalent to making (38), shown at the bottom of the page, full rank over the complex field. For clarity, in our notation
In [22], Mengali and Morelli extend Laurent’s results in [21] to M-ary CPM signals. They show that 4-ary CPM signals can be approximated as a superposition of related linear modula, an approximations, i.e., defining tion is given as (44) where (45)
(39) Further specializing the results to stream can be written as5
(46)
, the modified data (40) (41)
(47) with (48)
and are the codeword matrices before the where is the modulo-2 addition defined on PAM mapper and . Since is the BPSK mapping and , are linear transformations of the binary inputs , applying BPSK rank criterion [16] implies if all in are full rank on the binary field, then nonzero elements of see (42), shown at the bottom of the page, is full rank over the complex field by the binary rank criterion. Since ..
(43)
.
is also full rank over the complex it follows directly that field. A direct result of the above theorem is that if a space–time code satisfies BPSK rank criterion, its Laurent approximation 5For clarity j without subscript will denote (m; i) element of j .
p01 while
(m) denotes the
j
.. .
.. .
(49)
elsewhere. (50) It is easy to verify that for the commonly used CPM scheme (e.g., RC, REC family) with the practical memory length (e.g., ), the principal pulse shapes , , and are linearly independent in , then we can have the following proposition. shown in (B.5) is full rank on the Proposition 2: If and full diversity is achieved. complex field, Proof: Given in Appendix B. This proposition leads directly to two theorems which give design rules for 4-ary CPM.
.. .
.. .
.. .
(38)
.. .
(42)
ZHANG AND FITZ: SPACE–TIME CODE DESIGN WITH CONTINUOUS PHASE MODULATION
Theorem 5: Let be a linear space–time code over with . Suppose that every nonzero code word satisfies QPSK binary rank criterion, then the linear modulation and approximation for any 4-ary CPM with based on the space–time code achieves full spatial diversity . , Proof: If and are QPSK mapping rotated by , where is and the addition operated on modulo-4. Since are linear transformations of the 4-ary inputs they are full diversity by the rank criterion in [16]. The on rotation does not change rank property and consequently using Proposition 2 gives that full spatial diversity is achieved. Theorem 6: Let be a linear space–time code over with . Suppose that the binary projection of every satisfies BPSK rank criterion, then the nonzero code word linear modulation approximation for any 4-ary CPM with based on space–time code achieves full spatial diversity . , then Proof: If and are , where and BPSK mappings rotated by are the binary projections of the original information and is the modulo-2 addition. These BPSK mapping achieve full diversity by the rank criterion in [16]. The rotation does not change rank property and consequently using Proposition 2 gives that full spatial diversity is achieved. Comparing with the results in Section IV, we can see that the theorems given in this section consider partial response CPM formats with arbitrary number of transmit antennas. Partial response CPM usually achieves better bandwidth efficiency than the full response CPM and is often used in the real system. Since the linear approximation captures the main energy in the signal, it is highly probable that full rank of the linearized signal will not result in linear deficiency of the actual signal. For instance, are develrank criteria for binary 2RC and 2REC with oped both in an exact manner and an approximate manner. This, in turn, demonstrates the approximation tends not to change the validity of the rank criteria. The validity of these design criteria is also demonstrated in the subsequent simulation results. VI. SIMULATION RESULTS In this section, we present some simulation results to verify the above theorems. In each simulation, the frame length is chosen to be 130 and there is no receive diversity. Each spatial channel is modeled as independent and Rayleigh fading. A. Performance Curve for Theorem 4 For Theorem 4 we provide the performance curve of GMSK with two, three, and four transmit antennas. The phase smoothing function used in GMSK is given as [19]
789
Fig. 3. Frame-error rate of GMSK in quasi-static fading.
(52) is chosen to be 0.3. The simuwhere the bandwidth and lated space–time codes are delay diversity, ( is the standard representation of a convolutional code). Their corresponding generator matrices are given as
and (53)
They are full rank over binary field and according to [16, Th. 14], the nonzero codewords generated by these matrices are full rank over . Therefore, applying our Theorem 4, they should guarantee full diversity with GMSK. The performance curves in Fig. 3 show that full diversity is achieved with the diversity level being . B. Performance Curve for Theorem 2 and 5 For Theorem 2 and Theorem 5, we provide the performance . RC stands curve of 4-ary 1RC with modulation index for the phase smoothing function being the raised cosine pulse shape [19]. The simulated space–time codes are delay diversity, eight-state TSC code (give in [4]) and eight-state BBH (given in [14]). The generator matrix of 8-state TSC code is (54)
(51)
whose row-based indicant is full rank on and the code satisfies QPSK binary rank criterion. The eight-state BBH code is a
790
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Fig. 4. Frame-error rate of 4-ary 1RC with h
= 0 25 in quasi-static fading. :
Fig. 5. Frame-error rate of 4-ary 1REC with h = 0:5 in quasi-static fading.
dyadic construction with where the generator matrices of the two component codes are and
(55)
which are full rank over the binary field . According to [16, Th. 32] it satisfies the QPSK binary rank criterion. Applying our Theorem 2 and Theorem 5, these codes should give full , which can be observed from diversity with 1RC and the simulation results shown in Fig. 4. C. Performance Curve for Theorem 1 and Theorem 6 For Theorem 1 and Theorem 6 we provide the performance . REC curve of 4-ary 1REC with modulation index stands for the phase smoothing function being rectangular pulse shape [19]. The simulated space–time codes are delay diversity and eight-state BBH. As the binary projections of these codes meet BPSK binary rank criterion, full diversity is achieved. Fig. 5 shows simulated performance and the results confirm the theoretic analysis. D. Performance Curve to Verify Theorem 3 and Theorem 4 In Section VI-A, we provided simulation performance of GMSK to verify that the approximation does not change the rank property. Here, we give the performance curve with as another example. The of binary space–time codes we simulated are: delay diversity, and . Since all the codes satisfy BPSK binary rank criterion, full diversity can be observed through the performance curves in Fig. 6. VII. CONCLUSION In this paper, we have derived the design criterion to ensure full spatial diversity of space–time CPM system in quasi-static fading. Due to both the inherent memory and nonlinearity of CPM, it would be computationally prohibitive to use hand design or computer search to come up with space–time codes and
Fig. 6.
Frame-error rate of binary 2REC with h = 0:5 in quasi-static fading.
systematic procedure is desirable. Through linear decomposition of CPM signals and applying the rank criteria available in linear modulation we are able to identify some general code construction with CPM. A meaningful result is that with the GMSK used in GSM standard, if the binary data satisfies the BPSK binary rank criterion full spatial diversity is guaranteed. These general criteria greatly simplify the search for powerful space–time codes in CPM. For linear codes we only need to check the generator matrices instead of all pairs of code words. Simulation results are provided to verify these design criteria and they demonstrate that full diversity is achieved as long as the space–time codes meet the rank criteria. APPENDIX A PROOF OF PROPOSITION 1 Sufficient and necessary conditions of Proposition 1 will be proven by contradiction. for all unless 1) If , then is full rank.
ZHANG AND FITZ: SPACE–TIME CODE DESIGN WITH CONTINUOUS PHASE MODULATION
.. .
Proof: Assume nonzero vector
.. .
is not full rank, then there exists some which satisfy
791
.. .
(B.5)
If we look at the first symbol period for , (B.1) implies
and note that
(A.1) is the th column of signal matrix where the th row of (A.1) is expressed as
. Using (18),
(A.2) Multiply each row of (A.1) by have
and sum them together, we
(A.3) Equation (A.3) and the characteristics of CPM imply
(B.2) (A.4)
which provides the desired contradiction. is full rank, then for all 2) If unless . for some nonzero vector Proof: Assume . Consider a linear combination of the column vectors of , the th component of this resultant vector can be expressed as
(A.5) By assumption (A.5) can be made to be zero and, therefore, is not full rank which provides the desired contradiction.
Since
, and are linearly independent over , it follows directly from equation (B.2) that (B.3)
Proceeding with induction, we have for (B.4) Equation (B.4) implies that (B.5), shown at the top of the page is not full rank over the complex field and provides the desired contradiction. ACKNOWLEDGMENT The authors would like to thank Prof. H. E. Gamal for his insightful discussions about the rank criteria, especially with regard to Theorem 3.
APPENDIX B PROOF OF PROPOSITION 2 Proof: Again, we prove the proposition by contradiction. with at least one , then there exists Assume which satisfy some nonzero vector
(B.1)
REFERENCES [1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecomm., vol. 10, pp. 585–595, Nov.–Dec. [Online]. Available: http://mars.bell-labs.com/cm/ms/what/mars/papers/proof. [2] 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, pp. 311–335, Mar. 1998. [3] A. Wittneben, “A new bandwidth efficient transmit antenna modulation diversity scheme for linear digital modulation,” in Proc. Int. Conf. Communications, 1993, pp. 1630–1634. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [5] 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. IEEE Vehicular Technology Conf., vol. 1, Atlanta, GA, 1997, pp. 136–140. [6] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [7] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes for orthogonal designs,” IEEE Inform. Theory, vol. 45, pp. 1456–1466, July 1999.
792
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
[8] D. Aktas and M. P. Fitz, “Computing the distance spectrum of space-time trellis codes,” in Proc. Wireless Communications and Networking Conf., Chicago, IL, Sept. 2000, pp. 51–55. [9] E. Biglieri and A. M. Tulino, “Designing space-time codes for large number of receiving antennas,” Electron. Lett., vol. 37, pp. 1073–1074, Aug. 2001. [10] H. El Gamal and M. P. Fitz, “Generalized maximum ratio combining for space-time decoding,” IEEE Trans. Commun., submitted for publication. [11] Z. Chen, J. Yuan, and B. Vucetic, “Improved space-time trellis coded modulation scheme on slow Rayleigh fading channels,” in Proc. International Conf. Communications, Helsinki, Finland, Mar. 11–14 2001, pp. 440–441. [12] J. Grimm, M. P. Fitz, and J. V. Krogmeier, “Further results in space-time coding for Rayleigh fading,” in Proc. 36th Annual Allerton Conf. Communication, Control, Computing, Monticello, IL, Sept. 1998, pp. 391–400. [13] J. Grimm, “Transmitter Diversity Code Design for Achieving Full Diversity on Rayleigh Fading Channels,” Ph.D. dissertation, Purdue Univ., Lafayette, IN, 1998. [14] S. Bäro, G. Bauch, and A. Hansmann, “Improved codes for space-time trellis-coded modulation,” IEEE Commun. Lett., vol. 4, pp. 20–22, Jan. 2000. [15] Q. Yan and R. S. Blum, “Optimum space-time convolutional codes for quasistatic slow fading channels,” in Proc. 2nd Wireless Communications and Networking Conf., Chicago, IL, Sept. 23–28, 2000. [16] A. R. Hammons, Jr., and H. El Gamal, “On the theory of space-time codes for PSK modulation,” IEEE Inform. Theory, vol. 46, pp. 524–542, Mar. 2000. [17] Y. Liu, M. P. Fitz, and O. Y. Takeshita, “A rank criterion for QAM space-time codes,” in Proc. Int. Symp. Information Theory, Sorrento, Italy, June 2000, p. 314. [18] T. Aulin, N. Rydbeck, and C. W. Sundberg, “Continuous phase modulation—Part I and part II,” IEEE Trans. Commun., vol. COM-29, pp. 196–225, Mar. 1981. [19] J. B. Anderson, T. Aulin, and C. W. Sundberg, Digital Phase Modulation Plenum, New York, 1986. [20] B. E. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inform. Theory, vol. 34, pp. 260–270, Mar. 1988. [21] P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses (AMP),” IEEE Trans. Commun., vol. 34, pp. 150–160, Feb. 1986. [22] U. Mengali and M. Morelli, “Decomposition of M-ary CPM signals into PAM waveforms,” IEEE Inform. Theory, vol. 41, pp. 1265–1275, Sept. 1995.
[23] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: Performance results,” IEEE J. Select. Areas Commun., vol. 17, pp. 451–459, Mar. 1999. [24] M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” in Proc. Wireless Communications and Networking Conf., New Orleans, LA, Sept. 1999, pp. 139–144. [25] X. Zhang and M. P. Fitz, “Space-time coding for Rayleigh fading channels in CPM system,” in Proc. 38th Annual Allerton Conf. Communication, Control and Computing, Monticello, IL, Oct. 2000.
Xiaoxia Zhang received the B.S. and M.S. degrees from the University of Science and Technology of China, Hefei, in 1994 and 1997, respectively, and the Ph.D degree from Ohio State University, Columbus, in 2002, all in electrical engineering. Her research interests include continuous phase modulation, space–time coding, iterative detection, and channel estimation in wireless communications. She is currently working with the Standards Group, Qualcomm Inc., San Diego, CA.
Michael P. Fitz received the B.E.E. degree (summa cum laude) from the University of Dayton, Dayton, OH, in 1983 and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, in 1984 and 1989, respectively. From 1983 to 1989, he worked as a Communication Systems Engineer for Hughes Aircraft and TRW Inc. In 1989, he ventured into academia and has been on Faculty at Purdue University, West Lafayette, IN, and The Ohio State University (OSU). He is currently a Professor at the University of California. His research is in the broad area of statistical communication theory and experimentation. His research group currently is interested in the theory of space–time modems and operates an experimental wireless wide area network testbed. Dr. Fitz is recipient of the 2001 IEEE Communications Society Leonard G. Abraham Prize Paper Award in the Field of Communications Systems.
