A Space–Time Code Design for CPM: Diversity ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009

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A Space–Time Code Design for CPM: Diversity Order and Coding Gain Alenka G. Zajic´, Member, IEEE, and Gordon L. Stüber, Fellow, IEEE

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Abstract—Sufficient conditions are derived under which -ary partial- and full-response continuous phase modulation (CPM) space–time (ST) codes will attain both full spatial diversity and optimal coding gain. General code construction rules are desirable due to the nonlinearity and inherent memory of the CPM signals which makes manual design or computer search difficult. Using a linear decomposition of CPM signals with tilted phase, we identify a rank criterion for -ary partial- and full-response CPM that specifies the set of allowable modulation indices. We also propose a coding gain design criterion. Optimization of the coding gain for CPM ST codes is shown to depend on the CPM frequency/phase shaping pulse, modulation index, and codewords. The modulation indices and phase shaping functions that improve the coding gain are specified. Finally, optimization of coding gain for ST-CPM and orthogonal ST-CPM codewords is discussed.

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Index Terms—Coding gain, continuous phase modulation (CPM), determinant criterion, linear decomposition, rank criterion, Rayleigh fading, space-time (ST) coding, spatial diversity, tilted phase.

I. INTRODUCTION

PACE–TIME (ST) coding transmits coded waveforms from multiple antennas to maximize link performance. Full spatial diversity is one design objective for ST codes, being , where and are the upper-bounded by the product number of transmit and receive antennas, respectively. Coding gain optimization is another design objective for ST codes. Many different ST codes have been developed for quasi-static Rayleigh-fading channels [1]–[19]. Tarokh et al. [1] devised the rank and determinant criteria for spatial diversity that optimizes the worst case pairwise error probability (PWEP) and presented some simple design rules that guarantee full spatial diversity for linear modulation schemes. Several ST code designs based on the rank and determinant criteria are proposed in [2]–[4]. In [5],

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the determinant criterion is strengthened by showing that the Euclidean distance must be optimized in order to optimize the product distance. Finally, in [6] is shown that different design criteria apply depending on the diversity order. For sufficient , the code performance is degrees of freedom governed by the minimum squared Euclidean distance of the , the rank and determinant criteria will code. For small govern the code performance. One of main difficulties encountered when deriving general design rules for ST codes is that the diversity and coding design criteria apply to the complex domain of baseband-modulated waveforms, whereas block code designs are usually conducted over finite fields. Hammons and Gamal [7] proposed rank criteria that ensure full diversity for ST codes with binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK). Liu et al. [8] generalized these rank criteria for higher order quadrature amplitude modulation (QAM) constellations. ST coding can also be applied to continuous phase modulated signals. Zhang and Fitz [9], [10] derived design criteria for ST-coded continuous phase modulation (CPM) (ST-CPM) on quasi-static fading channels and identified a rank criterion, but only for particular CPM schemes: full-response -ary CPM with , full-response -ary CPM with , partial-response binary CPM with , . and partial-response -ary CPM with Some attempts to optimize coding gain of ST-CPM have been made in [11]–[13]. In [11], [12], an orthogonal ST-CPM system , where different CPM schemes are used is proposed with on the two transmit antennas. This method attains full diversity and good coding gain, but requires bandwidth expansion due to a large CPM modulation index. In [13], an ST-CPM scheme is that uses different mapping rules on the proposed with two antennas to achieve full diversity and optimal coding gain. has not been considered. In However, the extension to conclusion, a general framework for ST-CPM is lacking both in terms of diversity order and coding gain. This paper derives sufficient conditions under which any -ary partial- and full-response ST-CPM arrangement will attain full spatial diversity for any . Design rules are specified for coding gain optimization. Paralleling the work of Mengali and Morelli [20], we first derive a linear decomposition of CPM signals with tilted phase. The tilted phase is time-invariant and simplifies receiver processing [21]. However, we must stress that this paper is not about minimizing complexity, but rather to provide a general framework for ST-CPM. Based on our linear decomposition, we propose a rank criterion for -ary partialand full-response CPM that eventually defines a set of optimal modulation indices for any candidate CPM scheme (excluding

Manuscript received May 23, 2006; revised nulldate. Current version published July 22, 2009. This work was prepared through collaborative participation in the Collaborative Technology Alliance for Communications and Networks sponsored by the U.S. Army Research Laboratory under Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The material in this paper was presented in part at the IEEE International Conference on Communications, Glasgow, Scotland, U.K., June 2007. A. G. Zajic´ was with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. She is now with the Naval Research Laboratory, Washington DC 20375 USA. Gordon L. Stüber is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Communicated by H. Boche, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2009.2023755

0018-9448/$25.00 © 2009 IEEE

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009

Rayleigh fading, and independent zero-mean complex additive white Gaussian noise (AWGN). The received signal can be represented in the convenient vector form (3)

Fig. 1. Block diagram for ST-coded CPM system.

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multi-h CPM). We also propose a coding gain design criterion for ST-CPM. Maximization of the coding gain for ST-CPM depends not only on the codewords as in linear modulation, but also on the frequency/phase shaping function. Since the computer search for the codes and phase shaping functions that maximize the coding gain is a computationally challenging problem, a coding gain optimization criterion that improves (or maximizes) the coding gain is developed. Finally, we discuss the optimization of ST-CPM and orthogonal ST-CPM as special cases. The remainder of the paper is as follows. Section II describes ST-CPM on a quasi-static fading channel. Section III derives the linear decomposition of CPM signals with tilted phase. Section IV presents our design criteria for -ary partial- and fullresponse ST-CPM. Section V presents several examples and simulation results verifying the developed ST-CPM rank and coding gain design optimization criteria. Section VI concludes the paper.

is the vector where of transmitted signals, is the vector of the code symbols assigned to th transmit antenna, is the vector of received signals, contains the noise samples that are independent zero-mean complex Gaussian random variper dimension, is ables with variance is the symbol the matrix of complex channel fading gains, denotes the matrix transpose operation. energy, and In some embodiments, the ST-CPM receiver employs maximum-likelihood sequence detection (MLSD) as implemented with the Viterbi algorithm. The bit-error rate performance of MLSD is typically evaluated by upper-bounding the pairwise error probability for any two ST codewords , and , defined and denote the CPM vectors correas in (2). Let be sponding to matrices and , respectively. Finally, let the matrix of correlation functions of the differential CPM signals received at the different antennas [9]

II. ST-CPM SYSTEM MODEL

.. .

.. .

.. .

(4)

This section describes ST-CPM on a quasi-static fading transmit channel. We consider an ST-CPM system with antennas and receive antennas. As shown in Fig. 1, the ST information encoder uses a block code to encode blocks of codeword vectors that symbols into lengthmatrix in the following manner: are mapped onto an codeword

where differential CPM signals are defined as for and denotes the symbol duration. For a quasi-static flat Rayleigh-fading channel, the pairwise error probability has the following, asymptotically tight, upper bound [1]:

(1)

is mapped to the

(5)

matrix

.. .

..

.

.. .

(2)

where is the code symbol assigned to th transmit antenna at time epoch . The outputs of the ST encoder are streams of symbols, that are input to separate tilted-phase CPM modulators that drive the antennas. Due to the tilted phase representation, the ST encoder outputs can be directly input to the CPM modulators without the need for additional modulation mapping as is the case if an excess-phase CPM modulator is used. The -modulated signals transmit antennas. are simultaneously transmitted from The signal at each receiver antenna is a noisy superposition transmitted signals, each affected by quasi-static flat of the

where is the rank of matrix are the nonzero eigenvalues of , and is their geometric mean. From (5), we can conclude that ST-CPM has a direct analogy to ST-coded linear modulation. The rank criterion and the determinant criterion used for ST linear modulation are directly applicable to ST-CPM, the only difference being that CPM has the “signal” matrix as in (4). III. LINEAR DECOMPOSITION OF CPM WITH TILTED PHASE Laurent [22] showed that a binary CPM signal can be decomposed into a linear combination of pulse amplitude modulated (PAM) waveforms. Mengali and Morelli [20] extended Laurent’s CPM signal decomposition to -ary CPM signals. These linear decomposition approaches [22], [20] use the CPM excess phase, but as mentioned previously, a decomposition based on the time-invariant CPM tilted phase is desirable. Paralleling the

ZAJIC´ AND STÜBER: A SPACE–TIME CODE DESIGN FOR CPM

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work of Mengali and Morelli [20], we derive a linear decomposition of CPM signals with tilted phase. The CPM tilted-phase baseband complex envelope is [21] (6) where

, and

The equality in (11) can be shown by applying the Euler transwith and , formation on sine functions and by replacing respectively. as shown in (12), also shown at the bottom of Define the page.Therefore, when (13)

(7)

From (10) and (13), the CPM tilted-phase complex envelope on the interval becomes

is the tilted phase, is the modulation index, is the information sequence with elements , and is chosen from the -ary alphabet in (7) is the symbol duration. The term (14) (8)

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The phase shaping pulse is defined by , is the frequency shaping pulse of length such where for . The CPM waveform has full that response when and partial response when . To derive the CPM tilted-phase decomposition, first note that . Since some integer exists such that varies in the range has the radix- representation , where . Hence, the tilted phase in (7) can be rewritten as

Closely following the algorithm in [20], we obtain the complex envelope of -ary partial-response CPM signals with tilted phase as (15)

where and symbols are

and the Laurent functions are defined below. The Laurent functions

(16)

(9)

Then, the CPM tilted phase complex envelope has the form shown in (10) at the bottom of the page. The next step replaces the partial-response term associated th bit in (10) by an equivalent sum of two terms, with the such that only the second term depends upon the th bit. The partial-response term in (10) associated with the th bit of satisfies symbol

where tion of the index and functions The symbols

are coefficients in the binary representa, are defined as in (12). are

(17)

(11)

(10)

for for otherwise.

(12)

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009

where are coefficients in the binary representation of the information symbol . The integer , used in (15)–(17), and satisfies is chosen from the set , where . Finally, integers , used in (15)–(17), are chosen to satisfy and , where are du-

are coefficients in the binary representation of the symbols , are coefficients in the binary representation and . Note that can only of the index . assume values from the set ST-CPM vector can be From (15)–(18), the written as given in (19) at the bottom of the page, where and . Then, the differential ST-CPM vector for two ST codewords, and , is given in the matrix shown in (20), also at the bottom of the page. To simplify further nomatrix of accumulative values tations, we define the , where , and as . Furthermore, we denote the matrices and , in (19) and (20) as can be rewritten respectively. Now, the ST-CPM vector as

rations of the functions and

An outline for the derivation of (15)–(17) is presented in Appendix A. IV. DESIGN CRITERIA FOR ST-CPM This section first derives sufficient conditions under which -ary partial- and full-response ST-CPM arrangement any will attain full spatial diversity for any . Then, design rules are specified for coding gain optimization. Optimization of the coding gain for ST-CPM is shown to depend on the CPM frequency/phase shaping pulse, modulation index, and codewords. The modulation indices and phase shaping functions that optimize the coding gain are specified. Finally, optimization of coding gain for ST-CPM and orthogonal ST-CPM codewords is discussed. The CPM modulator inputs in Fig. 1 are elements from the matrix defined in (2), while the outputs are elements , from the vector are defined in (15) and . where the signals , where and are relatively prime Assume that integers. For -ary, partial/full-response, ST-CPM codes, we define

(21)

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whereas the differential ST-CPM vector can be rewritten as

(22)

Using (4) and (22), the matrix of correlation functions of the differential ST-CPM signals, , can be written as

(23)

where mitian operation.

denotes the Her-

A. Full-Diversity Design Criterion

(18)

where

and

The ST-CPM signals achieve full diversity if the matrix in (23) has full rank for any two ST codewords. In [9], is shown has full rank iff the elements of the differthat the matrix ential ST-CPM vector in (22) are linearly independent. Here, we show that a sufficient condition for achieving full dihave full rank for any two versity is that the matrices

.. .

.. .

..

.

..

.

(19)

.. .

.. .

(20)

ZAJIC´ AND STÜBER: A SPACE–TIME CODE DESIGN FOR CPM

ST codewords , and the following lemma.

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. To show this result, we first introduce

Lemma 1: Suppose that all the functions , for and , are collected to form the vector

The components of the vector

are linearly independent.

The proof is shown in Appendix B. Using the results from Lemma 1, we show the following. have Lemma 2: If the complex matrices has linfull rank, then the differential ST-CPM vector early independent elements, i.e., the ST-CPM system achieves full spatial diversity .

(27) are drawn from , and not all of elements of the vector [15]. We will prove this by contrathem are divisible by diction. Assume that it is possible to find such a vector . Since , from Lemma 16 and the proof of Theorem 4 in [15], we have

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The proof is shown in Appendix C. Note that the Laurent functions do not affect the full-diversity design criterion. However, they affect the coding gain, as discussed in the next subsection. Next we determine conditions under which the matrices have full rank. Observe that each -ary codeword , defined in (2), has the form

where denotes modulo- summation. Since rows in the accuare obtained using linear nonsingular mulative matrices , which do not change the transformations of the rows in rank property, the accumulative matrices also have full rank. From the injection property of these linear transformathen . tions, if have full rank for any Next, we show that the matrices two different accumulative matrices and . First, we note that are matrices over the cyclotomic number field , . Let denote the ring of algewhere . The matrices have full rank if it is braic integers in such impossible to find a nonzero vector that

(28)

and

(24)

where

denotes the operation

(29)

.

Theorem 3 (Rank Design Criterion): Denote as a linear ST code over the commutative ring of integers , with . Suppose that all nonzero modulohave different nonzero modulo- projections codewords and that matrices have full rank over the binary -ary CPM field . Then for any partial- or full-response (where , and and are scheme with relatively prime integers), the ST code achieves full spatial diversity . Proof: Earlier, we have defined the accumulative matrices as

(25)

where

are coefficients in the binary representation of the symbols are coefficients in the binary representa, and . The tion of the index is modulo- projection of the matrix (26)

. Since , where from the code linearity follows that the accumulative matrices also have full rank. Hence, (27) cannot be satisfied, which leads to contradiction. The previous theorem showed that if the ST code satisfies have full rank for the CPM rank criterion, the matrices and . Lemma any two different accumulative matrices 2 proved that the elements of the differential ST-CPM vector are linearly independent if matrices have full rank. has full rank iff the elements In [9], is shown that the matrix are linearly indepenof the differential ST-CPM vector dent. Consequently, the overall ST-CPM system achieves full diversity.

Remark 1: The CPM rank criterion requires that different codewords have different modulo- projections because the linear transformation used in (25) to obtain the accumulative matrices only preserves the rank of the lowest bit code matrix . Hence, if there is no will have full rank even though guarantee that matrices codewords and have full rank. The higher bit matrices can be used to optimize coding gain, as discussed in the sequel. Also note that if all modulo- projections are zero, the rank design criterion applies to modulo- projections, i.e., . Remark 2: The proposed rank criterion for -ary CPM , is similar to the BPSK with modulation index rank criterion [7] and, hence, it provides both necessary and

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sufficient conditions for full spatial diversity, whereas the proposed rank criterion for -ary CPM with modulation indices provides only sufficient conditions for full spatial diversity. Nonetheless, the CPM rank criterion is a useful design tool for ST-CPM.

and the matrices semi-identity matrices (i.e.,

Remark 3: To derive the CPM rank criterion we used the linear decomposition of CPM signals with tilted phase. The CPM rank criterion applies to the original tilted-phase or excess-phase CPM waveform as well.

Reasoning Behind the Proposition: From Theorem 4 follows that the first step toward coding gain optimization is to as a semi-identity design the matrix matrix. Note that the PAM pulse shaping functions in can be separated from the transmitted symbols and considered as part of the channel impulse response [24]. This allows us to separate the influence of the pulse shaping functions and codewords on the coding gain and optimize and separately. The the matrices will be optimized if they are designed to be matrices , where is the semi-identity matrices, identity matrix and . Then, the matrix simplifies to . The matrix will be optimized if it is designed to be a semi-identity matrix constrained on its trace. There is no unique solution for this problem. To simplify the problem, we choose a solution , where is the identity matrix and . Then, it follows that the coding can be improved if the matrices and gain are designed to be semi-identity matrices with and , maximized product of traces what was our claim.

are the and product of traces

B. Optimization of Coding Gain Once full diversity is guaranteed, the next objective is to maximize the coding gain, , over all pairs of distinct codewords and , defined by geometric-mean of the nonzero , i.e., eigenvalues of the matrix (30)

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where denotes the determinant operation. From (23), the determinant can be written as

are designed to be and , where and identity matrices) with maximized and .

(31)

Equation (31) shows that maximization of the coding gain for ST-CPM codes is more difficult than that for linearly modulated ST codes, because the coding gain is not only a function of codewords and , but also a function of the phase shaping pulses . The following theorem introduces the used in the vectors coding gain design criterion for ST-CPM codes.

C. Improvement of Coding Gain Through Phase Shaping Pulses

Theorem 4: (Coding Gain Design Criterion) The coding gain is maximized if the positive–definite Hermitian matrix is a semi-identity matrix with maximized diagonal elements (i.e., constrained on the trace ). Proof: This claim can be proven by observing that the maare different from zero only for coefficients trices (this will be proven in Section IV-C) and using the results in [23, Example 2.B].

In this subsection, we first show that the matrices are equal to zero. Then, we investigate conditions under which the can be constructed as semi-identity matrices. A matrices is difficult because funcgeneral analysis of the matrices depend on the memory length . Hence, we analyze tions for cases of practical importance . The matrices results show that the matrices are semi-identity matrices for the raised cosine frequency shaping function with memory and the modulation indices for length . Finally, the modulation indices should be chosen to maximize the traces . are nonzero in the From (16), observe that functions . Then, the matrix time interval defined in (23) has the form given in (32) at the bottom of the page, where denotes complex conjugate operation and the are defined in (16). Appendix D shows that functions , matrices for an arbitrary phase shaping function

To find the codes, phase shaping pulses, and modulation in) that dices (i.e., the terms in matrix jointly maximize the coding gain, computer search is required. However, in this paper, we are interested in defining a theoretical framework that leads to coding gain improvement or optimization, as discussed below.

Proposition 5: (Coding Gain Improvement) The coding can be improved (or optimized) if gain

.. .

..

.

.. .

(32)

ZAJIC´ AND STÜBER: A SPACE–TIME CODE DESIGN FOR CPM

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.. .

..

.. .

.

(33)

in (32) becomes exare zero matrices. Then, the matrix pression (33) shown at the top of the following page.In general, has equal diagonal elements, i.e., the matrix

, matrices , are semifunction identity matrices. , because Previous reasoning cannot be applied to matrix for . Hence, we need to evaluate all integers the integrals

Since all diagonal elements of the matrix are equal, we just is a diagonal matrix. Howneed to define conditions when is difficult because ever, a general analysis of the matrices depend on the memory length . the Laurent functions Hence, we will analyze matrices for cases of practical im. portance , 1) Full-Response CPM: For full-response CPM in (33) becomes (34), shown at the bottom of the matrix the page, where the Laurent functions can be written as

for particular phase shaping functions. Using (35) and can be written as the integral

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,

(35) will be

Since all diagonal elements are equal, the matrix semi-identity if

(37)

as derived in Appendix E. In general, . We have ob, where served that these integrals have solutions in the form is a constant that depends on the modulation index and . Hence, the integrals will be phase shaping function minimized if the constant is made as small as possible. Furthermore, the following proposition shows that for commonly and . used phase shaping functions Proposition 6: For the phase shaping functions [25]

and

(38)

We start evaluation of these integrals, by evaluating the products

and with adequate selection of parameters . For the raised cosine (1RC) frequency shaping , with the modulation indices , where function , the integrals and are zero.

(36)

. Following similar reasoning as in Apfor pendix D, we can conclude that the product is always zero. A similar argument can be used to show that . Hence, for an arbitrary phase shaping

.. .

The proof is shown in Appendix F.

Remark 4: If the rectangular pulse (1REC) is selected as the frequency shaping function (i.e., has and ), and are not always zero. However, the integrals for minimum-shift keying (MSK) modulation (1REC with

..

.

.. .

(34)

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), , because MSK has orthogonal carriers with minimum frequency separation. 2) Partial-Response CPM: For become the Laurent functions

partial-response CPM, (40)

(39) where For indices

. having at least one coefficient only on the time interval . Since there is no overlap among functions and for , matrices are semi-identity for arbiare trary phase shaping functions. For indices for which not semi-identity for an arbitrary phase shaping function, the integrals

are defined in (18) and is the where elements squared Euclidean distance between the code matrices and . The maximization of is not straightforward because elements in matrices and are obtained as linear combinations of elements in the codebecomes words and . Using (25), the trace of the matrix

(41) where functions

for integers

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and

are defined as for , and . From (41) follows that the trace of the matrix over all pairs of codewords will be optimized if the squared minimum Euclidean distances for are maximized. By observing that each has the form matrix of accumulative values

, and

(42)

and

for should be minimized. Furthermore, the following proposition shows when the integrals , and are zero. Proposition 7: For the raised cosine (2RC) frequency shaping function, , with the modulation indices , where , the integrals , and are zero. The proof is shown in Appendix G.

D. Improvement of Coding Gain Through ST-CPM Codewords As shown in Proposition 5, the coding gain can be improved if matrices are semi-identity with maximized trace . In this section, we derive constraints on ST-CPM codewords that lead to trace maximization and construct orthogonal ST-CPM codewords (i.e., an ex). ample of codewords with semi-identity matrices is equal to 1) Trace Maximization: The trace

where denotes operation and by using the results in [26], the minimum trace of the matrix over all can be written as pairs of codewords

(43)

where is the minimum Hamming distance over all code matrices and . From (43) follows that the over all pairs of codewords trace of the matrix will be optimized if all minimum Hamming distances are maximized. When the codewords cannot be designed , then to maximize all minimum Hamming distances they should be designed to maximize the minimum Hamming that contain most distances associated with the matrices of the signal energy. Finally, we note that a design approach relying on the trace maximization with the rank constraint on the codeword matrices can lead to coding gain improvement. 2) Orthogonal ST-CPM Codewords: Since orthogonal space-time block codes satisfy the semi-identity requirement can be designed as orthogonal in Proposition 5, matrices space-time (OST) codewords. We will start with the orthogonal code for two transmit antennas proposed by Alamouti [18]. The should be equal to elements in the elements in the matrix Alamouti’s codeword, i.e., (44)

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where and are complex numbers from the modulation aland should be chosen to satisfy phabet. Code elements

and

(45) and (46) (48) where the modulation index is and for . The other two code elements and should be selected to maximize coding gain, i.e., . For more than two transmit antennas, orthogonal ST codes are unsuitable for CPM modulation, because orthogonal ST codes have some zero elements in the code matrices whereas symbols are nonzero. Quasi-orthogonal ST codes may be used can be designed using a similar instead. The code matrices algorithm as for orthogonal ST codes. V. EXAMPLES AND NUMERICAL RESULTS

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Several ST-CPM codewords are constructed using our rank and coding gain design criteria. Simulation results are presented to verify proposed design criteria. Each spatial channel is modeled as being independently Rayleigh faded. All simulations use . The receiver is designed as in [21] with a frame length of . As pointed out earlier, we used the linear decomposition of CPM signals with tilted phase to derive the rank and coding gain criteria, but these criteria also apply to the original tilted-phase or excess-phase CPM waveforms. Depending on the CPM waveform that is transmitted, reduced-complexity receivers can be used, but they are beyond the scope of this paper. The first example uses a full-response raised cosine (1RC) frequency shaping function. Two ST codewords are con-ary CPM ST code that satisfies structed from a as the rank design criterion. We construct codewords , where denotes the inverse of defined in (41), are binary codewords the function ST codes , all codewords have from linear full rank over , and all codewords satisfy . Note that any method for constructing CPM ST codes that satisfies the rank design criterion can be used. Some of the methods for constructing binary ST codes with full rank are described in [7] and [14]. According to the rank design criterion, the binary and can be arbitrary selected. We codewords first construct binary codewords and the same way as codewords and , and use them as a benchmark to measure further coding gain improvement. Following the method described in [14], we use as a zero of the primitive polynomial over and we use the generator to construct benchmark codewords matrix

According to the CPM rank criterion, these two codewords will achieve full spatial diversity, if modulation index takes . We now verify this statevalues from the set ment for each in the set. . To verify that codewords (47) and (48) First, let achieve full diversity, we need to check if all functions in the , defined in (22), are linearly independifferential matrix dent. This will be satisfied if components of the vector are linearly independent and the matrices have full rank. From (35), we calculate functions . Results are presented in the first column of Table I from which we conclude that they are linearly independent. The first column in Table II shows the , where . Obmatrices serve that all matrices have full rank, implying that this pair of codewords achieves full spatial diversity. . Functions remain the same as Second, let in Table I. The second column in Table II shows the recalcu. Since all matrices have full rank, this pair lated matrices of codewords achieves full diversity and also satisfies rank criterion proposed by Zhang and Fitz [9]. it can be similarly verified that full diFinally, for versity is achieved. Since this pair of codewords achieves full spatial diversity, the next objective is to improve their coding gain. According to Proposition 5, the coding gain will be improved if all are designed as semi-identity matrices and the the is maximized, where determinant . From (34), we calculate matrices , and determinants for modulation index . The results are presented in the second and third columns of Table I. Obtained results confirm theoretical results de, are rived in Subsection IV-C. The semi-identity matrices as expected. Finally, the trace obtained using codewords and , is equal to . The minimum trace of this ST-CPM code is . This result is obtained from (43), using the fact that the code proposed in [14] has the minimum Hamming distance . Since the rank design criterion only requires distinct matrices and to have full rank, the higher bit matrices and can be used for coding gain improvement. In this example, optimization of binary matrices and is ineffective and, hence, these mabecause the modulation index is . trices do not have impact on obtained values in matrices Also, we keep matrices and unchanged, since they satisfy the rank criterion. Using the binary ST code proposed in [16] with elementary polynomial

(47)

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M = 8 L = 1, AND = 1 4

codewords

;

h

=

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TABLE I MODULATION FUNCTIONS FOR 1RC,

and minimum Hamming distance for and , we construct the improved codewords

Further enhancement of the coding gain can be obtained if and are constructed not only to have full rank, matrices but also to optimize coding gain. Using the binary ST code proand , we obtain the posed in [16] for codewords optimized codewords

(49)

and

(51)

and

(50)

The fourth column of Table II shows the matrices obtained using codewords and . Observe that all matrices still have a full rank. The trace obtained using codewords and is equal to

However, the minimum trace of this ST-CPM code is still .

(52)

The fifth column of Table II shows the matrices obtained using codewords and . The trace obtained and , is equal to using codewords

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M=8

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TABLE II CODING GAIN OPTIMIZATION FOR FULL-RESPONSE ST-CPM CODES WITH

Fig. 2. Frame-error rate of -ary ST-CPM with 1RC and h

= 1=4 in quasi-static fading.

The minimum trace of this ST-CPM code is increased to . Fig. 2 compares the performance curves obtained for full, 1RC, CPM with response -ary, transmit antennas. The codewords used to obtain the second, third, and fourth curve are constructed to satisfy only the rank design criterion. The fifth curve is obtained using the improved codewords. Fig. 2 plots the frame-error rate versus the signal-to-

, where denotes the received energy per noise ratio ( bit) and shows that full diversity and coding gain improvement are achieved when the ST codes meet both the rank and trace maximization criteria. The second example uses a partial-response raised cosine (2RC) frequency shaping function. The ST codewords are -ary CPM ST code , where codechosen from a words are constructed as , where

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Fig. 4. Frame-error rate of (2

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Fig. 3. Frame-error rate of 16-ary ST-CPM with 2RC and h = 1=4 in quasi-static fading.

2 2) 8-ary ST-CPM with 1RC and h = 1=4 in quasi-static fading.

denotes the inverse of the function defined are binary codewords from linear ST in (41), have full rank over , and all codes , all codewords satisfy . codewords Fig. 3 compares the performance curve obtained for partialresponse -ary CPM signals with 2RC frequency shaping func, and transmit antennas. As tion, in the first example, the codewords used to obtain the second, third, and fourth curve are obtained using the binary ST code described in [14]. The codewords used to obtain the fifth curve are obtained using the binary ST code proposed in [16]. Fig. 3 shows that full diversity and coding gain improvement are achieved when the ST codes are constructed to satisfy the rank and trace maximization criteria. The last example uses a full-response 1RC frequency shaping -ary function. The ST codewords are chosen from a are constructed CPM ST code , where codewords , where denotes the inverse as defined in (41), are binary codeof the function words from linear ST codes described in [14]

, where is a zero of with the generator matrix over . All the primitive polynomial have full rank over and all codewords codewords satisfy . The performance curves for this -ary, , CPM ST code with transmit 1RC, antennas are plotted in Fig. 4. Results show that full diversity is obtained using this ST code. The minimum trace of . this ST-CPM code is The coding gain can be improved if the codewords are constructed as in (44), using the orthogonal code for two transmit antennas [18]. The minimum trace of this ST-CPM code is . We refer to this design as orthogonal ST full-response CPM (OST-FCPM) design. Further enhancement of the coding gain can be obtained if the codewords are constructed using the super-orthogonal two-state trellis ST code for two transmit antennas proposed in [17]. The minimum trace of this ST-CPM code is . We refer to this design as super-orthogonal ST full-response CPM (SOST-FCPM) design. Fig. 4 compares our OST-FCPM and SOST-FCPM

ZAJIC´ AND STÜBER: A SPACE–TIME CODE DESIGN FOR CPM

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Fig. 5. Frame-error rate of 8-ary OST-CPM with Gaussian frequency shaping function (BT = 0:25), full-response rectangular, and full-response raised cosine frequency shaping function and h = 1=4 in quasi-static fading.

Fig. 6. Frame-error rate of 8-ary OST-CPM with Gaussian frequency shaping function (BT = 0:25), partial-response rectangular, and partial-response raised cosine frequency shaping function and h = 1=4 in quasi-static fading.

designs with OST-FCPM design [12] and ST-FCPM design with mapping scheme [13], respectively. Results show that our SOST-FCPM design has similar performance as ST-FCPM with mapping scheme [13] and performs better than the OST-FCPM design [12] and our OST-FCPM design. Fig. 4 also shows that full diversity and coding gain improvement are achieved when the ST codes are constructed to satisfy the rank and coding gain improvement criteria. In Section IV-C is shown that the raised cosine frequency shaping function satisfies the coding gain improvement criterion. Figs. 5 and 6 illustrate how much the coding gain is sacrificed if other frequency shaping functions are used instead of the raised cosine frequency shaping function. The results

-ary orthogonal ST code and are obtained using the modulation index . In Fig. 5, the second curve is obtained using the Gaussian frequency shaping function (GSP, frequency shaping function used in GMSK) with normalized filter , the third curve is obtained using the bandwidth full-response rectangular frequency shaping function (1REC), and the fourth curve is obtained using the full-response raised cosine shaping function (1RC). Results show that the full-response raised cosine shaping function improves the coding gain for approximately 0.89 dB compared to the Gaussian frequency shaping function and approximately 0.57 dB compared to the full-response rectangular frequency shaping function. In Fig. 6, the second and the third curve are obtained using the rectangular

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009

where (12), and

is defined in (8), functions . Symbols

are defined as in ,

are (55)

are coefficients in the radix- representawhere tion of . in (53) is presented as a product The output signal of two sums; that makes our further analysis difficult. To simfollowing plify equations, we modify the output signal the algorithm proposed by Mengali and Morelli [20]. We start representation of an integer from the interval from radix-

(56) where . Then, for each pair , and use it as a subscript for we choose the corresponding defined in (54). As the next step, we form the functions vector , where elements of the are durations of the functions . Then, for a vector given , we choose the -tuples

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frequency shaping function with (3REC) and (2REC), respectively. The fourth curve is obtained using the Gaussian frequency shaping function (GSP) with normalized . Finally, the fifth and the sixth filter bandwidth curves are obtained using the raised cosine shaping function (3RC) and (2RC), respectively. Results show with that the Gaussian frequency shaping function with performs similarly to the raised cosine shaping function with . The raised cosine shaping function with improves the coding gain for approximately 0.47 dB compared to the Gaussian frequency shaping function and approximately 0.86 dB compared to the rectangular frequency shaping function with . Finally, note that the full-response raised cosine shaping function outperforms the partial-response raised cosine shaping for approximately 0.34 dB. This is exfunction with pected result because all matrices , defined in (23), are exactly diagonal matrices for full-response frequency shaping pulses, where as for partial-response frequency shaping pulses, these matrices are semi-diagonal (nondiagonal elements are much smaller than diagonal ones), which decreases the coding gain. VI. CONCLUSION

This paper derived sufficient conditions under which -ary partial- and full-response CPM ST codes will attain both full spatial diversity and improved coding gain. Using a linear decomposition of CPM signals in a tilted-phase representation, we have identified the rank criterion for the -ary partial- and full-response CPM that specifies the set of allowable modulation indices. We have also proposed a coding gain design criterion. It is shown that the optimization of the coding gain for ST-CPM depends on selection of phase shaping pulses, modulation indices, and codewords. We have specified the set of allowable modulation indices and phase shaping functions which can be used toward ST-CPM coding gain improvement (or maximization) and we have established rules for ST-CPM and OST-CPM codeword optimization. Several examples and simulation results show that full spatial diversity and optimal coding gain are achieved for ST-CPM systems that meet the proposed CPM rank criterion and the coding gain design criterion.

that have integer components which satisfy , and . It can be shown that there are such -tuples [20], where is equal to . This result implies that index can take only . values in the interval in (53) can be modified to Now, the signal (57)

where and symbols

and the Laurent functions are defined as

(58)

and

APPENDIX A THE OUTLINE FOR DERIVATIONS OF (15)–(17)

(59)

Similar to Laurent’s decomposition [22], -ary partial-response CPM signals with tilted phase can be presented as (53)

where functions Functions are

and symbols

respectively, where

are defined in (54),

are de-

. Finally, by substituting fined in (55), and (54) in (58) and using the equality , we obtain the expression in (16). Similarly, by substituting (55) into (59), we obtain the expression in (17).

are defined below. APPENDIX B PROOF OF LEMMA 1 (54)

Proof: We will prove this lemma by contradiction. Asare linearly sume that not all functions in the vector

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independent. Then, there exists some nonzero complex vector that satisfies (64)

(60) has nonzero value only over the Note that every function , where . interval Hence, for an arbitrary time interval , (60) can be modified to

where

are elements of the matrices for . Since all functions are linearly independent (as shown in Lemma 1), it follows that (65)

(61) From (16), observe that functions ferent combinations of functions (61) can be rewritten as . For

for

and have dif, defined in (12). Then,

, (62) reduces to

for each and (65) implies that the matrices contradicts our assumption.

APPENDIX D PROOF OF EQUATION

(62) (63)

To show that the matrices evaluate the products

are equal to zero, we first

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for nonzero coefficients , . When , the right term in (63) is equal to zero, because . . This The left term in (63) is equal to equality is satisfied only if . Similarly, when , the right term in (63) is equal to zero, . Following a similar argument as above, because . The fact that and it follows that causes a contradiction. Therefore, for , all are linearly independent. functions Using similar reasoning as above, we now show that all are linearly independent for an arbitrary functions in . For , (62) simplifies . This equality is satisfied only if to . Similarly, for , . Following a sim(62) simplifies to ilar argument as above, it follows that . For , (62) simplifies . Since to , the equality is satisfied only if . Similarly, for , (62) simplifies to and it follows that . Using similar reasoning (or induction) for other time intervals, it follows that coeffiare zero which cients are causes a contradiction. Therefore, all functions linearly independent for an arbitrary .

. Equation do not have full rank, which

(66)

where the functions are defined in (12), , and . Note that the integers and are chosen to satisfy and functions

, where

Then, at least one integer . Since

we assume that

and

are the durations of the

and at least one integer

cannot be zero at the same time,

. Hence, at least one integer

and at least one pair of integers and satisfies . Without loss of generality, we can assume that modified to

for

. Then, (66) can be

APPENDIX C PROOF OF LEMMA 2

Proof: We will prove this lemma by contradiction. Assume are that the elements of the differential ST-CPM vector not linearly independent. Then there exists some nonzero comthat satisfies plex vector

(67) Note that the function tions the time intervals

.. .

and at least one of the funcare nonzero in and

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, respectively. Since at least one , these two functions cannot be nonzero at the same time. Hence, the . A similar argument can product is always be used to show that for . Hence, , matrices for an arbitrary phase shaping function are zero matrices.

Note that the same result can be obtained for other values of parameters and . If the raised cosine function (1RC) is selected as the phase and modulation index is , where shaping function , the integrals are equal to

APPENDIX E DERIVATION OF EQUATION (37) Using (35) and

, the integral

can be written

as (71)

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Using the asymptotic expansion of integrals [29], (71) can be simplified to yield

(72)

By noting that and that all functions are real functions, we can conclude inside the integrals that integrals are also equal to zero.

(68)

APPENDIX G PROOF OF PROPOSITION 7

Proof: Proof of this proposition begins by evaluating the integral for all integers equal to zero. The intecan be written as gral

APPENDIX F PROOF OF PROPOSITION 6

Proof: The proof of this proposition starts by evaluating the with the phase shaping function defined in (38). integral Choosing parameters and , the integral is equal to

(69)

Also, if we choose parameters , the integral index

and the modulation is equal to

(70)

(73)

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17

For the raised cosine function (2RC) and modulation index , where , the integral is equal to

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[9]

[10]

(74)

[11]

Using the asymptotic expansion of integrals, (74) can be rewritten as

[12]

[13]

[14] [15] [16] [17]

(75)

[18]

is equal to By noting that the value of the integral and that all a complex conjugate value of the integral functions inside the integral are real functions, we can conclude are also equal to zero. Similarly can be that integrals and for other combinations shown that integrals and integrals and of integers for all are equal to zero.

[19] [20]

[21] [22]

ACKNOWLEDGMENT

[23]

The authors would like to thank the anonymous reviewers whose feedback helped improve the quality of this paper.

[24]

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Rayleigh fading channels in CPM system,” in Proc. 38th Annu. Allerton Conf. Communication, Control, and Computing, Monticello, IL, Oct. 2000. H. Lu and P. V. Kumar, “Rate-diversity tradeoff of space-time codes with fixed alphabet and optimal constructions for PSK modulation,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2747–2751, Oct. 2003. H. Lu and P. V. Kumar, “A unified construction of space-time codes with optimal rate-diversity tradeoff,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1709–1730, May 2005. H. Lu, Y. Wang, P. V. Kumar, and K. M. Chugg, “Remarks on spacetime codes including a new lower bound and an improved code,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2752–2757, Oct. 2003. H. Jafarkhani and N. Seshadri, “Super-orthogonal space-time trellis codes,” IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 937–950, Apr. 2003. S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. U. Mengali and M. Morelli, “Decomposition of -ary CPM signals into PAM waveforms,” IEEE Trans. Inf. Theory, vol. 41, no. 5, pp. 1265–1275, Sep. 1995. B. E. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 260–270, Mar. 1988. P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses,” IEEE Trans. Commun., vol. 34, no. 2, pp. 150–160, Feb. 1986. A. Assalini and S. Pupolin, “A unified SNR-dependent analysis of space-time codes in fading channels,” IEEE Trans. Inf. Theory, vol. 53, no. 9, pp. 3021–3034, Sep. 2007. J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. Amsterdam, The Netherlands: Kluwer Academic, 2004.

M

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[25] T. Aulin and C. Sundberg, “Continuous phase modulation—Part I: Full response signaling,” IEEE Trans. Commun., vol. COM-29, no. 3, pp. 196–209, Mar. 1981. [26] G. Caire and E. Biglieri, “Linear block codes over cyclic groups,” IEEE Trans. Inf. Theory, vol. 41, no. 5, pp. 1246–1256, Sep. 1995. [27] G. L. Stüber, Principles of Mobile Communication 2e. Amsterdam, The Netherlands: Kluwer, 2001. [28] G. K. Kaleh, “Simple coherent receivers for partial response continuous phase modulation,” IEEE J. Sel. Areas Commun., vol. 7, no. 9, pp. 1427–1436, Dec. 1989. [29] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers , 2nd ed. Berlin, Germany: Springer, 1999.

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Alenka G. Zajic´ (S’99–M’08) received the B.Sc. and M.Sc. degrees from the School of Electrical Engineering, University of Belgrade, Belgrade, Serbia, in 2001 and 2003, respectively. She received the Ph.D. degree in electrical and computer engineering from the Georgia Institute of Technology, Atlanta, in 2008. From 2001 to 2003, she was a Design Engineer for Skyworks Solutions Inc., Fremont, CA. Recently, she was awarded the Naval Research Laboratory Postdoctoral Fellowship and currently she is a Research Scientist at the Naval Research Laboratory, Washington DC. Her research interests are in wireless communications and applied electromagnetics. Dr. Zajic´ received the Best Paper Award at ICT 2008, the Best Student Paper Award at WCNC 2007, and was also the recipient of the Dan Noble Fellowship in 2004, awarded by Motorola Inc. and the IEEE Vehicular Technology Society for quality impact in the area of vehicular technology. PLEASE NOTE THAT HIS TRANSACTIONS DOES NOT PUBLISH CONTRIBUTORS’ PHOTOS.

Gordon L. Stüber (S’81–M’82–SM’96–F’99) received the B.A.Sc. and Ph.D. degrees in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1982 and 1986, respectively. In 1986, he joined the School of Electrical and Computer Engineering, Georgia Institute of Technology, ATLANTA, where he holds the Joseph M. Pettit Chair in Communications. His research interests are in wireless communication systems and communications signal processing. He has graduated 25 Ph.D. students, and has over 250 refereed publications in journals and conferences in these areas. He is author of the textbook Principles of Mobile Communication (Kluwer Academic, 1996; 2nd edition 2001). Prof. Stüber was corecipient of the Jack Neubauer Memorial Award in 1997 for the best systems paper published in the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He became an IEEE Fellow in 1999 for contributions to Mobile Radio and Spread Spectrum Communications. He received the IEEE Vehicular Technology Society James R. Evans Avant Garde Award in 2003 recognizing his contributions to theoretical research in wireless communications. In 2007, he received the IEEE Communications Society WTC Recognition Award (2007) “for outstanding technical contributions in the field and for service to the scientific and engineering communities.” He is an IEEE Communication Society Distinguished Lecturer (2007–2008). He served as Technical Program Chair for the 1996 IEEE Vehicular Technology Conference (VTC’96), Technical Program Chair for the 1998 IEEE International Conference on Communications (ICC’98), General Chair of the Fifth IEEE Workshop on Multimedia, Multiaccess and Teletraffic for Wireless Communications (MMT’2000), General Chair of the 2002 IEEE Communication Theory Workshop, and General Chair of the Fifth International Symposium on Wireless Personal Multimedia Communications (WPMC’2002). He is a past Editor for Spread Spectrum with the IEEE TRANSACTIONS ON COMMUNICATIONS (1993–1998), and a past member of the IEEE Communications Society Awards Committee (1999–2002). He is currently an elected member on the IEEE Vehicular Technology Society Board of Governors (2001–2003, 2004–2006, 2007–2009), and an elected Member-atLarge on the IEEE Communications Society Board of Governors (2007–2009). He received the IEEE Vehicular Technology Society Outstanding Service Award in 2005. PLEASE NOTE THAT HIS TRANSACTIONS DOES NOT PUBLISH CONTRIBUTORS’ PHOTOS.