Two Designs of SpaceTime Block Codes Achieving Full Diversity with Partial Interference Cancellation Group Decoding
Abstract
A partial interference cancellation (PIC) group decoding based spacetime block code (STBC) design criterion was recently proposed by Guo and Xia, where the decoding complexity and the code rate tradeoff is dealt when the full diversity is achieved. In this paper, two designs of STBC are proposed for any number of transmit antennas that can obtain full diversity when a PIC group decoding (with a particular grouping scheme) is applied at receiver. With the PIC group decoding and an appropriate grouping scheme for the decoding, the proposed STBC are shown to obtain the same diversity gain as the ML decoding, but have a low decoding complexity. The first proposed STBC is designed with multiple diagonal layers and it can obtain the full diversity for twolayer design with the PIC group decoding and the rate is up to symbols per channel use. But with PICSIC group decoding, the first proposed STBC can obtain full diversity for any number of layers and the rate can be full. The second proposed STBC can obtain full diversity and a rate up to with the PIC group decoding. Some code design examples are given and simulation results show that the newly proposed STBC can well address the rateperformancecomplexity tradeoff of the MIMO systems.
Diversity techniques, spacetime block codes, linear receiver, partial interference cancellation.
I Introduction
Spacetime (ST) coding is a bandwidthefficient transmission technique that can improve the reliability of data transmission in MIMO wireless systems [2, 3]. Orthogonal spacetime block coding (OSTBC) is one of the most attractive ST coding approaches because the special structure of orthogonality guarantees a full diversity and a simple (linear) maximumlikelihood (ML) decoding. The first OSTBC design was proposed by Alamouti in [2] for two transmit antennas and was then extended by Tarokh et. al. in [3] for any number of transmit antennas. A class of OSTBC from complex design with the code rate of was also given by Tarokh et. al. in [3]. Later, systematic constructions of complex OSTBC of rates for or transmit antennas for any positive integer were proposed in [4, 5, 6]. However, the OSTBC has a low code rate not more than for more than two transmit antennas [7].
To enhance the transmission rate of the STBC, various STBC design approaches were proposed such as quasiOSTBC [8, 9, 10, 12, 13, 14, 16, 18, 17, 19] and algebraic number theory based STBC [20, 21, 22, 23, 24, 25, 26, 27, 28]. The quasiOSTBC increases the code rate by relaxing the orthogonality condition on the code matrix, which was originally proposed in [8], [9], and [10], independently. Due to the group orthogonality, the ML decoding is performed pairwise or groupwise with an increased complexity compared to the singlesymbol decoding. In [15, 17, 16], quasiOSTBC was studied in the sense of minimum decoding complexity, i.e., a real pairwise symbols decoding. In [18, 19, 17], the pairwise decoding was generalized to a general groupwise decoding. The decoding for these codes is the ML decoding and their rates are basically limited by that of OSTBC. The algebraic number theory based STBC are designed mainly based on the ML decoding that may have high complexity and even though some nearML decoder, such as sphere decoder [29] can be used, the expected decoding complexity is still dominated by polynomial terms of a number of symbols which are jointly detected [30].
To reduce the large decoding complexity of the high rate STBC aforementioned, several fastdecodable STBC were recently proposed [31][32]. The STBC proposed in [31] achieves a high rate and a reduced decoding complexity at the cost of loss of full diversity. The fastdecodable STBC in [32] can obtain full rate, full diversity and the reduced ML decoding complexity, but the code design is limited to and MIMO transmissions only. Another new perspective of reducing the decoding complexity was recently considered in [34] and [35] to resort to conventional linear receivers such as zeroforcing (ZF) receiver or minimum mean square error (MMSE) receiver instead of the ML receiver to collect the full diversity. The outage and diversity of linear receivers in flatfading MIMO channels were studied in [33], but no explicit code design was given to achieve the full diversity when the linear receivers are used. Based on the new STBC design criterion for MIMO systems with linear receivers, Toeplitz STBC [34] and overlappedAlamouti codes [35] were proposed and shown to achieve the full diversity with the linear receivers. Recently, some other new designs of STBC with linear receivers were proposed [36, 37, 38]. However, the code rate of STBC achieving full diversity with linear receivers is upper bounded by one. Later, Guo and Xia proposed a partial interference cancellation (PIC) group decoding scheme [39] which can be viewed as an intermediate decoding approach between the ML receiver and the ZF receiver by trading a simple singlesymbol decoding complexity for a high code rate larger than one symbol per channel use. Moreover, in [39] an STBC design criterion was given to achieve full diversity when the PIC group decoding is applied at the receiver. The proposed PIC group decoding in [39] was also connected with the successive interference cancellation (SIC) strategy to aid the decoding process, referred to as PICSIC group decoding. A few code design examples were presented in [39], but a general design of STBC achieving full diversity with the PIC group decoding remains an open problem.
In this paper, we propose two designs of STBC which can achieve full diversity with the PIC group decoding for any number of transmit antennas. The first proposed STBC have a structure of multiple diagonal layers and for each diagonal layer there are exactly coded symbols embedded, being equal to the number of transmit antennas, which are obtained from a cyclotomic lattice design. Indeed, each diagonal layer of the coded symbols can be viewed as the conventional rateone diagonal STBC [40, 41]. The code rate of the proposed STBC can be from one to symbols per channel use by adjusting the codeword length, i.e., embedding different number of layers in the code matrix. With the PIC group decoding the code rate of the first proposed fulldiversity STBC can be only up to symbols per channel use, i.e., for two layers. For more than two layers embedded in the codeword, the code rate is increased at the cost of losing full diversity with the PIC group decoding. However, with the PICSIC group decoding, the proposed STBC with arbitrary number of layers can obtain full diversity and the code rate can be up to .
The second proposed STBC is designed with three layers of information symbols embedded in the codeword and the PIC group decoding can be performed in three separate groups accordingly. Without loss of decoding complexity compared to the first proposed STBC, the second proposed STBC can achieve full diversity and a code rate larger than . Note that the code rate for the first proposed fulldiversity STBC with PIC group decoding can not be above . In the PIC group decoding of the proposed STBC, every neighboring columns of the equivalent channel matrix are clustered into one group.
This paper is organized as follows. A system model of ST transmission over MIMO channels with the PIC group decoding is introduced in Section II. In Section III, a design of high rate STBC with the PIC group decoding is proposed, which contains multiple diagonal layers of coded symbols. For a particular code design with two diagonal layers, the full diversity with the PIC group decoding is proved. For the code with PICSIC group decoding, the full diversity is shown for any number of diagonal layers. Several fulldiversity code design examples are given in Section IV. In Section V, another design of high rate STBC with the PIC group decoding is proposed, which can achieve full diversity with three layers. Simulation results are presented in Section VI. Finally, in Section VII, we draw our conclusions.
Notations: Column vectors (matrices) are denoted by boldface lower (upper) case letters. Superscripts and stand for transpose and conjugate transpose, respectively. denotes the field of complex numbers. denotes the identity matrix, and denotes the matrix whose elements are all . is the vectorization of matrix by stacking the columns of on top each other.
Ii System Model and PIC Group Decoding
In this section, we first briefly describe the system model and then describe the PIC group decoding proposed in [39].
Iia System Model
We consider a MIMO transmission with transmit antennas and receive antennas over block fading channels. The received signal matrix is
(1) 
where is the codword matrix, transmitted over time slots, is a noise matrix with independent and identically distributed (i.i.d.) entries being circularly symmetric complex Gaussian distributed , is the channel matrix whose entries are also i.i.d. with the distribution , denotes the average signaltonoise ratio (SNR) per receive antenna and is the normalization factor to ensure that the average energy of the coded symbols transmitting from all antennas during one symbol period is . The realization of is assumed to be known at the receiver, but not known at the transmitter. Therefore, the signal power is allocated uniformly across the transmit antennas.
Definition 1 (Code Rate)
Let be the number of independent information symbols per codeword , selected from a complex constellation . The code rate of the STBC is defined as symbols per channel use. If , the STBC is said to have full rate, i.e., symbols per channel use.
In this paper, we consider that information symbols are coded by linear dispersion STBC as
(2) 
where is the linear STBC matrix.
To decode the transmitted sequence at the receiver, we need to extract from . This can be done by as follows. By substituting (2) into (1), we get
(3) 
Then, by taking vectorization of the matrix we have
(4)  
where , , , and is an equivalent channel matrix,
(5) 
with the th column , .
For a ZF receiver, the estimate of the transmitted symbol sequence is,
(6) 
where . Equivalently, it can be written as the singlesymbol decoding as follows,
(7) 
where denotes the th row of .
For an ML receiver, the estimate of that achieves the minimum of the squared Frobenius norm is given by
(8) 
In the ML decoding, computations of squared Frobenius norms for all possible codewords are needed and therefore result in prohibitively huge computational complexity when the length of the information symbols vector to be decoded is large. In the following, we give a metric to evaluate the computational complexity of the ML decoding, which is the same as the one shown in [32, Definition 2].
Definition 2 (Decoding Complexity)
The decoding complexity is defined as the number of squared Frobenius norms that should be computed in the decoding process.
With the above definition, we have the following two remarks.
Remark 1
The decoding complexity of the ZF detection is , i.e., times of the cardinality of the signal constellation. It is equivalent to the singlesymbol decoding complexity.
Remark 2
The decoding complexity of the ML detection is , i.e., the complexity of the full exhaustive search of all information symbols drawn from the constellation .
We next describe the PIC group decoding studied in [39].
IiB PIC Group Decoding
Define index set as
where is the number of information symbols in . We then partition into groups: with
where is the cardinality of the subset . We call a grouping scheme. For such a grouping scheme, we have
Define
(9)  
(10) 
With these notations, (4) can be written as
(11) 
Suppose we want to decode the symbols embedded in the group . The PIC group decoding first implements linear interference cancellation with a suitable choice of matrix in order to completely eliminate the interferences from other groups [39], i.e., , and . Then, we have
(12)  
where the interference cancellation matrix can be chosen as follows [39],
(13) 
in case
(14) 
has full column rank. If does not have full column rank, then we need to pick a maximal linear independent vector group from and in this case a projection matrix can be found too [39].
Afterwards, the symbols in the group are decoded with the ML decoding algorithm as follows,
(15) 
The above PIC group decoding is connected to some of the known decodings as in the following remarks.
Remark 3 (ML and PIC Group Decoding)
For one special case of , the grouping scheme is with . From (13), we have . Then, the PIC group decoding is equivalent to the ML decoding where all information symbols are jointly decoded.
Remark 4 (ZF and PIC Group Decoding)
For the special case of , the grouping scheme is i.e., every single symbol is regarded as one group. Then, the PIC group decoding is equivalent to the ZF decoding where every single symbol is separated from all the other symbols and then decoded.
Remark 5 (ZF, ML and PIC Group Decoding)
The PIC group decoding with can be viewed as an intermediate decoding approach between the ML decoding and the ZF decoding. Alternatively, the ML decoding and the ZF decoding can both be regarded as the special cases of the PIC group decoding corresponding to and , respectively.
Remark 6 (PIC Group Decoding Complexity)
For the PIC group decoding, the following two steps are needed: the group zeroforcing to cancel the interferences coming from all the other groups as shown in (12) and the group ML decoding to jointly decode the symbols in one group as shown in (15). Therefore, the decoding complexity of the PIC group decoding should reside in the above two steps. Note that the interference cancellation process shown in (12) mainly involves with linear matrix computations, whose computational complexity is small compared to the ML decoding for an exhaustive search of all candidate symbols. Therefore, to evaluate the decoding complexity of the PIC group decoding, we mainly focus on the computational complexity of the ML decoding within the PIC group decoding algorithm. According to Definition 2, the ML decoding complexity in the PIC group decoding algorithm is . It can be seen that the PIC group decoding provides a flexible decoding complexity which can be from the ZF decoding complexity to the ML decoding complexity .
Remark 7 (PICSIC Group Decoding)
In [39], an SICaided PIC group decoding algorithm, namely PICSIC group decoding was proposed. Similar to the BLAST detection algorithm [42], the PICSIC group decoding is performed after removing the alreadydecoded symbol set from the received signals to reduce the interference. If each group has only one symbol, then the PICSIC group will be equivalent to the BLAST detection.
IiC STBC Design Criterion with PIC Group Decoding
The performance of a decoding algorithm for a wireless communication system is related to the diversity order. If the average probability of a detection error for communication over a fading channel usually behaves as:
where is a constant and is called the diversity order of the system. For an MIMO communication system, the maximum diversity order is , i.e., the product of the number of transmit antennas and the number of receiver antennas. In order to optimize the reception performance of the MIMO system, a full diversity is usually pursued which can be achieved by a proper signal transmission scheme or data format (e.g., STBC). In [3], the “rankanddeterminant criterion” of STBC design was proposed to maximize both the diversity gain and the coding gain of the MIMO system with an ML decoding. Recently, in [39] an STBC design criterion was derived to achieve full diversity when the PIC group decoding is used at the receiver. In the following, we cite the main result of the STBC design criterion proposed in [39].
Proposition 1
[39, Theorem 1] [FullDiversity Criterion under PIC Group Decoding]
For an STBC with the PIC group decoding, the full diversity is achieved when

the code satisfies the full rank criterion, i.e., it achieves full diversity when the ML receiver is used; and

are linearly independent vector groups for any .
In [39], the STBC achieving full diversity with PIC group decoding were proposed for and transmit antennas. However, a systematic code design of the fulldiversity STBC with PIC group decoding remains an open problem.
Proposition 2
[39, p.4374] [FullDiversity Criterion under PICSIC Group Decoding]
For an STBC with the PICSIC group decoding, the full diversity is achieved when

the code satisfies the full rank criterion, i.e., it achieves full diversity when the ML receiver is used; and

at each decoding stage, , which corresponds to the current tobe decoded symbol group , the remaining groups corresponding to yet uncoded symbol groups are linearly independent vector groups for any .
Iii A Design of STBC with PIC Group Decoding
In this section, we first propose a systematic design of highrate STBC which has a rate up to symbols per channel use and achieves full diversity with the ML decoding. The systematic design of the STBC is structured with multiple diagonal layers. Then, we prove that the proposed STBC with two diagonal layers can obtain full diversity with the PIC group decoding and the code rate can be up to symbols per channel use. Finally, we prove that the proposed STBC with any number of diagonal layers can obtain full diversity with PICSIC group decoding and the code rate can be up to symbols per channel use.
Iiia Encoding Technique
Our proposed spacetime code , i.e., in (1), is of size (for any given , and ) and will be transmitted from antennas over time slots. Let . The symbol stream (composed of complex symbols chosen from QAM constellation and then scaled by ) is first parsed into symbol vectors (). Each symbol vector is linearly precoded by an matrix , which is a chosen constellation rotation matrix. Next, the vector is used to form the spacetime code matrix , in which the th descending diagonal from left to right is the diagonal form of .
The resulting transmitted code matrix is given by
(16) 
where the th descending diagonal from left to right, denoted by is given by
(17) 
and the information symbol vector is given by
(18) 
for .
Proposition 3
The proposed STBC in (16) has asymptotically full rate when the block length is sufficiently large.
In the codeword in (16), a total number of independent information symbols are encoded into the codeword , which is then transmitted from antennas over time slots. The code rate of transmission is therefore
(19) 
For a very large block length , it can be seen that the rate of the proposed ST coding scheme approaches symbols per channel use, i.e. the full rate.
IiiB Choice of Rotation Matrix
In [41], the rotation matrix was designed for diagonal STBC to achieve the full diversity gain and the optimal diversity product. With the optimal cyclotomic lattices design for transmit antennas, from [41, Table I] we can get a set of integers and let . Then, the optimal lattice is given by [41, Eq. (16)]
(20) 
where with and are distinct integers such that and are coprime for any .
Example 1: For transmit antennas we can choose and according to [41, Table I]. Then, in order to ensure that and are coprime for any we can obtain , , . When , the signal constellation is located on the equal literal triangular lattice. When , can be and can be , and in this case the signal constellation is located on the square lattice.
Example 2: For transmit antennas we can select and . Then, , , , .
The cyclotomic design of the matrix is vital for the design of the algebraic STBC. In the following, we show some properties of the matrix that will be used later for our design.
Property 1
Property 2
Every entry of the matrix in (20) is nonzero.
This property is obvious from (20).
IiiC Achieving Full Diversity with ML Decoding
We show the main result of the proposed STBC when an ML decoding is used at the receiver, as follows.
Theorem 1 (Full Diversity with ML Decoding)
Consider a MIMO transmission with transmit antennas and receive antennas over block fading channels. The STBC as described in (16) achieves full diversity under the ML decoding.
[Proof of Theorem 1] In order to prove that the ST code in (16) can obtain full diversity under ML decoding, it is sufficient to prove that achieves full rank for any distinct pair of ST codewords and .
For any pair of distinct codewords and , there exists at least one index () such that , where and are related to and from (17), respectively. Let denote the minimum index of vectors satisfying . Then, for any index with , it must have . Define as the difference between symbols and . Then, from (16) can be expressed as
(21) 
where for . This is because for , it exists . Due to the suitably chosen constellation rotation matrix in (20), must have nonzero entries for any . Then, the matrix has full rank.
The full rankness of can be examined similar to that for the Toeplitz code (or delay diversity code) [34] by checking if the columns of are linearly independent. Specifically, we establish with . First, we examine the th equation in and get . Because for , . Then, we examine the th equation and get . Immediately, . Likewise, we examine the th equation until th equation, and we can get . Therefore, all columns of are linearly independent and has full rank.
This property will be used in next section in the proof of the full diversity property under the PIC group decoding.
IiiD Achieving Full Diversity with PIC Group Decoding when
We show the main result of the proposed STBC when a PIC group decoding with a particular grouping scheme is used at the receiver, as follows.
Theorem 2 (Full Diversity with PIC Group Decoding when )
Consider a MIMO transmission with transmit antennas and receive antennas over block fading channels. The STBC as described in (16) with two diagonal layers (i.e., ) is used at the transmitter. The equivalent channel matrix is . If the received signal is decoded using the PIC group decoding with the grouping scheme where for , i.e., the size of each group is equal to the number of transmit antennas , then the code achieves the full diversity. The code rate of the fulldiversity STBC can be up to symbols per channel use.
In order to prove Theorem 2, let us first introduce the following lemmas.
Lemma 1
Consider the system as described in Theorem 2 with and the STBC as given by (16),

the equivalent channel matrix can be expressed as
(22) where
(23) 
When , and are linearly independent vector groups as long as , where .
A proof of Lemma 1 is given in Appendix I.
Lemma 2
Consider the system as described in Theorem 2 and the STBC as given by (16). For the equivalent channel matrix , are linearly independent vector groups for when if and only if are linearly independent vector groups for when .
The proof of Lemma 2 is straightforward and is also the same as what is mentioned in [39].
[Proof of Theorem 2] As shown in Proposition 1, a codeword with PIC group decoding can obtain the full diversity if
 1)

achieves the full diversity with the ML receiver, and
 2)

are linearly independent vector groups as long as .
For the proposed code in (16) with , the first condition is satisfied as shown in Theorem 1. The second condition is satisfied as shown in Lemma 1 for and Lemma 2 for , respectively. Therefore, the code in (16) with can obtain full diversity with the PIC group decoding provided that the grouping scheme is where for .
The code rate of the fulldiversity STBC with the PIC group decoding can be derived by substituting and into (19) as
For a large number of transmit antennas, the rate approaches to (but not larger than) symbols per channel use.
Corollary 1
The decoding complexity of the PIC group decoding of the proposed STBC with the grouping scheme as described in Theorem 2 is .
Remark 8
The decoding complexity of the proposed STBC with the PIC group decoding is equivalent to the ML decoding of independent information symbols jointly. As shown in (19), the code rate of the proposed STBC in (16) for a given can be increased by embedding larger number of groups in the codeword, i.e., increasing the value of . It is noteworthy to mention that the increase of the code rate does not result in the increase of the decoding complexity.
IiiE Achieving Full Diversity with PICSIC Group Decoding
For the proposed STBC with any number of layers and the PICSIC group decoding we have the following results.
Theorem 3 (Full Diversity with PICSIC Group Decoding)
Consider a MIMO transmission with transmit antennas and receive antennas over block fading channels. The STBC as described in (16) with diagonal layers is used at the transmitter. The equivalent channel matrix is . If the received signal is decoded using the PICSIC group decoding with the sequential order and with the grouping scheme being where for , i.e., the size of each group is equal to the number of transmit antennas , then the code achieves the full diversity. The code rate of the fulldiversity STBC can be up to symbols per channel use.
The proof of this theorem is simple. Observing that and the group is linearly independent from the groups for any and where is given by (23), according to Proposition 2 the full diversity can be easily proved. The detailed proof is omitted.
Iv Code Design Examples
In this section, we show a few code design examples. We denote the code constructed by (16) for given parameters: the number of transmit antennas, the block length of the code, and the number of groups to be decoded in the PIC group decoding. For notational brevity, we only show the equivalent channel of the proposed codes for MISO systems.
Iva For Two Transmit Antennas
Consider a code for 2 transmit antennas with 3 time slots. According to the code structure (16), we have
(24) 
where and . The constellation rotation matrix can be chosen as
where and with [39].
The code rate of the code is . In fact, this code is equivalent to the one proposed in [39, Section VI  Example 1].
The equivalent channel of the code is given by
(25) 
The grouping scheme for the PIC group decoding is and . It can be seen that and are linearly independent. Then, the code can obtain full diversity with the PIC group decoding.
IvB For Four Transmit Antennas
For given , the code achieving full diversity with the PIC group decoding can be designed as follows,
(26) 
This code has a code rate of and two groups to be decoded. The equivalent channel of the code is
(27) 
where is given by
(28) 
with being the ()th entry of the matrix for . The grouping scheme for the PIC group decoding is and . It can be seen that the groups and are linearly independent to each other. Then, the code can obtain full diversity with the PIC group decoding.
Consider time slots. We can get
(29) 
The code rate of the code is which has the same rate as the one proposed in [39, Section VI  Example 2]. The equivalent channel of the code is
(30) 
where is given by (28). Because the groups and are linearly independent to each other. Then, the code can obtain full diversity with the PIC group decoding.
Moreover, we can also design the code for with layers (i.e., ) as follows,
(31) 
The code rate of the code is and the equivalent channel is given by
(32) 
where
It can be proved that the groups , , and are not linearly independent groups. Therefore, according to Proposition 1, the code cannot achieve the full diversity with the PIC group decoding.
However, the code can obtain full diversity with PICSIC group decoding. This is because is linearly independent from and , and is linearly independent from . According to Proposition 2, with PICSIC group decoding and a proper decoding order or , the code can achieve the full diversity.
IvC For Five Transmit Antennas
For given and , the code is designed as follows,
(33) 
The code rate of the code is . The equivalent channel is
(34) 
where is the rotation matrix of size .
The grouping scheme for the PIC group decoding of is and . It can be seen that the groups and are linearly independent to each other. Then, the code can obtain full diversity with the PIC group decoding.
V Another Design of STBC with PIC Group Decoding
Notice that the code design in (16) can only achieve the full diversity with PIC group decoding for two diagonal layers (r.f. Theorem 2) and the code rate is not larger than symbols per channel use. With () diagonal layers in the code (16), the rate can be increased but the independence among channel groups in (22) is not satisfied, thereby may lose the full diversity gain. In this section, we propose a new code design which can achieve full diversity with PIC group decoding and a rate above .
Va Code Design
For ( is an integer), our proposed STBC for transmit antennas is given by
(35) 
where the symbol vector is given by
is an constellation rotation matrix given by (20) and is the information symbol vector.
For , our proposed STBC is given by
For , our proposed STBC is given by