Generalized Silver Codes
For an transmit, receive antenna system ( system), a full-rate space time block code (STBC) transmits complex symbols per channel use. The well known Golden code is an example of a full-rate, full-diversity STBC for 2 transmit antennas. Its ML-decoding complexity is of the order of for square -QAM. The Silver code for 2 transmit antennas has all the desirable properties of the Golden code except its coding gain, but offers lower ML-decoding complexity of the order of . Importantly, the slight loss in coding gain is negligible compared to the advantage it offers in terms of lowering the ML-decoding complexity. For higher number of transmit antennas, the best known codes are the Perfect codes, which are full-rate, full-diversity, information lossless codes (for ) but have a high ML-decoding complexity of the order of (for , the punctured Perfect codes are considered). In this paper111Part of the content of this manuscript has been presented at IEEE ISIT 2010 and another part at IEEE Globecom, 2010., a scheme to obtain full-rate STBCs for transmit antennas and any with reduced ML-decoding complexity of the order of , is presented. The codes constructed are also information lossless for , like the Perfect codes and allow higher mutual information than the comparable punctured Perfect codes for . These codes are referred to as the generalized Silver codes, since they enjoy the same desirable properties as the comparable Perfect codes (except possibly the coding gain) with lower ML-decoding complexity, analogous to the Silver-Golden codes for 2 transmit antennas. Simulation results of the symbol error rates for 4 and 8 transmit antennas show that the generalized Silver codes match the punctured Perfect codes in error performance while offering lower ML-decoding complexity.
Anticommuting matrices, ergodic capacity, full-rate space-time block codes, low ML-decoding complexity, information losslessness.
I Introduction and Background
Complex orthogonal designs (CODs) , , although provide linear Maximum Likelihood (ML)-decoding, do not offer a high rate of transmission. A full-rate code for an MIMO system transmits independent complex symbols per channel use. Among the CODs, only the Alamouti code for 2 transmit antennas is full-rate for a MIMO system. A full-rate STBC can efficiently utilize all the degrees of freedom the channel provides. In general, an increase in the rate tends to result in an increase in the ML-decoding complexity. The Golden code  for 2 transmit antennas is an example of a full-rate STBC for any number of receive antennas. Until recently, the ML-decoding complexity of the Golden code was reported to be of the order of , where is the size of the signal constellation. However, it was shown in ,  that the Golden code has a decoding complexity of the order of for square -QAM. Current research focuses on obtaining high rate codes with reduced ML-decoding complexity (refer to Sec. II for a formal definition). For 2 transmit antennas, the Silver code, named so in , was first mentioned in  and independently presented in  along with a study of its low ML-decoding complexity property. It is a full-rate code with full-diversity and an ML-decoding complexity of the order of for square -QAM. Its algebraic properties have been studied in  and  and a fixed point fast decoding scheme has been given in . For 4 transmit antennas, Biglieri et. al. proposed a rate-2 STBC which has an ML-decoding complexity of the order of for square -QAM without full-diversity . It was, however, shown that there was no significant reduction in error performance at low to medium SNR when compared with the previously best known code - the DjABBA code . This code was obtained by multiplexing Quasi-orthogonal designs (QOD) for 4 transmit antennas . In , a new full-rate STBC for system with an ML-decoding complexity of was proposed and was conjectured to have the non-vanishing determinant (NVD) property. This code was obtained by multiplexing the coordinate interleaved orthogonal designs (CIODs) for 4 transmit antennas . These results show that codes obtained by multiplexing low complexity STBCs can result in high rate STBCs with reduced ML-decoding complexity and by choosing a suitable constellation, there won’t be any significant degradation in the error performance when compared with the best existing STBCs. Such an approach has also been adopted in  to obtain high rate codes111Fast decodable STBCs have been constructed in -, but these codes are not full-rate in general, and make use of near ML-decoding algorithms. from multiplexed orthogonal designs. More recently, full-rate STBCs with an ML-decoding complexity of the order of and a provable NVD property for the system have been proposed in  and .
In general, it is not known how one can design full-rate STBCs for an arbitrary number of transmit and receive antennas with reduced ML-decoding complexity. It is well known that the maximum mutual information achievable with an STBC is at best equal to the ergodic capacity of the MIMO channel, in which case the STBC is said to be information lossless (see Section II for a formal definition). It is known how to design information lossless codes  for the case where . However, when the only known code in literature which is information lossless is the Alamouti code, which is information lossless for the system alone. It has been shown in ,  and  that when , self-interference of the STBC (a formal definition of self interference is given in Section II) has to be minimized for maximizing the mutual information achieved with the STBC. Not much research222The full-rate STBCs in , designed for , are not linear dispersion codes. They are based on maximal orders and use spherical shaping due to which the encoding and decoding complexity is extremely high. The STBCs in , also designed for , use the concept of restricting the number of active transmit antennas to be no larger than the number of receive antennas, and so, the mutual information analysis for these codes is very difficult. These STBCs are diversity-multiplexing gain tradeoff (DMT) optimal but are associated with a very high ML-decoding complexity. has been done on designing codes that allow a high mutual information when . In this paper, for , we systematically design full-rate STBCs which have the least possible self-interference and the lowest ML-decoding complexity among known full-rate STBCs for and consequently allow higher mutual information than the best existing codes (the Perfect codes with puncturing , ), while for , the proposed STBCs are information lossless like the comparable Perfect codes. We call these codes the generalized Silver codes since, analogous to the silver code and the Golden code for 2 transmit antennas, the proposed codes have every desirable property that the Perfect codes have, except the coding gain, but importantly, have lower ML-decoding complexity. The contributions of the paper are:
We give a scheme to obtain rate-1, 4-group decodable codes (refer Section II for a formal definition of multi-group decodable codes) for through algebraic methods. The speciality of the obtained design is that it is amenable for extension to higher number of receive antennas, resulting in full-rate codes with reduced ML-decoding complexity for any number of receive antennas, unlike the previous constructions - of rate-1, 4-group decodable codes.
Using the rate-1, 4-group decodable codes thus constructed, we propose a scheme to obtain the generalized Silver codes, which are full-rate codes with reduced ML-decoding complexity for transmit antennas and any number of receive antennas. These codes also have the least self-interference among known comparable STBCs and allow higher mutual information with lower ML-decoding complexity than the comparable punctured Perfect codes for the case , while being information lossless for . In terms of error performance, by choosing the signal constellation carefully, the proposed codes have more or less the same performance as the corresponding punctured Perfect codes. This is shown through simulation results for 4 and 8 transmit antenna systems.
The paper is organized as follows. In Section II, we present the system model and the relevant definitions. The criteria for maximizing the mutual information with space time modulation are presented in Section III and our method to construct rate-1, 4-group decodable codes is proposed in Section IV. The scheme to extend these codes to obtain the generalized Silver codes for higher number of receive antennas is presented in Section V. Simulation results are discussed in Section VI and the concluding remarks are made in Section VII.
Notations: Throughout, bold, lowercase letters are used to denote vectors and bold, uppercase letters are used to denote matrices. Let X be a complex matrix. Then, and denote the Hermitian and the transpose of X, respectively and unless used to denote indices or subscripts, represents . The entry of X is denoted by while and denote the trace and determinant of X, respectively. The set of all real and complex numbers are denoted by and , respectively. The real and the imaginary part of a complex number are denoted by and , respectively. denotes the Frobenius norm of X, denotes the vector norm of a vector x, and and denote the identity matrix and the null matrix, respectively. The Kronecker product is denoted by and denotes the concatenation of the columns of X one below the other. For a complex random variable , denotes the mean of and denotes the mean of , a function of the random variable . The inner product of two vectors x and y is denoted by . For a set , . Let and be two sets such that . Then denotes the set of elements of excluding the elements of . For a complex variable , the operator acting on is defined as
The can similarly be applied to any matrix by replacing each entry with , , resulting in a matrix denoted by . Given a complex vector , is defined as . It follows that for , and , the equalities and hold.
Ii System Model
We consider the Rayleigh block fading MIMO channel with full channel state information (CSI) at the receiver but not at the transmitter. For MIMO transmission, we have
where is the codeword matrix whose average energy is given by , is a complex white Gaussian noise matrix with i.i.d. entries (complex normal distribution with zero mean and unit variance), is the channel matrix with the entries assumed to be i.i.d. circularly symmetric Gaussian random variables , is the received matrix and is the signal-to-noise ratio at each receive antenna.
(Code rate) Code rate is the average number of independent information symbols transmitted per channel use. If there are independent complex information symbols (or real information symbols) in the codeword which are transmitted over channel uses, then, the code rate is complex symbols per channel use ( real symbols per channel use).
(Full-rate STBCs) For an MIMO system, if the code rate is complex symbols per channel use, then the STBC is said to be full-rate.
Assuming ML-decoding, the metric that is to be minimized over all possible values of codewords S is given by
(ML-Decoding complexity) The ML decoding complexity is measured in terms of the maximum number of symbols that need to be jointly decoded in minimizing the ML decoding metric.
For example, if the codeword transmits independent symbols of which a maximum of symbols need to be jointly decoded, the ML-decoding complexity is of the order of , where is the size of the signal constellation. If the code has an ML-decoding complexity of order less than , the code is said to have reduced ML-decoding complexity.
(Generator matrix) For any STBC that encodes real symbols (or complex information symbols), the generator matrix is defined by 
where S is the codeword matrix, is the real information symbol vector.
A codeword matrix of an STBC can be expressed in terms of weight matrices (linear dispersion matrices)  as
Here, , are the complex weight matrices of the STBC and should form a linearly independent set over . It follows that
Due to the constraint that , we have, . Choosing for all , we have
(Multi-group decodable STBCs) An STBC is said to be -group decodable  if its weight matrices can be separated into groups , , , such that
(Self-interference) For an STBC given by , the self-interference matrix  is defined as
(Punctured Codes) Punctured STBCs are the codes with some of the symbols being zeros, in order to meet the full-rate criterion.
For example, a codeword of the Perfect code for 4 transmit antennas  transmits sixteen complex symbols in four channel uses and has a rate of 4 complex symbols per channel use. If this code were to be used for a two receive antenna system which can only support a rate of two independent complex symbols per channel use, then, eight symbols of the Perfect code can be made zeros, so that the codeword transmits eight complex symbols in four channel uses. These eight symbols correspond to the two layers  of the Perfect code.
Equation (1) can be rewritten as
where , called the equivalent channel matrix. is given by , with being the generator matrix as in Definition 4.
(Ergodic capacity) The ergodic capacity of an MIMO channel is 
With the use of an STBC, the maximum mutual information achievable is 
It is known that . If , the STBC is said to be information lossless. If the generator matrix G is orthogonal (from Definition 4, this case arises only if and the STBC is full-rate, i.e, ), the STBC is information lossless.
Iii Relationship between weight matrices and the maximum mutual information
Capacity can be achieved with the use of continuous inputs with Gaussian distribution. If one were able to use continuous Gaussian distributed inputs in practice, using the V-blast scheme would suffice, since diversity is irrelevant. But in practice, one has to use finite discrete inputs, and diversity becomes an important aspect, necessitating the use of full-diversity STBCs. Even though we considered the limited block length scenario for space-time coding as a standalone scheme, in practice, one would also have an outer code and coding would be done over large block lengths to go close to capacity. In such a scenario, the maximum mutual information that an STBC allows becomes an important parameter for the design of STBCs. It is preferable to use STBCs which allow mutual information as close to the channel capacity as possible. It has been shown that if the generator matrix is orthogonal, the maximum mutual information achievable with the STBC is the same as the ergodic capacity of the MIMO channel , . For the generator matrix to be orthogonal, a prerequisite is that the number of receive antennas should be at least equal to the number of transmit antennas. When , only the Alamouti code has been known to be information lossless for the MIMO channel. In , by using the well known matrix identities and , an expansion of the ergodic MIMO capacity in SNR was obtained as
with . The first two coefficients can easily be checked to be and . On a similar note, can also be expanded in SNR as , where
Let . It is straightforward to check that , where . Hence,
In , it was argued that typical discrete input schemes fail to achieve capacity at the third order in the expansion of the mutual information and hence, should be maximized. From (5), it is clear that to maximize , the following criteria should be satisfied.
Hurwitz-Radon Orthogonality: as many of should be equal to as possible, for .
Tracelessness: should be traceless, for all .
In fact, the first criterion, which is equivalent to minimizing the self-interference, is already clear from (4), where it can be observed that a larger number of zero entries of contributes to a lower value of the trace of . Hence, to design a good STBC with a high mutual information when , one should have as many as possible weight matrix pairs satisfying Hurwitz-Radon (HR) orthogonality. We would, of course, like all the weight matrices to satisfy HR-orthogonality, but there is a limit to this number  which, except for the Alamouti code, is much lesser than , the number of weight matrices of a full-rate STBC when . It can easily be checked that for the Alamouti code, . It is known that for a rate-1 code for , one cannot have all the full-ranked weight matrices mutually satisfying HR-orthogonality. For such STBCs, the minimum self-interference is achieved if the STBCs are -group decodable, with as large as possible. At present, the best known rate-1 low complexity multi-group decodable codes are the 4-group decodable codes for any number of transmit antennas , , . These codes are not full-rate for . If one were to require a full-rate code, the codes in literature , ,  are not suitable for extension to higher number of receive antennas, since their design is obtained by iterative methods. In the next section, we propose a new design methodology to obtain the weight matrices of a rate-1, 4-group decodable code by algebraic methods for transmit antennas. These codes can be extended to higher number of receive antennas to obtain full-rate STBCs with lower ML-decoding complexity and lower self-interference than the existing designs.
Iv Construction of Rate-1, 4-group decodable codes
We make use of the following theorem, presented in , to construct rate-1, 4-group decodable codes for transmit antennas.
 An linear dispersion code transmitting k real symbols is -group decodable if the weight matrices satisfy the following conditions:
Table I illustrates the weight matrices of a -group decodable code which satisfy the above conditions. The weight matrices in each column belong to the same group.
In order to obtain a rate-1, 4-group decodable STBC for transmit antennas, it is sufficient if we have matrices satisfying the conditions in Theorem 1. To obtain these333These STBCs can be obtained elegantly using the theory of Clifford Algebra but to make the paper accessible to a wider group of readers, we have preferred to make use of simple concepts from matrix theory without reference to Clifford Algebra., we make use of the following lemmas.
 If and invertible complex matrices of size , denoted by , anticommute pairwise, then the set of products with along with forms a basis for the dimensional space of all matrices over .
The proof is provided for the sake of completeness. Assume that in the set of products , , along with , at most elements are linearly independent over , for some . So,
Noting that anticommutes with but commutes with each of , , , premultiplying each term of (6) by results in a new equation with the coefficients negated for those terms in (6) containing . Adding this new equation to (6) yields another equation containing fewer summands than (6), leading to a contradiction. So, , which proves the theorem.
If all the mutually anticommuting matrices , are unitary and anti-Hermitian, so that they square to , then the product with squares to .
which proves the lemma.
Let be anticommuting, anti-Hermitian, unitary matrices. Let and with and . Let . Then the product matrix commutes with if exactly one of the following is satisfied, and anticommutes otherwise.
and are all odd.
The product is even and is even (including 0).
When , we note that
and when , we have . Now,
Case 1) Since and are all odd, = 1.
Case 2) The product is even and is even (including 0). Hence = 1.
From Theorem 1, to get a rate-1, 4-group decodable STBC, we need 3 pairwise anticommuting, anti-Hermitian matrices which commute with a group of Hermitian, pairwise commuting matrices. Once these are identified, the other weight matrices can be easily obtained. From , one can obtain pairwise anticommuting, anti-Hermitian matrices and the method to obtain these is presented here for completeness. Let
and . The anti-Hermitian, pairwise anti-commuting matrices are
Henceforth, , refer to the matrices obtained using the above method. For a set , define as
We choose , and to be the three pairwise anticommuting, anti-Hermitian matrices (to be placed in the top row along with in Table I. Consider the set , the cardinality of which is . Using Lemma 2 and Lemma 3, one can note that consists of pairwise commuting matrices which are Hermitian. Moreover, it is clear that each of the matrices in the set also commutes with , and . Hence, , which has cardinality is also a set with pairwise commuting, Hermitian matrices which also commute with , and . The linear independence of over is easy to see by applying Lemma 1. Hence, we have 3 pairwise anticommuting, anti-Hermitian matrices which commute with a group of Hermitian, pairwise commuting matrices. Having obtained these, the other weight matrices are obtained from Theorem 1. To illustrate with an example, we consider and show below how the weight matrices are obtained for the rate-1, 4-group decodable code.
Iv-a An example -
Let denote the 6 pairwise anticommuting, anti-Hermitian matrices. Choose , and to be the three anticommuting matrices required for code construction. Let
The 16 weight matrices of the rate-1, 4-group decodable code for 8 antennas are as shown in Table II. Each column corresponds to the weight matrices belonging to the same group. Note that the product of any two matrices in the first group is some other matrix in the same group.
Iv-B Coding gain calculations
Let , where denotes the codeword difference matrix. Let , where and are the real symbols encoding codeword matrices S and , respectively. Hence,
Note that because of the nature of construction of the weight matrices, we have
Further, since the code is 4-group decodable,
All the weight matrices in the first group are Hermitian and pairwise commuting and the product of any two such matrices is some other matrix in the same group. It is well known that commuting matrices are simultaneously diagonalizable. Hence,
where is a diagonal matrix. Since is Hermitian as well as unitary, the diagonal elements of are . The following lemma proves that is traceless.
Let be unitary, pairwise anticommuting matrices. Then, the product matrix , with the exception of , is traceless.
It is well known that for any two matrices A and B. Let A and B be two invertible, anticommuting matrices. Then, . So,
Similarly, it can be shown that . By applying Lemma 3, it can be seen that any product matrix , excluding , anticommutes with some other invertible product matrix from the set . Hence, from (7), we can say that every product matrix except is traceless.
From the above lemma, except identity is traceless. Hence, has an equal number of ’1’s and ’-1’s. In fact, because of the nature of construction of the matrices , the product matrices , for even , and the product matrix are always diagonal (easily seen from the definition of , ). Hence, all the weight matrices of the first group excluding are diagonal, with the diagonal elements being . Since these diagonal matrices also commute with and , the diagonal entries are such that for every odd , if the entry is 1(-1), then, the entry is also 1(-1, respectively). To summarize, the properties of , are listed below.
In view of these properties,