IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

5485

A Low ML-Decoding Complexity, Full-Diversity, Full-Rate MIMO Precoder K. Pavan Srinath and B. Sundar Rajan, Senior Member, IEEE

Abstract—Precoding for multiple-input multiple-output (MIMO) antenna systems is considered with perfect channel knowledge available at both the transmitter and the receiver. For two transmit antennas and QAM constellations, a real-valued precoder which is approximately optimal (with respect to the minimum Euclidean distance between points in the received signal space) among real-valued precoders based on the singular value decomposition (SVD) of the channel is proposed. The proposed precoder is obtainable easily for arbitrary QAM constellations, unlike the known complex-valued optimal precoder by Collin et al. for two transmit antennas which is in existence for 4-QAM alone and is extremely hard to obtain for larger QAM constellations. The proposed precoding scheme is extended to higher number of transmit antennas on the lines of the precoder for 4-QAM by Vrigneau et al. which is an extension of the complex-valued optimal precoder for 4-QAM. The proposed precoder’s ML-decoding complexity as a function of the constellation is only while that of the precoder is size . Compared to the recently proposed - and -precoders, the error performance of the proposed precoder is significantly better while being only marginally worse than that of the precoder for 4-QAM. It is argued that the proposed precoder provides full-diversity for QAM constellations and this is supported by simulation plots of the word error probability for 2 2, 4 4 and 8 8 systems. Index Terms—Diversity gain, low ML-decoding complexity, multiple-input multiple-output (MIMO) precoders, singular values, word error probability.

I. INTRODUCTION AND BACKGROUND

M

ULTIPLE-INPUT multiple-output (MIMO) antenna systems have evoked a lot of research interest primarily because of the enhanced capacity they provide compared with that provided by the single antenna point-to-point channel. transmit antennas and reMoreover, for a system with system), the maximum diversity gain (a ceive antennas ( definition of diversity gain is given in Section II of this paper) . achievable with coherent detection has been shown to be For MIMO systems with the channel state information available Manuscript received January 14, 2011; revised May 11, 2011 and July 04, 2011; accepted July 09, 2011. Date of publication July 25, 2011; date of current version October 12, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiqi Gao. This work was supported in part by the DRDO-IISc Program on Advanced Research in Mathematical Engineering. The work of B. S. Rajan was supported by the INAE Chair Professorship. The authors are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012 India(e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2162957

only at the receiver (CSIR), suitably designed space-time block codes (STBCs) [1] provide full-diversity. Full-rate transmission independent is said to occur if at least information symbols are transmitted in every channel use [2]. Full-rate STBCs achieving full-diversity have also been proposed [3], [4]. However, all full-rate, full-diversity STBCs are characterized by a high ML-decoding complexity (refer to Section II for a formal definition of ML-decoding complexity). In general, decoding full-rate STBCs requires jointly decoding symbols. MIMO systems with full channel state information at the transmitter (CSIT) or partial CSIT have been extensively studied in literature. From an information-theoretic perspective, capacity is an important parameter for MIMO systems and waterfilling [5] can be employed to achieve capacity with a Gaussian codebook. From a signal processing point of view, the error performance of MIMO systems using finite constellations is one of the important parameters for system design and several precoding1 schemes have been proposed in this regard. Maximal ratio transmission was introduced in [6] to achieve full-diversity while maximizing the signal-to-noise ratio (SNR) by precoding at the transmitter and equalizing at the receiver for transmission of a single symbol per channel use. Subsequently, the use of precoding and equalizing matrices at the transmitter and the receiver, respectively, was employed in [7] to maximize the SNR at the receiver, but this scheme resulted in low-rate transmission. Several works on optimal linear precoders and decoders have been done for the minimum mean square error (MMSE) criterion [8]–[11]. Since these precoders are linear and optimal for MMSE decoding, the decoding complexity is very low and full-diversity is also achieved, but the error performance is worse than that for ML-decoding. Other non-ML-decoding techniques include lattice-reduction based techniques [12] which provide full-rate transmission with possibly full-diversity, but lattice-reduction itself involves a high complexity for large MIMO systems. Extensive research has also been done on MIMO systems with limited feedback to the transmitter about the channel from the receiver (see, for example, [13] and references therein). In this paper, we consider MIMO systems with full CSIT. The channel state information could be either sent to the transmitter by the receiver (when there are separate frequency bands for uplink and downlink transmission) or the transmitter could estimate the channel if it is reciprocal, like in a time division duplexing (TDD) system, by receiving pilot signals from the receiver. In literature, to the best of our knowledge, there is no known precoding technique 1Precoding

is also referred to as “transmit beamforming.”

1053-587X/$26.00 © 2011 IEEE

5486

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

that achieves all the three attributes—full-rate, full-diversity, and low ML-decoding complexity (“low ML-decoding complexity” is a relative term and in this paper, it is used to mean joint decoding of at most 2 complex symbols). Almost all the popular precoding techniques employing ML-decoding use the singular value decomposition (SVD) of precoder for 4-QAM the MIMO channel [14]. The [15], an extension of the complex-valued optimal2 precoder [16] to higher number of transmit antennas, has been shown to perform very well for 4-QAM, beating all other linear precoding and decoding schemes based on the MMSE criterion, and ML-decoding involves jointly decoding two complex symbols only. However, this precoder exists in literature for 4-QAM alone and is very hard to obtain for larger QAM constellations since it involves a numerical search over 3 parameters. Recently, - and -precoders have been proposed in [17] as rivals for the precoder. The -precoder has been shown to offer an ML-decoding complexity of (this can be brought down to by the same decoding scheme as for our precoder, which is explained in Section V-B) while the -precoder has an ML-decoding complexity which is invariant with respect to the constellation size . The disadvantage with the -precoder is that it loses out to the precoder in error performance for 4-QAM and it is not known if an explicit expression for the precoding matrix can be obtained for larger QAM constellations. The -precoder (which uses a two-dimensional constellation), although explicitly obtainable for constellations of any size , loses out in error performance to the precoder since it has not been optimized for error performance. In literature, all the aforementioned low ML-decoding complexity precoders have been claimed to offer by the authors a diversity gain of precoder (but we argue out in Section VI that the has full-diversity for 4-QAM). Concerned by the limitations of each of the low ML-decoding complexity precoders, we first propose a real-valued precoder that is approximately optimal (we explain in Section III why the precoder is for “approximately optimal”) among real-valued precoders based on the SVD of the channel, and then extend it to higher number of transmit antennas using an approach similar to that in [15]. The ML-decoding complexity offered by our precoder as a is shown to be for -QAM. For function of 4-QAM, the proposed precoder has only a marginally poorer precoder, but has lower error performance than the ML-decoding complexity. For larger QAM constellations, precoder. When it is easily obtainable, unlike the compared with the - and -precoders, it has a much better error performance (for a summary of the comparison of various low ML-decoding complexity schemes, see Table III). The main contributions of the paper are as follows: 1) We propose a novel scheme to obtain an SVD-based, real-valued precoder that is approximately optimal among real-valued precoders for 2 transmit antennas and any -QAM. The method of obtaining this precoder is different from the one taken to obtain the complex-valued 2Throughout this paper, unless otherwise stated, optimality is with respect to the minimum Euclidean distance between points in the received signal space.

optimal precoder for 2 transmit antennas [16], and is easily applicable for any -QAM, unlike that in [16]. 2) We extend this real-valued precoder to higher number of transmit antennas and argue out that our precoding scheme offers full-diversity with ML-decoding. The simulation plots of the word error probability for 2 2, 4 4, and 8 8 systems support our claims about full-diversity. 3) The ML-decoding complexity of the proposed precoder is shown to be for square as a function of . The rest of the paper is organized as follows. Section II gives the system model, the relevant definitions and some known results which are needed for our precoder design. The method to obtain the proposed precoder for two transmit antennas is presented in Section III and its extension to higher number of transmit antennas is presented in Section IV. The complexity of the proposed precoder is analyzed in Section V while Section VI deals with the achievable diversity gain with the proposed precoder. Simulation results are given in Section VII and concluding remarks constitute Section VIII. Notations: Throughout the paper, bold, lowercase letters are used to denote vectors and bold, uppercase letters are used to denote matrices. For a complex matrix , the Hermitian, the transpose and the Frobenius norm of are denoted by , and , respectively. The element of a vector is denoted by , the entry of is denoted by , denotes the trace of , and implies that is a diagonal matrix with as the diagonal entries. The set of all real numbers, complex numbers, and integers are denoted by , , and , respectively. The real and imaginary parts of a complex-valued vector are denoted by and , respectively, denotes the absolute value of a complex number and denotes the cardinality of the set . The identity matrix and the sized null matrix are denoted by and , respectively. For a complex denotes the expectation of while random variable , implies that has the complex normal distribution with zero mean and unit variance. Unless used as a and for a funcsubscript or to denote indices, represents , and denote that value of tion which minimizes and maximizes , respectively. For any real number , denotes the largest integer smaller than , denotes the smallest integer larger than , denotes the operation that rounds off to the nearest integer and gives the sign of , the latter two operations given by if otherwise . if otherwise. The Gamma function and the Q-function of and , respectively, and given as

are denoted by

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

Let

5487

and

be two real-valued functions. Then, as if and only if there and a real number exists a positive constant such that for all , . For a multivariate, real-valued function , we as write if and only if there exists a posand real numbers itive constant such that for all , . We write as if and only if

For a real variable , the unit step function if , and if .

is defined as

II. SYSTEM MODEL We consider an MIMO system with full CSIT and CSIR. The channel is assumed to be quasi-static and flat with Rayleigh fading. The channel is modeled as SNR

(1)

is the received vector, is the where channel matrix, is the precoded symbol vector, and is the noise vector. The entries of and are i.i.d. circularly symmetric complex Gaussian random variables with zero mean and variance 0.5 per real dimension. In (1), the scalar SNR is the average signal-to-noise ratio at each receive antenna and is constrained such that . The precoded symbol vector can be defined as , where is the precoding matrix with , and is the symbol vector, with its entries taking values independently from a signal constellation having an average energy of units. The rate of transmission is independent symbols per channel use. In this paper, we assume that is even. Note that in this model, the variable scalar which defines the average signal-to-noise ratio at each receive antenna is SNR while is a constant. For example, for a standard -QAM with for some positive integer , . Let , obtained upon the SVD of , with and being unitary matrices. is such that if and if , where is a diagonal matrix given by , with being the nonzero singular values of placed in the nonincreasing order on the diagonal. Let the precoding matrix be given as (2) where

. Now, (1) can be written as SNR

(3)

and , with where the distribution of being the same as that of . The ML-decoding rule seeks to find that which minimizes the metric given by SNR

(4)

Clearly, the error performance of the system depends on the choice of and . From (2), it is evident that the design of the precoding matrix amounts to designing . Henceforth in this paper, is referred to as precoder and the constellation is assumed to be an -QAM, where for some positive integer . Definition 1 (Full-Diversity Precoder): In a MIMO system, if at a high SNR, the average probability that a transmitted symbol vector is wrongly decoded is given by SNR , where stands for “is approximately equal to”, then, and are called the diversity gain (or diversity order) and the coding gain of the system, respectively. For a MIMO system with precoding, if , then, we call the precoder a full-diversity precoder. Definition 2 (ML-Decoding Complexity): The ML decoding complexity is measured in terms of the number of computations involved in minimizing the ML-decoding metric given in (4) and is a function of the constellation size . If at most complex symbols are required to be jointly decoded, the ML-decoding complexity as a function of is said to be . We make use of the following known results which are needed for our purpose. Theorem 1: [19]For a scalar channel modeled by SNR , where , and is a nonnegative random variable whose probability density function (PDF) is such that as , the average symbol error probability (SEP) , which is given by SNR , is such that as SNR SNR

SNR

where is a fixed positive constant depending on the constellation, is another constant defining the marginal PDF of and SNR is the dependent instantaneous SEP. If , then, SNR is the average signal-to-noise ratio at the receiver and the diversity gain and the coding gain are given as

It is known that are the nonzero eigenvalues of , denoted in the nonincreasing order by , . The following theorem gives the expression for the first-order expansion of the marginal PDF of as . Theorem 2: [20] Let the entries of the matrix be i.i.d. complex Gaussian random variables with zero mean and unit variance. The first-order expansion of the marginal PDF of the largest eigenvalue of the complex central Wishart

5488

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

matrix is given by as , , with and being positive constants. Let , where . The following theorem, first stated in [21], gives a sufficient condition for achieving full-diversity with ML-decoding in precoded MIMO systems. Theorem 3: [21] For such that for any nonzero value of , the diversity gain of the system is . Proof: The instantaneous probability that a transmitted symbol vector is falsely decoded to some other vector is given by SNR

Let

(5)

, with

. So, the

that a transmitted vector probability is upper bounded as

is falsely decoded

SNR

plexity is . This is because the independent symbols are entangled in the decoding metric but the real part of the symbol vector can be independently decoded from the imaginary part since is real-valued. In [21], complex-valued precoders are used to achieve full-diversity with an ML-decoding complexity of . III. SVD-BASED, APPROXIMATELY OPTIMAL, REAL-VALUED PRECODER FOR In this section, we propose a real-valued precoder for 2 transmit antennas and QAM constellations. The precoder is approximately optimal among the SVD based real-valued precoders for QAM constellations. The primary advantage of this precoder over the complex-valued optimal precoder [16] is that it is much easier to find the entries of the precoder for larger constellations since it has only 2 parameters that need to be searched for, while the complex-valued precoder has 3 parameters. Without loss of generality, we consider 2 receive antennas and 2 transmit antennas, for which in (3) can be expressed as

(6)

where , the largest singular value of . Assuming that all the symbol vectors taking values from are equally likely to be transmitted, the average instantaneous word error probability (WEP), dependent on , is given by (7) Using (6) in (7) SNR

where . Let

and

. Clearly, and (9)

which is the square of the minimum Euclidean distance between any two points in the received signal space in two dimensions. From (5), the optimal precoder is given by , which may or may not be unique. was obtained for 4-QAM as follows. In [16], Using SVD, can be written as , where is a unitary matrix of size 2 2 and

SNR where erage WEP

. So, from Theorem 1 and Theorem 2, the avas SNR is given by SNR

SNR

(8)

where

with

being a positive constant such that as . Note that in obtaining , we have used the fact that and . Since , and from (8), the diversity gain achieved by the system is . Obtaining such that is not difficult. Choosing to be (for ) or (for ) for QAM constellations, where is the rotated lattice generator matrix with a nonzero product distance, as presented in [22], ensures that the diversity gain is . For a square QAM constellation of size , the ML-decoding com-

(10) For QAM constellations, because of the symmetry associated with the constellation, , , and . It was shown in [16] that can be taken to be identity without affecting the optimality. Using numerical search, the optimal values for , and were found out for 4-QAM. However, there are two major obstacles when this method is used for larger QAM constellations. First, numerical search becomes practically hard for larger constellations due to the fact that there are three parameters to be searched for. Second, numerical searches do not give a closed form expression for the optimal angles and the method employed in [16] to obtain closed form expressions for the optimal angles for 4-QAM is not amenable for application to larger QAM constellations. Due to these limitations, we look for a real-valued optimal precoder given by

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

5489

which also naturally offers lower ML-decoding complexity (this is elaborated in Section V-B). In general, a real-valued precoder can be expressed as where can be taken to be identity without affecting optimality and

Note that there are only two parameters to be searched for. Our approach towards finding the optimal precoders is also based on numerical search, but the method to obtain closed form expressions for the optimal angles is novel and easily applicable for any . However, since this method is based on numerical search, it is not known if the angles are exactly optimal. Finding the exactly optimal values of and as a function of involves an exhaustive search over the range of and , which is practically impossible. However, a numerical search, with and varying in very small increments, gives the values of and , which we denote by and , respectively, such that is nearly equal to , with being the optimal real-valued precoder. For this reason, we call our precoder for an approximately optimal real-valued precoder, which means a real-valued precoder that is approximately optimal among all real-valued precoders. A square QAM signal set (not necessarily Gray coded) of size is given by

, where is a very small fraction of the order of . Based on the observations on the numerical search results, the interval that takes values from can be viewed as a disjoint union of a finite number of subintervals such that remains constant over each subinterval. Mathematically, can be expressed as (14) , are constants, is the finite number of where subintervals of , is the value of at which changes from to with and , is the length of the subinterval and . The search results also reveal that cannot be expressed as a weighted sum of shifted step functions and hence a closed form expression needs to be obtained analytically. To obtain this, we first obtain as follows. A. Calculating For , in order to obtain and as given by (13), the entries of take values . Let from be such that

(11) where PAM constellation of size

is a . Let

and with

, let

(15) The numerical searches performed for 5 QAM constellations—4-/16-/64-/256-/1024-QAM reveal that 1) there are two distinct pairs for which (15) is satisfied when , where is as defined in (14). These are and . Also in this range of . 2) There are three distinct pairs for which (15) is satisfied when , . Let

(12) where, for our numerical search, we have taken , , with being the increment size taken to be 0.001 radians for our searches. Let (13) We note that for

,

. Hence, we only need to search for and for which is obtained. Note that this simplification of the search to only a -PAM is possible since is real-valued. This is another huge advantage over the complex-valued precoder are which does not enjoy this benefit. Henceforth, and used to denote the approximately optimal angles of and . Due to our choice of the increment size, it can be safely said that

So, for

, we have , solving which we obtain

solution, which is

. The other , is ruled out since

it has been observed that for . For , , we have , where , , and are the three pairs for which (15) is satisfied. Solving them, we arrive at (16)–(17), shown at the bottom of the next page. Equating (16) and (17), we obtain

(18)

5490

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

where

where

and as explained in Section III-B, given by , for (20) and (21), we obtain

and

are constants for

Equation (18) has been observed to have only one solution in the range . This solution gives .

(22) Using (19) and (22)

B. Calculating As aforementioned, obtain for and (17) that hence

for

,

. In order to , we note from (16) is constant in that range of and

The value of

(19) where

. Solving

is given by the right-hand side (RHS) of (16) [or (17)].

as defined in (12) for , is given by (23)

where

C. Calculating Having obtained and , we proceed to find the exact values of , as follows. For convenience, let [given by (19)] for . Since is discontinuous at , where it makes a transition from to , we have

where the pairs and

and

satisfy (15) for , respectively. So, we have

(20)

(21)

with any of the pairs satisfying (15). Table I presents the values of for different values of for 4-QAM, 16-QAM, 64-QAM, 256-QAM, and 1024-QAM. The value of the constants and the corresponding pairs for which (15) is satisfied are also tabulated. Except for the case of 4-QAM, the values presented in Table I are the approximately optimal values rounded off to the fourth decimal. This has been done since it is very cumbersome to express them in the exact form. All angles are expressed in radians. Noting the values of for 4-QAM, it is natural to believe that the angles tabulated are optimal for 4-QAM. Also, it can be noted that for every subsequent larger constellation, differs from its corresponding values for the lower-sized constellation only at low values of , meaning which the numerical search need not be done over the entire range of as the size of the constellation increases. The plots of as a function of for different unnormalized QAM constellations are given in Fig. 1. The curves for 256- and 1024-QAM appear to coincide, since

(16) (17)

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

5491

TABLE I APPROXIMATELY OPTIMAL VALUES OF

Fig. 1. as a function of ious QAM constellations.

AND

for the proposed precoder for var-

they differ only at extremely low values of . In Fig. 2, the plots3 of for the precoder, the proposed precoder, the -precoder, and the -precoder are given for with the same power constraint for all the precoders as for our precoder. As was expected, the precoder has the best values of over the entire range of while our precoder has better values of than the - and -precoders. Fig. 3 shows the plots of for our precoder, the -precoder and the 3In

all the plots, the use -QAM, while the , as defined in [17].

precoder, our precoder and the -precoder -precoder uses a two-dimensional codebook of size

FOR

VARIOUS QAM CONSTELLATIONS

Fig. 2.

comparison for

.

-precoder for . Note that for low values of , our precoder and the -precoder have identical , which is because both transmission schemes are effectively the same in this range of . The -precoder has increasingly lower values of than that of our precoder and the -precoder at higher values of . It is clear from the plots that the -precoder is expected to have better error performance than the -precoder only for ill-conditioned channels, i.e., for low values of . IV. EXTENSION OF THE PRECODER FOR For the case of two transmit antennas, it is possible to obtain SVD-based, approximately optimal precoders (com-

5492

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

we choose to use the same method of pairing for all constellations as for 4-QAM. For the subsystem associated with the subchannel-pairing, , , and (24) where and are as defined in the previous section and without the subscript (refer to (23)) and depend on . Proceeding on the lines of the proof of Theorem 3, the instantaneous WEP is upper bounded as SNR and

where

Fig. 3.

comparison for

(25)

.

plex-valued precoder for 4-QAM, real-valued precoder for any -QAM). Such precoders are defined by two or three parameters depending on whether the precoder is real-valued or complex-valued, respectively. However, such an approach cannot be taken for the case of since even for , a real-valued optimal precoder would be defined by as many as 5 parameters, ruling out the possibility of a computer search even for 4-QAM. So, a more practical way of obtaining a precoder with a reasonable error performance is to pair two subchannels together and use the precoding scheme given in Section III for that pairing. This results in subsystems, each subsystem associated with a pairing. For a subsystem with the subchannel-pairing (the and the subchannels are paired), the channel coefficient is . Also, for a system consisting of independent subsystems, the error performance of the system is dictated by the subsystem with the worst error performance. Since , it is natural to expect a reasonable error performance by pairing the and the subchannels, . The precoder would then have an ’ ’ structure. In fact, this method of pairing has been proved to be optimal for the precoder for 4-QAM [15] and the steps of the same proof can be applied to show the optimality of the pairing for the proposed precoder for 4-QAM. However, finding the optimal pairing for other QAM constellations is an open problem and

..

.

with and being the precoder obtained using the subchannel-pairing, each pairing resulting in a subsystem for which the precoding scheme proposed in Section III is used. So,

,

given by (24). Observe that the with scaling factor of has been used to take into account the constraint that . Since the values of can be computed at the transmitter, we can enhance the error performance of the precoder by premultiplying the precoding matrix with a power control matrix such that (26) where is a constant and the power constraint on is such that . Due to this power constraint, from (26), we obtain

where is obtainable from (24). Hence, the proposed precoder is given by (27), shown at the bottom of the page.

..

. (27)

..

.

..

.

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

5493

B. ML-Decoding Complexity

In (27) (28) with and being the approximately optimal values obtainable from (18) and (19), respectively, both depending on and . For example, for a 4 4 system using 4-QAM signaling, if for a particular channel realization, and , then, from Table I, ,

,

,

,

, and . The upper bound on

Processing the received signal involves performing the to obtain [refer to (3)] with a complexity operation of . Subsequently, ML-decoding is performed. As explained in the previous section, the structure of the precoding matrix for the proposed precoder, as given by (27), enables the whole system (transmitting independent symbols denoted by ) to be viewed as independent subsystems, each transmitting 2 symbols and . The subsystem, , is associated with the subchannel-pairing. Let and . Considering the ML-decoding metric given by (4), it is clear that SNR

the instantaneous WEP is now given as SNR

SNR (29)

where (30) It can easily be checked that SNR

SNR

and hence, the upper bound in (29) is lower than that in (25). Therefore, the use of the power control matrix enhances error performance. A similar approach of using a power control matrix has been taken in [15] for 4-QAM, but since we need to have explicit values of , applying this scheme precoder with larger constellations is not feafor the sible. Structurally, the precoder and the -precoder differ from (27) in that for the precoder, is optimized using an additional parameter [as shown in (10)], while for the -precoder, is optimized with and . V. COMPLEXITY ANALYSIS OF THE PROPOSED PRECODER A. Encoding Complexity Preprocessing before transmission of the symbols involves calculating the SVD of the channel matrix , determining the precoder matrix based on the singular values of the channel and finally obtaining the matrix . The complexity of calculating the SVD of is . The complexity is measured in terms of the number of multiplications involved. Since has the ’ ’ structure, obtaining it involves a complexity of while calculating is associated with a complexity of . Hence, the overall encoding complexity is dictated by the complexity involved in the calculation of SVD, which is . This complexity is the same for all SVD based linear precoders, including the precoder and the -precoder.

where and is as given by (28). So, the symbols associated with each subsystem can be decoded independently from the rest of the symbols. Hence, the ML-decoding complexity of the overall system as a function of the constellation size is dictated by the ML-decoding complexity of a subsystem. We make use of the following lemma to analyze the ML-decoding complexity of a subsystem which naturally gives the ML-decoding complexity of the proposed precoder. Lemma 1: For symbols and taking values from , the symbol takes values from . Proof: Firstly, represents the standard, unnormalized -QAM constellation as given by (11). Let denote the -QAM constellation scaled by . So, the distance between any two adjacent signal points on the same vertical or horizontal line of is . Now, the constellation given by (31) can be viewed to be obtained by replacing every element of by the entire constellation such that the origin of is the signal point being replaced. This is illustrated in Fig. 4, where and denote two adjacent signal points of which are replaced by the entire constellation . The resulting signal points, which are a part of , are denoted by stars. Hence, has signal points and a QAM structure, and the distance between adjacent points on the same vertical or horizontal line is 2. Therefore, is an -QAM. The following theorem gives the ML-decoding complexity (as a function of the constellation size ) of the subsystem of the proposed precoding scheme . Theorem 4: For the subsystem, with , the following claims hold. 1) The ML-decoding complexity is .

when

5494

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

with

To obtain , instead of using a 2-dimensional real sphere decoder, we do the following. For each possible value of , the corresponding value of is evaluated as (33) Fig. 4. Part of an -QAM constellation obtained from two stellations of unequal average energy.

-QAM con-

TABLE II PROBABILITY THAT NO SEARCH IS REQUIRED FOR EACH SUBSYSTEM DIFFERENT MIMO SYSTEMS

where that OF

2) The ML-decoding complexity is the same as that of a real scalar channel when , with no exhaustive search over all the signal points required. Proof: These claims are proved below. Case 1: . In this case, the decoded signal vector is SNR

SNR

SNR

SNR

and pair that minimizes

is given by

SNR

So, there are only searches (for possibilities for ) involved in minimizing the ML-metric. The operation shown on the R.H.S of (33) quantizes to its nearest possible value for a fixed . This is made possible due to the structure of -QAM which is a Cartesian product of two -PAM constellations. The same method can be applied to obtain . So, the ML-decoding complexity as a function of is . Using the quantizing operation shown in (33), it can be similarly shown that the ML-decoding complexity for any subsystem of the precoder and the -precoder are respectively and , although in literature, the corresponding ML-decoding complexities are claimed to be and . Case 2: . From Table I and also as was pointed out earlier, for , we have and . This means subchannel and the that transmission is made only on the received signal of interest, with regard to (3), can be expressed SNR , , where as

(32)

is the average energy of an -QAM and , with and obtained on the QR-decomposition of . Since and are realvalued, can be written as , where where

SNR

and values from

, where and take . From Lemma 1, takes values4 from . So, in the first step, is decoded to obtain by quantizing, where and are given by

4From a bit error rate point of view, it is advisable to transmit the symbol alone on the first virtual subchannel, with taking values from a Gray coded . This is because the constellation given by (31) is not Gray coded. However, with a view of minimizing the word error rate, transmission of , with and taking values from , is as good a strategy as transmitting alone, with taking values from a Gray . coded

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

5495

TABLE III COMPARISON OF LOW ML-DECODING COMPLEXITY PRECODING SCHEMES

From , is decoded to obtain given by

, with

and

(34) and

is decoded to obtain , with given by

and

(35) Note that the operations shown in (34) and (35) together perform the inverse of the function given by

for . Therefore, decoding and requires no exhaustive search over the signal points of the constellation. It is crucial to note that the advantage of not having to search over any of the signal points when is unique to the proposed real-valued precoder and not obtainable for the precoder [15] for 4-QAM. Overall, the ML-decoding complexity of the proposed precoder as a function of is while the ML-decoding complexity as a function of and is given by since there are subsystems (The entire decoding process including performing the operation has a complexity of ). However, it must be noted that the symbols belonging to some subsystems can be decoded for most channel realizations without the

need for a search over the signal points of the constellation. In Table II, by simulating channel realizations, we have tabulated the probability that ML-decoding for subsystems of the proposed precoding scheme for 2 2, 4 4 and 8 8 MIMO systems can be done without searching over any of the signal points for 4- and 16-QAM. Table III gives a comparison of the various low ML-decoding complexity precoding schemes. VI. DIVERSITY GAIN The precoder, the -precoder and the -precoder have all been shown to guarantee a diversity gain equal to . Recall that the condition in Theorem 3 is only a sufficient condition for achieving full-diversity gain equal to . It is not necessary that be such that , for , to achieve full-diversity. This can be seen by noting that for and 4-QAM, our precoder does not satisfy the condition when but still gives full-diversity. This is proved in the following lemma. Lemma 2: The proposed precoder offers full-diversity, i.e., a diversity gain equal to for . Proof: Consider the precoder given by

which is the full-diversity rotation matrix [22] in 2 dimensions and has the highest nonzero product distance among all 2 2 sized orthogonal matrices. This precoder, which we call the lattice precoder for , has full-diversity from Theorem 3. Clearly, for any value of is greater for our precoder than that for the lattice precoder, since our precoder is approximately optimal among real-valued precoders. So, our precoder has better error performance than the lattice precoder.

5496

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

CHARACTERISTICS

OF

TABLE IV FOR DIFFERENT MIMO SYSTEMS

Hence, our precoder too offers full-diversity, like the lattice precoder for . From Lemma 2, Theorem 2, and Theorem 3, one would be inclined to believe that for , the subsystem of the proposed precoding scheme (associated with the subchannel-pairing) has a diversity gain of , with , in which case the diversity gain of the whole system would be the minimum of the diversity gains of all the subsystems, i.e., . In fact, the diversity gains of systems using the precoder and the -precoder have been claimed to be due to this reason. It must be noted that the power control matrix plays an important role in the error performance of our precoder (also the precoder for 4-QAM), as explained in Subsection IV. Before we analyze the achievable diversity gain of the system with the proposed precoding scheme, the following important observation needs to be made about . Since , we have

indicate that there exists a such that is lower bounded by . So, from (29) SNR

SNR

SNR

SNR where, as used throughout the paper, and is a non-zero positive constant depending on . From Theorem 1, we obtain, as SNR

. Conse-

for and therefore, . From Fig. 1, we can conclude that over a large part of the interval , , for . As a result of this fact, it is not guaranteed that for , due to which even without the use of , the overall diversity gain of the system might be higher than (this holds true even for the -precoder). With the use of for our proposed precoder, the channel dependent instantaneous WEP is dependent on , as seen in (29). Let

quently,

and be the probability that . In Table IV, we have tabulated the values of , which is the minimum value of obtained on simulations for channel realizations, and , which is again calculated by simulating channel realizations, for different MIMO systems. In the table, we observe that for and for , is always greater that 1. This can be attributed to the fact that for higher values of , the ratio of to is very low and the corresponding value of is also very low. For such systems, we can safely say that the full-diversity gain equal to is achieved (since is associated with a diversity gain of ). For other systems, the simulations results in Table IV

, i.e.,

SNR

SNR

where

(36) with being a constant in the expression for the marginal PDF of , as defined in Theorem 2. Therefore, the overall diversity gain of the system is . Note that in (36), and define the coding gain—higher the value of and , better the error performance. Obtaining analytically is an open problem. The values in Table IV are only indicative of what the actual is likely to be. For example, for the 16 16 system with 64-QAM, is likely to be greater than 1. Thus, we have argued out that our precoding scheme provides full-diversity and since the precoder has a slightly better performance than our precoder for 4-QAM, it can be argued to have full-diversity too. This claim is supported by the WEP plots for different MIMO systems, shown in the following section. VII. SIMULATION RESULTS For all simulations, we consider the Rayleigh fading channel with perfect CSIT and CSIR. We consider three MIMO systems—2 2, 4 4 and 8 8 MIMO systems. For the 2 2 MIMO system, the rival precoders for our precoder are the

SRINATH AND RAJAN: FULL-RATE MIMO PRECODER

Fig. 5. WEP comparison for 2

2 MIMO systems employing 4-/16-/64-QAM.

precoder and the -precoder. We have left out the -precoder since it has been shown in [17] to have an error performance comparable with that of the -precoder for 4-QAM while for 16-QAM, it is not expected to beat the -precoder, as can be inferred from Fig. 3. The constellations employed are 4-QAM, 16-QAM, and 64-QAM. For 16-QAM and 64-QAM, the precoder is not considered since it is very hard to obtain and not explicitly stated in literature. Fig. 5 shows the plots of the word error probability (WEP) as a function of the average SNR at each receive antenna for the 2 2 system. As expected, the precoder has the best error performance for 4-QAM, marginally beating our precoder which in turn significantly beats the -precoder. For 16-QAM and 64-QAM, our precoder clearly beats the -precoder. For 4 4 and 8 8 systems, we also consider the Lattice precoder, which is the orthogonal matrix with the largest known nonzero product distance for real dimensions, and given explicitly in [22]. From Theorem 3, this precoder offers full-diversity. The plots of the WEP for the 4 4 system and the 8 8 system are respectively given in Figs. 6 and 7. The plots indicate that the precoder and our proposed precoder offer full-diversity, since they beat the full-diversity achieving Lattice precoder, except for the case of the 8 8 MIMO system with 64-QAM, where the Lattice precoder enjoys coding gain over the proposed precoder but has a similar slope. It must be noted that the Lattice precoder has an ML-decoding complexity of . Our precoder significantly outperforms precoder has the best error the -precoder, while the performance for 4-QAM, marginally beating our precoder, but this is at the expense of ML-decoding complexity. VIII. DISCUSSION For systems with full CSIT, we have proposed a real-valued precoder for which, for QAM constellations, is approximately optimal among all real-valued precoders based on the SVD of the channel matrix. This precoder is extended to

5497

Fig. 6. WEP comparison for 4

4 MIMO systems employing 4-/16-/64-QAM.

Fig. 7. WEP comparison for 8

8 MIMO systems employing 4-/16-/64-QAM.

higher number of transmit antennas. The advantage of the proposed precoder over the precoder is that it is much easier to obtain for larger QAM constellations and it also has lower ML-decoding complexity. The proposed precoder handsomely beats the -precoder in error performance. Our precoding scheme is argued to offer full-diversity with QAM constellations. It would be interesting to design low ML-decoding complexity, full-rate, full-diversity precoders for more realistic scenarios, like for systems with imperfect CSIT or partial CSIT. REFERENCES [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space time codes for high date rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [2] E. Biglieri, Y. Hong, and E. Viterbo, “On fast-decodable space–time block codes,” IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 524–530, Feb. 2009.

5498

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

[3] B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, high-rate space-time block codes from division algebras,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2596–2616, Oct. 2003. [4] F. Oggier, G. Rekaya, J. C. Belfiore, and E. Viterbo, “Perfect space time block codes,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3885–3902, Sep. 2006. [5] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. New York: Wiley, 2006. [6] T. K. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, no. 10, pp. 1458–1461, Oct. 1999. [7] P. Stoica and G. Ganesan, “Maximum-SNR spatial-temporal formatting designs for MIMO channels,” IEEE Trans. Signal Process., vol. 50, no. 12, pp. 3036–3042, Dec. 2002. [8] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2198–2206, Dec. 2001. [9] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, and H. Sampath, “Optimal designs for space-time linear precoders and decoders,” IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1051–1064, May 2002. [10] P. Rostaing, O. Berder, G. Burel, and L. Collin, “Minimum BER diagonal precoder for MIMO digital transmission,” Signal Process., vol. 82, no. 10, pp. 1477–1480, Oct. 2002. [11] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE. Trans. Signal Process., vol. 51, no. 9, pp. 2381–2401, Sep. 2003. [12] C. Windpassinger and R. F. H. Fischer, “Low-complexity near-maximum-likelihood detection and precoding for MIMO systems using lattice reduction,” in Proc. Inf. Theory Workshop (ITW’03), Paris, France, Apr. 4, 2003. [13] D. J. Love and R. W. Heath, Jr, “Multimode precoding for MIMO wireless systems,” IEEE Trans. Signal Process., vol. 53, no. 10, Oct. 2005. [14] G. Raleigh and J. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, no. 3, pp. 357–366, Mar. 1998. [15] B. Vrigneau, J. Letessier, P. Rostaing, L. Collin, and G. Burel, “Extension of the MIMO precoder based on the minimum Euclidean distance: A cross-form matrix,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 2, pp. 135–146, Apr. 2008. [16] L. Collin, O. Berder, P. Rostaing, and G. Burel, “Optimal minimum distance based precoder for MIMO spatial multiplexing systems,” IEEE Trans. Signal Process., vol. 52, no. 3, pp. 617–627, Mar. 2004. [17] S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO precoding with X- and Y-codes,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3542–3566, Jun. 2011. [18] E. Viterbo and J. Boutros, “Universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory., vol. 45, no. 5, pp. 1639–1642, Jul. 1999. [19] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1389–1398, Aug. 2003.

[20] L. G. Ordonez, D. P. Palomar, A. P. Zamora, and J. R. Fonollosa, “High-SNR analytical performance of spatial multiplexing MIMO systems with CSI,” IEEE Trans. Signal Process., vol. 55, no. 11, pp. 5447–5463, Nov. 2007. [21] H. J. Park, B. Li, and E. Ayanoglu, “Multiple beamforming with constellation precoding: Diversity analysis and sphere decoding,” in Proc. Inf. Theory Appl. (ITA 2010), San Diego, CA, Feb. 5, 2010, pp. 1–11. [22] Full Diversity Rotations [Online]. Available: http://www1.tlc.polito.it/ viterbo/rotations/rotations.html

K. Pavan Srinath received the B.Eng. degree in electronics and communication from B. M. Sreenivasiah College of Engineering, Bangalore, India, and the M.Eng. degree in telecommunication from the Indian Institute of Science, Bangalore, in 2005 and 2008, respectively. From September 2005 to June 2006, he was with Robert Bosch India limited, Bangalore. Currently, he is a working towards the Ph.D. degree with the Department of Electrical Communication Engineering, Indian Institute of Science. His primary research interests include wireless communication, space-time coding, and coding for wireless relay networks.

B. Sundar Rajan (S’84–M’91–SM’98) was born in Tamil Nadu, India. He received the B.Sc. degree in mathematics from Madras University, India, the B.Tech. degree in electronics from Madras Institute of Technology, and the M.Tech. and Ph.D. degrees in electrical engineering from the Indian Institute of Technology, Kanpur, in 1979, 1982, 1984, and 1989, respectively. He was a faculty member with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, from 1990 to 1997. Since 1998, he has been a Professor with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore. His primary research interests include space-time coding for MIMO channels, distributed space-time coding and cooperative communication, coding for multiple-access, relay channels, and network coding with emphasis on algebraic techniques. Dr. Rajan is an Associate Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY, an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and an Editorial Board Member of the International Journal of Information and Coding Theory. He served as a Technical Program Co-Chair of the IEEE Information Theory Workshop (ITW’02), held in Bangalore, in 2002. He is a Fellow of Indian National Academy of Engineering, a Fellow of the National Academy of Sciences, India, and a recipient of the IETE Pune Center’s S.V.C Aiya Award for Telecom Education in 2004. He is a Member of the American Mathematical Society.