WO2013026164A1

WO2013026164A1 - System and method for decoding space-time code matrices

Info

Publication number: WO2013026164A1
Application number: PCT/CA2012/050579
Authority: WO
Inventors: Young Gil Kim; Norman C. Beaulieu
Original assignee: The Governors Of The University Of Alberta
Priority date: 2011-08-23
Filing date: 2012-08-22
Publication date: 2013-02-28

Abstract

Systems and methods for decoding matrices, such as Matrix C in the WiMAX standard, are provided. An exhaustive search and zero-forcing (ES-ZF) decoder and an exhaustive search and nulling canceling (ES-NC) decoder are provided for uncoded systems. The computational complexity of the ES-ZF decoder for Matrix C decoding is shown to be the same as the complexity of the ZF decoder for Matrix B decoding with twice the number of receive antennas times the complexity of the maximum likelihood (ML) decoder for Matrix B decoding with twice the number of receive antennas. Matrix C can be implemented in a 2x2 multiple-input multiple- output (MIMO) system using the ES-NC decoder with reduced complexity compared to ML decoding with no performance loss. For coded systems, decoders employing double pruned trees using zero-forcing (DPT-ZF) algorithm or nulling canceling (DPT- NC) algorithm are provided. The DPT-NC decoder is shown to be implemented in a 2x2 MIMO coded system with reduced complexity compared to the Max-Log decoding, with no performance loss, by computer simulations.

Description

SYSTEM AND METHOD FOR DECODING SPACE-TIME CODE MATRICES Field

The application relates to systems and methods for decoding space- time code matrices, for example Matrix C defined in the IEEE 802.16e-2005

Standard.

Background

Multiple-input multiple-output (MIMO) techniques based on using multiple antennas at both transmitter and receiver can provide spatial diversity, multiplexing gain, and interference suppression, as well as make various tradeoffs among them. These techniques have been incorporated in all of the recently developed wireless communications system specifications including the IEEE

802.1 1 n standard for local area networks and the IEEE 802.16e-2005 standard for mobile broadband wireless access systems [1]. Mobile WiMAX systems are based on the scalable OFDMA mode of IEEE 802.16e-2005 specifications and use a subset of the different options. Regarding MIMO options, the WiMAX Forum has specified two mandatory profiles for use on the downlink. One of them is based on the space-time code (STC) proposed by Alamouti for transmit diversity [2]. Denoted in the WiMAX standard as Matrix A, this code provides perfect second-order diversity when used with a single receive antenna and fourth-order diversity when used with two receive antennas. However, it is only half-rate, because it only transmits two symbols using two time slots and two transmit antennas. The other profile, denoted Matrix B in the WiMAX standard, is spatial multiplexing (SM), which uses two transmit antennas to transmit two independent data streams. This scheme is full-rate, but it does not benefit from any diversity gain at the transmitter, and at best, it provides second-order diversity with two receive antennas.

Experience has shown that the desired quality of service cannot always be achieved using Matrix A or Matrix B. For future evolutions of the WiMAX standard, it is highly desirable to include a new code combining the respective advantages of the Alamouti code and spatial multiplexing, while avoiding their drawbacks. Such a code actually already exists in the IEEE 802.16e-2005 specifications, where it is referred to as Matrix C [3]. The Matrix C code is known to be one of the best STCs of size 2x2. However, this code has a high decoding complexity which grows with the fourth power of the modulation order for maximum likelihood (ML) decoding. This decoding complexity has likely hindered the adoption of Matrix C by the industry, despite its superior performance over Matrix A and Matrix B.

Summary

According to one aspect of the present invention, there is provided a method of decoding a transmitted M_tx P space-time code matrix where M_t is a number of transmit antennas > 2 and P is a number of time slots > 1 , that is a function of K symbols where K > 2, the method comprising: using M_r receive antennas, receiving an M_rx P matrix, where M_r≥ 1 ; fixing at least one of the K symbols to each possible permution of the at least one of the K symbols; for each possible permutation of the at least one symbol, decoding the remaining symbols; using exhaustive search, finding the permutation of at least one symbol and the decoded remaining symbols that minimizes an objective function. In some

embodiments, P > 2.

According to another aspect of the present invention, there is provided a method of processing a transmitted M_tx P space-time code matrix where M_t is a number of transmit antennas > 2 and P is a number of time slots > 1 , that is a function of K symbols modulated using an M-ary modulation format, M > 2, K > 2 the method comprising: using M_r receive antennas, receiving an M_rx P matrix, where M_r > 1 ; generating at least two pruned trees, by for each pruned tree: a) fixing a respective at least one of the K symbols to each possible permution of the at least one of the K symbols; and b) for each possible permutation of the at least one symbol, decoding the remaining symbols; the at least two pruned trees being such that the at least two prune trees collectively include at least one node that has a one in the b^th bit of the j^th symbol, and at least one node that has a zero in the bth bit of the jth symbol, for b = 1 ,2, ...,log₂M, and j=1 K; for the b^th bit of the j^th symbol, for b

= 1 ,2, ...,log₂M, and j=1 , ...,K, determining a respective soft decision using the at least two pruned trees. In some embodiments, P > 2.

Further embodiments provide a computer readable storage medium having stored thereon instructions for execution by a computer to execute one of the above summarized methods or any of the other methods described or claimed herein.

Further embodiments provide a receiver that comprises at least one antenna and a space-time transmit diversity decoder configured to implement any of the methods summarized above or described or claimed herein.

Further embodiments provide a space-time transmit diversity decoder configured to implement one of the methods summarized above or any of the methods described or claimed herein.

Brief Description of the Drawings Embodiments of the disclosure will now be described with reference to the attached drawings in which:

Figure 1 is a block diagram of a system for decoding space-time code matrices;

Figure 2 is a flowchart of a method of decoding space-time code matrices;

Figure 3 is a block diagram of another system for decoding space-time code matrices in which error correction coding is present;

Figure 4 is a flowchart of a method of decoding space-time code matrices where error correction coding has been performed; Figures 5A and 5B show examples of a tree and a pruned tree respectively; and

Figures 6 and 7 show example performance results.

Detailed Description The embodiments set forth below represent the necessary information to enable those skilled in the art to practice the claimed subject matter and illustrate the best mode of practicing such subject matter. Upon reading the following description in light of the accompanying figures, those skilled in the art will

understand the concepts of the claimed subject matter and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims.

Moreover, it will be appreciated that any module, component, or device exemplified herein that executes instructions may include or otherwise have access to computer readable storage medium or media for storage of information, such as computer readable instructions, data structures, program modules, or other data. A non-exhaustive list of examples of computer readable storage media include magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, optical disks such as CD-ROM, digital versatile disks (DVD) or other optical storage, volatile and non-volatile, removable and non-removable media implemented in any method or technology, RAM, ROM, EEPROM, flash memory or other memory technology. Any such computer storage media may be part of the device or accessible or connectable thereto. Any application or module herein described may be implemented using computer readable/executable instructions that may be stored or otherwise held by such computer readable storage media.

Referring now to Figure 1 , shown is a MIMO system. On the transmit side, generally indicated at 10, a set of modulation signals s_t, s₂, s₃ and s₄ , 22, are input to a space-time transmit diversity encoder 24. In the illustrated example, the space-time transmit diversity encoder operates using Matrix C. The output of the space-time diversity encoder 24 is transmitted over two transmit antennas 26,28. On the receive side, generally indicated at 12, there are M_r receive antennas 30. These are connected to a space-time transmit diversity decoder 32 that produces a set of decoded symbols 34 that includes s_x ,s₂ ,s₃ and s₄ . Matrix C is defined in the WiMAX standard as [3]

where s₁ ,s₂ ,s₃ ,s₄ are -ary quadrature amplitude modulation (QAM) signals with average symbol energy E_s , r = _> and j = V-T . Since the complexity of Matrix C decoding is compared with that of Matrix B decoding, Matrix B is given as

The average bit energy is denoted by E_b , which is equal to E71og₂ . The received signal is given by

R = HC + n (3) where H is the M_r x 2 channel gain matrix whose entries h_lm represent the gain from the m th transmit antenna to the /th receive antenna. The gains h_lm are assumed to be independent and identically distributed (iid) circularly symmetric complex

Gaussian random variables with unit variance. It is assumed that the vector n is a M_t x 2 additive white Gaussian noise (AWGN) vector with variance N₀/2 per dimension. The symbol || · || is used for vector two-norm. Of course it should be clearly understood that the decoders and methods provided herein may be used in systems where one or more of these assumptions is not entirely valid; this may have an effect upon the performance of the decoders and methods. The system shown in Figure 1 assumes two transmit antennas and M_r receive antennas, where M_r > 1 . More generally, a system can be implemented using M_t transmit antennas, where M_t≥2 . With these generalizations, the decoders described herein can be applied to Golden code decoding, and to MIMO systems generally.

In the above example, the Matrix C is employed, this being a 2 x 2 matrix that is transmitted over two antennas and over two time slots. More generally, an M_t x P space-time code matrix can be employed where again, M_t is the number of transmit antennas and P is the number of time slots used to transmit the matrix, where P > 1 . In the illustrated example, Matrix C has been employed, and this matrix is a function of four symbols, denoted as s_t , s₂, s₃ and s₄ in the above equation. More generally, the matrix is a function of K symbols, where K > 2.

It should be clearly understood that a system may include many more components than those depicted in Figure 1 . For example, there may be other components between the space-time transmit diversity encoder 24 and the transmit antennas 26, 28. Similarly, there may be other components between the receive antennas 30 and the space-time transmit diversity decoder 32.

Referring now to Figure 2, shown is a flowchart of a method of decoding a transmitted M_t x P space-time code matrix, where M_t is a number of transmit antennas and P is a number of time slots, that is a function of K symbols. The method begins in block 2-1 using M_r antennas, to receive a M_r x P matrix. In block

2-2, at least one of the K symbols is fixed to each possible permutation of the at least one symbol. In block 2-3, for each possible permutation of the at least one symbol, the remaining symbols are decoded. In block 2-4, using an exhaustive search, the permutation of the at least one symbol and the decoded remaining symbols that minimizes an objective function is found. It is noted that in block 2-3, the decoding of the remaining symbols can be performed using any appropriate method. Specific examples include zero forcing decoding, null cancelling decoding and minimum mean square error decoding. However, it is to be clearly understood that other methods can be employed. Given that the received matrix is a function of K symbols, and at least one of these are fixed, the decoding of the remaining symbols involves decoding at most K-1 symbols. As an example, if L symbols are fixed, the K-L symbols are decoded and then the L symbols will be determined by exhaustive searching.

Because of this, typically a method is used to decode the K-L symbols that is simpler than would be required to decode all K symbols. In the detailed examples provided below, two of the K symbols are fixed, but more generally at least one of the K symbols can be fixed .

In the examples below, the pruned trees are generated by fixing the values of s₁ and s₄ to be every possible permutation of these two symbols and then determining s₂ and s₃ using another method, such as zero forcing decoding or null cancelling decoding. More generally, at least one symbol is fixed to each possible permutation, and the remaining symbols are determined using another method, and then a selection is made by exhaustive searching.

Exhaustive Search and Zero Forcing Decoder for Matrix C Decoding for

Uncoded System with One Receive Antenna

In a first example, an exhaustive search and zero-forcing (ES-ZF) decoder with one receive antenna for uncoded systems is provided. The received signal can be written as

{h_ls_l + h_l jrs₄ + h₂s₂ - h₂rs₃ ) + n_x (4)

{l rs₂ + f s₃ + h₂jrs₁ + h₂s₄) + n₂ (5) where R_x and R₂ are the received signal at the first and the second time slot, respectively, and and h₂ are the channel gain from the first and the second transmit antennas, respectively. Then, the values of s_x and s₄ are fixed to any symbol in the signal constellation, so one has

= (¾ + K^SA ) = I (k^rS2 + k^S3 ) + «2 ·

+ r VI + r

Defining _l = R_l

Given β_χ and β₂ , (6) can be solved using ZF decoding. The details of ZF decoding are well known to those skilled in the art. See for example [9], hereby incorporated by reference in its entirety. In a s ecific example of ZF decoding, multiplying the inverse of the

Symbols s₂ and s₃ can be detected, for example, using a slicer or threshold device, with the left hand side of Equation (7) being the input to the slicer or threshold device.

This ZF decoding step is repeated for every permutation of s_x and s₄ in the signal constellation. Using exhaustive search, s_x and s₄ are found that minimize

where S₂ and ¾ are symbols detected using the ZF decoder given s₁ and s₄ .

Equation (8) is a specific example of an objective function. Others can be used. The equation has been written for Matrix C, with K= four symbols, but can easily be generalized to apply to other system configurations. The example above uses squared Euclidean distance. This is used because of the assumption that the noise is Gaussian. The squared Euclidean distance is from the exponent of probability density function of the Gaussian distribution. If the noise is impulsive like the

Laplacian noise model, another objective function may be better, for example, the one-norm or absolute value or other nonlinear function.

Note that (6) can be viewed as a problem of detecting Matrix B with two receive antennas. Eq. (8) can then be viewed as the distance metric of Matrix B decoding with two receive antennas. Thus, the complexity of the ES-ZF decoder for Matrix C decoding with one receive antenna, Κ_ΕΊΊ__ΖΡ (Μ_Τ = 1 , C) , is the same as the complexity of the ZF decoder for Matrix B decoding with two receive antennas, K_ZF (M_I = 2, B) , times the complexity of the ML decoder for Matrix B decoding with two receive antennas, K_ML (M_I = 2, B) . That is,

ES-ZF ( _r = 1 , C) = ( _r = 2, 2?) · K_M, ( _r (9) Exhaustive Search and Null Cancelling Decoder for Matrix C Decoding for Uncoded System with One Receive Antenna

In a second example, an exhaustive search and null cancelling (ES-NC) decoder with one receive antenna for uncoded systems is provided. For the ES-NC decoder, an approach that is similar to the above-described exhaustive search and zero forcing decoder is employed, with an NC decoder used in place of the ZF decoder. Thus, NC decoding is used for detecting s₂ and s₃ in (6). If || ||>|| h₂ || , s₃ is detected first. Then, putting the detected s₃ into (6), s₂ is detected following NC decoding. The details of NC decoding are well known to those skilled in the art. See for example [5], hereby incorporated by reference in its entirety. Then, NC decoding is repeated for every s_l and s₄ in the signal constellation. Symbols s_l and s₄ are detected using exhaustive search using equation (8) or another objective function. For the ES-NC decoder, a similar complexity equality holds as was the case for the ES-ZF decoder. The complexity of the ES-NC decoder for Matrix C decoding with one receive antenna, K _CD (M_T = 1,C) , is the complexity of the NC decoder for Matrix B decoding with two receive antennas, K_NC(M_R = 2, B) , multiplied by the complexity of the ML decoder for Matrix B decoding with two receive antennas, K_ML (M_I = 2, B) . That is,

K_BS-_NC (Μ_Τ = 1 ,€) = K_NC (Μ_Τ = 2, Β) - (Μ_Τ = 2, Β). (10)

Exhaustive Search and Zero Forcing Decoder for Matrix C Decoding for Uncoded System with Two Receive Antennas

In another example, a ES-ZF decoder with two receive antennas for uncoded systems is provided. The received signal with two receive antennas can be written as

f

where R_LM and n_lm are the received signal and the noise signal, respectively, at the /th receive antenna and the m th time slot. Then, the values of s_x and s₄ are fixed to any symbol in the signal constellation, and one has Defining Ai ^na

(7^¹ + ^⁴ ) _^ ^2) can be written as

+ r

Given β_η , β_η ,β_2λ , and β₂₂ , (13) can be solved using ZF decoding as described previously. A Moore Penrose pseudo-inverse of the 2 x 4 matrix can be used. The detection of s₂ and s₃ is repeated by ZF decoding for each permutation of symbols s_l and s₄ in the signal constellation. Using exhaustive search, ^ and s₄ are detected as before. Note that (13) can be viewed as a problem of detecting Matrix B with four receive antennas. Thus, the complexity of the ES-ZF decoder for Matrix C decoding with two receive antennas, K_ES __ZF(M_I = 2,C) , is the complexity of the ZF decoder for

Matrix B decoding with four receive antennas, K_ZF(M_I = 4,B) , multiplied by the complexity of the ML decoder for Matrix B decoding with four receive antennas, K_ML(M_I = 4,5) . That is,

K_ES-_zA = 2,C) = Κ_ζρ(Μ_τ = 4,Β) - Κ_ΜΙ(Μ_τ = 4,B). (14) Exhaustive Search and Null Cancelling Decoder for Matrix C Decoding for Uncoded System with Two Receive Antennas

In another example, an ES-NC decoder with two receive antennas for uncoded systems is provided. For the ES-NC decoder, an approach that is similar to the above-described exhaustive search and zero forcing decoder is employed, with an NC decoder used in place of the ZF decoder. More specifically, NC decoding is used for detecting s₂ and s₃ in (13). For the ES-NC decoder, a similar complexity equality holds as holds for the ES-ZF decoder. The complexity of the ES-NC decoder for Matrix C decoding with two receive antenna, K_ES__NC(M_T = 2, C) , is the complexity of the NC decoder for Matrix B decoding with four receive antennas, Κ_Ν(:(Μ_τ = 4, B) , times the complexity of the ML decoder for Matrix B decoding with four receive antennas, K_ML(M_I = 4,B) , viz

= 2, C) = ¾_c( _r = 4,B) - K_ML(M_I = 4,B). (15)

Since the computational complexity of the ZF decoder and the NC decoder does not depend on the modulation order , the complexity of the ES-ZF decoder and the ES-NC decoder is proportional to M² , which comes from the exhaustive search operation. However, the complexity of the ML decoder for Matrix C decoding is proportional to M⁴ . A discussion on the complexity of the ZF decoder and the NC decoder is available in [6, p. 8]. Another advantage of the ES-ZF decoder and the ES-NC decoder for Matrix C decoding is that they can reuse the hardware already implemented for Matrix B decoding. For example, if NC decoder hardware for Matrix B decoding is implemented on a chip, then the additional hardware required for the ES-NC decoder for Matrix C decoding is only the ML decoder for searching M² candidates since the NC decoder hardware for Matrix B decoding can be reused. In Table 1 , computational complexities of the ML decoder, the ES-ZF decoder, and the ES-NC decoder are shown. The Matrix C decoding for uncoded systems can be thought of as a tree-searching problem. For the ML decoder, we have a tree with M⁴ leaf nodes as in Fig. 5A. For the ES-ZF and ES-NC decoders, we have a pruned tree with only M² leaf nodes as in Fig. 5B. This fact explains why the computational complexity can be reduced by adopting the ES-ZF decoder or the ES-NC decoder.

Table 1 : Computational complexity of ML decoding, ES-ZF decoding, and ES-NC decoding for uncoded systems

Matrix C Decoding for Coded Systems

A double pruned tree (DPT) -ZF decoder and a DPT-NC decoder for coded systems are provided. For coded systems, an error correction code is used as an outer code and Matrix C can be considered as an inner code. The system diagram for decoding for coded systems is shown in Figure 3. This is similar to Figure 1 , but there are additional blocks for error correction encoding and decoding on the transmitter and receiver side respectively, and the space-time transmit diversity decoder in the receiver will be different as detailed below. On the transmit side, generally indicated at 36, there is error correction encoder 41 connected to space-time transmit diversity encoder 44. There are two antennas 46 and 48. On the receive side, generally indicated at 38, there are a set of M_r receive antennas 50 that are connected to space-time transmit diversity decoder 52, the output of 54 of which is connected to an error correction decoder 56. It should be clearly understood that a system in accordance with Figure 3 may include additional components, and these may be interposed between the components shown in Figure 3. For example, there may be additional elements between the antennas and the space-time transmit diversity encoder/decoders and there may be other components between the error correction coder and the space-time transmit diversity encoder, or between the space-time transmit diversity decoder and the error correction decoder. The generalizations described for the Figure 1 embodiment concerning the number of transmit antennas, the number of receive antennas, the transmitted space-time code matrix used, the size of the transmitted matrix M_t x P, and the number of symbols K represented in the transmitted matrix also apply here.

Soft decisions which will function as soft-input for the outer error correction decoder can approximated as [8] min II R - HC_j)b ||² - min \\ R ^~ HC ^ ^ (18) b^^Cj,b ^^χ j,b } '^lCj,b^^Cj,b ^^χ j,b ^} where x_{J b} is the b bit in the / symbol in Matrix C, 7 = 1,2,3,4 , b = l ,2,...,log₂ , z_j° _b is a set of Matrix C's that have zero at the b^th bit in the f symbol and z) _b is a set of Matrix C's that have one at the b^th bit in the symbol. The equation has been written for Matrix C, with K= four symbols, but can easily be generalized to apply to other system configurations. More generally, other equations for the soft decisions can alternatively be used. By way of example, the metric below is the log-likelihood (LLR) of the b^th bit of /'' symbol, which is the optimum metric for soft decoding. Compared to (18), it provides up to 0.25 dB gain. This metric could be used instead of the metric of equation (18). It is noted that Eq. (18) is a dominant term approximation of the following metric:

As an example, if j = 1 or 4, then the terms in PT1 are added in both the numerator and the denominator. As an example, if j = 2 or 3, then the terms in PT2 are added in both numerator and the denominator. In this way, the number of terms in the numerator and the denominator is reduced by a factor of M² .

The signal-to-noise ratio (SNR) loss of the max-log approximation in (18) is around 0.25 dB over a large range of SNRs [8]. This max-log approximation will be referred to as Max-Log decoding. The Max-Log decoding for coded systems requires computation of (18) for / = 1,2,3,4 , b = 1,2,...,log₂ . For each j and 6 , the number of required comparisons is 2( ⁴/2 -l) since the number of possible Matrix

C's that have zero at the b^th bit in the ^h symbol is M⁴/2 . Note that Matrix C can be represented as a tree that has M⁴ leaf nodes since Matrix C consists of four -ary symbols.

In accordance with an embodiment of the invention, the number of computations of (18) is reduced through use of a double pruned tree approach. The double pruned tree approach uses two pruned trees with a reduced number of leaf nodes for use in the comparison operation (18). More specifically, using an algorithm such as the previously detailed ZF or NC algorithm, the tree of Matrix C can be pruned to a tree that has only M² leaf nodes. This is achieved as in previous embodiments by fixing s₂ and s₃ to certain values for each permutation of s₁ and s₄ . For such a pruned tree as in Fig. 5B, L(x_{J b}) in (18) for 7 = 1,4 exists for b = l, 2,..., log₂ . However, L(x_{J b}) in (18) for 7 = 2,3 may not exist for some 6 = 1,2,...,log₂ . Thus, another pruned tree is made so that L(x_{J b}) in (18) for 7 = 2,3 exists for all > = l ,2,...,log₂ . Using these two pruned trees, L(x_{J b}) in (18) are calculated. For each j and b , the number of required comparisons is 2( ²/2 -l) , which is much less than 2( ⁴/2 - l) of the Max- Log decoding. In Table 2, the computational complexities of the Max-Log decoder, the DPT-ZF decoder, and the DPT-NC decoder are shown. One can see that the number of comparisons of the DPT-ZF decoder and the DPT-NC decoder is reduced from the order of M⁴ to the order of M² by pruning the tree in Fig. 5A. Table 2: Computational complexity of Max-Log decoding, DPT-ZF decoding, and DPT-NC decoding for coded systems

As a working example, QPSK is used for s₁,s₂,s₃,s₄ and it is assumed that ZF decoding is employed to reduce the number of options, as in the previous embodiments.

First, a ZF decoder is used to determine two sets of candidate outputs - that can be represented as two pruned trees. For the QPSK case, each pruned tree has 4x4 leaf nodes. Each node corresponds to a set of (s₁,s₂,s₃,s₄), i.e. a Matrix C realization.

For a first set of candidate outputs (also referred to herein as a first pruned tree (PT1), s₁,s₄ could be any symbols in QPSK, and s₂,s₃ are determined using a ZF decoder given s_x,s₄. Since there are 16 permutations of s_l,s₄, this set of candidate outputs contains 16 candidates. The first set will contain entries that cover all of the permutations of s s₄. However, the set may not contain entries that cover all of the permutations of s₂,s₃. This is because there might be no leaf node that has zero at the b^th bit in the f symbol in PT1.

For a second set of candidate outputs (also referred to herein as a second pruned tree (PT2), s₂,s₃ could be any symbols in QPSK, and s₁,s₄ are determined using ZF decoder given s₂,s₃. Since there are 16 permutations of s₂,s₃, this set of candidate outputs contains 16 candidates. The second set of candidates contains entries that covers all of the permutations of s₂,s₃.

To calculate || R-HC-_b ||² for j = 1,4, a search is conducted

through the PT1. To calculate || R-HC_jb ||² for j = 2,3 , a search is conducted

through PT2. As an example, assume j = l. Then the search is conducted through

1

PT1. The cardinality of the set l,b is equal to 2x4x1 x 1. "2" comes from the fact that only half of QPSK symbols have one in the b^th bit for s_} . "1x1" comes from the fact that there is no freedom to choose s₂, s₃ since s₂,s₃ are already determined by ZF decoder given s₁₅s₄. Any QPSK symbol for s₄ can be chosen. That is why there is a "4".

Using ZF decoder, the number of candidate Matrix C realizations, which 1

is the cardinality of ^l,b , is reduced from "4x4" to "1x1", since s₂,s₃ (or s₁₅s₄) are determined by the ZF decoder.

In general, searching time for minimization in min || R -HC_jb ||² and min 11 R - HC_{j b} 11² is reduced by the factor of M² since the tree is pruned by the ZF decoder.

The following is an example of a step by step procedure for DPT-ZF decoder:

1) Given any s₁₅s₄, find s₂,s₃ using ZF algorithm. (Generate PT1 - this tree could also be referred to as the SiS₄-tree because this tree is used to find the metric of any bit of S_j or s₄.

2) Given any s₂,s₃ , find s₁₅ s₄ using ZF algorithm.

(Generate PT2 = - this tree could also be referred to as the s₂s₃-tree because this tree is used to find the metric of any bit of s₂ or s₃. ) 3) Calculate _min \\R-HC°_ib ||², where C° is from PT1 for y = 1,4 and C° is from PT2 for j = 2,3. Note that the minimization is done over

—^■M candidates ofC°

2 ^J'

4) Calculate _min || R-HC)_b ||² , where C)_b is from PT1 for j = 1,4 and C)_b is from PT2 for j = 2,3. 5) Calculate L(x_Jb) = min \\R-HC%\\²- min \\R-HC)_b\\²

Referring now to Figure 4, shown is a flowchart of a method of processing a transmitted M_tx P space-time code matrix where M_t is a number of transmit antennas > 2 and P is a number of time slots > 1, that is a function of K symbols modulated using an M-ary modulation format, M > 2, K > 2. The method begins in block 4-1 with using M_r receive antennas, receiving an M_rx P matrix, where M_r> 1. Next, in block 4-2, the method continues with generating at least two pruned trees, by for each pruned tree: a) fixing a respective at least one of the K symbols to each possible permution of the at least one of the K symbols; and b) for each possible permutation of the at least one symbol, decoding the remaining symbols. The pruned trees are such that they collectively include at least one node that has a one in the b^th bit of the j^th symbol, and at least one node that has a zero in the bth bit of the jth symbol, for b = 1 ,2, ...,log₂M, and j=1 K. The method continues in block

4-3 with, for the b^th bit of the symbol, for b = 1 ,2, ...,log₂M, and j=1 , ...,K,

determining a respective soft decision using the at least two pruned trees.

In the detailed example above, the at least two pruned trees consist of an Sis₄-tree and an S2S3-tree. The sis₄-tree has M² leaf nodes, as does does the s₂s₃-tree. An alternative is to use an Si S₂-tree and an s₃s₄-tree. Another alternative is to use an SiS3-tree and an s₂s₄-tree.

In another example in which only one symbol is fixed in a given tree, a set of four trees, namely an Si-tree, s₂-tree, S3-tree, s₄-tree can be employed. The Sr tree has M nodes. In the s tree, for example, s₂, s₃, s₄ symbols are determined by using another method (such as the zero forcing algorithm). For the metric of any bit of Si symbol, this is looked up in the Si-tree. For the metric of any bit of s₂ symbol, this is looked up in the s₂-tree, and so on.

In yet another example in which three symbols are fixed in a given tree, a set of trees including an SiS₂S3-tree, s₂S3S₄-tree, si s₂s₄-tree, Si S3S₄-tree can be used. The Si S₂S3-tree has ³ leaf nodes. For the metric of any bit of Si symbol, we have to look up Si S₂s₃-tree, SiS₂s₄-tree, and SiS₃s₄-tree. For the metric of any bit of s₂ symbol, we have to look up SiS₂S3-tree, Si S₂s₄-tree, and s₂S3S₄-tree. and so on. This may still reduce complexity especially when M is large. The above examples show how pruned trees can be used where one, two or three symbols are fixed out of four. More generally, as detailed above, when there are K symbols, at least one symbol is fixed and up to K-1 symbols, for each pruned tree. Numerical results

Numerical results for the BEP performance are provided for the sake of illustration only. These results may not necessarily be achieved in a practical realization. The results are obtained by computer simulation. The FFT size is chosen as 1024 which corresponds to a system bandwidth of 10 MHz in the mobile WiMAX OFDMA PHY specifications [3], [7]. The downlink Partial Usage of Subchannel (PUSC) permutation for a 10 MHz bandwidth was used in the simulations as in [7]. Jake's channel model is used in a Pedestrian B environment with a speed of 3 km/h in [7] in the simulations. In Fig. 6, the BEP (bit error probability) vs. E_b/N₀ for M_R = 1, 2 and quadrature phase shift keying (QPSK) signaling is shown. Observe that when there is one receive antenna, M_R = \ , the ES-ZF decoder gives up 4.2 dB in power gain to the ML decoder in return for its reduced complexity compared to the ML decoder for BEP = 10 ³ . However, the ES-NC decoder gives up only 2.1 dB in power gain to the ML decoder for BEP = 10 ³ when one receive antenna is used, while offering worthwhile reduced complexity. When two receive antennas are used, the ES-ZF decoder gives up only 1.4 dB to the ML decoder for BEP = 10 ³ . Significantly, the ES- NC decoder provides virtually the same performance as that of the ML decoder, at reduced complexity. This is an important result as it implies that Matrix C can be implemented in 2 x 2 MIMO systems, greatly improving performance over Matrix A and Matrix B, with reduced decoding complexity using the ES-NC decoder with no performance loss relative to optimal ML decoding.

In Fig. 7, the BEP vs. E_b/N₀ for Μ_τ = 1,2 and QPSK signaling is shown. A 1/2 convolutional code with a constraint length 7 that has the generator

polynomials 171 and 133 in octal format is used for coded system simulations with soft-input Viterbi decoding. For one receive antenna, the DPT-ZF decoder gives up 1.7 dB in power gain to the Max-Log decoder in return for its reduced complexity compared to the Max-Log decoder for BEP = 10 ³ . However, the DPT-NC decoder gives up only 0.8 dB in power gain to the Max-Log decoder for BEP = 10 ³ when one receive antenna is used, while offering worthwhile reduced complexity. When two receive antennas are used, the DPT-ZF decoder gives up only 0.1 dB to the ML decoder for BEP = 10 ³ . Significantly, the DPT-NC decoder provides virtually the same performance as that of the Max-Log decoder, at reduced complexity.

Numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the disclosure may be practiced otherwise than as specifically described herein.

References (All incorporated by reference in their entirety)

[1] S. Sezginer and H. Sari, ^"Full-rate full-diversity 2 x 2 space-time codes for reduced decoding complexity," IEEE Communications Letters, vol. 1 1 , No. 12, pp. 1 - 3, Dec. 2007. [2] S. M. Alamouti, ^"A simple transmit diversity technique for wireless communications," IEEE J. Sel. Areas Commun., vol. 16, pp. 1451-1458, Oct. 1998.

[3] IEEE 802.16e-2005: IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems - Amendment 2: Physical Layer and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands, Feb. 2006.

[4] J.-C. Belfiore, G. Rekaya, and E. Viterbo, ^"The Golden code: a 2 x 2 full rate space-time code with nonvanishing determinants," IEEE Trans. Inf. Theory, vol. 51 , No. 4, pp. 1432-1436, April 2005.

[5] G. B. Giannakis, Z. Liu, X. Ma, and S. Zhou, Space-Time Coding for Broadband Wireless Communications. Wiley, 2007.

[6] T. Kailath, H. Vikalo, and B. Hassibi, ^"MIMO Receive Algorithms," in Space-Time Wireless Systems: From Array Processing to MIMO Communications, (editors: H. Bolcskei, D. Gesbert, C. Papadias, and A. J. van der Veen), Cambridge University Press, 2005. [7] R. Kobeissi, S. Sezginer, and F. Buda, ^"Downlink performance analysis of full-rate STCs in 2 x 2 MIMO WiMAX systems," in Proc. IEEE Veh. Tech. Conf., pp. 1 -5, April 2009.

[8] C. Studer, A. Burg, and H. Bolcskei, ^"Soft-output sphere decoding:

algorithms and VLSI implementation," IEEE J. Sel. Areas Commun., vol. 26, pp. 290- 300, Feb. 2008. [9] C. Wang, E. K. S. Au, R. D. Murch, W. H. Mow, R. S. Cheng, and V. Lau, "On the Performance of the MIMO Zero-Forcing Receiver in the Presence of Channel Estimation Error," IEEE Transactions On Wireless Communications, Vol.6, No.3, March 2007, pp. 805 - 810.

Claims

WE CLAIM:

1 . A method of decoding a transmitted M_tx P space-time code matrix where M_t is a number of transmit antennas > 2 and P is a number of time slots > 1 , that is a function of K symbols, the method comprising: using M_r receive antennas, receiving an M_rx P matrix, where M_r> 1 ; fixing at least one of the K symbols to each possible permution of the at least one of the K symbols; for each possible permutation of the at least one symbol, decoding the remaining symbols; using exhaustive search, finding the permutation of at least one symbol and the decoded remaining symbols that minimizes an objective function.

2. A method of processing a transmitted M_tx P space-time code matrix where M_t is a number of transmit antennas > 2 and P is a number of time slots > 1 , that is a function of K symbols modulated using an M-ary modulation format, M > 2 the method comprising: using M_r receive antennas, receiving an M_rx P matrix, where M_r≥ 1 ; generating at least two pruned trees, by for each pruned tree: a) fixing a respective at least one of the K symbols to each possible permution of the at least one of the K symbols; and b) for each possible permutation of the at least one symbol, decoding the remaining symbols; the at least two pruned trees being such that the at least two prune trees collectively include at least one node that has a one in the b^th bit of the j^th symbol, and at least one node that has a zero in the bth bit of the jth symbol, for b = 1 ,2,...,log₂M, and j=1 , ...,K; for the b^th bit of the j^th symbol, for b = 1 ,2, ... ,log₂M, and j=1 K, determining a respective soft decision using the at least two pruned trees.

3. The method of claim 1 wherein:

K=4, such that the K symbols include a first two symbols and a second two symbols; generating at least two pruned trees comprises: generating a first pruned tree by: a) fixing the first two symbols of the K symbols to each possible permution of the first two of the K symbols; and b) for each possible permutation of the first two symbols, decoding the second two symbols; generating a second pruned tree by: a) fixing the second two symbols to each possible permution of the K symbols; and b) for each possible permutation of the second two symbols, decoding the first two symbols.

4. The method of claim 3 wherein the first two symbols are Si and s₄ and the second two symbols are s₂ and S3.

5. The method of claim 4 wherein for the b^th bit of the j^th symbol, for b = 1 ,2, ... ,log₂M, and j=1 K, determining a respective soft decision using the at least two pruned trees comprises: determining _mi_n || R - HC°_{i b} ||² , where C°_{i b} is from the first pruned

= J> 1 tree for j = 1,4 and C)_b is from the second pruned tree for j = 2,3 ; determining _mm || R - HC) _b ||² , where C) _b is from the first pruned tree for j = 1,4 and C) _b is from the second pruned tree for j = 2,3 ; determining soft decision

L{Xj b ) ⁼ min \\ R - HC°_{j b} \ - min HC) _b \^

where x_j>b is the b^th bit in the symbol in Matrix C , j = 1,2,3,4 , b = l,2,...,log₂ , _j°_b is a set of Matrix C's that have zero at the b^th bit in the f symbol and % _b is a set of Matrix C's that have one at the b^th bit in the symbol, where Matrix C is "Matrix C" from WiMAX standard or space-time code matrix.

6. The method of any preceding claim wherein the matrix is a Ml MO matrix .

7. The method of any preceding claim wherein the space-time code matrix is a Golden code.

8. The method of any one of claims 1 to 5 wherein the Matrix C is "Matrix C" as defined in the WiMAX standard.

9. The method of any preceding claim wherein for each possible permutation of the at least one symbol, decoding the remaining symbols comprises performing Zero Forcing decoding.

10. The method of any preceding claim wherein for each possible permutation of the at least one symbol, decoding the remaining symbols comprises performing null cancelling decoding.

1 1 . The method of any preceding claim wherein for each possible permutation of the at least one symbol, decoding the remaining symbols comprises performing minimum mean square error decoding.

12. The method of any preceding claim wherein M_t = 2, P = 2 and M_r = 1 .

13. The method of any one of claims 1 to 9 wherein M_t = 2, P = 2 and M_r = 2.

14, A computer readable storage medium having stored thereon

instructions for execution by a computer to cause the computer to execute the method of any one of claims 1 to 13.

15. A receiver comprising: at least one antenna; a space-time transmit diversity decoder configured to implement the method of any one of claims 1 to 13.

16. The receiver of claim 15 further comprising an error correction decoder.

17. A space-time transmit diversity decoder configured to implement the method of any one of claims 1 to 13.