Full-Diversity Space-Time Error Correcting Codes with Low-Complexity Receivers

2.1. Space-Time Code Design Criteria

Let us assume a coherent scenario and an optimal maximum likelihood STB code detection. The pairwise error probability (PEP) is defined as the probability of estimating a codeword at the receiver while has been sent. To minimize the pairwise error probability, the space-time code must fulfill the following constraints [18].

1. (i)

   The Rank Criterion. Maximize the minimum rank of the matrix .

    
2. (ii)

   The Determinant Criterion. Maximize the minimum product of the nonzero eigenvalues, ( ), of the matrix . This criterion maximizes the coding gain.

    

The maximum diversity advantage in this context is . Space-time codes that achieve such a diversity are called full diversity space-time codes [1, 18].

2.2. Definition of a STECC

A STECC is a space-time code defined as an symbol matrix based on the modulation of basic binary matrices , which can be grouped to form a binary matrix where each entry is a binary -tuple so that is the modulation efficiency. Matrix is a spatial rearrangement of a FEC codeword of . is called the inner code. Its length is equal to .

2.3. STECCs: Theoretical Analysis

We consider, in this subsection, the family of rectangular STECCs proposed in [16] and we verify that they do not fulfill the preceding criteria, and thus do not achieve full diversity.

The STECC in [16] is defined from an one-half coding rate systematic block code of length . Its codewords are given by

(2)

where , , represent the information bits and stands for the mod-2 addition. Let ( ) be the binary matrix associated to a space-time error correcting codeword. It is defined as [16]

(3)

where , , represents an information binary -tuple. Assuming a -ary quadrature amplitude modulation ( -QAM) where , the STECC codeword which corresponds to is defined by

(4)

where , , represents the -QAM symbol associated to the th binary -tuple denoted by and represents the -QAM symbol associated to the binary -tuple denoted by . To prove that the rank criterion is not realized, we verify that it is possible to find two STECC codewords such that their difference matrix has a rank of 1 inferior to the maximum possible rank .

Let , thus

(5)

Let thus . Let us denote by and the STECC codewords associated to and , respectively. If is odd, then the difference matrix has all entries of the second row equal to 0. In the other case ( is even), all the columns of are equal. Therefore, for all values of , the difference matrix has a rank equal to and as a consequence this family of STECCs does not achieve a full transmit diversity.

Furthermore, if we consider the minimum Hamming distance ( ) of the basic binary code defined in (2), one can see that and for , . For example, the Hamming weight of this basic binary codeword matrix for all is equal to 4. As a consequence, inner FEC codes defined for will not enable to increase the maximum time diversity achieved by the STECC.

In the following, we focus only on the cases and in order to illustrate the construction of full diversity STECCs.

3.1. Full-Diversity STECC Designs

In order to realize the full spatial diversity, we recall that the binary rank criterion is a sufficient but not necessary condition to guarantee a full spatial diversity [12]. We apply the unified construction proposed in [13] to ensure a full spatial diversity for high-order modulations.

In [12], the authors prove that if every nonzero codeword of a linear binary code matrix has a maximum rank over the binary field , then for a binary phase shift keying (BPSK) transmission, the STECC achieves full spatial diversity, that is, . Moreover, it is demonstrated in [13] that if a STECC achieves full spatial diversity for BPSK transmission then using the Lu and Kumar construction (unified construction), we can obtain a full diversity STECC for high-order modulations based on full rank linear binary code matrices.

It yields a sufficient but not necessary condition on the linear binary inner code matrix to guarantee full diversity STECCs.

3.2. Threaded Layering Approach

To maximize the STECC diversity, we consider the threaded layering approach [5]. We assume rectangular STECCs of size . To design full diversity STB code, the threaded layering approach consists in splitting information symbols into disjoint threads. The threads must be active over the transmission intervals. For each layer (thread) and at each transmission interval, symbols of this layer are transmitted. Threads use equally often the transmit antennas. To ensure a maximum diversity, each thread must achieve a maximum diversity when symbols corresponding to the other threads are put to zero and threads must be transparent to each other. This can be realized by affecting weighting numbers to each thread such that resulting threads span disjoint algebraic subspaces. These numbers are "Diophantine number".

In the case of STECCs, linear combinations are applied on binary elements which greatly relaxes the constraints to achieve full diversity STECCs. The number of threads is taken equal to the number of transmit antennas. The threaded layering set is defined, for , by

(7)

where denotes the mod- operation. Table 1 shows the threaded layering for and structures using 2 threads. We associate the information symbols (resp., redundancy symbols) to the first layer (resp., second layer). Let us denote by the information symbol vector and by the vector associated to the redundancy symbols. is an integer permutation matrix such that the STECC, built up from , , and a diophantine number, achieves a full spatial diversity ( must also keep properties of STECCs defined in (6), i.e., each entry of a space-time codeword matrix is composed of one modulation symbol).

3.2.1. Full-Diversity STECC

In this case, it was verified in [19] that a full diversity STECC can be defined by

(8)

where , whatever and (to ensure an energy efficiency). As the determinant of the difference codeword matrix depends on the value of , thus to maximize the coding gain we must carefully select the diophantine number. In [19], it was proved that is the optimal value for 4-QAM and 16-QAM. In this paper, by applying the binary rank criterion and the unified construction of [13] we verify that a full spatial diversity can be achieved without the necessity of a diophantine number ( ). For , (the same result can be obtained for the other possible 3-tuple) the associated linear binary code matrix is defined by

(9)

It clearly appears that the first and the second row of are linearly independent over the binary field . Thus for a BPSK transmission, the STECC constructed from achieves a full spatial diversity. Therefore, using the unified construction presented in [13], the STECC built up from this linear binary code matrix ensures a full spatial diversity for any order QAM modulation. We also note that this construction can be extended to phase shift keying (PSK) and pulse amplitude (PAM) modulations.

3.2.2. Full-Diversity STECC

The inner code corresponds to the extended Hamming code. Due to the fact that the all-one vector is a codeword, the binary rank criterion is false whatever the spatial arrangement into a binary matrix. We thus apply the threaded layering approach.

The binary matrix is

(10)

and the associated space-time error correcting codeword is

(11)

To ensure full diversity diophantine number is chosen such that and . In that case, the maximization of the coding gain yields an optimum value for depending of the modulation order. However, in practice the STECC will be serially concatenated to an outer FEC and it was shown in [20] that the asymptotic global coding gain is independent of the choice of the parameter provided that and . In [21], we proposed a way to construct full diversity STECCs defined from an inner half-rate invertible linear binary codes.

4.1. Reconstruction of a Product Code from a Concatenated STECC

As mentioned previously to achieve the optimal rate-diversity tradeoff (6) thus the maximum rate of inner codes are necessarily equal to in order to ensure a full diversity STECC. We consider systematic linear binary inner codes. Thus, to construct a full diversity STECC we need binary -tuples , that correspond to the information sequences and binary -tuples that correspond to the redundancy sequences. Thus for the redundancy sequences can be written as

(12)

where , and stands for mod-2 addition. It can be space-time formatted to provide space-time codes achieving full spatial diversity using the threaded layering approach presented in the previous section or the unified construction [13].

Let be a linear FEC code (outer code), where denotes the code length and its dimension. For the concatenated STECC, , represent information codewords and for , , is a linear combination of codewords defined by (12). After applying a sophisticated space-time formatter, we obtain space-time error correcting codewords , to be transmitted. The transmitter scheme is represented in Figure 1. An interleaver is used to protect the information against burst errors, and to benefit from the time diversity.

A product code is defined as a serial concatenation of two FEC codes denoted and . It consists in placing information bits in a array. is applied to encode rows of the array and the columns of the resulting array are encoded using . The obtained array is a codeword of the product code . Its minimum Hamming distance is the product of minimum Hamming distances of elementary constituent codes. In the case of concatenated STECCs, the FEC codewords , , form a product code represented as

(13)

where the inner code (resp., outer code) is applied along columns (resp., rows). We note that the linear outer FEC has no constraints on its rate neither on its length. Moreover, the information rate of the inner code can be increased by reducing the spatial diversity of the STECC according to the optimal rate-diversity tradeoff [13].

When full diversity algebraic STB codes (like the perfect STB codes, LDCs, …) are serially concatenated with linear FEC codes, the maximum order of diversity that can be achieved is equal to where is the minimum Hamming distance of the outer code. For concatenated full diversity STECCs, the maximum diversity order is equal to , where is the minimum Hamming distance of the inner code. Thus, concatenated STECCs are better adapted to a serial concatenation with linear outer forward error correcting codes.

4.2. Turbo Product Code Decoding for Concatenated STECC

The existence of a product code for concatenated STECCs enables a better information exchange between the inner code decoder, the outer code decoder and the STECC detector through the application of turbo product code decoding principles [22].

As it is summarized in Figure 2, the turbo product code decoding is based on the reliability information exchange between two elementary soft input soft output (SISO) decoders: the inner one and the maximum a posteriori (MAP) FEC decoder. Each elementary decoder benefits from two inputs: the soft values (intrinsic information) delivered by a detector, and the previous elementary decoder extrinsic information (a priori information). So as to ensure the convergence, weighting coefficients ( ) are applied to the a priori information generated by the FEC decoder.

Assuming a nonfrequency selective block fading channel and perfect synchronization, at each receive antenna, the observation is the superposition of attenuated symbols simultaneously transmitted from transmit antennas and a circular complex Additive White Gaussian Noise. To detect the symbols transmitted from a given antenna, one has to take into account the interference of symbols corresponding to the other antennas. Maximum likelihood detection is possible but its complexity may be high and depends on the modulation order. Linear equalization such as successive interference cancellation enables to reduce the computation cost [23–25]. It consists in providing the equalizer with the estimation of the interfering symbols and cancelling it in the observation. The equalizer consists of a feedforward filter and a feedback filter that can be optimized so as to minimize the mean square error at its output (in such case it is referred as MMSE interference canceller). In [17], we proposed for the concatenated STECC an iterative receiver structure composed by a MAP symbol detector over one symbol duration, which takes a priori soft information from the SISO turbo product code decoder. This structure exploits the channel diversity. But unfortunately, the complexity of MAP detector remains high especially for high-order modulations. To reduce the complexity of the detector, we propose an hybrid structure mixing the MMSE interference canceller and the turbo product code decoder. The resulting receiver scheme is given in Figure 3. The MMSE interference canceller does not take into account the STECC existence. It considers each symbol duration independently of the others and assumes that each received sample only depends on symbols simultaneously transmitted. It thus works as if symbols were transmitted using a spatial multiplexing scheme. The time correlation between transmitted symbols is taken into account in the product code decoder.

Complexity Analysis

In the MMSE turbo equalizer presented in [25], the MMSE interference canceller is the same as the one used in this paper. It is followed by an inner code decoder to take into account the STECC structure, an outer code decoder and another inner code decoder before going back to the equalizer. The structure proposed in this paper, based on the turbo product code decoding algorithm, has a lower complexity since one iteration consists of only one inner code decoding instead of two. Nevertheless, it outperforms the receiver given in [25] thanks to the identification of a turbo product code and to an optimization of the extrinsic information exchange (see Figure 3).

At the detection stage, the MMSE interference canceller is a linear filter, thus its complexity is determined by the size of matrices to be inverted. Moreover, the proposed receiver structure deals with concatenated STECCs as a spatial multiplexing scheme concatenated with an outer product code. Therefore, the MMSE interference canceller must invert a square matrix of dimension at each iteration. An STECC detection thus requires inversions of matrices of size , that can be carried out in parallel. Considering a similar transmitter and receiver scheme involving an usual full rate algebraic STB code (like perfect STB code, LDC, TAST code), we need one inversion of a matrix of size . Indeed, at the receiver, an algebraic STB code is equivalent to a spatial multiplexing scheme with receive antennas and transmit antennas. Concatenated STECCs thus enable lower complexity receivers compared to linear dispersion codes concatenated to the same outer code.

5.1. Performance of the Full-Diversity STECC

Figure 4 compares the performance of the full diversity STECC with those of the STECC [16] and the Alamouti scheme [2], without concatenation with an outer code.

In that case, a maximum likelihood (ML) detection is considered at the receiver and as space-time error correcting codewords (resp., Alamouti space-time codewords) are independent from each others, the BER performance remains the same for every multiple of (resp., 2). One can see that the BER slope of the full diversity STECC for and is equal to , which confirms the full spatial diversity of this structure. On the other hand, as it was expected by the theoretical analysis in the Section 2.3, the BER slope of the STECC is equal to which means that this transmission scheme cannot fully exploit the transmit diversity. For a gain of 5.3 dB is achieved at a BER of with the full spatial diversity STECC. Moreover, the full diversity STECC performs 0.5 dB worse compared to the Alamouti scheme which is satisfactory as it is obtained without taking into account the outer FEC code. When , the full diversity STECC benefits from the additional time diversity of the channel. For a , the gain over the Alamouti scheme is equal to 0.5 dB.

5.2. Optimization of the Concatenated STECC Iterative Receivers

In Figure 5, we compare the performance of the MMSE interference canceller with a SISO turbo product code decoder versus that of the MMSE turbo equalizer [25] for the concatenated full diversity STECC. Simulations are carried out for (roughly quasifast fading channel) and (slow fading channel), respectively, in order to observe the influence of the time diversity on the performance of the receiver structures. In addition to the computation cost reduction per iteration (one inner decoding instead of two per iteration), the hybrid scheme enables a SNR gain for a same BER value. Another advantage of product code identification is the stopping criterion [22] that can be applied to stop the iterative process as soon as a product codeword is detected, yielding a power consumption saving.

For a and for (resp., ) a gain roughly equal to 0.9 dB (resp., 0.8 dB) is offered by the proposed iterative receiver with respect to the existing one. This hybrid receiver scheme is thus selected.

5.3. Performance of Concatenated Full-Diversity STECCs

We compare for and the performance of the concatenated full diversity STECC with the concatenated STECC [16] in order to further highlight the advantage of optimizing STECCs from a space-time point of view.

In Figure 6, we compare the performance of the concatenated full diversity STECC versus the concatenated STECC for (the time duration to transmit one STECC codeword) and using 4-QAM Gray-mapped constellation. The higher the coherence time ( ), the more significant the spatial diversity effect on performance. For a BER = and for the proposed transmission scheme outperforms by 0.6 dB the one presented in [16]. The gain increases with . Indeed for and (resp., ), the gain becomes equal to 2.6 dB for a BER of (resp., 1.7 dB for a ). Note that the computation cost at the receiver side is the same for both schemes.

Figure 7 shows the effect of the error correction capability of the outer code on the performance measured in terms of BER. We consider a slow fading channel ( ). At a and using the -rate convolutional code as an outer code, a gain of 0.9 dB is offered by exploiting the spatial diversity. At the same BER and taking into account the -rate convolutional code as an outer code, the gain achieved by the concatenated full diversity STECC with respect to the suboptimal-diversity concatenated STECC is reduced to 0.7 dB. When the error correction capability of the FEC outer code increases, the time diversity becomes more influent than spatial diversity on performance. Concatenated STECCs exploit efficiently the time diversity. Furthermore, considering the scheme with the outer code , the gain observed in Figure 7 for is less than the gain observed in Figure 6 for . The information length is indeed higher in the scheme with .

5.4. Puncturing

A drawback of concatenated STECCs is their low coding rate. However, puncturing can be used to increase the system throughput. Unfortunately, puncturing the inner code induces a degradation of the transmit diversity. In order to guarantee a full spatial diversity, we apply puncturing on the outer code. We consider -rate and -rate punctured convolutional codes generated from the half-rate parent convolutional code .

In Figure 8, the outer code is punctured so as to get a coding rate of . For a BER of and , the gain of the full diversity scheme compared to the suboptimal one is equal to 1.2 dB. In Figure 9, the outer code coding rate is . For a BER of , the full diversity scheme outperforms the suboptimal one by 4 dB.

The outer code puncturing yields a decrease of its error-correction capability and thus a lower exploitation of the available time diversity. Maximization of the space-diversity is all the more essential.