- Research
- Open Access
An efficient MIMO scheme with signal space diversity for future mobile communications
- Zhanji Wu†^{1}Email author and
- Xiang Gao†^{1}
https://doi.org/10.1186/s13638-015-0301-x
© Wu and Gao; licensee Springer. 2015
- Received: 23 July 2014
- Accepted: 16 February 2015
- Published: 24 March 2015
Abstract
An efficient wireless transmission scheme with the signal space diversity (SSD) is proposed to improve the performance of multiple-input multiple-output (MIMO) systems in fading channels. By introducing the rotated modulation and space-time component interleaver, the proposed scheme jointly optimizes channel coding, modulation, and MIMO and can improve the link reliability and energy efficiency. An optimum spatial component interleaver is proposed to maximize the MIMO achievable rate. Based on the average mutual information (AMI)-maximization criterion, the optimal rotation angles of real-valued signal and complex-valued QAM signal are investigated for the MIMO scheme. For the iterative demapping and decoding (ID) scheme, a simple genetic algorithm (GA) to search binary convolution code (BCC) is also put forward to match the rotated modulation. Simulation results show that the optimized BCC-coded MIMO scheme with SSD-ID outperforms the turbo-coded MIMO scheme with bit-interleaved coded modulation (BICM)-ID by 1.4 dB signal-to-noise ratio (SNR) gain, while the new scheme has much lower complexity. So, the proposed scheme is simple, efficient, and promising for future wireless communication systems.
Keywords
- Quadrature amplitude modulation (QAM)
- Signal space diversity (SSD)
- Multiple-input multiple-output (MIMO)
- Iterative demodulation and decoding (ID)
- Extrinsic information transfer chart (EXIT)
1 Introduction
Wireless communications have made a great progress in the recent few years. By introducing more advanced technology, 5G will provide higher spectral efficiency, more spectrum resources, and more reliability to meet the growing demand for mobile traffic [1].
Bit-interleaved coded modulation (BICM) is a bandwidth-efficient coded modulation scheme which increases the time diversity in fading channels [2,3]. For its iterative version, BICM with iterative demapping and decoding (BICM-ID), the extrinsic information is transferred between the channel decoder and the soft-in-soft-out demapper, which is like the serial turbo decoder. Multiple-input multiple-output (MIMO) scheme is the extension of the coding theory on the space domain, so it is also named space-time coding (STC) [4]. Foschini proposed a layered space-time (LST) architecture to process multidimensional signals in the space domain [5]. The BICM-LST is a conventional spectral-efficient spatial multiplexing technology to deal with MIMO fading channels, and the BICM-threaded layered space-time (TLST) with a cyclic-shift spiral spatial interleaver is regarded as the most efficient method, because the cyclic-shift spatial interleaver introduces effective space diversity for the codeword on each layer [6]. In general, the BICM-LST can be viewed as the serial concatenation of the channel coding, modulation, and spatial layered multiplexing. Because BICM-LST exhibits a robust diversity performance on fading channels, it is widely deployed by wireless communication standards.
As for the bandwidth-efficient quadrature amplitude modulation (QAM), uncoded rotated multidimensional modulation schemes over independent Rayleigh fading channels were studied in [7] for the single-input single-output (SISO) scheme. Different from the other well-known diversity (time, frequency, code, space), it has an intrinsic modulation diversity, which is named signal space diversity (SSD). Through the combination of constellation rotation and component interleaver, the schemes can achieve very high modulation diversity, and the error performance over fading channels can approach that on the additive white Gaussian noise (AWGN) channels. SSD schemes for SISO system have been extensively researched. In [8], SSD is introduced to the BICM by the means of modifications to the QAM constellation mapper and demapper so as to improve the BICM performance of QAM constellations for broadcasting applications. In [9], a LDPC-coded SSD scheme for multi-level modulation was presented. N.F. Kiyani and J.H. Weber studied the rotated-MPSK SISO BICM-ID system [10,12], which focused on two-dimensional multiphase shift keying (MPSK) scheme. In [11], the performance analysis of BICM-ID with SSD in fading channels is presented. In [13], the extension of BICM-SSD schemes with a non-binary code was proposed. We also proposed coded orthogonal frequency division multiplexing (OFDM) systems with SSD in [14,15]. In [16], the schemes combining SSD with SISO-coded BICM and BICM-ID systems were investigated. It provided a new criterion for determining the optimal rotation angle by maximizing the average mutual information (AMI). For the optimization of BICM-ID system, it proved that SSD can mitigate the different-slope problem of the demapper’s extrinsic information transfer (EXIT) curve under different channels. However, finding well-matched channel codes for given labeling in BICM-ID system with SSD is still a big challenge.
The combination of signal rotation and space-time coding in MIMO system can effectively improve the diversity gain. In order to achieve full diversity, the quasi-orthogonal space time block codes (QOSTBC) with rotating the constellations of half of the complex symbols has been widely discussed in [17-20]. Some specific optimal rotation angles and corresponding optimization criterions for QAM and phase shift keying (PSK) constellations are provided. A rotation-based method that aims at maximizing the minimum distance in the space-time constellation is proposed in [17]. The proposed scheme shows good improvement of the codes compared to their non-rotated counterparts. In [18], the authors considered the design of rotated QOSTBC for the MISO system. The code designs are based mainly on the rank and the determinant criteria, and the optimal rotation angle π/6 can provide full diversity and the optimal coding gain. In [19], the authors proposed to design the signal constellations properly to ensure that the resulting quasi-orthogonal STBCs can guarantee to achieve the full diversity. The optimal rotation angles are determined by maximizing the diversity product. A novel method to exactly derive the coding gain of QSTBC as a function of the rotation angle and the minimum Euclidean distance of two-dimensional constellations is proposed in [20]. A coded MIMO scheme for block-fading channels was proposed in [21], which consists of a channel code and a space-time code. The space-time code is designed based on SSD technique, which allows full spatial multiplexing MIMO transmission and achieves full space diversity. In [22], the uncoded SSD scheme was extended to V-BLAST MIMO systems in order to achieve the maximum diversity gain without additional power or bandwidth consumption. An improved turbo-coded SSD scheme was proposed for MIMO-OFDM BICM system in [23], and the linear minimum mean square error (LMMSE) equalization is utilized for the non-ID MIMO detection. In general, the research of SSD technique in coded MIMO systems is still on the original stage. There are still many open problems. For instance, the optimal rotation angles in current research mainly depend on the maximum product distance introduced in [7]. Unfortunately, this criterion is only valid for the SISO system in high signal-to-noise ratio (SNR) region. As for the coded MIMO scheme, when powerful forward error-correction codes (FECs) are considered, actual SNR can be quite low. Hence, the angle values applied to uncoded SISO system do not lead to the best error performance for the coded modulation MIMO scheme. What is more, current research works mainly focus on local optimizations. For example, most proposed MIMO systems with SSD are only an extension of SISO-SSD system, and all are based on the conventional non-precoding transmitter. The channel coding, QAM modulation, and STC are independent with each other, which is just a straightforward serial concatenation. Hence, the performance of the BICM-LST is still rather far away from the MIMO fading channel capacity. For example, a near-capacity BICM-LST scheme was proposed in [24], which allows the iterative processing of the list sphere detection (LSD) and turbo decoding, but simulation results indicate that the gaps to the MIMO capacity are still more than 2 dB. As each individual optimization becomes mature so far, from the philosophy, it is high time to optimize these key technical elements jointly so as to improve the overall performance.
An improved coded MIMO system based on SSD is proposed for the jointly optimization of constellation rotation angle, spatial component interleaver, and the matching of channel coding and labeling, which is named joint coding and modulation diversity (JCMD), where the terminology ‘coding’ refers to both the channel coding and space-time coding. Firstly, in order to maximize the MIMO achievable rate, an optimum spatial component interleaver is proposed. Secondly, based on the AMI-maximization criterion, the optimal rotation angles of real-valued signal and complex-valued QAM signal are investigated for MIMO schemes, which are different from the SISO scheme. Thirdly, for the JCMD-ID scheme, a simple genetic algorithm (GA) to search binary convolution code (BCC) is put forward to match the rotated QAM modulation. Simulation results show that the optimized BCC-coded JCMD-ID MIMO scheme outperforms the turbo-coded BICM-ID MIMO scheme in [24] by 1.4 dB SNR gain, while the new scheme has much lower complexity.
Throughout this paper, we use bold letters to represent vectors or matrices. (·)^{ T } and (·)^{ H } represent transposition and conjugate transposition, respectively. SNR =E _{ s }/N _{0}, where E _{ s } denotes the average symbol energy per receive antenna and N _{0}=2σ ^{2} denotes the variance of the complex Gaussian noise.
The paper is organized as follows. An improved JCMD MIMO scheme is proposed in Section 2. Theoretical analysis about the achievable rate of JCMD-MIMO for rotated real-valued signals is given in Section 3. Based on the AMI analysis, the optimal rotation angles for JCMD and JCMD-ID MIMO systems are presented in Section 4. Section 5 introduces an outer convolutional code search method for the optimization of JCMD-ID system with optimal rotation angle. Simulation results are presented in Section 6 on fast fading channels. Concluding remarks are offered in Section 7.
2 System model
If perfect CSI is known, we prove that the reverse interleaver is better than other interleavers through the later theoretical analysis and computer simulations. If CSI is unknown at the transmitter, the cyclic-shift interleaver in Equation 3 can be used.
In order to make the fading of I component and that of the Q component as uncorrelated as possible in the time domain, after the spatial Q interleaving, Q components of the mapped symbols in each layer are interleaved through a time-domain pseudo S-random interleaver to reconstruct a new symbol vector \({{\mathbf {s}}_{k}} = \left [s_{k}^{1} \cdots {{s}_{k}^{{N_{L}}}}\right ]^{T}\), where \({s}_{k}^{l}\) denotes the kth symbol at the lth layer after the component interleaving. Afterwards, the symbols are mapped onto N _{ T } transmit antennas via the spatial precoding and then transmitted.
where \({N_{k}^{l}} (I)\) and \({N_{k}^{l}} (Q)\) are identically independently distributed (i.i.d.) Gaussian noise random variables with zero mean and variance of \(\sigma ^{2} = \frac {{N_{0} }}{2}\). For MIMO fading channels, \({\Lambda _{k}^{l}} (I)\) and \({\Lambda _{k}^{l}} (Q)\) are singular values of corresponding sub-channels. This is in net contrast with respect to SISO scheme where the fading coefficients are Rayleigh distributed. That means the modulation diversity of proposed scheme is further extended to the spatial dimension. By denoting \({\mathbf {X}} = \left [ {{\mathbf {X}}_{1},\ldots,{\mathbf {X}}_{N_{L}}} \right ]^{T}\), \({\mathbf {Y}} = \left [ {{\mathbf {Y}}_{1},\ldots,{\mathbf {Y}}_{N_{L}}} \right ]^{T}, {\mathbf {N}} = \left [ {{\mathbf {N}}_{1},\ldots,{\mathbf {N}}_{N_{L}}} \right ]^{T}\), and \({\mathbf {\Lambda }} = {\text {diag}}\left ({{\mathbf {\Lambda }}_{1},\ldots,{\mathbf {\Lambda }}_{N_{L}}} \right)\) representing a (2N _{ L }×2N _{ L }) diagonal matrix, the channel model in Equation 9 can be written in the matrix form as Y=Λ X+N, where \({\mathbf {X}}_{l} = \left [ {{X_{k}^{l}} (I),{X_{k}^{l}} (Q)} \right ]\), \({\mathbf {Y}}_{l} = \left [ {{Y_{k}^{l}} (I),{Y_{k}^{l}} (Q)} \right ]\), \({\mathbf {N}}_{l} = \left [ {{N_{k}^{l}} (I),{N_{k}^{l}} (Q)} \right ]\), and \({\mathbf {\Lambda }}_{l} = {\text {diag}}\left ({{\Lambda _{k}^{l}} (I),{\Lambda _{k}^{l}} (Q)} \right)\).
where \(\Omega _{k} \left ({{{\hat x}}} \right) = - \frac {{\left ({{y_{k}^{I}} - {\lambda _{k}^{I}} {{\hat x}}^{I}} \right)^{2} + \left ({{y_{k}^{Q}} - {\lambda _{k}^{Q}} {{\hat x}}^{Q}} \right)^{2} }}{{2\sigma ^{2} }}\)
For the JCMD system without the iterative demapping and decoding, A(c _{ i,k })=0. Finally, the decoder can utilize the extrinsic values to decode information bits.
In the transmitter, compared with the conventional BICM, the JCMD scheme introduces extra constellation rotation and Q-component interleavers. Constellation rotation does not increase the complexity, because the rotated symbol mapping can be implemented through look-up table operations as the same as the conventional modulation without rotation. Q-component interleavers also can be implemented by the low-complexity index-based look-up table operations.
In the receiver, the soft rotated demapping operation of JCMD system is the same as that of BICM system, which is shown in Equation 11. Q-component de-interleavers also can be implemented by the simple reverse index-based look-up table operations.
3 Theoretical analysis of the achievable rate for rotated real-valued signals
Firstly, since real-valued signals are elementary, we analyze the real-valued transmit signals in MIMO systems.
Lemma 1.
For any constant rank-2 MIMO fading channel with real-valued transmit signals, the constant achievable rate of JCMD-MIMO is not less than that of BICM-MIMO with the conventional uniform power allocation. If and only if the constant eigenvalues are identical, both of them are equal, otherwise the former is greater than the latter.
Proof.
where W is the channel bandwidth, σ ^{2} is the variance of AWGN. As we know, in order to achieve the capacity in the Equation 14, the transmit signals should be Gaussian distributed. Note that the rotation does not change the achievable rate for the BICM-MIMO scheme without the I/Q-component interleaver.
So, comparing Equation 14 with 15, it is easy to come to the conclusion: C _{2}≥C _{1}
If and only if ρ _{2}=ρ _{1}, C _{2}=C _{1}. □
Obviously, when θ=0, the achievable rate is minimum as Equation 14; and when \(\theta = \frac {\pi }{4}\), the achievable rate is maximum as Equation 15.
According to Lemma 1, we can come to the following theorem:
Theorem 1.
For any actual rank-2 MIMO fading channel with real-valued transmit signals, the ergodic achievable rate of JCMD-MIMO is greater than that of BICM-MIMO with the conventional uniform power allocation.
Proof.
For any actual rank-2 MIMO fading channel, the ergodic achievable rate is the mathematical average expectation of the above constant achievable rate over all possible channel eigenvalue realization. Generally speaking, the equal eigenvalue realization is only a small probability event. The case that the channel eigenvalues are different must be existed. According to Lemma 1, the ergodic achievable rate of JCMD-MIMO is greater than that of BICM-MIMO with the conventional uniform power allocation. □
where P is the total transmit power for L layers. Hence, the optimum problem of the achievable rate is to find the optimum \({\boldsymbol {\bar \zeta }}\) so as to maximize C _{ L }. We can reach the following theorem:
Theorem 2.
To maximize the achievable rate of rank-L JCMD-MIMO with a descending-order eigenvalue vector \({\boldsymbol {\bar \rho }} = \{ \rho _{1},\rho _{2},\ldots,\rho _{L} \} \), the optimum Q-component interleaver vector \(\overline {\boldsymbol {\omega }} = \{ \rho _{L},\rho _{L - 1},\ldots,\rho _{1} \}\), that is to say, \(\overline {\boldsymbol {\omega }} \) should be in ascending order, which is just in reverse order of \({\boldsymbol {\bar \rho }}\).
Proof.
Because ρ _{ i }>ρ _{ i+1} and \({\rho ^{\prime }_{i} > \rho ^{\prime }_{i + 1} }\).
So, M _{1}−M _{2}>0; Thus, \(\frac {{M_{1} }}{{M_{2} }} > 1\); and then, \(C_{L} ({\boldsymbol {\eta }}) - C_{L} ({\boldsymbol {\bar \rho ^{\prime }}}) > 0\).
That is to say, we find a counter-example η to have more achievable rate than \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\), which contradicts the assumption of maximum achievable rate \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\). Therefore, \(\overline {\boldsymbol {\omega }} \) should be in ascending order, which is just in reverse order of \({\boldsymbol {\bar \rho }}\). □
So, in Section 2, the reverse Q-component spatial interleaver is applied as Equation 2 so as to have the maximum achievable rate. In fact, if L is an even number, the rank-L JCMD-MIMO with the reverse Q-component spatial interleaver can be viewed as \(\frac {L}{2}\) parallel pairs of rank-2 JCMD-MIMO with eigenvalues {ρ _{ i },ρ _{ L+1−i }}, where \(i \in \left [1,\frac {L}{2}\right ]\). Likewise, if L is an odd number, it can be viewed as the parallel combination of one SISO fading channel with eigenvalue \(\left \{ \rho _{\frac {{L + 1}}{2}} \right \} \) and \(\frac {{L - 1}}{2}\) pairs of rank-2 JCMD-MIMO with eigenvalues {ρ _{ i },ρ _{ L+1−i }}, where \(i \in \left [1,\frac {{L - 1}}{2}\right ]\). Thus, the largest eigenvalue layer couples with the smallest eigenvalue layer, the second largest eigenvalue layer couples with the second smallest eigenvalue layer, and so on. Actually, BICM-MIMO is also one special case of JCMD-MIMO when \(\bar \zeta = \bar \rho \).
Theorem 3.
As for the achievable rate of a rank-L JCMD-MIMO with a descending-order eigenvalue vector \({\boldsymbol {\bar \rho }} = \{ \rho _{1},\rho _{2},\ldots,\rho _{L} \} \), the upper bound is \(C_{L} (\overline {\boldsymbol {\omega }})\), where \(\overline {\boldsymbol {\omega }}\) is in reverse order of \({\boldsymbol {\bar \rho }}\), and the lower bound is the BICM-MIMO achievable rate \(C_{L} ({\boldsymbol {\bar \rho }})\).
Proof.
According to Theorem 2, the maximum achievable rate is \(C_{L} ({\boldsymbol {\bar \omega }})\), we can get \(C \le C_{L} ({\boldsymbol {\bar \omega }})\). In addition, \(C_{L} ({\boldsymbol {\bar \rho }})\) is the minimum achievable rate, which can be proved by the similar math skill as follows.
Because ρ _{ i }>ρ _{ i+1} and \({\rho ^{\prime }_{i} < \rho ^{\prime }_{i + 1} }\).
So, M _{1}−M _{2}<0; Thus, \(\frac {{M_{1} }}{{M_{2} }} < 1\); and then, \(C_{L} ({\boldsymbol {\eta }}) - C_{L} ({\boldsymbol {\bar \rho ^{\prime }}}) < 0\).
That is to say, we find a counter-example η to have small achievable rate than \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\), which contradicts the assumption of minimum achievable rate \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\). Therefore, \(C_{L} ({\boldsymbol {\bar \rho }})\) is the minimum achievable rate. □
4 AMI analysis for complex-valued QAM signals
AMI analysis is an effective method to calculate its achievable rate. From Equation 28, because I(X;Y|Λ) is independent from labeling, it implies that the AMI of CM system is not related to the labeling. However, for the BICM system, the BICM-AMI strongly depends on the labeling. Extensive literature has proven that Gray labeling is optimal for the non-ID BICM system in the AWGN channel [25]. Monte Carlo simulation techniques are useful tools to get the expectation of a complex function by ergodicity of the random variables [26]. The expectation operations in Equations 28 and 29 can be evaluated by using the Monte Carlo simulation techniques. Based on Equation 9, we can get the received symbol before the soft demapper by generating the random coded bits c _{ k } (corresponding to the modulated symbol \({x_{k}^{l}}\) on each layer), the random fading coefficient \({\lambda _{k}^{l}}\) (SVD of i.i.d. Rayleigh-distributed random channel matrix H _{ k }), and the i.i.d. Gaussian noise random variable \({n_{k}^{l}}\). Thus, by using the Monte Carlo simulation techniques, we can estimate the expectation values of Equations 28 and 29 by ergodicity of (c _{ k }, H _{ k }, \({n_{k}^{l}}\)), where P(y _{ l }|x _{ l },λ _{ l }) can be calculated as Equation 12. As for complex-valued QAM signals, the optimum rotation angle usually is different from the value of the real-valued signals, and we can get it by maximizing AMI.
5 Code design for JCMD-ID system
For the non-ID BICM system, Gray mapping has been proved to be optimal. Based on the optimal rotation angle obtained by maximizing BICM-AMI and Gray mapping, the non-ID JCMD system only needs to consider the optimization of channel codes to achieve excellent performance. For the JCMD-ID system, besides the optimal rotation angle, it is also very crucial to choose a pair of a well-matched labeling and an outer channel code by some joint optimization.
The function T _{1} can not be expressed in a closed form. For a given value of the input mutual information I _{ A1}=I(c _{ k };A(c _{ k })) (0≤I _{ A1}≤1) to the demodulator, to compute \(T_{1} (I_{A_{1} },SNR,\theta)\phantom {\dot {i}\!}\), the distributions \({p_{E({c_{k}})}}\left ({\xi |{c_{k}} = x} \right)\) are most conveniently determined by the Monte Carlo simulation (histogram measurements) proposed in [27,28]. The convenient method has been verified to allow an accurate prediction of the SNR-decoding threshold with low complexity [27].
Demapper-matched code design is very crucial for the JCMD-ID MIMO system with rotation. In order to approach the capacity, based on EXIT chart, we propose an optimization method of outer channel codes to match with a given demapper for the JCMD-ID system. We choose the BCC as the channel code. For a given SNR, if two EXIT curves of the demapper and the decoder do not intersect, the iterative decoding can converge, otherwise it cannot converge. Thus, the SNR which makes the two EXIT curves tangent is the SNR convergence threshold, which is also called pinch-off SNR. The objective of JCMD-ID optimization is to find the outer channel code that has the lowest SNR convergence threshold to match with the demapper.
where G(N _{Reg}) denotes the generator polynomial of BCC with N _{Reg} registers. Num is the number of selected statistical samples. \(\overline {\text {SNR}}\) is the pinch-off SNR. For \(\frac {1}{N_{\text {out}}}\) rate BCC, the objective is to find the optimum G _{Opt.}(N _{Reg}) with the lowest pinch-off SNR from \(\left ({2^{N_{\text {Reg}} + 1} - 1} \right)^{N_{\text {out}}}\) generator polynomial candidates. However, the exhaustive searching of a channel code to well match the demapper’s EXIT curve is trivial, especially for N _{Reg} is large.
GA is an efficient optimization algorithm, which is stochastic search techniques based on the mechanism of natural selection and natural genetics [31]. In GA, a genetic representation is required for the individuals in a population. Generator polynomials of BCC \({\mathbf {G}}(N_{\text {Reg}})=\left [{\mathbf {g}}_{1},{\mathbf {g}}_{2},\ldots,{\mathbf {g}}_{N_{\text {out}}}\right ]_{2}\) inherently provides a (N _{Reg}+1)×N _{out}-length binary string \({\mathbf {S}}_{g}=<{\mathbf {g}}_{1},{\mathbf {g}}_{2},\ldots,{\textbf {g}}_{N_{{\text {out}}}}>\phantom {\dot {i}\!}\). g _{ i } is the (N _{Reg}+1)-length binary generator polynomial corresponding to ith (1≤i≤N _{out}) output. Based on the genetic algorithm, an optimization method is proposed as follows.
Step 1: Initial population. Set the current iterative number (number of generations) N _{pop}=0, the number of candidate generator polynomials (population size) N _{ g }, the maximum iterative number N _{max}, crossover probability P _{ c }, and mutation probability P _{ m }. N _{ g } binary strings \(\mathbf {S}_{g}^{i}=< \mathbf {g}_{1}^{i},\mathbf {g}_{2}^{i},\ldots,\mathbf {g}_{N_{\text {out}}}^{i}>\phantom {\dot {i}\!}\) that correspond to the candidate polynomials are randomly initialized, which are denoted by a set \({{\mathbf {C}}^{N_{\text {pop}}}}\phantom {\dot {i}\!}\), where \(\mathbf {g}_{1}^{i},\mathbf {g}_{2}^{i},\ldots,\mathbf {g}_{N_{\text {out}}}^{i} \in \left [ {1,{2^{{N_{\text {Reg}}} + 1}}} \right)\), 1≤i≤N _{ g }.
N _{ g } individuals are selected to breed a new generation with probability proportional to the fitness value. The probability that \({{\mathbf {S}}_{g}^{i}}\) is selected is \(P(i) = \frac {{f\left ({{\mathbf {S}}_{g}^{i}} \right)}}{{\sum \limits _{k = 1}^{{N_{g}}} {f\left ({{\mathbf {S}}_{g}^{k}} \right)} }}\). Based on roulette wheel selection (RWS) algorithm, the tth (1≤t≤N _{ g }) individual selection follows the steps below.
A. Generate a uniform random number χ(t), χ(t)∈[0,1].
B. If \(\sum \limits _{i = 0}^{k - 1} {P(i)} \le \chi (t) < \sum \limits _{i = 1}^{k} {P(i)} (1 \le k \le {N_{g}})\), P(0)=0, \({{\mathbf {S}}_{g}^{k}}\) is selected.
Step 3: Crossover. For two adjacent selected individuals, a random number κ _{ c } from the range [0,1] is generated. Only when κ _{ c }<P _{ c }, the crossover operator is carried out. The crossover point is selected randomly. All bits beyond that point in either string are swapped between the two parent individuals, and then two children individuals are obtained.
Step 4: Mutation. For each individuals, a random number κ _{ m } from the range [ 0,1] is generated. Only when κ _{ m }<P _{ m }, the mutation operator is carried out through one bit flip at random mutation position.
Step 5: Judgment. N _{pop}=N _{pop}+1. After step 2∼4, a new population \({{\mathbf {C}}^{N_{\text {pop}}+1}}\phantom {\dot {i}\!}\) is formed. If N _{pop}<N _{max}, then go to step 2, otherwise stop. The generator polynomial with the lowest pinch-off SNR in \({{\mathbf {C}}^{N_{\text {pop}}+1}}\phantom {\dot {i}\!}\) is chosen as the optimum one.
Using the method above, a rate-half BCC code with N _{Reg}=5 is optimized for the natural labeling 16QAM JCMD-ID system. Based on the AMI analysis in Section 4, the optimal rotation angle for 0.5 rate 16QAM JCMD-ID system is 45°. The optimal generator polynomial is [63,32] _{8}. As shown in Figure 11, the natural labeling demapper’s EXIT curve with 45° rotation keeps a narrow open tunnel with that of BCC decoder, while its non-rotation curve intersects with the decoder, which shows the effect of rotation to the ID system. The other two labeling demappers’ EXIT curves with and without rotation always intersects with the decoder, so the Gray and reference labelings are not suitable for the BCC decoder. Therefore, the natural labeling with 45° rotation matches well with the [63,32] _{8} BCC code and has the best performance.
6 Simulation result
6.1 Results of non-ID JCMD system on fast fading channels
For the non-ID JCMD MIMO system, the optimal Gray labeling and powerful DVB-T2 LDPC coding are used to achieve excellent performance [32]. The optimal rotation angle is obtained by maximizing BICM-AMI in Figure 9. The size of coded block N is 64,800 bits. For the LDPC decoder, the log-belief propagation (BP) algorithm with 30 maximum iterations is utilized. In order to ensure the fairness of the comparison, the SVD precoding is implemented for both the conventional BICM MIMO system and proposed JCMD MIMO system.
SNR gains and gaps to the capacity for JCMD 4×4 MIMO systems
Parameters | Gains compared with | Gaps to the JCMD/Gaussian |
---|---|---|
BICM system (dB) | input Shannon limit (dB) | |
BPSK, 1/2-rate | 2.2 | 1.0/1.5 |
QPSK, 1/2-rate | 0.6 | 1.1/1.9 |
QPSK, 3/4-rate | 3.9 | 0.7/2.3 |
6.2 Results of JCMD-ID system on fast fading channels
For the JCMD-ID 16QAM MIMO system, the optimal rotation angle is obtained by maximizing CM-AMI in Figure 9, and it is 45° at 1/2 rate for 4×4 MIMO systems. In order to confirm our optimization method for JCMD-ID MIMO system, simulations are carried out with proposed [63,32] _{8} BCC-coded JCMD-ID scheme on fast fading channels for 4×4 MIMO systems. The size of coded block N=64,800 bits and the maximal global iterative number is 30. The powerful 1/2 rate DVB-T2 LDPC-coded Gray-labeling BICM and BICM-ID schemes with the same block size are also simulated as the reference, and the same ideal SVD precoding method is used for the conventional BICM and BICM-ID schemes and the proposed JCMD-ID scheme.
For the 4×4 MIMO system with natural labeling, the optimized BCC [63,32] _{8} coded JCMD-ID scheme with 45° rotation and reverse Q interleaver exhibits excellent performance, which is only 1.3 and 1.1 dB away from the Gaussian-input capacity and JCMD-ID limit, respectively. It can obtain about 2.9 dB SNR gain compared with the BCC-coded BICM-ID scheme and JCMD-ID without rotation, which coincides with the above EXIT analysis. In addition, the scheme with the reverse interleaver obtains 0.4 dB SNR gain at BER =10^{−5} compared with that with cyclic-shift Q interleaver, which also proves the above analysis. Furthermore, the optimized JCMD-ID scheme also outperforms the DVB-T2 LDPC-coded BICM-ID scheme and the turbo-coded BICM-ID scheme in [24] by 0.9 and 1.4 dB gains, respectively.
Average decoding complexity for each information bit
Operation | Optimized BCC | Turbo used in [ 24 ] | DVB-T2 LDPC |
---|---|---|---|
Additions | 331 | 816 | 997.5 |
Max ops. | 158 | 288 | 0 |
Look-ups | 0 | 0 | 735 |
7 Conclusions
A high-spectral-efficient JCMD scheme over MIMO fading channels is proposed. By jointly optimizing the component interleaver, the rotation modulation, and the BCC code, this scheme exhibits excellent performance. An optimum spatial component interleaver is proposed to maximize the achievable rate. For real-valued signals, we prove that the achievable rate of JCMD MIMO is greater than that of the conventional BICM MIMO scheme and \(\frac {\pi }{4}\) is the optimal rotation angle. For the rotated QAM, the optimal rotation angles are investigated for the MIMO system according to the maximizing AMI criterion. For the JCMD-ID MIMO system, a simple GA-based search algorithm of BCC generator polynomials is also proposed to match the rotation modulation. Simulation results prove that this new scheme can significantly outperform the conventional turbo-coded BICM-ID scheme over MIMO fading channels by 1.4 dB SNR gain, while it has much lower complexity. In a word, the proposed JCMD scheme is simple, efficient, and robust for the future wireless communication systems.
Declarations
Acknowledgements
This work is sponsored by the National Natural Science Fund (61171101), National Great Science Specific Project (2009ZX03003-011-03) of People’s Republic of China and 2014 Doctorial Innovation Fund of BUPT (CX201426) and the Fundamental Research Funds for the Central Universities.
Authors’ Affiliations
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