An efficient MIMO scheme with signal space diversity for future mobile communications

Based on the BICM-TLST scheme, a system model of N _L -layer MIMO-JCMD-ID is shown in Figure 1. Perfect channel state information (CSI) is assumed to be known at both the transmitter and the receiver. In Figure 1, the iterative feedback processing is depicted in dashed lines. Without loss of generality, a rank- L N _R ×N _T MIMO system with L nonzero eigenvalues is assumed, where N _R and N _T are the number of receive and transmit antennas, respectively, and N _L ≤L≤ min{N _R ,N _T }. In the transmitter, K information bits B=(b ₁,b ₂,…,b _K ) ^T are encoded and interleaved to yield the coded bit sequence C=(c ₁,c ₂,…,c _N ) ^T . Afterwards, m-tuple coded bits are mapped to a complex symbol x _k =x _k (I)+j·x _k (Q), which is chosen from a 2 ^m -ary rotated QAM constellation set \(\chi = \left \{{\hat x}_{1}, {\hat x}_{2},\ldots,{\hat x}_{2^{m}} \right \}\) according to some optimal angle. Each symbol x _k has one Q-component x _k (Q) and one I-component x _k (I). The rotated mapped symbol sequence is first mapped onto N _L layers in a round robin manner. Afterwards, the conventional cyclic-shift spatial interleaver of TLST is used for different symbols on N _L layers to exploit both the space and time diversity as the following, because the cyclic-shift spatial interleaver allows the codewords to be distributed on all layers:

$$ {w}_{k}^{l} = {x}_{k}^{i}, \ \ \ \ \ l = (i + k -2)\bmod {N_{L}} + 1, $$

(1)

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.

Firstly, since real-valued signals are elementary, we analyze the real-valued transmit signals in MIMO systems.

Proof.

A simple rank-2 MIMO case with two eigenvalues ρ ₁ and ρ ₂ is illustrated in Figure 2. Due to the well-known SVD, BICM-MIMO can be viewed as two parallel fading channels with two eigenvalues ρ ₁ and ρ ₂ for spatial layer 1 and layer 2, respectively. Two fading amplitude coefficients of layer 1 and layer 2 are the corresponding singular values \(\sqrt {\rho _{1}} \) and \(\sqrt {\rho _{2}} \), respectively, as shown in the left half of Figure 2. Thus, given the same transmit power \(\frac {P}{2}\) on each layer for one real-valued symbol, the received symbol power of layer 1 and that of layer 2 are \(\frac {{\rho _{1} P}}{2}\) and \(\frac {{\rho _{2} P}}{2}\), respectively, where P is the total transmit power for two layers. So, according to Shannon’s theory, the achievable rate of BICM-MIMO is shown as the following:

$$ \begin{aligned} C_{1} &= \frac{W}{2} \cdot \log_{2} \left[ \left({1 + \frac{{\rho_{1} P}}{{2\sigma^{2} }}} \right)\left({1 + \frac{{\rho_{2} P}}{{2\sigma^{2} }}} \right) \right]\\&= \frac{W}{2} \cdot \log_{2} \left({1 + \frac{{\rho_{1} + \rho_{2} }}{{2\sigma^{2} }}P + \frac{{\rho_{1} \rho_{2} }}{{4\sigma^{4} }}P^{2}} \right), \end{aligned} $$

(14)

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.

For the JCMD-MIMO scheme, we consider the \(\frac {\pi }{4}\)-rotated real-valued transmit signal, which is rotated by \(\frac {\pi }{4}\) compared with the conventional real-valued signal. Due to the orthogonal modulation, the transmit powers of I component and that of Q component on each layer are both \(\frac {P}{4}\). For JCMD-MIMO, after the spatial Q-component de-interleaver at the receiver, the fading amplitude coefficient of I component is different from that of Q component in one symbol, that is to say, one is \(\sqrt {\rho _{1}} \), and another is \(\sqrt {\rho _{2}} \). Thus, the received power of I component is also different from that of Q component, that is to say, one is \(\frac {{\rho _{1} P}}{4}\), and another is \(\frac {{\rho _{2} P}}{4}\). So, the total received symbol power is \(\frac {{(\rho _{1} + \rho _{2})P}}{4}\) both for layer 1 and layer 2. Thus, JCMD-MIMO can be viewed as two parallel fading channels with identical fading amplitude coefficient \(\sqrt {\frac {{\rho _{1} + \rho _{2} }}{2}} \) for both layer 1 and layer 2, as shown in the right half of Figure 2. In the receiver, after the phase compensation, the received signal also can be viewed as the real-valued signal with the power \(\frac {{(\rho _{1} + \rho _{2})P}}{4}\). Therefore, the achievable rate of JCMD-MIMO is shown as the following:

$$ \begin{aligned} C_{2} &= \frac{W}{2} \cdot \log_{2} \left[ {1 + \frac{{(\rho_{1} + \rho_{2})}}{{4\sigma^{2} }}P} \right]^{2} \\&= \frac{W}{2} \cdot \log_{2} \left[ {1 + \frac{{\rho_{1} + \rho_{2} }}{{2\sigma^{2} }}P + \frac{{(\rho_{1} + \rho_{2})^{2} }}{{16\sigma^{4} }}P^{2}} \right]. \end{aligned} $$

(15)

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:

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:

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 \(\bar \zeta = \bar \rho \).

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.

Figure 3 shows how 2×2 MIMO CM-AMI varies with the rotation angle θ for real-valued binary phase shift keying (BPSK) symbols. Since I(X;Y|Λ)=I(C _k ;Y|Λ), the I _CM and I _BICM are the same for BPSK. Both for low and high SNR, e.g., at SNR =−5 dB and SNR =−15 dB, the optimal rotation angles are always θ= 45°, which coincides with our analysis in Section 3.

Figures 4 and 5 show how 4×4 MIMO CM-AMI and BICM-AMI vary with the rotation angle θ for QPSK and 16QAM, respectively. The AMIs of proposed systems with two kinds of spatial Q-component interleavers (reverse and cyclic-shift) are plotted in the same figures. Examples of low SNR and high SNR are presented. In Figures 4 and 5, for both spatial Q interleaving algorithms, the optimal rotation angles just have slight difference in high SNR. For CM-AMI, the optimal angle of QPSK at SNR =−3 dB is 45°, while the optimal angle at SNR = 11 dB is about 30° and 29° for the cyclic-shift interleaver and the reverse interleaver, respectively. For BICM-AMI, the optimal angle of QPSK at SNR =−3 dB is 0°, while the optimal angle at SNR = 11 dB is about 26° and 27° for the cyclic-shift interleaver and the reverse interleaver, respectively.

Based on the AMI maximization criterion, the optimal angle corresponds to the maximum AMI value. Figure 6 shows the CM-AMI and BICM-AMI curves that correspond to the optimal angles in Figures 4 and 5 for 4×4 MIMO systems. In Figure 6, CM-AMI and BICM-AMI for the reverse interleaver described in Equation 2 are always not less than that of the cyclic-shift interleaver described in Equation 3 both for QPSK and 16QAM. These observations concide well with Theorem 2.

Based on maximizing AMI, we can also obtain the relationship between optimal angle and SNR. The optimal angles for CM and BICM systems vary with SNR for QPSK and 16QAM, which is plotted in Figure 7. The optimal rotation angle of CM is always bigger than that of BICM. When SNR increases, the optimal angle of CM becomes smaller, but the optimal angle of BICM becomes bigger.

For a given modulation order m, according to the AMI value I corresponding to optimal rotation angle in Figure 6, the optimal code rate for a given SNR can be calculated as \(R= \frac {I}{{m \cdot N_{L} }}\). Therefore, we can get the relationship between the optimal code rate R and SNR, which is shown in Figure 8.

Thus, for a given operating SNR, we can obtain the optimum rotation angle in Figure 7 and the optimal code rate in Figure 8. Furthermore, we can get the relationship between code rate R and optimal rotation angle, which is shown in Figure 9 for QPSK and 16QAM. It provides a good reference to select an optimal rotation angle for a given code rate. For instance, for code rate = 0.5, the optimal angles for 4×4 MIMO-BICM QPSK and 16QAM are 18° and 0°, respectively. According to Figures 7 and 9, at low SNR or low code rate, non-rotation is the best for BICM, which implies that the channel coding dominates the BICM performance, while 45° rotation is the best for CM, which indicates that the rotation symmetry affects the CM performance. However, at high SNR or high code rate, the signal space diversity dominates the performance both for BICM and CM, so BICM-AMI can approach CM-AMI, and the optimum angle of BICM with the optimum reverse spatial interleaver is also close to that of CM with the same interleaver. Note that the rotation angle is just related to the code rate and modulation. So, in practice, the rotation symbol mapper can be implemented through a look-up table that is calculated and stored in advance for a given modulation order and code rate, which is the same as the conventional symbol mapper. Hence, it does not introduce additional processing complexity and delay.

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].

Qiuliang et al. [16] verified that the SSD technique can affect the demapper’s EXIT curve for the SISO BICM-ID SSD system. For the MIMO system, we analyze the effect of the SSD technique to different demapper labelings. Different 16QAM labelings (natural, Gray, and reference [16]) are shown in Figure 10. For the 1/2-rate coded 16QAM 4×4 MIMO system, the corresponding 16QAM EXIT curves with 45° rotation and non-rotation are depicted in Figure 11. The doping technique is used for error-floor removal [30] and the doping rate P=100. For Natural and Gray labelings, the demappers’ EXIT curves with rotation have a larger slope than that without rotation. For the referenced labeling, the demapper’s EXIT curve with rotation is always above the curve without rotation.

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.