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*.

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}. □

Furthermore, we will prove that \(\frac {\pi }{4}\) is the optimum rotation angle for real-valued transmit signals in the JCMD-MIMO scheme. Let us consider a general *θ*-rotated real-valued transmit signal, which is rotated by *θ* compared with the conventional real-valued signal. Thus, the transmit power of I component and that of Q component on layer 1 are \(\frac {P}{2}\cos ^{2} \theta \) and \(\frac {P}{2}\sin ^{2} \theta \), respectively, and the total transmit power on layer 1 is also \(\frac {P}{2}\). Therefore, the received power of I component and that of Q component are \(\frac {{\rho _{2} P}}{2}\cos ^{2} \theta \) and \(\frac {{\rho _{1} P}}{2}\sin ^{2} \theta \), respectively. So, the total received symbol power on layer 1 is \(\frac {{(\rho _{1} \sin ^{2} \theta + \rho _{2} \cos ^{2} \theta)}}{2}P\). Likewise, the total received symbol power on layer 2 is \(\frac {{(\rho _{2} \sin ^{2} \theta + \rho _{1} \cos ^{2} \theta)}}{2}P\). Thus, we can get the following achievable rate:

$$ {\fontsize{7.8pt}{9.6pt}\selectfont{\begin{aligned} {} C(\theta) &\,=\, \frac{W}{2} \!\!\cdot\! {\log_{2}}\!\left\{ \!\left[\! {1 \,+\, \frac{{\left({\rho_{1}}{{\sin }^{2}}\theta \,+\, {\rho_{2}}{{\cos }^{2}}\theta \right)}}{{2{\sigma^{2}}}}P} \!\right] \!\!\cdot\!\! \left[\!\! {1 \,+\, \frac{{\left({\rho_{1}}{{\cos }^{2}}\theta \,+\, {\rho_{2}}{{\sin }^{2}}\theta \right)}}{{2{\sigma^{2}}}}P}\! \right]\! \right\} \\ &\,=\, \frac{W}{2} \cdot {\log_{2}}\left[1 \,+\, \frac{{{\rho_{1}} \!+\ {\rho_{2}}}}{{2{\sigma^{2}}}}P \,+\, \left({{\rho_{1}}{\rho_{2}} + \frac{{{{\sin }^{2}}(2\theta){{({\rho_{1}} - {\rho_{2}})}^{2}}}}{4}} \right)\frac{{{P^{2}}}}{{4{\sigma^{4}}}} \right]. \\ \end{aligned}}} $$

((16))

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. □

Generally speaking, assuming a rank-*L* MIMO with a descending-order eigenvalue vector \(\boldsymbol {\bar {\rho }} = \{\rho _{1},\rho _{2},\ldots,\rho _{L}\} \), Q-component interleaver only changes the order of Q components on *L* layers to another eigenvalue vector \(\boldsymbol {\bar {\zeta }} = \{ \zeta _{1},\zeta _{2},\ldots,\zeta _{L} \} \), where \(\boldsymbol {\bar {\zeta }}\) is just another arrangement order of \(\boldsymbol {\bar {\rho }}\) corresponding to the output order of Q-component spatial interleaver. Hence, due to the orthogonal modulation, JCMD-MIMO can be viewed as *L* parallel fading channels with an eigenvalue vector \(\boldsymbol {{\bar \upsilon }}=\frac {{{\boldsymbol {\bar \rho + \bar \zeta }}}}{{{2}}}\). So, the achievable rate of rank-*L* JCMD-MIMO is shown as the following:

$$ \begin{aligned} C_{L} ({\boldsymbol{\bar \zeta}}) &= \frac{W}{2} \cdot \sum\limits_{i = 1}^{L} {\log_{2} \left[ {1 + \frac{{(\rho_{i} + \xi_{i})}}{{2\sigma^{2} L}}P} \right]} \\ &= \frac{W}{2} \cdot \log_{2} \left[ {\prod\limits_{i = 1}^{L} {\left({1 + \frac{{(\rho_{i} + \xi_{i})}}{{2\sigma^{2} L}}P} \right)}} \right], \end{aligned} $$

((17))

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*.

Assuming someone claims to find a non-increasing-order vector \({\boldsymbol {\bar \rho ^{\prime }}}\) other than \(\overline {\boldsymbol {\omega }}\) to have maximum achievable rate \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\), we must can find a pair of \(\left \{ \rho ^{\prime }_{i},\rho ^{\prime }_{i + 1} \right \} \) from \({\boldsymbol {\bar \rho ^{\prime }}}\) to satisfy \(\rho ^{\prime }_{i} > \rho ^{\prime }_{i + 1} \). And then, we can construct a new vector \({\boldsymbol {\eta }} = \left \{ \rho ^{\prime }_{1},\rho ^{\prime }_{2},\ldots,\rho ^{\prime }_{i - 1},\rho ^{\prime }_{i + 1},\rho ^{\prime }_{i,} \rho ^{\prime }_{i + 2},\ldots,\rho ^{\prime }_{L} \right \}\), which only changes the order of \(\left \{ \rho ^{\prime }_{i},\rho ^{\prime }_{i + 1} \right \} \) from \({\boldsymbol {\bar \rho ^{\prime }}}\). So, we can compute the difference of \(\frac {{C_{L} ({\boldsymbol {\eta }}) - C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})}}{{W/2}}\),

$$ {\fontsize{7.6}{6}{\begin{aligned} {} \frac{{C_{L} ({\boldsymbol{\eta}}) \,-\, C_{L} ({\boldsymbol{\bar \rho^{\prime}}})}}{{{W/2}}} &= \sum\limits_{k = 1}^{L} {\log_{2}\! \left[\! {1 \,+\, \frac{{(\rho_{k} \,+\, \eta_{k})}}{{2\sigma^{2} L}}P} \right]} \,-\, \sum\limits_{k = 1}^{L} {\log_{2}\! \left[ {1 \,+\, \frac{{\left(\rho_{k} \,+\, \rho^{\prime}_{k} \right)}}{{2\sigma^{2} L}}P} \right]} \\ &= \log_{2} \left[ {1 + \frac{{(\rho_{i} + \eta_{i})}}{{2\sigma^{2} L}}P} \right] + \log_{2} \left[ {1 + \frac{{(\rho_{i + 1} + \eta_{i + 1})}}{{2\sigma^{2} L}}P} \right] \\ &\quad- \log_{2} \!\left[ {1 + \frac{{\left(\rho_{i} + \rho^{\prime}_{i}\right)}}{{2\sigma^{2} L}}P} \right] \!- \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right] \\ &= \log_{2}\! \left[ {1 \,+\, \frac{{\left(\rho_{i} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right] + \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i} \right)}}{{2\sigma^{2} L}}P} \right] \\ &\quad- \log_{2}\! \left[ {1 \,+\, \frac{{\left(\rho_{i} + \rho^{\prime}_{i} \right)}}{{2\sigma^{2} L}}P} \right] \,-\, \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right]. \\ \end{aligned}}} $$

((18))

Let a constant \(c = \frac {P}{{2\sigma ^{2} L}}\), and then,

$$ \frac{{C_{L} ({\boldsymbol{\eta }}) - C_{L} ({\boldsymbol{\bar \rho^{\prime}}})}}{{{W/2}}} = \log_{2} \frac{{M_{1} }}{{M_{2}}}, $$

((19))

where

$$ \begin{aligned} M_{1} &= 1 + c\left(\rho_{i} + \rho^{\prime}_{i} + \rho_{i + 1} + \rho^{\prime}_{i + 1} \right)\\ &\quad+ c^{2} \left(\rho_{i} + \rho^{\prime}_{i + 1} \right)\left(\rho_{i + 1} + \rho^{\prime}_{i} \right) > 0, \end{aligned} $$

((20))

$$ \begin{aligned} M_{2} &= 1 + c\left(\rho_{i} + \rho^{\prime}_{i} + \rho_{i + 1} + \rho^{\prime}_{i + 1} \right)\\ &\quad+ c^{2} \left(\rho_{i} + \rho^{\prime}_{i} \right)\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right) > 0. \end{aligned} $$

((21))

So, the difference

$$ M_{1} - M_{2} = c^{2} (\rho_{i} - \rho_{i + 1})(\rho^{\prime}_{i} - \rho^{\prime}_{i + 1}). $$

((22))

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.

Assuming someone claims to find a non-descending-order vector \({\boldsymbol {\bar \rho ^{\prime }}}\) other than \(\overline {\boldsymbol {\rho }} \) to have minimum achievable rate \(C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})\), we must can find a pair of \(\left \{ \rho ^{\prime }_{i},\rho ^{\prime }_{i + 1} \right \} \) from \({\boldsymbol {\bar \rho ^{\prime }}}\) to satisfy \(\rho ^{\prime }_{i} < \rho ^{\prime }_{i + 1} \). And then, we can construct a new vector \({\boldsymbol {\eta }} = \left \{ \rho ^{\prime }_{1},\rho ^{\prime }_{2},\ldots,\rho ^{\prime }_{i - 1},\rho ^{\prime }_{i + 1},\rho ^{\prime }_{i,} \rho ^{\prime }_{i + 2},\ldots,\rho ^{\prime }_{L} \right \}\), which only changes the order of \(\left \{ \rho ^{\prime }_{i},\rho ^{\prime }_{i + 1} \right \} \) from \({\boldsymbol {\bar \rho ^{\prime }}}\). So, we can compute the difference of \(\frac {{C_{L} ({\boldsymbol {\eta }}) - C_{L} ({\boldsymbol {\bar \rho ^{\prime }}})}}{{{W/2}}}\),

$$ {\fontsize{7.8}{6}{\begin{aligned} {} \frac{{C_{L} ({\boldsymbol{\eta }}) \,-\, C_{L} ({\boldsymbol{\bar \rho^{\prime}}})}}{{{W/2}}} &= \sum\limits_{k = 1}^{L} {\log_{2}\! \left[\! {1 \,+\, \frac{{(\rho_{k} \,+\, \eta_{k})}}{{2\sigma^{2} L}}P} \right]} \,-\, \sum\limits_{k = 1}^{L} {\log_{2}\! \left[\! {1 \,+\, \frac{{\left(\rho_{k} \,+\, \rho^{\prime}_{k} \right)}}{{2\sigma^{2} L}}P} \right]}\\ &= \log_{2} \left[ {1 + \frac{{(\rho_{i} + \eta_{i})}}{{2\sigma^{2} L}}P} \right] \,+\, \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \eta_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right] \\ &\quad- \log_{2} \left[ {1 \,+\, \frac{{\left(\rho_{i} + \rho^{\prime}_{i} \right)}}{{2\sigma^{2} L}}P} \right] \,-\, \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right]\\ &= \log_{2} \left[ {1 \,+\, \frac{{\left(\rho_{i} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right] \,+\, \log_{2} \left[ {1 + \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i} \right)}}{{2\sigma^{2} L}}P} \right] \\ &\quad- \log_{2} \left[ {1 \,+\, \frac{{\left(\rho_{i} \,+\, \rho^{\prime}_{i} \right)}}{{2\sigma^{2} L}}P} \right] \,-\, \log_{2} \left[ {1 \,+\, \frac{{\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right)}}{{2\sigma^{2} L}}P} \right]. \\ \end{aligned}}} $$

((23))

Let a constant \(c = \frac {P}{{2\sigma ^{2} L}}\), and then,

$$ \frac{{C_{L} ({\boldsymbol{\eta }}) - C_{L} ({\boldsymbol{\bar \rho^{\prime}}})}}{{{W/2}}} = \log_{2} \frac{{M_{1} }}{{M_{2} }}, $$

((24))

where

$$ \begin{aligned} M_{1} &= 1 + c\left(\rho_{i} + \rho^{\prime}_{i} + \rho_{i + 1} + \rho^{\prime}_{i + 1} \right)\\ &\quad+ c^{2} \left(\rho_{i} + \rho^{\prime}_{i + 1} \right)\left(\rho_{i + 1} + \rho^{\prime}_{i} \right) > 0, \end{aligned} $$

((25))

$$ \begin{aligned} M_{2} &= 1 + c\left(\rho_{i} + \rho^{\prime}_{i} + \rho_{i + 1} + \rho^{\prime}_{i + 1} \right)\\ &\quad+ c^{2} \left(\rho_{i} + \rho^{\prime}_{i} \right)\left(\rho_{i + 1} + \rho^{\prime}_{i + 1} \right) > 0. \end{aligned} $$

((26))

So, the difference

$$ M_{1} - M_{2} = c^{2} (\rho_{i} - \rho_{i + 1})\left(\rho^{\prime}_{i} - \rho^{\prime}_{i + 1} \right). $$

((27))

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. □