In this section, we will propose a polar-coded STTD system model in which the estimation of channel information is considered. We assume that the CDI of Rician MIMO channel is known at the transmitter and perfect estimation of CSI is known at the receiver. DEs will be also proposed for polar-coded STTD systems with different numbers of antennas.

### Model of polar-coded STTD system

In Rician fading MIMO channel, the model of each point-to-point channel is comprised of a multiplicative gain coefficient and an additive noise. The gain coefficient follows the Rician distribution \(h_{ij}~\sim ~\frac {h_{ij}}{\sigma _{h}^{2}}\exp \big (-\frac {h_{ij}^{2}~+~\beta ^{2}}{2\sigma _{h}^{2}}\big)I_{0}\big (\frac {h_{ij}\beta }{\sigma _{h}^{2}}\big)\), where *I*_{0}(·) is the zero-order modified Bessel function of the first kind. *β* is the amplitude of the specular signal component which can be calculated from the Rician *K*-factor as \(\beta ~=~\sqrt {2\sigma _{h}^{2}K}\), in which *K* is a pure number. The Rician *K*-factor is often expressed in decibels (dB) in practice, which will be applied in the simulations in this paper. The additive noise of a point-to-point channel model follows the Gaussian distribution \(\mathcal {N} \left (0,\sigma _{n}^{2}\right)\).

As shown in Fig. 2, a polar-coded STTD system model is proposed. It can be regarded as a kind of concatenated coding, where the polar codes are the outer codes and the STC codes are inner codes. The outer polar codes are generated based on transmitter’s CDI, then sent to the STC encoder. The output concatenated codes after passing through the Rician fading MIMO channels are obtained by the receive antennas. The received signals are combined by the STC combiner, then decoded by the polar code decoder. The SC, SCL, and other decoders for polar codes can be adopted in the system.

### Density evolution for polar-coded STTD system with two transmit antennas and one receive antenna

Now, we focus on the DE for polar-coded STTD system to guide the construction of polar codes in the system. We assume that there are two transmit antennas and one receive antenna in the proposed polar-coded STTD system. Alamouti’s scheme is adopted to construct STC. In the system, the output of the STC combiner can be expressed as,

$$ y_{i}=\left(|h_{11}|^{2}+|h_{21}|^{2}\right)x_{i}+h_{11}^{\ast} n_{11}+h_{21}n_{12}^{\ast}, $$

(8)

where *x*_{
i
} is any modulated polar code bit. Each combined signal *y*_{
i
} can be regarded as an output of a fading channel where a polar code bit *x*_{
i
} traverses. Thus, the polar-coded STTD system can be equivalent to a single fading transmission channel for each polar code bit. The gain coefficient of the equivalent channel can be expressed as \(\mathop {\sum }\limits _{q}|h_{q}|^{2}\). The i.i.d variable |*h*_{
q
}|^{2} can be regarded as a square sum of two nonzero mean Gaussian variables, so the variable \(\mathop {\sum }\limits _{q}|h_{q}|^{2}\) follows a noncentral chi-square distribution

$$ {}f_{h} \!\left(\! h\,=\,\mathop{\sum}\limits_{q=1}^{n}|h_{q}|^{2}\! \right)\,=\,\frac{h^{(n\,-\,1)/2}}{2\sigma_{h}^{2}\beta^{n\,-\,1}}\exp\!\left(\,-\,\frac{h\,+\,\beta^{2}}{2\sigma_{h}^{2}}\right)I_{n\,-\,1}\!\left(\!\sqrt{\frac{\beta^{2}h}{\sigma_{h}^{4}}}\right)\!{,} $$

(9)

where *I*_{n − 1}(·) is the *n* − 1-order modified Bessel function of the first kind. For 2 × 1 polar-coded STTD system, the gain coefficient of the equivalent channel is expressed as *h* = |*h*_{11}|^{2} + |*h*_{21}|^{2}. Now, each output *y*_{
i
} can be regarded as a Gaussian random variable with mean *h* and variance \(h\sigma _{n}^{2}\). The conditional PDF of *y*_{
i
} can be expressed as \(p(y_{i}|x_{i},h)~=~\frac {1}{\sqrt {2\pi h\sigma _{n}^{2}}}\exp {\left (-\frac {(y_{i}~-~hx)^{2}}{2h\sigma _{n}^{2}}\right)}\), and the initial LLR is obtained by

$$ L_{W}=\ln{\frac{p(y_{i}|x_{i}=-1,h)}{p(y_{i}|x_{i}=1,h)}}=-\frac{2y_{i}}{\sigma_{n}^{2}}\qquad. $$

(10)

The initial LLR *L*_{
W
} is a Gaussian variable with mean \(2h/\sigma _{n}^{2}\) and variance \(4h/\sigma _{n}^{2}\). Now, we can derive the PDF of the initial LLR by using the distribution of *h*. We assume that each *h*_{
ij
} is a normalized Rician fading factor. Substituting *n* = 2 to function (9), we can obtain the distribution of *h*. Finally, the PDF of the initial LLR can be expressed as

$$ {{\begin{aligned} \mathbf{a}_{W}(z)\,=\,\int_{0}^{\infty}\frac{\sigma_{n}}{\sqrt{8\pi}\beta}\exp \!\left(\! {-}\frac{\left(z-\frac{2}{\sigma_{n}^{2}}h\right)^{2}}{8h/\sigma_{n}^{2}}-h-\beta^{2} \!\right)\!I_{1}(\sqrt{4\beta^{2}h})\mathrm{d}h {.} \end{aligned}}} $$

(11)

Thus, densities of subchannels can be calculated by (4). Corresponding probabilities of incorrect messages can also be calculated. Then, the information set can be selected.

Now, we focus on the symmetry of the proposed DE, which is a key property when using DE in symmetric channels. The symmetry of DE is defined in [22]. If the density **a**_{
W
}(*z*) satisfies

$$ \mathbf{a}_{W}(z)=\mathbf{a}_{W}(-z)\exp(z), $$

(12)

the density **a**_{
W
}(*z*) is symmetric. When substituting (11) to (12), we find that the equation is true and the initial density for 2 × 1 polar-coded STTD system is symmetric.

### Density evolution for polar-coded STTD system with two transmit antennas and two receive antennas

We now analyze the polar-coded STTD system corresponding to Rician fading 2 × 2 MIMO channel. The output of the STC combiner can be expressed as

$$ \begin{aligned} y_{i}&=\left(|h_{11}|^{2}+|h_{21}|^{2}+|h_{12}|^{2}+|h_{22}|^{2}\right)x_{i}\\ &\quad+h_{11}^{\ast} n_{11}+h_{21}n_{12}^{\ast}+h_{12}^{\ast} n_{21}+h_{22}n_{22}^{\ast}.\\ \end{aligned} $$

(13)

Now, the gain coefficient of the equivalent channel is expressed as *h* = |*h*_{11}|^{2} + |*h*_{21}|^{2} + |*h*_{12}|^{2} + |*h*_{22}|^{2}. Each output *y*_{
i
} can still be regarded as a Gaussian random variable with mean *h* and variance \(h\sigma _{n}^{2}\). Substitute *n* = 4 to function (9) to get the distribution of *h*. The PDF of the initial LLR is as follows

$$ {{\begin{aligned} \mathbf{a}_{W}(z)\,=\,\int_{0}^{\infty}\!\frac{\sigma_{n}h}{\sqrt{8\pi}\beta^{3}}\exp\!\left(\!-\frac{\left(z\,-\,\frac{2}{\sigma_{n}^{2}}h\right)^{2}}{8h/\sigma_{n}^{2}}-\!h-\!\beta^{2}\!\right)\!I_{3}(\!\sqrt{4\beta^{2}h})\mathrm{d}h. \end{aligned}}} $$

(14)

When substituting (14) to (12), we find that the initial density for 2 × 2 polar-coded STTD system is symmetric.

### Density evolution for polar-coded STTD system with four transmit antennas and two receive antennas

The generate matrix of STC for four transmit antennas is given as follows,

$$ \mathbf{G}_{4} =\left[ \begin{array}{llll} x_{1} & x_{2} & x_{3} & x_{4}^{\ast} \\ -x_{2}^{\ast} & x_{1}^{\ast} & -x_{4} & x_{3}^{\ast} \\ -x_{3}^{\ast} & x_{4} & x_{1}^{\ast} & -x_{2}^{\ast} \\ -x_{4}^{\ast} & -x_{3} & x_{2} & x_{1} \end{array}\right]. $$

(15)

We adopt this STC scheme in the 4 × 2 polar-coded STTD system. Now, the output of the combiner can be expressed as

$$ \begin{aligned} y_{i}\!&=\!(|h_{11}|^{2}\,+\,|h_{21}|^{2}\,+\,|h_{31}|^{2}\,+\,|h_{41}|^{2}\,+\,|h_{12}|^{2}\,+\,|h_{22}|^{2}\\ &\quad+\!|h_{32}|^{2}\,+\,|h_{42}|^{2})x_{i}\,+\,h_{11}^{\ast} n_{11}\,+\,h_{21}n_{12}^{\ast}\,+\,h_{31}n_{13}^{\ast}\\ &\quad+\!h_{41}^{\ast} n_{14}\,+\,h_{12}^{\ast} n_{21}\,+\,h_{22}n_{22}^{\ast}\,+\,h_{32}n_{23}^{\ast}\,+\,h_{42}^{\ast} n_{24}.\\ \end{aligned} $$

(16)

The distributions of gain coefficient and additive noise for each output signal *y*_{
i
} are same respectively. The gain coefficient of the equivalent channel is expressed as *h* = |*h*_{11}|^{2} + |*h*_{21}|^{2} + |*h*_{31}|^{2} + |*h*_{41}|^{2} + |*h*_{12}|^{2} + |*h*_{22}|^{2} + |*h*_{32}|^{2} + |*h*_{42}|^{2}. We can substitute *n* = 8 to function (9). Now, the PDF of the initial LLR can be expressed as

$$ {{\begin{aligned} {}\mathbf{a}_{W}(z)\,=\,\int_{0}^{\infty}\!\frac{\sigma_{n}h^{3}}{\sqrt{8\pi}\beta^{7}}\exp\!\left(\!-\frac{\left(z\,-\,\frac{2}{\sigma_{n}^{2}}h \right)^{2}}{8h/\sigma_{n}^{2}}-\!h-\!\beta^{2}\!\right)\!I_{7}(\!\sqrt{4\beta^{2}h})\mathrm{d}h. \end{aligned}}} $$

(17)

This initial density for 4 × 2 polar-coded STTD system is also symmetric obviously.