Optimal space-time coding under iterative processing

Abstract

In this paper, we deal with the design of a full-rate space-time block coding (STBC) scheme optimized for linear iterative decoding over fast fading multiple-input multiple-output (MIMO) channel. A general and simple coding scheme called diagonal threaded space-time (DTST) code is presented for an arbitrary number of transmit and receive antennas. Theoretical analysis shows that DTST code associated with linear iterative decoding tends towards full diversity performance while providing maximum MIMO multiplexing gain. Simulation results confirm the ability of DTST to outperform the state-of-the-art STBC and conventional spatial data multiplexing schemes under iterative processing.

Appendix

Let us first derive the general expression of received sample from antenna $j \in [1,N_{r}]$ at sampling time $p \in [1,T]$. From (3), we can write:

$$ [\mathbf{Y}]_{j,p} = \sum\limits_{i=1}^{N_{t}} [\mathbf{H}]_{j,i} [\mathbf{X}]_{i,p} + [\mathbf{W}]_{j,p}. $$

(53)

Let us now consider DTST encoding as presented in Section 4. By using (32) and setting $i=\lfloor l+p-2 \rfloor _{N_{t}}+1$, Eq. (53) may be rewritten for the DTST case as:

$$ [\mathbf{Y}]_{j,p} = \sum\limits_{l=1}^{L} \Big([\mathbf{H}]_{j,\lfloor l+p-2 \rfloor_{N_{t}}+1} \cdot [\mathbf{u}_{l}]_{p} \Big) + [\mathbf{W}]_{j,p} $$

(54)

where:

$$ [\mathbf{u}_{l}]_{p} = \frac{1}{\sqrt{L}} {\boldsymbol{\Theta}} \mathbf{x}_{l} $$

(55)

$$ =\frac{1}{\sqrt{L}}\sum\limits_{n=1}^{N_{t}} [\mathbf{\Theta}]_{p,n} [\mathbf{x}_{l}]_{n} $$

(56)

$$ =\frac{1}{\sqrt{L}}\sum\limits_{n=1}^{N_{t}} [\mathbf{\Theta}]_{p,n} \cdot x_{(l-1)N_{t}+n}$$

(57)

We can see that $[\mathbf {u}_{l}]_{p}$ is only function of $[ x_{(l-1)N_{t}+1},..,x_{(l-1)N_{t}+N_{t}}]$. In other words $[\mathbf {u}_{l}]_{p}$ is not linked to $[\mathbf {u}_{l'}]_{p}$ for $l \neq l'$. As a result, (54) may be factorized as:

$$\begin{array}{rll} [\mathbf{Y}]_{j,p} &=& \frac{1}{\sqrt{L}} \sum\limits_{l=1}^{L} \sum\limits_{n=1}^{N_{t}} \Big([\mathbf{H}]_{j,\lfloor l+p-2 \rfloor_{N_{t}}+1}\\ &&\qquad\qquad\qquad \cdot [\mathbf{\Theta}]_{p,n} \cdot x_{(l-1)N_{t}+n}\Big)+ [\mathbf{W}]_{j,p}.\\ \end{array} $$

(58)

Since DTST codes are complex linear STBC, Eq. (5) can be invoked. This yields that Eq. (54) may be rewritten in the following form:

$$ [\mathbf{Y}]_{j,p} = \sum\limits_{k=1}^{Q} \big[\underline{\mathbf{H}}\big]_{(p-1)N_{r}+j,k} \cdot x_{k} + [\mathbf{W}]_{j,p}. $$

(59)

Let us write $k=(l-1)N_{t}+n$ for $l\in [1,L]$ and $n \in [1,N_{t}]$. By identifying (58) and (59), we obtain:

$$ \big[\underline{\mathbf{H}}\big]_{(p-1)N_{r}+j,(l-1)N_{t}+n} = \frac{1}{\sqrt{L}} [\mathbf{\Theta}]_{p,n} [\mathbf{H}]_{j,\lfloor l+p-2 \rfloor_{N_{t}}+1}. $$

(60)

The Euclidean norm of $\underline {\mathbf {h}}_{k}$ for $l\in [1,L]$ and $n\in [1,N_{t}]$ becomes:

$$ \|\underline{\mathbf{h}}_{k} \|^{2} = \sum\limits_{p=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}}\Big|\big[\underline{\mathbf{H}}\big]_{(p-1)N_{r}+j,(l-1)N_{t}+n}\Big|^{2}$$

(61)

$$=\frac{1}{L}\sum\limits_{p=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}} \Big|[\mathbf{\Theta}]_{p,n}\Big|^{2} \cdot \Big| [\mathbf{H}]_{j,\lfloor l+p-2 \rfloor_{N_{t}}+1}\Big|^{2}.$$

(62)

By construction of the precoding matrix $\mathbf {\Theta }$, we have

$$ \Big|[\mathbf{\Theta}]_{p,n}\Big|^{2} = \frac{1}{N_{t}} \quad \forall p,n \in [1,N_{t}]. $$

(63)

On the other side, we can notice:

$$ \sum\limits_{p=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}} \Big| [\mathbf{H}]_{j,\lfloor l+p-2 \rfloor_{N_{t}}+1}\Big|^{2} = \sum\limits_{i=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}} \Big| [\mathbf{H}]_{j,i}\Big|^{2}. $$

(64)

Finally, it yields:

$$\|\underline{\mathbf{h}}_{k} \|^{2} = \frac{1}{LN_{t}}\sum\limits_{i=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}} \Big| [\mathbf{H}]_{j,i}\Big|^{2}$$

(65)

$$= \frac{1}{LN_{t}}\sum\limits_{i=1}^{N_{t}} \sum\limits_{j=1}^{N_{r}} |h_{ij}|^{2}$$

(66)

$$=\frac{1}{L N_{t}}\operatorname{tr}\left[\mathbf{H}^{H} \mathbf{H}\right] $$

(67)