No-zero-entry full diversity space-time block codes with linear receivers

Abstract

In MIMO systems, Toeplitz codes, overlapped Alamouti codes (OACs), and embedded Alamouti codes (EACs) can achieve full diversity when linear receivers are used. However, these codes have a large number of zero entries in their codeword matrix. Due to the zero entries in the design, the peak-to-average power ratio (PAPR) is high, and the transmitting antennas need to be switched on and off imposing severe hardware constrains. To solve this problem, in this paper, we propose a design scheme for no-zero-entry full diversity space-time block codes (NZE-STBCs). New no-zero-entry Toeplitz codes (called NZE-TCs) are designed first, and are then combined/overlaid with orthogonal space-time block code, Alamotui code, to construct no-zero-entry overlapped Alamouti codes (NZE-OACs) and no-zero-entry embedding Alamouti codes (NZE-EACs). The full diversity properties of proposed NZE-TCs, NZE-OACs, and NZE-EACs with linear receiver are derived. Simulation results show that our proposed codes outperform the Toeplitz codes, OACs, and EACs under peak power constraint, while performing the same under average power constraint.

Appendixes

1.1 Appendix 1—Proof of Lemma 2

Firstly, we rewrite Toeplitz matrix $\mathbf{\mathcal{T}}\left(\mathbf{v},\mathrm{L},\mathrm{K}\right)$ given in (5) as

$$ \mathbf{\mathcal{T}}\left(\boldsymbol{v},L,K\right)={\left[\begin{array}{ccc}\hfill {\boldsymbol{T}}_1\hfill & \hfill {\boldsymbol{T}}_2\hfill & \hfill {\boldsymbol{T}}_3\hfill \end{array}\right]}^T $$

(45)

where T ₁ and T ₃ are matrices of size (K-1) × L constructed by the K-1 first rows and the K-1 last rows of the Toeplitz matrix $\mathbf{\mathcal{T}}\left(\boldsymbol{v},L,K\right)$, respectively, and T ₂ is a matrix of size (L-K + 1) × L constructed by the remaining rows of $\mathbf{\mathcal{T}}\left(\boldsymbol{v},L,K\right)$. Next, we rewrite NZE-Toeplitz matrix (10) as

(46)

where , 0 is the all-zero matrix of size (L-K + 1) × L. It is not difficult to check that

(47)

Hence, we have

(48)

The first inequality holds because ${\mathbf{\mathcal{T}}}^{\mathrm{H}}\mathbf{\mathcal{T}}$ and ℬ ₁ ^H ℬ ₁ are positive semidefinite matrices, and the second inequality comes from the result in Lemma 1. This means inequality (10) holds. The proof for inequality (11) is similar.

1.2 Appendix 2—Proof of Theorem 2

Lemma 3

With F(v,M,L) given in (24) and G(v,M,L) in (26), we always have

$$ det\left({\boldsymbol{G}}^{\mathrm{H}}\left(\boldsymbol{v},M,L\right)\boldsymbol{G}\left(\boldsymbol{v},M,L\right)\right)\ge det\left({\boldsymbol{F}}^{\mathrm{H}}\left(\boldsymbol{v},M,L\right)\boldsymbol{F}\left(\boldsymbol{v},M,L\right)\right) $$

(49)

A proof of Lemma 3 is similar as that of Lemma 2.

By considering the equivalent channel matrix ℋ of the NZE-OAC in (28) and using Lemma 3, we have

(50)

On the other hand, Theorem 2 in [22] has shown that

$$ det\left({\boldsymbol{F}}^{\mathrm{H}}\left(\boldsymbol{h},M,L\right)\boldsymbol{F}\left(\boldsymbol{h},M,L\right)\right)\ge c{\left\Vert \boldsymbol{h}\right\Vert}^{2L} $$

(51)

with any nonzero h, and c is a positive constant independent of h.

From (50) and (51), we conclude that matrix ℋ ^H ℋ is nonsingular for any nonzero h. ℋ is the equivalent channel matrix of the NZE-OACs given in (28).