## 3 Properties of Different STBCs Signals
### Fourth STBC Option, \(\alpha=c_{3}\)  
Likewise, for \(\alpha=c_{3}\), we write  
\[\begin{split}\mathrm{E}\Big{[}\overline{w}_{q}^{(\alpha=c_{3})}(n )\overline{w}_{q^{\prime}}^{(\alpha,c_{3})}(n+\tau)\Big{]}=\\ \left\{\begin{array}{cc}-2\sigma_{s}^{2}&\text{for}\ (q=0,q^{ \prime}=3,\tau=1),(q=3,q^{\prime}=0,\tau=1),\\ &(q=0,q^{\prime}=1,\tau=3),(q=1,q^{\prime}=0,\tau=-3),\\ &(q=0,q^{\prime}=2,\tau=3),\text{and}\ (q=1,q^{\prime}=0,\tau=-3),\\ -\sigma_{s}^{2}&\text{for}(q=0,q^{\prime}=3,\tau=3),(q=3,q^{\prime}=0,\tau=-3),\\ &(q=1,q^{\prime}=2,\tau=3),(q=2,q^{\prime}=1,\tau=-3),\\ &(q=0,q^{\prime}=2,\tau=7),\text{and}\ (q=2,q^{\prime}=0,\tau=-7),\\ \sigma_{s}^{2}&\text{for}(q=1,q^{\prime}=2,\tau=1),(q=2,q^{\prime}=1,\tau=-1),\\ &(q=0,q^{\prime}=2,\tau=5),(q=2,q^{\prime}=0,\tau=-5),\\ &(q=2,q^{\prime}=3,\tau=5),(q=3,q^{\prime}=2,\tau=-5),\\ &(q=1,q^{\prime}=2,\tau=5),(q=2,q^{\prime}=1,\tau=-5),\\ &(q=1,q^{\prime}=2,\tau=7),\text{and}\ (q=2,q^{\prime}=1,\tau=-7)\\ 2\sigma_{s}^{2}&\text{for}(q=0,q^{\prime}=2,\tau=5),(q=2,q^{\prime}=0,\tau=-5),\\ &(q=2,q^{\prime}=3,\tau=5),\text{and}\ (q=3,q^{\prime}=2,\tau=-5).\end{array}\right.\end{split} \tag{13}\]Using (3), (5), and (13) in (6), we demonstrate  
\[U^{(a=c_{3})}(\tau)=\left\{\begin{array}{ll}-\frac{1}{4}\sigma_{s}^{2}\left(h_{ \mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(3,2)}-h_{\mathcal{SD}}^{(3,1)}h_{ \mathcal{SD}}^{(0,2)}\right)&\text{for $\tau=\pm 1$,}\\ +\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(1,1)}h_{\mathcal{SD}}^{(2,2 )}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(1,2)}\right)\\ -\frac{1}{4}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(1,2 )}-h_{\mathcal{SD}}^{(1,1)}h_{\mathcal{SD}}^{(0,2)}\right)&\text{for $\tau=\pm 3$,}\\ -\frac{1}{4}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(0,2)}\right)\\ -\frac{1}{4}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(3,2)}-h_{\mathcal{SD}}^{(3,1)}h_{\mathcal{SD}}^{(2,2)}\right)\\ -\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(3,2)}-h_{\mathcal{SD}}^{(3,1)}h_{\mathcal{SD}}^{(0,2)}\right)\\ -\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(1,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(1,2)}\right)\\ \frac{1}{4}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(0,2)}\right)&\text{for $\tau=\pm 5$,}\\ +\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(0,2)}\right)\\ +\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(3,2)}-h_{\mathcal{SD}}^{(3,1)}h_{\mathcal{SD}}^{(2,2)}\right)\\ +\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(1,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(1,2)}\right)\\ +\frac{1}{4}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(3,2)}-h_{\mathcal{SD}}^{(3,1)}h_{\mathcal{SD}}^{(2,2)}\right)\\ -\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(0,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(0,2)}\right)&\text{for $\tau=\pm 7$,}\\ +\frac{1}{8}\sigma_{s}^{2}\left(h_{\mathcal{SD}}^{(1,1)}h_{\mathcal{SD}}^{(2, 2)}-h_{\mathcal{SD}}^{(2,1)}h_{\mathcal{SD}}^{(1,2)}\right)&\text{.}\end{array}\right. \tag{14}\]  
It is worth mentioning that this property is applied to any group of STBC options in which \(U^{(a)}(\tau)\) has separating peaks for each code.