stage and noise pre-whitening as the second stage of preprocessing at the receiver resulting in two independent subsystems [@SharmaPapadias03], one of which $$\nonumber
    \tilde{\mathbf{y}}_o = \underbrace{\left[%
\begin{array}{rr}
  \beta    &     \jmath\beta   \\
  \epsilon & -\jmath\epsilon \\
\end{array}%
\right]}_{\mathbf{H}_E^{Q}} \left[\begin{array}{c}
      x_1 \\
      x_3 \\
    \end{array}\right] + \tilde{\mathbf{n}}_o \;,$$ is only a function of the elements of $\mathbf{x}$ with odd index, and the other one is only a function of the elements of $\mathbf{x}$ with even index, $$\nonumber
    \tilde{\mathbf{y}}_e = \underbrace{\left[%
\begin{array}{rr}
  \beta    &     \jmath\beta   \\
  \epsilon & -\jmath\epsilon \\
\end{array}%
\right]}_{\mathbf{H}_E^{Q}} \left[\begin{array}{c}
      x_4 \\
      x_2 \\
    \end{array}\right] + \tilde{\mathbf{n}}_e \;,$$ where $\mathbf{H}_E^{Q}$ is the $2\times 2$ equivalent channel for QSTBC, $\beta = \sqrt{\frac{\lambda+\alpha}{2}}$, $\epsilon = \sqrt{\frac{\lambda-\alpha}{2}}$, $\lambda = \sum_{i=1}^{n_R}\sum_{j=1}^{n_T} |h_{i,j}|^2$, and $\alpha =
\sum_{i=1}^{n_R}2\mathrm{Im}(h_{i,1}^*h_{i,3}+h_{i,4}^*h_{i,2})$. Both subsystems can now be detected separately, which reduces the complexity of the receiver significantly.

In order to get the best performance with respect to error rates and a decoupled system with scalar input and scalar output as in the case of OSTBC, the columns of $\mathbf{H}_E^{Q}$ have to be orthogonal. However, the probability that this occurs for $\mathbf{H}_E^{Q}$ is zero.

For orthogonality, it follows from the scalar product of the columns of $\mathbf{H}_E^{Q}$ that $\alpha$ has to 