4.2 Geometric representation of modulation signals

Dimension of the signal set is 1 with $E_1=s_{11}^2$ and $E_2=s_{21}^2$ .

${\psi }_m(t)=\frac{s_m(t)}{\sqrt{E_s}}$ where $E_s=\int_0^T s_m(t)^2\,d t=\frac{A^{2}T}{4}$

$s_1=\left(\begin{array}{c}\sqrt{E_s}\\ 0\\ 0\\ 0\end{array}\right)$ , $s_2=\left(\begin{array}{c}0\\ \sqrt{E_s}\\ 0\\ 0\end{array}\right)$ , $s_3=\left(\begin{array}{c}0\\ 0\\ \sqrt{E_s}\\ 0\end{array}\right)$ , and $s_4=\left(\begin{array}{c}0\\ 0\\ 0\\ \sqrt{E_s}\end{array}\right)$

Set of 4 equal energy biorthogonal signals. $s_1(t)=s(t)$ , $s_2(t)=s^{\perp }(t)$ , $s_3(t)=-s(t)$ , $s_4(t)=-s^{\perp }(t)$ .

The orthonormal basis ${\psi }_1(t)=\frac{s(t)}{\sqrt{E_s}}$ , ${\psi }_2(t)=\frac{s^{\perp }(t)}{\sqrt{E_s}}$ where $E_s=\int_0^T s_m(t)^2\,d t$

$s_1=\left(\begin{array}{c}\sqrt{E_s}\\ 0\end{array}\right)$ , $s_2=\left(\begin{array}{c}0\\ \sqrt{E_s}\end{array}\right)$ , $s_3=\left(\begin{array}{c}-\sqrt{E_s}\\ 0\end{array}\right)$ , $s_4=\left(\begin{array}{c}0\\ -\sqrt{E_s}\end{array}\right)$ . The four signals can be geometrically represented using the 4-vector of projection coefficients $s_1$ , $s_2$ , $s_3$ , and $s_4$ as a set of constellation points.