# Given the following signalling elements design the optimum receiver?

Given the following signaling elements, \begin{align} g_1(t) &= A \quad \text{for} \quad -T/4 < t \leq 0 \\ g_2(t) &= A \quad \text{for} \quad 0 < t \leq T/4 \\ g_3(t) &= A \quad \text{for} \quad -T/2 < t \leq 0 \\ g_4(t) &= A \quad \text{for} \quad 0 < t \leq T/2 \\ \end{align} I have deduce the orthonormal basis to be, \begin{align} \omega_1(t) &= 1 \quad \text{for} \quad -T/2 < t \leq -T/4 \\ \omega_2(t) &= 1 \quad \text{for} \quad -T/4 < t \leq 0 \\ \omega_3(t) &= 1 \quad \text{for} \quad 0 < t \leq T/4 \\ \omega_4(t) &= 1 \quad \text{for} \quad T/4 < t \leq T/2 \\ \end{align}

And finally I have the vector representation of each signaling element which is given by, \begin{align} \begin{bmatrix} g_1 \\ g_2 \\ g_3 \\ g_4 \end{bmatrix} = \begin{bmatrix} 0 & A & 0 & 0 \\ 0 & 0 & A & 0 \\ A & A & 0 & 0 \\ 0 & 0 & A & A \\ \end{bmatrix} \begin{bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ \omega_4 \end{bmatrix} \end{align} Giving me, \begin{align} v_1 &= \begin{bmatrix} 0 & A & 0 & 0 \end{bmatrix} & v_2 &= \begin{bmatrix} 0 & 0 & A & 0 \\ \end{bmatrix} \\ v_3 &= \begin{bmatrix} A & A & 0 & 0 \\ \end{bmatrix} & v_4 &= \begin{bmatrix} 0 & 0 & A & A \\ \end{bmatrix} \end{align} I am not sure how to use this information to design an optimum receiver. Should I translate the vectors into a QPSK constellation or should I change to orthonormal basis into a 2-dimensional basis?