I'm trying to resolve an excercise of telecomunication that ask me to crate an ortonrmale base for this set of waveform. $\ s_{1,2,3,4} (t) = \pm Acos(2 \pi f_0t+\varphi_0) \pm Asin(2 \pi f_0t+\varphi_0)$ $\displaystyle s_{5,6,7,8} (t) = \pm 3Acos(2 \pi f_0t+\varphi_0) \pm Asin(2 \pi f_0t+\varphi_0)$
It's clear to me that the 8 waveforms are depending with each other and they are linear combinations of $ \cos(2 \pi f_0t+\varphi_0)$ e $ \sin(2 \pi f_0t+\varphi_0)$, so, the orthonormal base will be of dimensions 2.
Now my problem is I'm not able to find the coefficients for the orthonormal base. According to my book, the results are
$\ \varphi_1 = \sqrt {2/T} cos(2 \pi f_0t+\varphi_0)$ $\ \varphi_2 = - \sqrt {2/T} sin(2 \pi f_0t+\varphi_0)$
where T is the symbol period $\ 0<t<T$
Someone can explain to me a general process to resolve this type of exercise?