How to find orthonormal basis function in the following digital communication problem?
Let's call your $4$ signals $f_0$, $f_1$, $f_2$ &amp; $f_3$. 
Just by eyeballing you could have a basis of two function where 

b0 = f0, for 0 &lt; t &lt; Ts/2, 0 otherwise and
b1 = f0, for Ts/2 &lt; t &lt; Ts, 0 otherwise and


So you basically have a one first half ramp up in the first half and one second half ramo down. The signals can be expressed simply as

f0 =  b0 + b1
f1 = -b0 - b1
f2 =  b0 - b1
f3 = -b0 + b1


So it's clearly a base. It's also orthogonal since $b_0  b_1$ is zero for all $t$. It's probably not normal but could be easily made so by proper scaling.