How to find orthonormal basis function in the following digital communication problem? I'm trying to express the following set of signals on orthonormal basis functions. These signals are to be transmitted using a 4-ary modulation scheme.How do i write ψ1(t)and ψ2(t) for the same. Also how to find orthonormal function set for given set of signals.
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.