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 $f_0$, $f_1$, $f_2$ & $f_3$. Just by eyeballing you could have a basis of two function where
b0 = f0, for 0 < t < Ts/2, 0 otherwise and
b1 = f0, for Ts/2 < t < 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.
Even more simpler than @Hilmar's answer, the signals $f_0$ and $f_2$ are orthogonal just by eyeballing the pictures and recalling basic ideas about integration giving the area under the curve. Thus, the signals are $f_0, -f_0, f_2, -f_2$ with respect to the orthogonal basis $\{f_0, f_2\}$. The basis could be made a orthonormal basis (unit-energy basis signals) if absolutely needed.
Interestingly, the above description shows that the four signals are a bi-orthogonal signal set -- a $4$-QAM signal set rotated by $\pi/4$ in signal space -- for which the symbol error probabilities and bit error probabilities have been studied and documented throroughly. The bi-orthogonality is not as easily noticed with the basis used in Hilmar's answer.
As a minor note, if we use the Gram-Schmidt orthogonalization procedure (as suggested by @MattL in a comment) starting with $f_0$ and continuing with $f_1, f_2, f_3$, we end up with $\{f_0, f_2\}$ as the orthogonal basis for this set of $4$ signals.
