# 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.

• You need to apply the Gram-Schmidt algorithm, which is probably described in the same textbook the problem came from. If you have a specific problem applying the algorithm, please add it to your question.
– MBaz
Commented Mar 5, 2015 at 12:51
• Hey! Thanks. This is a problem I found in old notes of my senior. Can you pls name the book you are referring to? B. Sklar? Commented Mar 5, 2015 at 12:58
• I'm sure it's in Sklar, but I don't have it at hand right now. It's also in Wikipedia.
– MBaz
Commented Mar 5, 2015 at 14:37
• Here is a note (hopefully accessible world-wide) that I wrote a long time ago describing the Gram-Schmidt process. Commented Mar 5, 2015 at 14:57

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.