I was given this set of 3 formulas and asked to find a complete set of orthonormal basis for it:
$$s1(t) = u(t) − u(t − 1)$$
$$s2(t) = u(t − 2) − u(t − 3)$$
$$s3(t) = u(t) − u(t − 3)$$
I'm not sure how to find the complete set of orthonormal basis for these graphs, honestly I am not even certain of the meaning of "complete orthonormal basis". The problem comes from an engineering class but I believe it is more mathematical than anything else so I'm asking here. I don't need a solution but I would like to know how to go about solving this kind of problem on my own (the logic behind it and the process).
as asked for in the comments, I am not sure about the inner product space but u(t) is the unit step function as seen here: https://www.tutorialspoint.com/signals_and_systems/signals_basic_types.htm basically a signal with y = 1 and varying length across x
Just guessing here, but it appears that: S1 = 1 from 0 to 1; S2 = 1 from 2 to 3; and S3 = 1 from 0 to 3.
So S3 overlaps S1 and S2. So you can subtract S1 and S2 from S3 to create a new coordinate S*
S* = S3-S2-S1 = 1 from 1 to 2
Thus S1, S2 and S* have nothing in common, hence orthonormal. Again, guessing.