Re: Orth in time and freq
Orthogonality of 2 signals (functions) means, as you've written, that their inner product is equal to zero. Your formula is almost right, but the complex conjugate sign must be considered:
I = int(x1•x2*) = 0. If this is true, 2 functions are orthogonal.
It's necessary to define the orthogonality interval. Usually it's considered to be inf, but sometimes it is shrank to finite boundaries. If orthogonality is valid on both finite and infinite intervals, these functions have the double orthogonality
In frequency domain orthogonality is checked the same way:
I = (1/2pi)* int(X1•X2*) = 0, where X1 and X2 are the Fourier transforms of x1 and x2 respectively.
According to Parseval formula int(x1•x2*) = (1/2pi)*int(X1•X2*) = 0. That is, orthogonality may be checked either in time or frequency domains.
If 2 signals don't overlap in any of these domains, they are definetely orthogonal. In all the other cases we need to check it. Very often 2 signals overlap on time span and frequency span, but they are orthogonal (Haar, Walsh functions, etc.)
With respect,
Dmitrij