🚀 go-pugleaf

u7811837@ccsun9.cc.nctu.edu.tw () writes:

>----------------------------------------------------------------------------
>Dear reders ,

> I am looking for real functions A(x) and B(x) which satisfy both  the following constraints :

> (a) < A(x-k), A(x-l) > = delta(k,l). 
>     < B(x-k), B(x-l) > = delta(k,l).
>     < A(x-k), B(x-l) > = 0. 

> (b) < A"(x-k), B"(x-l) > = 0.

> where the symbol <.,.> stands for the inner product, and delta(.,.) stands
> for the kronecker delta function. k and l are integers. A" and B" are defined
> as the second derivatives of functions A and B.
Assuming that this relations have to hold for integers only
(and the delta function is the integer one),
AND ignoring that you used a convolution in your definition, instead
of a correlation ( <A(k-x), A(x-l)> = 0, we can ignore this iff A is even),
apart from condition (b) this is just the Nyquist criteria. Now if
< A"(x-k), B"(x-l) > = 0, this means
< A(x-k), B(x-l), delta""(x) > = 0, which apart from the four time
differentiation is the same as Nyquist condition above. If you have any harmonic
signals (frequency multiplex), differentiating 4 times will not make any
difference, so the usual frequency multiplex signals will do.

Also, any orthogonal signals with a duration of strictly less than 1
will have an interval of zero about the critical sampling points in
their correlation functions, and will thus also qualify.