I find myself working with a time series of length n composed of sparse frequency data.
I would like to extend this to a time series of length m > n that has the same sparse frequencies at the same amplitude.
This seems quite possible by a basis conversion between the two frequency spaces, but I have been unable to determine how.
Is there such a method? Thank you for your time.
This is the poster answering the question:
Turns out the most simple answer is typically the right answer.
If I have a time series x of length n and want one y of length m > n having the same frequencies at the same amplitudes and phases then the operation is much better done in real space than inverted space.
Because X, the Fourier transform of x, describes only frequencies with periods that are integer divisors of n, one can simply extend the length of x as if x is periodic. This will maintain those frequencies at the same amplitude and phase.
To wit, for 0 < i < m: y[i] = x[i % n] will create y with the desired properties. D'oh!