Meaning of sample points in nonuniform discrete Fourier transform
The nonumiform discrete Fourier transform is defined by the following formula: $${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-2\pi ip_{k}\omega _{n}},\quad 0\leq k\leq N-1,}$$ where $p_0,...p_{N-1}\in[0,1]$ are so called sample points and ${\displaystyle \omega _{0},\ldots ,\omega _{N-1}\in [0,N]}$ are frequencies.
I now want to use the formula to calculate the Fourier series at so called nonequispaced points. That means, my points have a variable space between each other. An example:
There exist three types of nonuniform discrete Fourier transforms. For my use case, the second type (NUDFT-II) seems to fit the best, which uses uniform frequencies with $\omega_n = n$ and nonuniform sample points $p_k$, as it evaluates "a Fourier series at nonequispaced points" (Wikipedia).
I could, however, not find any explanation regarding how to choose the sample points $p_k$, or what those points actually stand for.
In the discrete Fourier transform $p_k$ equals $n/N$. Does this mean, that in the nonuniform discrete Fourier trasnform of type II it equals the distance of the current point $n$ to the beginning of the original signal divided by the length of the signal, hence describing "the percent of the time we've gone through" (as described here)?
I now want to use the formula to calculate the Fourier series at so called nonequispaced points. That means, my points have a variable space between each other. An example:
There exist three types of nonuniform discrete Fourier transforms. For my use case, the second type (NUDFT-II) seems to fit the best, which uses uniform frequencies with $\omega_n = n$ and nonuniform sample points $p_k$, as it evaluates "a Fourier series at nonequispaced points" (Wikipedia).
I could, however, not find any explanation regarding how to choose the sample points $p_k$, or what those points actually stand for.
In the discrete Fourier transform $p_k$ equals $n/N$. Does this mean, that in the nonuniform discrete Fourier trasnform of type II it equals the distance of the current point $n$ to the beginning of the original signal divided by the length of the signal, hence describing "the percent of the time we've gone through" (as described here)?
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