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2

Not really. Welch's method is specifically designed to estimate spectral density and discards phase information. IFT and FT are almost identical and have the exact same complexity. If you want to do an IFT, just do an IFT unless you have a really good reason not to.

2

The FFT is faster than the naive DFT through matrix-vector products because it can reuse many intermediate results. However, for inputs this sparse, there's really not much that can be re-used, even in the best case. So, the most efficient way here is probably to work straight forward: Allocate the space ($N=7408800$ values) for your output (initialize with ...

4

FFTW is a nice library. Real forward, real backward, complex forward, complex backward. Refill the input arrays when executing a new plan. Works for me, see documentation via the link graciously supplied by MM for more details. Extracted from working code: //--- Have FFTW do the DFT via a FFTW Plan fftw_plan thePlan; /* //--- Real DFT ...

5

Is the backwards transform just an IFFT (for forward transform)? Yes. Compare the "what FFTW actually computes", here

2

Depends on who's practicing and what they are doing. You should know that it is mathematically equivalent to doing interpolation using the discrete sinc function (aka Dirichlet kernel, or alias sinc). As your N gets large, this approaches the normalized sinc function. Personally, I'm likely to use cubic interpolation, see Multi-channel audio upsampling ...

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