# Extrapolation after taking inverse Fourier transform

Suppose I have 2000 point(time series) signal and I take 512 point FFT, after this some operation is done in frequency domain. Now if I take IFFT to get 512 points in time domain, Is there a way to map the output 512 IFFT points to the original time domain 2000 points.

Inverse of this problem would be 400 point(time series) data and 512 point fft, and finally 512 IFFT points, here we can use Overlap and Add method to get 400 point output, is there any similar method for above problem.

An FFT possesses linear features. Let us use a linear analogy. If one tells you that the sum of two numbers is $a+b=3$, what can you say about these numbers? Not much. More generally, $n$ linear equations on $m$ variables is underdetermined when $n<m$: you do not have enough information to retrieve the variables without more conditions. This can be experimented with a very simple DFT case.

Let us assume the vector $x = [1,-1]^T$. A simple two-point DFT is defined by the matrix:

$$\begin{bmatrix} +1 & 1\\ 1 & -1 \end{bmatrix}$$

Its products with $x$ yields either $0$ (for the DC) or $2$ (up to a $\sqrt{2}$ Fourier definition scaling) at Nyquist. Can you recover the two elements from $x$ from only a single coefficient $0$ or (xor) $2$?

Probably not, without further assumptions, or on very specific signals. This happens whenever you have less DFT points than samples. There is no bijection between two finite sets of different sizes. This is indeed one definition of infinity: a set is infinite if it can be put in bijection with a proper subset.

However, using sparsity, positivity properties, there are frameworks to retrieve (at least approximately) the data.

Assuming you used a 512 point rectangular window on your 2000 original points of data, you need to know something additional about the other 1488 points and how they are related or correlated to the 512 points that were analyzed by your 512 point FFT if you want to do any "mapping". Otherwise, those 1488 points could be completely independent and of any value, no matter what mapping you might try.

In general, no. Consider the matrix form of the DFT. It will be $512 \times 2000$, which is not in general invertible.