the inverse discrete Fourier transform

If the forward DFT is taking a chord apart into its separate notes, the inverse DFT is the reverse: handed a recipe of frequencies and how loud each one is, it adds those pure tones back together to reconstruct the original sound. It is the synthesis step that guarantees nothing was lost — analysis and re-synthesis return you exactly to where you started.

Given the N frequency coefficients X_0, ..., X_{N-1}, the inverse DFT rebuilds the time samples by x_n = (1/N) * sum over k from 0 to N-1 of X_k * e^(+2*pi*i*k*n/N). Notice the two differences from the forward transform: the exponent sign flips from minus to plus (the spinning arrows now turn the other way), and a 1/N factor scales the result. That 1/N is the bookkeeping that makes the round trip exact: do the DFT, then the inverse DFT, and you recover the original x_n to within rounding error. Whichever normalization convention a library uses — 1/N on the inverse, 1/sqrt(N) split symmetrically, or 1/N on the forward — the only iron rule is that the forward and inverse factors multiply to 1/N.

The inverse DFT is what makes the whole 'transform, work, transform back' strategy useful. You move into the frequency domain because some operation is trivially easy there — filtering by zeroing unwanted frequencies, convolution by simple multiplication, solving certain differential equations by dividing — and then the inverse DFT carries the result back to the time or space domain you actually care about. It costs the same as the forward transform, so the fast Fourier transform speeds it up identically; the same butterfly machinery runs with the sign and scaling adjusted.