Purpose of inverse Fourier transform as well as IDFT – inverse DFT is to recover original signal from it’s sine and cosine Fourier coefficients. Schematically it is presented in figure bellow.
Fourier Transform decompose analog signal into sum of sine and cosine signals on frequencies that are multiplies of main frequency. Now, original signal can be recovered with following definition
In here, F(X) is signal in frequency domain. Upper expression means that in order to reconstruct the original signal from it’s sine and cosine coefficients, we only need summarize samples of it’s sine and cosine components.
The more coefficients we use in the signal reconstruction, recovered signal is closer to original one. Maximal number of sine (or cosine) coefficients that we can obtain from set of N given samples is N/2. However if we accept smaller accuracy in signal reconstruction, we can use smaller number of coefficients in signal calculus and compress the signal that way.