Why FFT time domain after 0, by FFT transform is the frequency domain interpolation?

The question should be understood in this way:

0 after the DfT (FFT is a fast DFT algorithm), in fact, the formula has not changed, the change is only the frequency domain terms (such as: 0 before the FFT calculation is m*2*pi/m at the frequency domain value, and 0 after the n*2*pi/n at the frequency domain value), M is the original DFT length, n becomes the length of the complement 0. Will (-PI,PI) from the original m parts into N parts, if the complement 0 before and after these frequency domain values are drawn in the coordinates, wherein the m*2*pi/m and n*2*pi/n coincident parts, it corresponds to the frequency domain value (the transformed value) is unchanged, and in the original M part of the (n-m) portion of the component, That is, in the frequency domain more than (n-m) part interpolation, so that understanding is clear.

Complementary 0 Advantages have two:
One is to make the integer power of 2 data points so that the FFT can be used
Second, the original data played the role of interpolation, on the one hand to overcome the "fence" effect, so that the appearance of the spectrum smooth, on the other hand, due to the data truncation caused by the frequency domain leakage, there may be some difficult to confirm the spectral peaks in the spectrum (see "Digital signal Processing" Textbook 147 page Figure 6-13), After 0, it is possible to eliminate this phenomenon.

FFT complement 0:

The spectral resolution of the N-point DfT is 2π/N. The Fence effect section indicates that more frequency points can be observed by filling 0 (see digital signal Processing, 148 pages), but this does not mean that 0 can improve the true spectral resolution. This is because X[n] is actually a sequence of primary values sampled by X (t), and the X ' [n] cycle extension obtained by X[n] 0 is not the same as the original sequence, nor is it a sample of X (t). Therefore, the spectrum of different discrete signals is already. For the DfT of X ', which is 0 to M, it can only be said that its resolution 2π/m has only a computational significance and is not a real, physical spectrum. The increase of spectral resolution can only be achieved by increasing the time-domain effective sampling length under the condition that the sampling theorem is satisfied (see "Digital Signal Processing" Textbook page 146), and the complement 0 is not the valid data of time domain signal.