Sense of zeropadding in a time domain I have the task related to Radon transform which contains a subtask which uses resampling by means of DFT. 

Let's consider the non-periodical discretized signal (Fig.1) (for example the string of pixels) having 515 pixels length. In my implementation for resampling contains following steps:


Cyclic left shift (Fig.2).
Add zeros to the center in order to the length of signal became 2^n (in our case 1024-515 = 509 zeros we must have add) (Fig. 3).
Get DFT from this signal (Fig. 4).
Cyclic right shift. (for shifting low frequencies to the center) (Fig.5)


Fig.1


Fig.2


Fig.3


Fig.4


Fig.5


The main question:

Why we must perform cyclic shift of the signal and add zeros exactly in the center? (I assumed what this made the signal periodic)
Zeropadding makes interpolation DFT spectrum, is it correct? (I asked and someone says what it is not quite so)
Maybe someone can explain in simple way what happens with signal after zeropadding.

I have made some experiments in a Matlab and found out that any other sequence of actions can not give required result.

Now let's consider two cases:

a) (THIS CORRECT VARIANT) We has the non-periodical discretized signal (for example the string of pixels) which will be cyclic shifted to left and filled zeros in the center after that will be obtained DFT from this and to shift it back.


b) We has the non-periodical discretized signal (for example the set strings of pixels) which will be filled zeros from left and right  after that will be obtained DFT from this.



What the difference these DFT spectrums?

I have read some books but not found the answer of this zeropadding's case. It seems this can be found only by own experience.

Answer in book:

A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, 2001 on page 25
Shifting the data points (fftshift) and zero-padding the exact center of the FFT aperture has the property that all the even (symmetric) components with reference to the center of the original data set end up in the real part of the complex FFT result, and all the odd components end up in the imaginary part. e.g. the evenness to oddness ratio is preserved, which allows phase (with reference to the center of the window) to be easily interpolated. 

Being able to interpolate phase is important in the case of a zero-padded FFT because the zero-padding interpolates the spectral magnitude as well.  Since zero-padding results in an interpolated FFT result, that means that any original non-interpolated FFT result points will need to be interpolated from the zero-padded result.  Without an FFT shift of both the non-padded and zero-padded data, these interpolated results will be different from the non-interpolated FFT results (in phase).

This technique is a simple (homework or quiz level) result of the FT property that a shift in one domain is a complex frequency modulation in the other domain.