All that is happening when you zero-pad the input signal prior to a DFT, is that you are interpolating the frequency domain representation.
For example, you might get a certain shape as the output of your absolute magnitude of the DFT, and this shape has 10 points. If you zero-padded the input of your DFT to say, 100 points, you will get the same shape as before, but it will have more points, and will look smoother. That is all a zero-padding does for you. One may ask what type of relationship, and this relationship is simply a sinc-interpolation relationship.
A picture is worth a thousand words, so here you go:
As you can see, the shape of the DFT output (magnitude in this case) remains the same, but the granularity increases.
People will usually zero-pad for a variety of reasons:
- Peak picking: A smoother DFT output allows a peak-picking algorithm to gain more accuracy as to the frequency of the actual peak.
- Interpolation in the frequency domain for further processing downstream
- Linear Convolution: If linear convolution is sought, then performing it as a multiplication of two spectra in the frequency domain means that the DFT outputs must be of length $N+M-1$, where $N$ is the length of the first signal, and $M$ is the length of the second signal it is being convolved with.