Definition of the DFT / FFT Bin Size When reconstructing the frequency domain for an FFT, what is the most self-consistent way to do this -- i.e. how is it best to define the bin width, $\Delta f$?

For example, previously I thought it a simple matter of saying,
$$\Delta f = span / N_{points,FFT}$$ where the span is the size of the window you want to look at and $N_{points,FFT}$ is the number of points that should appear in the final FFT result. This makes very clear sense especially when you deal with some FFT spectrum analyzer where you set the span and the number of points in the final FFT (some times called the number of FFT lines) can also be set but tends to have some upper value for that device. 

However I have also seen definitions attached to the sampling rate for the incoming transient signal to be FFT'd, as $$\Delta f = f_{sr} / N_{points,FFT}$$ or even as $$\Delta f = 2 f_{sr} / N_{points,FFT}$$ depending on how you define $N_{points,FFT}$. I.e. the number of points that goes into making the FFT or the number of points that will appear in the final FFT result because half the spectrum is thrown away due to mirroring.

Maybe it doesn't matter and it just depends on the way you choose to define things. But if anyone has any insight that would be helpful!



I am adding a little more to clarify my confusion.

Suppose you have some FFT spectrum analyzer that can process at maximum 2048 points in a time transient. That then means The final FFT would have 1024 points -- as we discard the mirror image. I then (again this is all in the context of making an FFT with some FFT spectrum analyzer device) say I want to set my span of this FFT to $50\ \rm{Hz}$. This would give a resultant FFT with a bin width of $\Delta f = 50 / 1024 = 0.049\ \rm{Hz}$. To me this must imply that the sample rate, $f_{sr}$, must vary according to the desired span. 

So is this is all question of what your free parameters are, whether the maximum number of points in a transient signal is fixed, or the maximum sample rate you can take, and so on?
Let's assume we have a sample rate of $f_s=10 kHz$ and FFT size of $N=1000$


Your bin spacing is $\delta f = 10 Hz$. It's simply the sample rate divided by the FFT size. That's all there is to it.
Keep in mind that the FFT requires both the time domain and the frequency domain signal to be perodic. Most signals are not, so there needs to be some sort "accommodation" that's specific to the needs for your application.
A real-valued sine wave with a frequency that is an integer multiple of the bin spacing will only show up in two bins symmetric around zero. I.e. a 100 Hz sine wave will show up in bins +10 and -10 (assuming we index the bins from $[-N/2,N/2-1]$
A complex exponential  with a frequency that is  integer multiple of the bin spacing will only show up in one bin. I.e. a +100 Hz complex exponential will show up bin +10.
A cosine wave with a frequency that's NOT an integer multiple of the bin spacing (e.g. 105Hz) will spread out over all bins. The bulk of the energy will be in the neighborhood of 105 Hz but there will be energy in ALL bins. This is a direct consequence of bullet 2): the FFT assumes periodicity with $N/f_s$. In essence the FFT doesn't calculate the spectrum of the sine wave, but the spectrum of a signal that consists of 1000 sample long snippet of a 105Hz cosine that gets repeated infinite times. At the splice point there is a massive discontinuity and that causes the energy to spread over all bands
In order to minimize this spreading effect, one can apply a window, that tapers down the edges of the time domain frame and reduces the discontinuity. On the other hand, the window also smears out the energy in the frequency domain, so you can't get any sharp lines any more.
Many people think that the FFT will give you an "exact" representation of the signal's frequency content. That's not the case unless the time domain signal is either periodic with the FFT length or if it's finite in time. For all other signals the results will depend heavily on the exact process and parameters chosen.


EDIT: reworded item 7 per the comment