As has already been rightly pointed out, this is a property of the discrete Fourier transform. To supplement jheindel's answer, I will try to give a taster of the mathematics behind this. The conventional NMR books typically don't cover this in sufficient detail, so for a proper treatment it is a good idea to look in engineering books, specifically on signal processing. MIT OCW also has an excellent course.
We start our journey at the Fourier series, instead of the transform. Consider a continuous-time Fourier series. (Obviously, NMR signals are discrete-time signals, and we will revisit that at the end of this post.) The point here is to express a periodic time-domain function, $f(t)$, as a sum of complex exponentials (or equivalently, sinusoids, although we will stick with the exponentials here). If this function has a period of $T_0$, then its Fourier series representation is
$$f(t) = \sum_{k = -\infty}^{\infty} c_k \exp\left(\frac{\mathrm{i} 2\pi kt}{T_0}\right) = \sum_{k = -\infty}^{\infty} c_k \exp(\mathrm{i}k\omega_0 t)$$
where we define the fundamental frequency $\omega_0 = 2\pi/ T_0$. The Fourier coefficients are given by
$$c_k = \frac{1}{T_0}\int_{T_0} f(t) \exp\left(-\frac{\mathrm{i} 2\pi kt}{T_0}\right) \,\mathrm{d}t.$$
We can plot these coefficients in the frequency domain. Each complex exponential $\exp(\mathrm{i}k\omega_0 t)$ is associated with a frequency $k\omega_0$. Since $k$ is restricted to be an integer, we only have discrete values of the frequency, and so the Fourier series can only be graphically represented as a series of points with associated heights $c_k$. In other words, we have a discrete function $F[\omega]$ which is only defined for $\omega = k\omega_0$.
(n.b. this is just for illustration purposes; the coefficients are not meant to be drawn accurately.) Now imagine the effect of padding your periodic time-domain function with extra regions between each period where $f(t) = 0$. That would increase the period $T_0$ and decrease the fundamental frequency $\omega_0$. So our discrete function $F[\omega]$ becomes more and more closely spaced.
In the limit of infinite padding, we have $T_0 \to \infty$ and $\omega_0 \to 0$, such that the function $F[\omega]$ actually becomes a continuous version $F(\omega)$. The sum becomes an integral and what we have is now a Fourier transform.
What's important to notice here is that if you draw the Fourier transformed function $F(\omega)$ over the Fourier series coefficients $F[\omega]$, you will find that it fits perfectly over them. In other words, the Fourier series is a frequency-domain sample of the Fourier transform. This only holds true if your time-domain function doesn't get stretched or otherwise modified, so this framework is not directly capable of dealing with infinitely long time-domain functions, but it suffices since our time-domain FIDs have finite durations.
So, when we pad the periodic function with zero-valued regions, we are not fundamentally changing the frequency-domain information that is present. All we are doing is taking more closely spaced samples of the Fourier transform $F(\omega)$, and this comes about only because the samples have a spacing of $\omega_0 = 2\pi/T_0$.
This is exactly analogous to zero-filling in NMR. Imagine that our NMR spectrum is (one period of) the periodic function at the beginning. When you zero-fill an NMR spectrum, you are not actually generating any new information about the peaks that are present. If the (continuous) Fourier transform $F(\omega)$ does not display a small coupling, then no amount of zero-filling your spectrum is going to reveal that small coupling.
Broadly speaking, the same considerations apply to discrete-time signals. The biggest thing to clear up is that the discrete Fourier transform (DFT) is not analogous to the continuous-time Fourier transform. Instead, the DFT is analogous to the continuous-time Fourier series, the sole difference being that it operates on a discrete-time signal. The discrete-time analogue of a continuous-time Fourier transform is instead called the discrete-time Fourier transform (DTFT).
This terminology is partly to blame for the confusion surrounding zero-filling. When you zero-fill a FID and "Fourier transform" it, you are not actually changing its Fourier transform (in the continuous-time sense); you are just calculating a more closely spaced Fourier series.
To sum up and to make the analogy clear:
- In continuous time, as you pad a periodic function with zero-valued regions, the Fourier series becomes more closely spaced, tending towards the Fourier transform.
- In discrete time, as you pad a periodic function with zeroes, the DFT becomes more closely spaced, tending towards the DTFT.
