For a causal time domain signal, the Fourier Transform will be complex with the imaginary part as the Hilbert Transform of the real part.
This may be clearer by noting the following additional properties of the Fourier Transform:
- The FT of a non-causal real even waveform ($f(t) = f(-t)$) will be
all real.
- The FT of a non-causal real odd waveform ($f(t) =
-f(-t)$) will be all imaginary.
- The FT of a non-causal imaginary odd waveforrm will be all real.
- The FT of a non-causal imaginary even waveform will be all imaginary.
It is easy to prove all of the above by observing waveforms as rotating phasors of constant magnitude in time on a complex IQ plane, since the FT is the mangitude of each of those phasors at the position in frequency based on their rate of rotation. (In fact the Fourier Series Expansion is specifically to decompose any arbitrary single valued analytic function into such rotating phasors).
With those properties in mind, you can extend that to causal and anti-causal functions by adding or subtracting the cases listed above: Causal and anti-causal functions are the sum of an even and odd functions. Understanding this provides more insight into many other Fourier Transform properties.
As far as the first statement given about the interesting relationship between the real and imaginary components; if it is still not clear how this occurs it may be easier for some to swap the time and frequency domains and consider what occurs: In the time domain if you add the Hilbert Transform of a real signal as an imaginary component the result is a one-sided spectrum. A one-sided spectrum is the equivalent in the frequency domain to what a causal or anti-causal signal is in the time domain. Stated another way, a causal signal is a one-sided time-domain signal.
Consider the very simple case of a cosine which is a two sided spectrum with a positive negative frequency. This is also apparent in Euler's Identity where we see the two rotating phasors I was mentioning above, one rotating counter-clockwise (positive frequency) and one rotating clockwise (negative frequency):
$$cos(\omega t) = \frac{1}{2}e^{j\omega t} + \frac{1}{2}e^{-j\omega t}$$
Now consider the identity given as:
$$e^{j\omega t} = cos(\omega t) + j sin(\omega t)$$
$sin(\omega t)$ is the Hilbert Transform of $cos(\omega t)$, and $e^{j\omega t}$ is the single sided spectrum referred to earlier.
So we see that the following relationship holds:
$$f(t) = a(t) + j H\{a(t)\}$$
Where $f(t)$ is a function with a single-sided Fourier Transform, and $H\{\}$ is the Hilbert Transform.
Similarly by swapping time and frequency domain the following relationship will hold:
$$F(\omega) = A(\omega) + j H\{A(\omega)\}$$
Where here $F(\omega)$ is the Fourier Transform of a causal time domain function $f(t)$. The resulting $f(t)$ which has a single-sided Fourier Transform is referred to as the "analytic signal" and it also has the property that it will be minimum phase: it's Laplace transform will have all poles and zeros in the left half plane (and similarly for discrete-time systems the z-transform will have all poles and zeros inside the unit circle). A minimum phase system will have a single-sided frequency spectrum but not all single-sided frequency spectrums are minimum phase systems: We can cascade a minimum phase system with an all-pass filter that only modifies phase resulting in a mixed-phase or maximum-phase system, but since this only modifies phase in the spectrum and doesn't add new frequencies, the spectrum will still be single-sided.
For further details on this see:
FFTs of a complex signal - separating the real and imaginary parts
