While the Fourier transform gets all the attention, there are other transforms that engineers and mathematicians use to transform signals from one form to another. Sometimes you use a transform to make a signal more amenable to analysis. Other times, you do it because you want to manipulate it, and the transform is easier to change than the original signal. [Electroagenda] explains the Hilbert transform, which is often used to generate single-sideband signals.
The math behind the transformation is pretty hairy. However, if you understand the Fourier transformer, you can multiply the Fourier transform by -i sgn(ω), but that isn’t really going to help you much in a practical sense. If you don’t want to bog down in the math, skip immediately to section two of the post. That’s where it focuses more on the practical effect of the transform. You can think of the transform as a function that produces a 90 degree phase shift with a constant gain. For negative frequencies, the rotation is 90 degrees and for positive frequencies, the shift is negative.
Section 3 shows how mixing a signal with its own Hilbert transform can produce single sideband signals. Typically, a signal is transformed, and the result is multiplied by j (the square root of negative one). When you mix this with the original signal, the negative parts cancel out, while the positive frequencies reinforce each other. If you prefer, you can subtract to get the opposite effect and, thus, the opposite sideband.
There are practical concerns. You must approximate the Hilbert transform, and that will require a filter that has a delay. You’ll need an equalizing delay in the main signal so that the parts that mix together are from the same input time. It also means the phase isn’t as clean as you expect from the theoretical model. If you want to model it all in Matlab, you might find this post enjoyable. If you want a more ham radio take on the same material, check out [K6JCA’s] article on the topic, or watch [ZL2CTM’s] video on the topic below.
If you aren’t ready to swim on the deep end of the signal processing pool, maybe start with some spreadsheets. Once you have a good grip on how IQ can demodulate and modulate, you’ll have an easier time with the Hilbert transform.