Summary of A QUICK LOOK AT THE HILBERT TRANSFORM
This article explores the Hilbert transform, a signal processing tool used to generate single-sideband signals by applying a 90-degree phase shift. It explains how mixing a signal with its Hilbert-transformed version cancels negative frequencies while reinforcing positive ones. The text highlights practical implementation challenges, such as the need for filtering delays and equalization to maintain signal alignment, and suggests using spreadsheets or simulation tools like Matlab for learning.
Parts used in the Hilbert Transform Project:
- Hilbert transform filter
- Equalizing delay component
- Mixing circuitry
- Matlab simulation environment
- Spreadsheets
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.
Source: A QUICK LOOK AT THE HILBERT TRANSFORM
- What is the primary function of the Hilbert transform?
It produces a 90 degree phase shift with constant gain to help generate single-sideband signals. - How does mixing a signal with its Hilbert transform affect frequencies?
The process causes negative parts to cancel out while positive frequencies reinforce each other. - Can you subtract signals instead of adding them?
Yes, subtracting produces the opposite effect and results in the opposite sideband. - Why is an equalizing delay required in practical implementations?
A filter introduces delay, so an equalizing delay ensures the mixed parts originate from the same input time. - Does the practical phase match the theoretical model perfectly?
No, the phase is not as clean as expected due to practical approximation concerns. - Which software tools are recommended for modeling this process?
You can use Matlab or start with spreadsheets to understand the concepts before diving deeper.
