[HN Gopher] Hilbert Transform
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Hilbert Transform
Author : topsycatt
Score : 94 points
Date : 2023-07-10 13:39 UTC (9 hours ago)
(HTM) web link (electroagenda.com)
(TXT) w3m dump (electroagenda.com)
| galcerte wrote:
| I find it incredibly funny that this appears a couple of days
| after I finished my master's thesis. I realized I made a bit of a
| blunder a week before handing it in: I thought the Hilbert
| transform gets rid of the aliasing when a real signal's bandwidth
| reaches below the frequency origin and into negative frequencies,
| which it doesn't. The bandwidth "folding" is still there, but the
| negative frequency components are gone (which I did know). Since
| I got rid of the negative frequencies because of the symmetry of
| real signals about the frequency origin, there wasn't any point
| in doing the Hilbert transform. Thankfully I checked all the
| preprocessing steps I carried out in the thesis, and weeded out
| this unnecessary step.
|
| In my defense, please do bear in mind that I have not had a
| thorough education in signal processing, physics degrees don't
| usually have courses like this. I know Hilbert spaces much more
| well than I do the Hilbert transform.
| HappySweeney wrote:
| Years ago I wrote a genetic program to maximize stock market
| gains, and as primitives I used the indicators and signals from
| TA-Lib. The Hilbert Transform Phasor was the clear winner, though
| the returns were too meagre for me to continue with the project.
| denial wrote:
| Singular integrals are a lovely topic, especially from a Fourier
| analytic perspective. Taking the Fourier transform, F(H(f))(x) =
| -i * sgn(x) * F(f)(x), which implies H^2 = - I. H has the Fourier
| multiplier -i * sgn(x). The Riesz transforms R_j are a higher
| dimensional generalization of the Hilbert transform with Fourier
| multipliers -i * x_j/|x_j|, which leads to the nice property sum
| R_j^2 = -I.
| ipunchghosts wrote:
| Are ML companies hiring for peeps that do dsp and ML? I'm looking
| for a new gig and do both!
| galcerte wrote:
| I just so happen to be working on that at my company, but sadly
| we do not seem to be hiring more for this position, and I don't
| think you'd like how low salaries are in southern Europe.
| dartvox wrote:
| [flagged]
| rocho wrote:
| These low-effort GPT comments should be removed as spam.
| ipunchghosts wrote:
| Good chatgpt
| janandonly wrote:
| I am unsure if this is a bunch of ChatGPT generated pseudo
| sciency sounding stuff, or actual useful and real information?
|
| I guess either way it's written too terse for me to understand
| CamperBob2 wrote:
| Agreed, this article doesn't add any real insight over the
| references it cites, it just regurgitates the math.
| stagger87 wrote:
| You will come across Hilbert transforms in a signals and
| systems or DSP course, which you would take in an EE/ME
| undergrad. A lot of the work in this space was done by
| mathematicians which I think contributes to the abstraction
| level at which it's taught. It can be useful and is definitely
| real.
| electroagenda wrote:
| The text attempts to summarize the Hilbert Transform for
| electronics engineers in telecoms applications just going to
| the point.
|
| It was written by me, and I am human.
|
| I think I got my purpose. But obviously the text will be
| useless for many people and very useful for others. Anyway,
| please take into account the context and the topic of the blog.
| tverbeure wrote:
| Over the years, I've googled on and off about the Hilbert
| Transform, but never came even close to understanding where
| it could be useful. Your explanation is the first one that
| makes sense, though I'll need to reread it a couple of times
| to really get it. And hopefully, this will help me understand
| the more rigorous material too.
|
| So thanks! It was very useful for me! And ignore the
| naysayers.
| electroagenda wrote:
| Thank you so much for your feedback!
| cozzyd wrote:
| Another useful application of the Hilbert transform is through
| the Kramers-Kronig relations. Basically, for a causal signal
| (e.g. an impulse that turns on), the real and imaginary parts of
| the Fourier transform are related by a Hilbert transform. This is
| very useful, for example, when trying to extract a time-domain
| impulse from imperfect frequency-domain measurements, for example
| with a network analyzer. Rather than attempting to inverse
| Fourier transform the frequency domain measurements, which will
| tend to produce an unphysical mess due to measurement errors,
| it's often better to take either just the real or imaginary part
| and synthesize the other one. Even better would be to try to find
| the nearest causal waveform to what is measured, but that's a bit
| more difficult...
|
| Another application is finding the minimum phase response given
| an amplitude response.
| ajot wrote:
| During my materials science PhD, I've encounteres this transform
| as a way to process and smooth out electrochemical impedance
| spectrograms, with the Z-HIT algorithm[0]. As far as I know, only
| one or two EIS appliance manufacturers include it in their
| software, unfortunately not the one I was using.
|
| [0] https://en.wikipedia.org/wiki/Z-HIT
| kurthr wrote:
| I always thought of the Hilbert Transform and Z-HIT as the
| complex Laplace like substitute for Fourier and DFT analysis.
| Having never used them or SSB in particular, is that an fair
| analysis or am I missing something important?
| ipunchghosts wrote:
| The signal model for Laplace is x(t)*exp(alpha t) where as
| the signal model for HT is x(t) only. Another way to say it
| is that Laplace models the signal envelope as an exponential
| explicitly while decomposing the remaining signal into
| frequency components. In Fourier space, the exponential is
| "rolled into" the frequency components.
| nimish wrote:
| This is the rigorous basis for a lot of electrical engineering
| shortcuts like phasors and active/reactive/real/apparent power
| analysis.
|
| Makes a hell of a lot more sense when it isn't just pulled out of
| the ether.
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