[HN Gopher] Is the frequency domain a real place?
___________________________________________________________________
Is the frequency domain a real place?
Author : zdw
Score : 205 points
Date : 2024-04-07 04:52 UTC (18 hours ago)
(HTM) web link (lcamtuf.substack.com)
(TXT) w3m dump (lcamtuf.substack.com)
| wyldfire wrote:
| If you want to view paradise /
|
| Simply look around and tune it. /
|
| Anything you want to - tune it. /
|
| Want to change the channel? nothing to it.
|
| There is no SDR /
|
| To compare with foreign number stations /
|
| Casting there, you'll be free /
|
| If you tune to ninety eight point three
| ranger_danger wrote:
| where can one who is bad at math, learn more about using the
| frequency domain to do such "neat tricks" as it relates to
| programming and electronics?
| hgomersall wrote:
| The trite response is "get better at maths". I honestly don't
| mean that facetiously - the whole understanding that underpins
| "neat tricks" and how to use them is "maths". Once you can use
| said neat tricks, you are better at maths.
|
| So, I would suggest a better question is: where can I find an
| introduction to this stuff that is better aligned with my
| current understanding and knowledge. To that end, here's a
| great intro book: https://www.dspguide.com/
|
| I read that cover to cover when I was 17 and I would not have
| been described as particularly mathematically gifted.
| logicziller wrote:
| GNU Radio is a great tool to get started quickly.
| jnation wrote:
| Search up 2d Fourier demo
| ssl-3 wrote:
| Just play around with audio tools, so you can get to throw
| ideas at a thing and get results that you can hear?
|
| The things you can hear scale pretty well all the way down to
| DC, and all the way up to daylight (or beyond, I suppose).
|
| After spending a few (or dozens, or hundreds) of hours
| tinkering with audio-range frequencies for fun, then maybe
| you'll have a real place with which to associate the maths that
| are involved in electronics and programming, and having that
| place might make that math (and whatever is needed to
| accomplish it) feel lot more worthwhile to actually-learn.
| fl7305 wrote:
| > might make that math (and whatever is needed to accomplish
| it) feel lot more worthwhile to actually-learn.
|
| I agree. Start with small hobby projects using what you
| already know.
|
| They should be things where you get immediate feedback, like
| graphics or sound. That makes them very fun to work and
| iterate on. It's addicting when you see/hear the output, make
| a change in a few seconds, and immediately see/hear the
| results of your change. That's the stuff that keeps you up
| all night at the computer.
|
| You'll start hitting barriers where you need more math to get
| better results. Then you'll be much more motivated to start
| learning, and it will be easier to retain the knowledge when
| you actually use it in practice.
|
| For instance, use Python to output audio samples. Start with
| simple sine wave tones, colored noise, and work your way up
| to implementing simple FIR and IIR filters to modify input
| audio like voice and music. Use Audacity to see the change
| between input and output as a spectrogram.
|
| If you're into music, use an existing Python library to read
| MIDI files with songs you like. Generate audio output files
| for those songs. First with just sine waves for the notes,
| then you can start emulating digital and analog synths. Write
| code that takes sampled audio files and emulate different
| guitar sound effect pedals or tube amplifiers.
| GuB-42 wrote:
| This article is actually good at that.
|
| Don't look too much at the Greek letters, look at the tables
| and code instead. A DCT is just a bunch of multiplications,
| additions and a lookup table. The approach to derive the Walsh-
| Hadamard transform is actually very computer sciencey. It is
| made using a recursive algorithm, it then takes advantage of
| the fact that a+b==b+a to reorder the rows. There is even a
| trick that uses a bitwise AND.
|
| As a programmer I find it much easier to start from here:
| tables, loops, and simple operations like multiplication and
| addition. And when I finally understand how a computer does the
| thing, maybe go back to the maths to see if I can get more
| insight.
|
| In the end, the only appropriate answer is "get good at maths".
| The question is how to achieve that. And if you have a
| programmer mindset like I do (and I guess like many people
| here) and struggle with maths, I recommend trying the bottom up
| approach. Write the code, have it actually show visuals and
| play sounds, play with the parameters and see what changes and
| what doesn't change.
|
| The other part is overcoming the language barrier, I am still
| struggling with that. Mathematicians use barbaric name like
| eigenvectors for "stuff that won't change", they call
| "integration" an addition in a loop and write it with a weird
| sigma thing, or a weird "S" thing when there are a lot of
| iterations over very small numbers. But in the end, it is not
| that different from reading Perl :)
| bjornsing wrote:
| During undergrad physics / math I came to the conclusion that
| knowing the value of a function f(x) at infinity many points x is
| equivalent to knowing the frequency content of f at infinity many
| frequencies. Both representations are equally "real" in a
| philosophical sense. Some problems are easier to solve in one
| representation than the other.
| zeehio wrote:
| Fully agree. Switching from the time domain to the frequency
| domain is just like switching coordinate systems.
|
| If you have a signal with a single narrow peak in the time
| domain you can represent it in a very compact or sparse way
| just using a delta at the position of a peak. If you try to
| represent it in the frequency domain you will not find such
| compact representation. Similarly if you have a sinusoidal
| signal in the time domain, you won't get a compact
| representation there, but you will get it in the frequency
| domain where you just have a couple of deltas.
|
| Time and frequency are two ways of representing the same thing.
| Sometimes it's easier to represent something in one domain,
| sometimes it's easier in the other domain.
|
| It can be proven that anything bounded in the time domain will
| be unbounded in the frequency domain and viceversa. So compact
| stuff in one domain always spreads when represented in the
| other domain.
|
| Beautifully, quantum mechanics tell us that position and
| momentum are conjugated variables (like time and frequency in
| the example above) and therefore if something has a bounded
| position (we know where it is) its momentum will be unbounded
| (we won't know its speed), and viceversa.
|
| That's the main idea of Heisenberg's uncertainty principle.
| vivzkestrel wrote:
| sometimes i think that our existence in the phyiscal world
| subject to limitations of spacetime is the time domain and after
| death, we get out of the time domain into frequency domain aka
| the soul, the reason perhaps we cannot prove of the existence of
| ghosts or soul is because we are using instruments and techniques
| of the time domain to measure entities in the frequency domain,
| just a shower thought
| ShamelessC wrote:
| https://youtu.be/peUe-82Qgs4
| dsego wrote:
| You are not the only one, spacetime is the "headset".
| https://youtu.be/MDDbsUr7KNU
| meowface wrote:
| I know I may get downvoted for this - and I know pre-emptively
| whining about downvotes is incredibly lame and deservedly
| frowned upon - and I know self-awareness of that fact doesn't
| make it any better - but: I'm sorry, this is just woo-woo. A
| replier linked Donald Hoffman, who also espouses a woo-woo
| theory about root reality being the realm of consciousness.
|
| There're not only no empirical but no theoretical grounds to
| believe anything like this. The mind is almost certainly wholly
| defined by the physical processes of the brain, in the same
| spacetime realm all other known physical processes reside in.
| danhau wrote:
| I think you're digging into a philosophy question with the
| wrong shovel. You're not wrong, but OP is debating something
| that is fundamentally unmeasurable (as I understand them).
| Science can only help us understand things within spacetime.
| Anything beyond that is philosophy.
|
| I believe this is basically just the dualism vs. materialism
| debate on consciousness. Consciousness is a fascinating
| topic. There's plenty of paradoxes or thought experiments to
| fry your brain on. It's not just about the electrochemical
| processes in the brain. It's about identity, the continuity
| thereof, etc.
| wruza wrote:
| _There's plenty of paradoxes or thought experiments to fry
| your brain on_
|
| Is there a good collection of these in one place?
| addicted wrote:
| This is just more woo woo.
|
| Just because something is unmeasurable doesn't mean it
| cannot be proven wrong.
|
| Just because something is derived from philosophy doesn't
| mean it cannot be proven to be wrong in the real world.
|
| > Science can only help us understand things within
| spacetime. Anything beyond that is philosophy.
|
| Even if one takes this statement to be correct, it doesn't
| imply that any specific philosophical idea about "beyond
| space time" is correct.
|
| And frankly even the "philosophy" of an after life can
| easily be dispensed with. There's absolutely no reason to
| suggest an after life, or a duality between the body and
| "identity" exists other than "we would like to believe so".
|
| It's not just awful science but also bad philosophy.
| kazinator wrote:
| Science can help us understand things within spacetime.
| Mathematics and logic within abstract, but rigorously
| defined spaces. Anything beyond that is philosphy,
| asthetics, politics, religion, ...
| non-chalad wrote:
| If not theoretical grouds, surely there are hypothetical
| grounds? Now to find a way to falsify, and thus prove them...
| mgaunard wrote:
| Why would you be downvoted for speaking facts?
|
| Also why should you restrain from pointing out weaknesses in
| other people's comments due to the fear of negative karma?
| Karma is meant to be burned.
| radomir_cernoch wrote:
| From a different point of view... The cochlea is a "real"
| "implementation" of Fourier transform
| (https://www.britannica.com/science/sound-physics/The-ear-as-...)
| __loam wrote:
| It's easy to forget how grounded in physics biology is. When I
| was in college, we had an issue where our stem cell lines were
| differentiating into bone. Turns out, the hardness of the
| environment is a signal stem cells can transduce, and the hard
| dish was telling them they were supposed to be bone cells.
| praptak wrote:
| Hmm, this mechanism can only add more bone to existing bone,
| right?
|
| So how do the first bone cells know to start becoming bone?
| empiricus wrote:
| probably a transient signal (in the rna-protein soup) that
| happens during embryonic development.
| __loam wrote:
| Which came first, the chicken or the egg?
| omnicognate wrote:
| The cochlea actually supports the point the article makes, as
| while it does transform to the frequency domain it doesn't do
| (or even approximate) a Fourier transform. The time->frequency
| domain transform it "implements" is more like a wavelet
| transform.
|
| Edit: To expand on this, to interpret the cochlea as a fourier
| transform is to make the same mistake as thinking eyes have
| cone cells that respond only to red, green or blue light. The
| reality is that each cell has a varying reponse to a range of
| frequencies. Cone cells have a range that peaks in the low,
| medium or high frequency area and tails off at the sides.
| Cochlear hair cells have a more wavelet-like response curve
| with secondary peaks at harmonics of their peak response
| frequency.
|
| Caveat: I'm not an expert in this, only an enthusiastic
| amateur, so I eagerly await someone well-akshuallying my well-
| akshually.
| willis936 wrote:
| Sure but the article asks the question about the frequency
| domain generally then constrains itself to Fourier
| transforms. Fourier has a lot of baggage from making large
| assumptions. Transforms like wavelet and laplace are closer
| to "real world" because of fewer non-physical assumptions and
| have actual physical implementations. It doesn't get much
| more real than seeing it with your own eyes.
| quibono wrote:
| > Transforms like wavelet and laplace are closer to "real
| world" because of fewer non-physical assumptions and have
| actual physical implementations.
|
| Could you expand on this a bit please? Especially as it
| relates to the Laplace transform.
| weinzierl wrote:
| For the cone cells we have excellent empirical data about the
| response curves. Do ypu know if there is public data for the
| chochlea hair cells?
| adrian_b wrote:
| Any kind of discrete Fourier transform, and also any device
| that generates the Fourier series of a periodic signal, even
| when done in an ideal way, must have outputs that are
| generated by a set of filters that have "a varying response
| to a range of frequencies".
|
| Only a full Fourier transform, which has an infinity of
| outputs, could have (an infinite number of) filters with an
| infinitely narrow bandwidth, but which would also need an
| infinite time until producing their output.
|
| So what you have said does not show that the eye cone cells
| do not perform a Fourier transform (more correctly a partial
| expansion in Fourier series of the light, which is periodic
| in time at the time scales comparable to its period).
|
| The right explanation is that the sensitivity curves of the
| eye cone cells are a rather poor approximation of the optimal
| sensitivity curves of a set of filters for analyzing the
| spectral distribution of the incoming light (other animals
| except mammals have better sensitivity curves, but mammals
| have lost some of them and the ancestors of humans have re-
| developed 2 filters for red and green from a single inherited
| filter and there has not been enough time to do a job as good
| as in our distant ancestors).
| SeanLuke wrote:
| I'm not certain the secondary peaks would matter very much
| though? It seems to me that maybe the most useful model would
| be not a wavelet transform but some form of DCT?
|
| At any rate, the point is that the frequency domain matters a
| lot, since our brain essentially receives sound data
| converted to the frequency domain in the first place...
| waveBidder wrote:
| this is one example of a whole family of orthogonal Wavelet
| transforms, that let you trade off how between frequency and
| scale resolution.
| nyanpasu64 wrote:
| > To turn the Hadamard matrix in the nicely-ordered flavor
| showcased earlier, we need to sort the rows based on their
| sequency. I'm not aware of an algorithm more elegant than
| counting the number of zero crossings
|
| By staring at the matrix, I guessed a pattern and algorithm
| already known according to
| https://en.wikipedia.org/wiki/Walsh_matrix:
|
| > The sequency ordering of the rows of the Walsh matrix can be
| derived from the ordering of the Hadamard matrix by first
| applying the bit-reversal permutation and then the Gray-code
| permutation:
| j16sdiz wrote:
| This is disappointing.
|
| The title prompted a science (or, maybe, philosophy of science)
| question, but the article give only applications.
|
| Maths don't have to be real to be useful.
| phkahler wrote:
| Yeah, I was hoping for something more along the lines of "are
| wave functions real".
| hcks wrote:
| To answer your question they aren't
| bowsamic wrote:
| How do you know? PBR theorem justifies the idea that the
| wavefunction is ontological rather than epistemological
| em500 wrote:
| The mundane answer to the philosophical question is that the
| frequency domain is not any less or more "real" than the time
| domain; they're different mathematical abstractions of change.
| Just like base-10 numbers are not any more or less "real" than
| base-2 or base-8. Carthesian coordinate locations are not any
| more or less "real" than polar coordinate locations. Some
| representations are just more convenient for specific
| calculations than others (mostly by simplifying specific
| calculations).
| a-dub wrote:
| no. not unless you throw the phases away.
| taneq wrote:
| Hah, nice.
| inatreecrown2 wrote:
| too complex for me.
| amitport wrote:
| The Hadamard transform IS a discrete Fourier transform (on the
| Boolean group (mod 2Z)^n)
| rendall wrote:
| As an aside _frequency domain_ sounds like a great name for a
| live music venue.
| cjk2 wrote:
| What about the s-domain?
| 01100011 wrote:
| Is "two" real? Is any math "real"? Is software "real"? You don't
| have to be "real" to be useful.
| mjfisher wrote:
| I used to wonder if imaginary numbers were "real" (in the
| common sense, rather than the mathematical one). This only
| intensified after I learnt that they're required in quantum
| mechanics to explain any physical behaviour at all.
|
| Now I consider them to be just as "real" as the integers -
| which is not at all. Both just human-invented concepts with no
| fundamental physical basis.
|
| As you point out - useful, though!
| adrian_b wrote:
| I do not believe that it is possible to claim that integer
| numbers or imaginary numbers do not have a fundamental
| physical basis. The small integer numbers are not invented by
| humans, because many other animals can count up to some small
| number (like 5 or 6).
|
| Both are abstractions. That means that they are properties of
| real physical objects, which are obtained by ignoring all the
| other properties of those physical objects that are
| irrelevant in the context of the application.
|
| Therefore an abstract property is an equivalence class of
| physical objects, where all their other properties are
| ignored, so they are equivalent if they have the same value
| for the property of interest.
|
| Non-negative integer numbers are equivalence classes of
| collections of physical objects, integer numbers are
| equivalence classes of pairs of such collections.
|
| The imaginary unit is the equivalence class of all rotations
| by a right angle in the 2-dimensional space. Humans, like
| many animals, have an innate ability to recognize right
| angles, like also certain small numbers, so looking around
| you can perceive as easily all imaginary units like all
| numbers 3.
|
| The complex numbers are the equivalence classes of all
| geometric transformations of the 2-dimensional space that can
| be decomposed in rotations and similarities (a subset of the
| affine transformations). In contrast, the 2-dimensional
| vectors are the equivalence classes of all translations of
| the 2-dimensional space (another subset of the affine
| transformations).
|
| All the things that are equivalent from the point of view of
| an integer number or a complex number, so they are the basis
| from which such numbers are abstracted, are things that you
| can see with your own eyes in the physical world (similarity
| transformations appear in optical projections, e.g. in the
| shadows of physical objects, and the eyes are based on them).
| Arech wrote:
| Much earlier than quantum mechanics, they are extremely
| useful in electrical engineering. And even before that
| (historically) - their 4D generalization, quaternions, are
| extremely useful to describe 3D rotations.
| 77pt77 wrote:
| Complex numbers are also useful to describe rotations on
| the plane, and conformal transformations more generally.
| dotnet00 wrote:
| I think "real" in the common sense is just too vague to
| really apply to things as abstract as numbers. 'Real' numbers
| seem real in a common sense because counting is a universal
| human experience. Adding in the imaginary plane allows for
| "counting" systems which have 2 interconnected components
| like waves, which also turn out to occur very often when you
| look at nature a bit more closely. Thus why you can derive
| all of trigonometry from Euler's formula. It just isn't a
| universal human experience to look at nature that closely
| yet.
| 77pt77 wrote:
| > I learnt that they're required in quantum mechanics to
| explain any physical behaviour at all.
|
| You can do QM without complex numbers as people are used to
| use them.
|
| But it gets really awkward really fast.
| kazinator wrote:
| Is it a place? Yes. A moving wave of some kind, propagating
| through space, has a certain wavelength, which is related to the
| frequency. The wavelength is a distance between wave crests, and
| those have a time-dependent location as the wave propagates. When
| the wave is periodic, the concept of frequency summarizes the
| state of a large number of the wave crests existing at equally-
| spaced locations.
|
| We can have a standing wave, e.g. vibrating string. Frequencies
| then translate to concrete place in space, such as the nodes
| where the string appears to be still, (like the exact middle if
| it is excited with the second harmonic).
| usgroup wrote:
| By way of reductio, I think one could make the same argument for
| the non uniqueness of the time domain and presumably anything
| else admitting to an isomorphism.
| sieste wrote:
| This reminds me of a great conversation I had on a whiteboard
| during my masters in the dynamical systems group:
|
| "So energy is pumped into the system on the left, and is
| dissipated over here on the right"
|
| "But the system is rotationally invariant, there is no left and
| right"
|
| "I meant in frequency space"
|
| "Oh I thought you were talking in real space"
|
| "ARE YOU STUPID, WHO THE HELL THINKS IN REAL SPACE??!?"
| bmacho wrote:
| Do you call each other stupid during masters, or is this
| heavily paraphrased?
| sieste wrote:
| It was tongue in cheek.
| ska wrote:
| Sadly I've heard much worse, not in jest.
|
| Some academic communities are pretty dysfunctional.
| datascienced wrote:
| This is XKCD worthy! Send it to Randall
| jdthedisciple wrote:
| Wait a second-
|
| is there even such a thing as left and right in frequency
| space?
|
| It's an abstract representation, I'd it doesn't have any
| relationship to spatial dimensions in terms of left right up
| down
| sieste wrote:
| it was a sketch of the spectrum with frequencies along the x
| axis, increasing towards the right.
| hinkley wrote:
| The amount of disdain I've radiated at white tower academia
| over my life time could power a small island nation for a
| decade.
| takd wrote:
| Mathematically, the Fourier transform is "simply" a way of
| representing time signals in a certain orthogonal vectorial
| basis. Vectors in an ordinary sense, e.g. a displacement vector
| on Earth's surface can also be represented in several orthogonal
| bases: one basis could, for example, be two vectors pointing
| North and East; another could be a vector pointing along a
| certain road and one perpendicular to it. There is nothing
| inherently special about any of these bases, one could draw maps
| according to any of these two or many other conventions.
| (Orthogonal basis vectors are not even necessary, only
| convenient.)
|
| The interesting thing about time-dependent signals (or any
| "pretty" function, really) is that they live in an infinite-
| dimensional vector space, which is hard to imagine; but (besides
| some important technicalities) the math works mostly the same
| way: signals as infinite-dimensional vectors can be represented
| in a lot of bases. One representation is the Fourier transform,
| where the basis vectors are harmonic functions. The "map" showing
| the shape of a signal as a combination of infinitely many
| harmonic functions -- i.e. the frequency domain -- is just as
| real as any other map with different basis vectors, e.g. the
| Walsh-Hadamard transform mentioned in the article. And,
| crucially, the original time-domain representation is also just
| one map showing us the signal, though it is often the most
| natural to us.
| weinzierl wrote:
| Excellent answer and I am sure you are aware of this, but like
| to point out:
|
| _" Mathematically, the Fourier transform is "simply" a way of
| representing time signals in a certain orthogonal vectorial
| basis."_
|
| Not just _time signals_ but any piecewise continuous and
| differentiable as well as Dirichlet integrable function. This
| has many applications, just a few examples from the top of my
| head: image processing, solving differential equations, fast
| multiplication.
|
| I'd also like to add that from a mathematical point of view
| these transforms are "lossless" in the sense that the
| transformed function has the _exact_ same information as the
| original and you can get back the _exact_ original even if all
| you have is the transform.
|
| I feel this often gets lost when people approach the Fourier
| transform from a more engineering perspective, not at least
| because we often do the transform to throw away unwanted
| information, like certain frequency components.
|
| In the end it really is just one of many perspectives to look
| at a function.
| quibono wrote:
| > I feel this often gets lost when people approach the
| Fourier transform from a more engineering perspective, not at
| least because we often do the transform to throw away
| unwanted information, like certain frequency components.
|
| That was my problem as well. My first introduction to Fourier
| transforms was through more of an engineering lens. I
| remember having trouble with the _inverse_ Fourier transform.
| I was OK with a Fourier inverse of an already transformed
| function but I wasn't quite sure what that would mean when
| applied to a non-transformed, "regular" function.
| takd wrote:
| As a related aside, the terms "cepstrum" and the
| "quefrencies" [1] (c.f. spectrum and frequencies) sound so
| hilarious that when I first heard about them I was
| convinced it was some kind of prank.
|
| [1] https://en.wikipedia.org/wiki/Cepstrum
| cmehdy wrote:
| This does read like a joke, I had never heard of it
| either and I'm wondering if many people do use this at
| all..
|
| Operations on cepstra are labelled quefrency analysis (or
| quefrency alanysis[1]), liftering, or cepstral analysis.
| It may be pronounced in the two ways given, the second
| having the advantage of avoiding confusion with kepstrum.
| aswanson wrote:
| It's used in speech signal processing & seismic signal
| analysis.
| ljosifov wrote:
| Almost all speech recognizers until this latest crop (of
| end-to-end DL NN ASR) operated on cepstral coefficients
| (and their delta-s and delta-delta-s) as their feature
| vector.
| pas wrote:
| > [...] I wasn't quite sure what that would mean when
| applied to a non-transformed, "regular" function.
|
| Have you gained some intuition/understanding for this?
|
| I tried a few inputs in WolframAlpha, but unless I manually
| type in the integral for the inverse transform there's not
| even a graph :) (and I have no idea whether it's even the
| same thing without putting a `t` in the exponent and
| wrapping it in an f(t) = ... )
|
| https://www.wolframalpha.com/input?i=integral+%28sin%28x%29
| +...
| weinzierl wrote:
| Not parent (but GP) and intuition can mean many things
| but what helped me was keeping in mind:
|
| Every continuous periodic function turns into a discrete
| aperiodic one when transformed. Works both ways.
|
| Continuous aperiodic stays continuous aperiodic. Discrete
| periodic stays discrete periodic.
| sharpneli wrote:
| Inverse fourier transform of a non transformed signal gives
| you basically the fourier transform with some changes (I
| can't remember which, were the numbers conjugates or
| something?). Applying it the second time gives you same
| result as if you'd do the forward direction transform
| twice.
|
| If you apply fourier transform 4 times you get your
| original function back. You can think of it as 90 degree
| rotation. Inverse transform just rotates it in the opposite
| direction.
|
| The rotation analog is not even too far fetched as
| fractional fourier transform allows you to do an arbitrary
| angle rotation.
| araes wrote:
| Having never heard of this for the Fourier Transform,
| needed to read.
|
| F0: original signal
|
| F1: frequency domain signal
|
| F2: reverse time signal
|
| F3: inverse fourier signal
|
| F4: original signal
|
| Also, has further weird applications I've never heard of
| with "Fractional Fourier Transforms" [1] which can
| apparently result in smooth smears of time -> frequency
| domain [2].
|
| [1] https://en.wikipedia.org/wiki/Fractional_Fourier_tran
| sform
|
| [2] https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stev
| encys.j...
| Salgat wrote:
| A fourier transform basically gives you an infinite number
| of sine waves with different amplitudes/phases at every
| frequency. If you add them all back together (the inverse
| fourier transform), you get back your original signal.
| Audio compression in this case would just be excluding the
| sine waves that are too high frequency too hear when you
| add them all back. I always hate how people try to make the
| fourier transform sound more complex than it actually is
| (and yes there is more nuance to compression than this, but
| this is just the basic idea).
| computerfriend wrote:
| I wonder how Fourier transformations are taught in
| engineering courses, because the idea that it could be
| "lossy" is strange and not obvious to me. It has an inverse
| after all.
| aswanson wrote:
| A square wave would take an infinite number of sinusoidal
| waves to perfectly reconstruct. To approximate it, one
| truncates the coefficients. This is done in any engineering
| application where memory isn't infinite. Which is all of
| them.
| mkaic wrote:
| Of course, a discrete, finite _sampling_ of a square wave
| at a set of points in time only requires a finite number
| of coefficients to perfectly reconstruct.
| bisby wrote:
| Which is the point. A discrete fourier transform can
| never recreate a square wave unless you have infinite
| samples. It can only recreate the finite sampled signal
| (which is only an approximation of the square wave, and
| not a real square wave).
|
| This means that sure, the Fourier transform itself isn't
| lossy (garbage in, garbage out) but Fourier transforms
| would be used in contexts where loss are introduced. If I
| have a real perfect square wave, and I want to a take a
| fourier transform of it, the sampling is going to
| introduce loss, so to associate sampling losses with the
| transform itself is fair. Real square wave ran through a
| DFT program on my computer is going to spit out an
| approximation of a square wave -- loss.
| davidgay wrote:
| The good news of course is that if you were sampling a
| real signal, then that signal was not actually a perfect
| square wave. So the fact that you can't (re)construct a
| perfact square wave is somewhat moot...
| bisby wrote:
| Generally yes, but it's a perfectly reasonable assumption
| that a natural source could generate a signal that is
| beyond the bounds of what we can record. Any real signal
| generated by a computer is going to fit within the
| constraints of what we can generate, but inevitably
| something like a whale, or a quasar or something will
| generate a wave that will be lossy.
|
| But also, the question this is all responding to was
| effectively "why would engineers associate Fourier
| transforms with loss" and the answer is simply "because
| the techniques used in calculating most Fourier
| transforms are going to inherently put a frequency limit
| and anything beyond that will be lost or show up as an
| artifact". Engineers work with real world constraints and
| tend to be hyper aware of those constraints even if they
| often don't matter.
| bisby wrote:
| In my engineering courses, Fourier transforms were taught
| in the context of discrete fourier transforms. Because
| sampling is a thing that matters (computer audio is
| discrete data points, not an actual wave).
|
| The Fourier transform of a discrete signal repeats in the
| frequency domain. For example, [1, -1, 1] could be a sine
| wave with the exact half of the sampling frequency going
| from 1 to -1 back to 1 once .... Or it could be a sine wave
| with double the sampling frequency that is actually going
| from 1 to -1 to 1 to -1 all within the gap of the first 2
| samples. Or it could be 3x the sampling frequency, 4x the
| sampling frequency, etc. The solution is to only keep the
| part of the transform that is below the Nyquist limit,
| because we don't have a sampling rate accurate enough to
| measure the higher frequencies, so just assuming they dont
| exist. This also means that if the source signal WAS in
| fact 4x the sampling frequency, we will see a spike at 1/2
| the sampling frequency in the fourier transform, and when
| we re-create the signal, it will be completely wrong.
|
| So unless you have analog hardware for measuring the
| Fourier transform (or are working purely in a non-physical
| mathematical domain, like "i have a sine wave" which can be
| perfectly represented), you are naturally going to be
| taking discrete samples of a signal to measure the Fourier
| transform, which means you are going to be losing any part
| of the signal that doesn't adhere to sampling rates.
|
| Because my engineering courses were so heavily focused on
| digital signal processing, when I hear "fourier transform"
| i immediately think of "discrete fourier transform" and
| loss is immediately applicable.
| aj7 wrote:
| The DFT has quite severe limitations that do not appear in
| the Fourier transform. In particular, the Nyquist criteria
| that there be zero signal energy for ALL frequencies above
| half the sampling frequency can only be approximated, and
| must be accomplished BEFORE sampling, i.e. in the time
| domain.
| OJFord wrote:
| > one basis could, for example, be two vectors pointing North
| and East; another could be a vector pointing along a certain
| road and one perpendicular to it.
|
| And there's no requirement that they be perpendicular is there?
| The second just needs 'some amount of perpendicular', North and
| North-East for example? Since any [n, e] can also be described
| as [(1-sqrt(2)*e)*n, sqrt(2)*e] in the latter. (I think that's
| right, but my main point is you can do it, not the particular
| value there, and if that's way off I'll blame the fever.)
| Jensson wrote:
| You typically want an orthonormal basis though, but yes you
| don't need it.
| datascienced wrote:
| If you can walk this way / and this way | then you can do /
| minus | to get your orthogonal -
| OJFord wrote:
| Yes exactly.
| eigenspace wrote:
| I used to think of it like another basis too, but nowadays I
| think this basis analogy is a bit fraught, or at least not the
| whole story.
|
| In particular, for multidimensional spaces, the usual
| multidimensional Fourier transform only really works if you
| have a flat metric on that space (I.e. no curvature). That's a
| bit of a warning signal given that our universe itself is
| curved.
|
| There was some very interesting work recently where it was
| shown how to generalize Fourier series to certain hyperbolic
| lattices [1], and one important outcome of that work is that
| the analog of the Fourier space is actually higher dimensional
| than the position space.
|
| Furthermore, the dimensionality of the 'Fourier space' in this
| case depends on the lattice discretization. One 2D lattice
| discretization may have a 4D frequency-like domain, and another
| 2D lattice might have a 8D frequency-like domain.
|
| [1] https://arxiv.org/abs/2108.09314 or
| https://www.pnas.org/doi/full/10.1073/pnas.2116869119
| Jensson wrote:
| > That's a bit of a warning signal given that our universe
| itself is curved.
|
| What does this has to do with whether they are a different
| basis for cases where we don't account for curvature? This
| seems completely irrelevant, sure the tool can't be used in
| some cases but it can be used as a basis change in other
| cases.
| mananaysiempre wrote:
| Not the whole story indeed, but you have to dive into
| representation theory somewhat to get more: the Fourier
| transform is more or less the representation theory of the
| (abelian) group of the translations of your space, thus the
| homogeneity requirement. The finite-lattice version[1] (a
| discretized torus, basically) may serve to hint what's in
| stock here.
|
| [1] https://www-
| users.cse.umn.edu/~garrett/m/repns/notes_2014-15... (linear
| algebra required at least to the degree that one is
| comfortable with the difference between a matrix and an
| operator and knows what a direct sum is)
| eigenspace wrote:
| If you like this topic, I strongly recommend you read the
| references I attached to my comment.
|
| In uniformly curved 2D hyperbolic spaces, it turns out that
| there is a higher dimensional _non-Abelian_ Fuchsian
| translation group defined on a higher genus torus.
| messe wrote:
| It's not an analogy. It's literally just another basis.
|
| > In particular, for multidimensional spaces, the usual
| multidimensional Fourier transform only really works if you
| have a flat metric on that space
|
| What the hell does the metric of space-time have to do with
| this? When computing a fourier transform, we're not working
| in 3+1 dimensional space-time, we're working in either an
| N-dimensional (in the discrete case) or \infty-dimensional
| (in the continuous case) vector space; while that term
| contains the word "space" they DO NOT, in this context, have
| anything to do with Euclidean space or the Pseudo-Riemannian
| manifold that GR treats space-time as.
| nextaccountic wrote:
| I wanted to know more about this too, and I hate to make
| meta comments, but I'm afraid your confrontational approach
| may make the other person think this conversation isn't
| worth the hassle
|
| Which would be a bad thing, reading this kind of
| conversation is what makes this site worthwhile
| eigenspace wrote:
| For what it's worth, I did reply, but I probably wouldn't
| have if you hadn't expressed frustration at the prospect
| of not reading further.
| eigenspace wrote:
| > What the hell does the metric of space-time have to do
| with this?
|
| Maybe calm down for a moment and try not being such a hot-
| headed ass. You seem to have missed the point entirely.
|
| I'm well aware that these functions can be described as
| vectors in an infinite dimensional Hilbert space.
|
| The problem I'm bringing up is that the domains of these
| functions (i.e. not the vector itself) typically have
| geometric properties we care about.
|
| The problem is that if one has a manifold with a non-
| trivial intrinsic geometry, then functions defined on that
| manifold cannot be faithfully Fourier transformed without
| losing pretty much all geometrically relevant information.
|
| It turns out that in some cases, there are generalizations
| of the Fourier transform of a function on a curved
| manifold, but in those cases, the domain of the transformed
| function is very different, typically having a higher
| dimensionality.
|
| This is particularly relevant and problematic in physics,
| where the Fourier transforms of functions on spacetime are
| really important and useful, but dont work in curved
| spacetimes.
|
| E.g. it's a big problem when doing QFT on a curved
| spacetime that one cannot separate positive frequencies of
| a field from negative frequencies.
| enaaem wrote:
| In the past astronomers believed in the geocentric model of the
| universe with epicycles. It was extremely accurate, and if more
| accuracy was needed they added more epicycles. It was a
| completely wrong model, but they unknowingly used the Fourier
| series as a function approximator.
| TeMPOraL wrote:
| It wasn't a _wrong_ model, it was just much more complex than
| needed, given better understanding of physics. Viewing the
| universe relative to stationary Earth is a perfectly fine
| exercise, even if it means you have to DFT the rest of the
| solar system for the math to work.
| HarHarVeryFunny wrote:
| Sure, nothing special about sine waves as basis functions for
| signal decomposition. Not necessarily the best either,
| depending on what you want to do.
|
| Still, as pertains to whether "the frequency domain is a real
| place", maybe sine waves are relevant as representing resonant
| frequencies of physical systems.
|
| There also seems to be something fundamental about the way
| multiple radio frequencies can simultaneously propagate through
| a vacuum as long as they are different frequencies.
| BeetleB wrote:
| Ok. Now what's the difference between the Fourier transform and
| the Fourier series?
|
| To me what you described sounds more like the Fourier series.
| zeristor wrote:
| A real place?
|
| There's an optics experiment I did, bloody fiddly, where a
| picture goes through some lenses, and there's a plane of the
| frequencies, and it goes through further lenses and is projected
| on a screen.
|
| By blocking out areas in the frequency plane, you can change the
| image. It was extremely fiddly so huge thanks to Dr Bruce
| Sinclair at St Andrew's.
|
| Physics lab work is where you get to see how things work,
| although if you go through the theory several months after you've
| done the lab work you're a bit lost.
| knolan wrote:
| Fourier optics. This is also how Schlieren works.
| Anotheroneagain wrote:
| _and there 's a plane of frequencies_
|
| Doesn't that always happen, which is why aperture limits
| resolution, why you get diffraction spikes with reflecting
| telescopes, and so on?
| teamonkey wrote:
| Upvoted by a fellow Bruce fan
| rasz wrote:
| you can do it in software
| https://imagemagick.org/Usage/fourier/#noise_removal
| thfuran wrote:
| In software, you can also make pictures of a lot of things
| that don't exist.
| dotnet00 wrote:
| Yep, that feature of getting the frequency domain
| representation through optics is pretty convenient for the
| various microscopies and spectroscopies performed at light
| sources.
| perlgeek wrote:
| Another interesting generalization of the DFT is the Lomb-Scargle
| transformation, which doesn't require a fixed measurement
| interval in the time domain.
|
| It is often used to determine the frequency of a period signal
| when you don't have fixed measurement interval, like in
| astrophysics.
|
| https://iopscience.iop.org/article/10.3847/1538-4365/aab766 has a
| general introduction, and
| https://docs.astropy.org/en/stable/timeseries/lombscargle.ht...
| is a nice intro to using it in the astropy library (in Python)
| hinkley wrote:
| I've been thinking a lot lately about how useful it might be to
| represent Prometheus data in a frequency domain, to visualize
| weekly, daily, and yearly access patterns in capacity planning.
| Autoscaling can avoid brownouts but it can't tell you what your
| annual budget should be. Or why.
|
| But Prometheus data isn't really a sampling interval. Even if
| each machine in you cluster is reporting on an interval, they
| aren't synchronized.
| jhfdbkofdchk wrote:
| Even if it is real, odds are it's complex.
| TheOtherHobbes wrote:
| More accurately, it's a complex place.
| weinzierl wrote:
| Which brings us to the question of whether the complex numbers
| are _real_ , despite them not being _real numbers_.
|
| This question has a long and fascinating history.
| jbottoms wrote:
| The scope of a domain is defined by its bounds. Expressed
| mathematically the bounds can be expressed using any variable
| type that can enclose an area or a field of elements or sets. The
| range of the bounds can extend to infinity. An abstract
| unspecifued domain may have no associated elements, sets or
| operators.
| zadwang wrote:
| Fourier basis is unique in that the complex exponential basis
| functions are the eigen vectors of the linear time invariant
| (LTI) systems. No other transform has this property. Many real
| world systems (circuits, communication channels, antennas, etc)
| are LTI. This property make sure for example, signals transmitted
| over different frequencies do not interfere. That is why Fourier
| transform is so useful and used instead of other transforms.
| There is also the connection with quantum physics, in using
| Fourier pair as wave functions of position and momentum, which
| other transforms don't have.
| nestes wrote:
| I'm surprised you're one of the only commenters to bring this
| up. I have an electrical engineering background -- for
| analysis, lots of systems are assumed to be either linear or
| very weakly nonlinear, and a lot of our signals are roughly
| periodic. Fourier transforms are a no-brainer.
|
| Convolution turns into multiplication, differentiation wrt time
| of the complex exponential turns into multiplication by
| j*omega. I don't know about you, but I'd rather do
| multiplication than convolution and time derivatives.
|
| As a corollary, once you accept "we use the Fourier
| representation because it's convenient for a specific set of
| common scenarios", the use of any other mathematical transform
| shouldn't be too surprising (for other problems).
| eigenspace wrote:
| I always found it so strange how the top comments on blogposts
| like this are trying to answer the 'question' without engaging
| with the article at all.
|
| Do they and the people upvoting them not realize that it's a blog
| post?
| krapp wrote:
| Few people here ever bother to read past the title.
| takd wrote:
| The question itself merits discussion, while the post did not
| exactly answer it.
| greatquux wrote:
| Sometimes the title itself generates a kind of excitement in
| your brain and you start answering it before you even read the
| blog post. And in this case as others have noticed, it's not
| exactly answered in the way some of us would.
|
| I do know I've thought about this, and even had one of those
| "moments of realization" on the drive home from my first
| lecture on DCTs, where I thought I could transform all of human
| history from time domain to frequency domain and how this would
| bring out certain patterns and truths that could not be
| understood otherwise. I swear I wasn't on drugs! (Though the
| lecture was in fact given by Prof. Marshall of Rutgers, the
| father of the lyricist for the rock band Phish, at whose
| concerts I have imbibed certain substances... but I digress.)
| The frequency domain is just as "real" as any other
| mathematical construction that can help us understand
| everything.
| cycomanic wrote:
| The article asks a very general/philosophical question, but then
| goes on to say the FD is not really that special because we can
| find other sets of orthogonal bases and transforms between them.
|
| I would argue that despite this fact the frequency domain and by
| extension the FT is special compared to many other transforms,
| because we can actually observe them in nature. Two examples: a
| lens will perform a 2D FT of an input image on a collimated beam,
| we can observe this with e.g. a screen. Second example, we can
| measure the wavelength (or frequency) of light by projecting the
| output of a grating or prism onto a ccd again a direct
| measurement of the FD (similar measurement can be done for RF
| waves).
| harpiaharpyja wrote:
| How real is the frequency domain you say? Well, it's complex...
| ptero wrote:
| The title is clickbait-y.
|
| TLDR: Fourier transform is an approximation. We can come up with
| other approximations, for example by square waves.
| tim333 wrote:
| Not really a place. I think the technical term is thing. You can
| argue it's a real thing but not very place like.
|
| Seahorse Valley is more place like if you want to debate the
| reality of these things
| https://www.mrob.com/pub/muency/seahorsevalley.html
| j7ake wrote:
| In music, it is common to think of the structures in frequency
| space rather than time space.
|
| For example, one thinks of tug C major triad as C E G, rather
| than mixed sound that comes out of C E G together.
| scottmsul wrote:
| Except sinusoids _are_ special in that they are natural solutions
| to the Helmholtz wave equation. There 's other problems too like
| square waves having infinite energy. This article might make
| sense to a mathematician or computer scientist but neglects the
| underlying physics of sound and waves.
| neltnerb wrote:
| Excellent point, lots and lots and lots and lots of physical
| objects are harmonic oscillators. That does have pretty
| fundamental grounding in physics.
|
| I can think of lots of other places I'd use fourier analysis
| (at least qualitatively as with doing diffusion modeling in my
| head) but you're right that sinusoids are more physically
| "real" whereas being possible to represent in any basis set is
| more "valid" if that makes any sense.
|
| Not quite sure what the right word is on this one, but I agree
| "real" kind of suggests real oscillators underlying the
| phenomena. Square waves are less physical because of
| discontinuities in both the signal and derivative; nature
| really doesn't care for discontinuities.
| whiterknight wrote:
| Sinusoids are also special because they are eigenfunctions of
| the derivative operator.
|
| The physics result is actually probably a consequence of that.
|
| At the end of the day the whole lesson of modern math is that
| its useful to view things from many perspectives.
| meatmanek wrote:
| Frequency domain also makes the math really easy for linear,
| time-invariant operations, which (approximately) describe a lot
| of systems that exist in nature.
|
| The Gibbs phenomenon, for example, falls out naturally from the
| IFT of a frequency response where all the frequencies above
| some cutoff are zero.
|
| I'm curious how the square wave frequency domain would describe
| the Gibbs phenomenon -- I think you'd have harmonics of the
| fundamental square frequency showing up as if the system were
| nonlinear.
| DasCorCor wrote:
| Can you book an AirBnB in the frequency domain?
| evrydayhustling wrote:
| Love this article but would like to dispute the author's notion
| of "real". In the post, he shows that the frequency domain is not
| _special_ , in the sense that there are infinitely other equally
| valid representations.
|
| But many places are real without being special - other than to
| those who make special use of them!
|
| I'd argue a real place is one that affords the operators that
| allow us to inhabit and interact there -- stuff like object
| permanency, adjacency and distance. If things can be organized
| and sustained there, are they not real?
|
| It's fun to imagine what kinds of structures can inhabit the
| frequency domain - or any other.
| raphlinus wrote:
| For the math aficionados in this thread, I have a frequency
| domain related set of ideas I'd like to develop into a more
| rigorous mathematical theory. Basically, represent a curve as
| Chebyshev polynomial: T_1 represents a line, T_2 represents an
| arc, T_3 an Euler spiral, etc. Smooth curves have rapidly
| decreasing Chebyshev coefficients, and this whole thing is
| potentially a lot easier to work with than Frechet distance,
| which is the usual error metric but very annoying.
|
| This is conceptual and theoretical, but potentially has immediate
| application for computing a better offset of a cubic Bezier, used
| for stroke expansion.
|
| If this sounds intriguing, a good starting point is the Zulip
| thread[1] I'm using to write down the ideas. I'd especially be
| interested in a collaborator who has the experience and
| motivation to coauthor a paper; I can supply the intuition and
| experimental approach, but the details of the math take me a long
| time to work out. (That said, I'm starting to wonder if engaging
| that slog myself might not actually be a good way to level up my
| math skills)
|
| [1]:
| https://xi.zulipchat.com/#narrow/stream/260979-kurbo/topic/E...
| xeyownt wrote:
| This is a great article. Wonderful easy explanations about
| something that is somewhat complex.
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