The Fourier series & rotating vectors in action (with i lurking) — more on the mathematical fabric of reality

The Fourier series is a powerful technique that can be used to break down any repeating waveform into sinusoidal components, based on integer number harmonics of a fundamental frequency:

Video:

This is already amazing, that by summing up harmonically related sinusoids (with suitable amplitudes and lagging) we can analyse any repeating waveform as a sum of components. This then extends to any non-repeating pulse, once we go to an integral, which brings in the idea of a continuous spectrum where some wave “energy” is found at every particular frequency in a band.

However, something subtler lurks: As the illustration based on clips from the video shows, a sinusoid can be seen as the projection of a rotating vector (= a phasor). Thus, what the Fourier series is doing is that it is adding up a set of harmonically related rotating vectors instant by instant, yielding the overall result of the periodic waveform (here, a square wave). And indeed, we can get the rotation by adding up two perpendicular oscillating vectors, one on oX, the other on oY. This then gives us two summed harmonic motions, which allows us to bring in the forces and inertia at work. (Where, yes, we can then go on to elliptical, parabolic, hyperbolic and even linear motion.)

Rotating vectors should ring a few bells for those who have been following our discussion recently.

Yes, we can start with the von Neumann construction of the natural counting numbers, N:

{} –> 0

{0} –> 1

{0,1} –> 2, etc

And from that define integers (Z), then rational numbers (Q) then reals, R, eventually the surreals:

That is interesting, but let us take the reals, R, and define a vector rotation operator i* so that it rotates R through a right angle anticlockwise pivoting on 0. So, any positive x in R is transformed to i*x, along i*R, going upwards. (Yes, the y-axis, oY.) Now, apply i* again, so we see i*i*x = – x, on the negative reals. That is, we have a natural interpretation of i, i.e. i^2 = -1, i is sqrt(-1). The rotating vectors approach makes for a more coherent understanding of so-called imaginary numbers. Thus, complex numbers, C are actually 2-d vectors that help us to do all sorts of interesting things, especially once they get to rotate.

We can go on from here, using power series to define a complex exponential form of the rotating vectors approach, which then yields the famous expression 0 = 1 + e^(i*pi).

All of this then comes back to the Fourier analysis, as the video demonstrates by showing rotating vectors added up tip to tail to form complex wave forms. Which, then points to how we can move freely between the time domain and the frequency domain to better understand phenomena in the real world.

Where, the underlying message clearly is that the logic of structure and quantity — mathematics — is deeply embedded in reality. And no, this is not flogging a dead horse, it is awakening human (as opposed to equine) understanding. END

8 Replies to “The Fourier series & rotating vectors in action (with i lurking) — more on the mathematical fabric of reality”

1. 1
kairosfocus says:

The Fourier series & rotating vectors in action (with i lurking) — more on the mathematical fabric of reality

2. 2
hazel says:

Nice math, and nice application to the physical world. I used to use a simple bicycle wheel on a stand to introduce the creation of the sine and cosine wave from rotational motion.

3. 3
kairosfocus says:

H, loss of net services. Praythee, tell me, how does hearing work, specifically the cochlea? KF

4. 4
kairosfocus says:

F/N: On how hearing creates a frequency domain transform of sound inputs, driving the onward processing, Wiki is a handy reference:

The stapes (stirrup) ossicle bone of the middle ear transmits vibrations to the fenestra ovalis (oval window) on the outside of the cochlea, which vibrates the perilymph in the vestibular duct (upper chamber of the cochlea). The ossicles are essential for efficient coupling of sound waves into the cochlea, since the cochlea environment is a fluid–membrane system, and it takes more pressure to move sound through fluid–membrane waves than it does through air; a pressure increase is achieved by the area ratio of the tympanic membrane to the oval window, resulting in a pressure gain of about 20× from the original sound wave pressure in air. This gain is a form of impedance matching – to match the soundwave travelling through air to that travelling in the fluid–membrane system . . . .

The perilymph in the vestibular duct and the endolymph in the cochlear duct act mechanically as a single duct, being kept apart only by the very thin Reissner’s membrane. The vibrations of the endolymph in the cochlear duct displace the basilar membrane in a pattern that peaks a distance from the oval window depending upon the soundwave frequency. The organ of Corti vibrates due to outer hair cells further amplifying these vibrations. Inner hair cells are then displaced by the vibrations in the fluid, and depolarise by an influx of K+ via their tip-link-connected channels, and send their signals via neurotransmitter to the primary auditory neurons of the spiral ganglion.

The hair cells in the organ of Corti are tuned to certain sound frequencies by way of their location in the cochlea, due to the degree of stiffness in the basilar membrane.[3] This stiffness is due to, among other things, the thickness and width of the basilar membrane,[4] which along the length of the cochlea is stiffest nearest its beginning at the oval window, where the stapes introduces the vibrations coming from the eardrum. Since its stiffness is high there, it allows only high-frequency vibrations to move the basilar membrane, and thus the hair cells. The farther a wave travels towards the cochlea’s apex (the helicotrema), the less stiff the basilar membrane is; thus lower frequencies travel down the tube, and the less-stiff membrane is moved most easily by them where the reduced stiffness allows: that is, as the basilar membrane gets less and less stiff, waves slow down and it responds better to lower frequencies. In addition, in mammals, the cochlea is coiled, which has been shown to enhance low-frequency vibrations as they travel through the fluid-filled coil.[5] This spatial arrangement of sound reception is referred to as tonotopy . . . . Not only does the cochlea “receive” sound, it generates and amplifies sound when it is healthy. Where the organism needs a mechanism to hear very faint sounds, the cochlea amplifies by the reverse transduction of the OHCs, converting electrical signals back to mechanical in a positive-feedback configuration. The OHCs have a protein motor called prestin on their outer membranes; it generates additional movement that couples back to the fluid–membrane wave. This “active amplifier” is essential in the ear’s ability to amplify weak sounds.[6][7]

The active amplifier also leads to the phenomenon of soundwave vibrations being emitted from the cochlea back into the ear canal through the middle ear (otoacoustic emissions) . . . .

Otoacoustic emissions are due to a wave exiting the cochlea via the oval window, and propagating back through the middle ear to the eardrum, and out the ear canal, where it can be picked up by a microphone. Otoacoustic emissions are important in some types of tests for hearing impairment, since they are present when the cochlea is working well, and less so when it is suffering from loss of OHC activity . . . .

The coiled form of cochlea is unique to mammals. In birds and in other non-mammalian vertebrates, the compartment containing the sensory cells for hearing is occasionally also called “cochlea,” despite not being coiled up. Instead, it forms a blind-ended tube, also called the cochlear duct. This difference apparently evolved in parallel with the differences in frequency range of hearing between mammals and non-mammalian vertebrates. The superior frequency range in mammals is partly due to their unique mechanism of pre-amplification of sound by active cell-body vibrations of outer hair cells. Frequency resolution is, however, not better in mammals than in most lizards and birds, but the upper frequency limit is – sometimes much – higher. Most bird species do not hear above 4–5 kHz, the currently known maximum being ~ 11 kHz in the barn owl. Some marine mammals hear up to 200 kHz. A long coiled compartment, rather than a short and straight one, provides more space for additional octaves of hearing range, and has made possible some of the highly derived behaviors involving mammalian hearing . . .

In short, sinusoidal frequency domain decomposition of sound waves is a key mechanical phenomenon exploited by our hearing system, leading to in effect a frequency domain transformation of the temporal pattern of compressions and rarefactions that we term sound. This is of course closely related to the patterns we explored and discovered using Fourier power series and integral analysis of oscillations and transient pulses.

Where, on the mechanical side, harmonic motion is tied to elastic and inertial behaviour. Which in turn is directly connected to a rotating vector analysis — leading straight to the complex exponential analysis that draws out the full power of complex numbers, form Z = R*e^i*wt, w being circular frequency 2* pi*f (in radians per second), f the cycle per second frequency. All of this ties back to the fundamental frequency cycles and integer-multiple frequency harmonic epicycles in the OP above.

Again, Mathematical study turns out to reflect quantities, structures and linked phenomena which are embedded in the fabric of our world.

KF

PS: Notice, not a few design subtleties?

PPS: The vocal tract, in effect a wind instrument, also exploits fundamentals and harmonics to create auditory, frequency-based patterns as well as transients.

5. 5
math guy says:

Allow me to point out that these sums of sine and cosine functions constitute the building blocks of the standard Hilbert space L2. Each element f of L2 is a (usually infinite) sum of different sines and cosines. But these functions f are the wave forms of quantum mechanics that model reality at the micro-level.

As BA77 often reminds us, the wave forms appear to “collapse” into measurements of reality only after choices are made by observers. QM is reminding us of Pythagorus’ dictum that “all is number” (or rather sums of sines and cosines).

6. 6
kairosfocus says:

MG, interesting observation. I for preference go to complex exponentials as characteristic functions for most things of interest but sinusoids are just as valid. The complex exponential vector z = R*e^iwt on polar axis 0X, captures the sine and cos picture in a power series based framework. Where, of course, cos wt + i* sin wt is giving us x and y axis components at right angles that then give the position vector from the origin as r — the hypotenuse — of length defined through pythagoras. KF

PS: Some notes: https://www.math.upenn.edu/~deturck/m426/notes02.pdf and a vid: https://www.youtube.com/watch?v=jgi8hbOmUmk Sadun goes on to introduce Fourier series in his next vid: https://www.youtube.com/watch?v=PTm1H40g-UI (I am thinking his channel is a trove of very well introduced Math: https://www.youtube.com/channel/UC3IZYipZ6A_Rs5v0GdwUM7Q/videos?sort=dd&view=0&shelf_id=1 I am looking at his over- the- shoulder- vid of partly covered sheets of paper approach as an interesting video-ised rendering of the chalk talk or transparency roll on an OHP approach also, cf. Khan Academy.)

7. 7
math guy says:

KF, I also prefer the standard basis of complex exponentials but “dumbed down” my observation for the benefit of readers who have limited math backgrounds.

8. 8
kairosfocus says:

MG,

Okay.

My thought is, it’s not hard to think in terms of vectors and rotations, which gets us to a lot of useful results, including a reasonable view of i.

The cos-sin view and magnitude angle to polar axis view is a next step, as is the concept of angular velocity, which can be demonstrated with a rotating wheel. Indeed, projecting to the axes (similar to the vid) brings out the sinusoids.

The power series representations can probably be motivated on Taylor or the like, or taken on faith by authority. In that context, the relationship between sin, cos and the complex exponential can be brought in.

Going beyond, I find it useful to see how standing waves on a string give a natural way to get to quantum states. The particle in a box context pops up and we are in the quantum world. Along the way, Bohr’s orbits may be useful.

All along, we should be observing how deeply embedded structure and quantity are in the physical world.

I think immersion in facts may well be the best antidote to the sort of nominalism that keeps cropping up.

Merely pointing out incoherence seems to have limited impact on today’s ultra-modernism indoctrinated mind. As in, they doubt the principle of explosion and think that their doubt trumps logic and evidence to the contrary.

As we have seen.

KF