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The Magic That Makes the World Work: Electromagnetic Fourier Transforms

Why a mind-boggling mathematical technique called Fourier transforms might be the most important discovery ever made. A discussion on light waves and electromagnetism.

What did you do after you woke up this morning?

Maybe you checked texts or Tinder on your cell phone. Perhaps you opened your laptop and checked work emails. If you were hungry, perhaps you microwaved some hot pockets for breakfast. You probably turned on the radio to some relaxing smooth jazz to start the day off right. Or maybe you simply looked out the window to enjoy the light of the morning sunrise kissing plants and chairs on your patio.

No matter what you did this morning, you interacted with light waves. You’re observing light waves everywhere you look. Every object you see is giving off visible light. However, the role of light waves in your life goes way beyond just this visible light.

If you interacted with your cell phone this morning, your cell phone was sending light waves back and forth with a cell tower to connect to the internet. If you hopped on your laptop, your laptop was sending light waves to your WiFi router to connect to the internet.

If you listened to the radio, your radio was picking up light waves broadcasted from the radio station tower. Even your microwaved hot pocket was made possible by concentrated light waves from the microwave oven.

Visible light, cell phones, WiFi, radio waves, microwaves, bluetooth, TV, radar, walkie talkie communication. Almost all airborne technology you can think of is the same fundamental thing: a light wave propagating through the air.

If these different technologies are all just light waves, how are they different from each other? Why do we have different labels, like “WiFi” or “radio” for the same underlying thing?

More importantly, how does this all work? As it turns out, Fourier transforms make all of this possible. It is not an understatement to say that they make the world go round!

To understand what Fourier transforms are and why they are so important, we need to understand more about light waves.

What is a Light Wave?

Fundamentally, a light wave is a stream of photons (massless particles) moving through space, carrying electrical and magnetic energy. Put another way, it is the electric and magnetic field oscillating at a point in space and time. This is why light waves are often called electromagnetic waves.

A fascinating side note is how light waves are different from sound waves. A sound wave is the vibration of particles in the medium the sound is moving through (for example air or water). This means that a light wave can move through a vacuum (emptiness) like outer space, whereas a sound wave cannot because there are no particles to vibrate.

A light wave has two main properties: amplitude and frequency. Amplitude is how strong the signal is, or the height of the wave. Frequency is how many waves pass through in any given second.

Image Source: ResearchGate

WiFi, radio, and microwaves are all simply just light waves with different frequencies!

Electromagnetic Spectrum

This brings us to the concept of the electromagnetic spectrum:

Image Source: nasa.gov

The higher the frequency of the light wave, the more dangerous it is. Light waves with frequencies at or above the ultraviolet (UV) range are called ionizing. This means that individual photons have the capability to break apart and damage DNA, which can cause cancer.

Once frequency drops below the UV range, light waves are called non-ionizing, meaning individual photons do not have the energy capability to damage DNA.

As you can see, the highest-frequency light waves are called gamma rays. These are very dangerous due to their extremely high frequency. Thankfully, Earth’s atmosphere blocks us from being exposed to them, otherwise we’d all be in trouble!

Next are x-rays, which can also be very dangerous. This is why you wear a metal plate over valuable body parts before getting x-rayed, and why the technician stands on the other side of a wall. Ultraviolet rays can also be damaging, and the sun gives off UV rays, so be sure to throw that sunscreen on!

As light wave frequency decreases from UV, we enter the visible light spectrum, ranging from the highest frequency blue light to the lowest frequency red light. It is called visible light because the human retina can only pick up light in this frequency range.

As frequency drops from here, light waves become infrared. A lot of nocturnal animals have evolved to see infrared waves because objects which give off heat (aka prey) emit infrared waves.

At even lower frequencies, we get into the microwave range of light. WiFi routers operate in this range, usually either at 2.4 or 5 GHz (gigahertz), which means 2.4 or 5 billion waves/second. Microwave ovens operate around 2.4 GHz.

Descending in frequency from there, cell phones generally use frequencies less than 1 GHz, in the 600-900 MHz (megahertz) range, which means 600-900 million waves/second. FM (frequency modulation) radio stations broadcast signals at frequencies ranging from 88-108 MHz. This is what the “99.9” means on your dial: the station broadcasts light waves at a frequency of 99.9 MHz.

Finally, at the very bottom of the spectrum are the lowest-frequency light waves, AM (amplitude modulation) radio waves. These generally range from 550-1720 kHz (kilohertz), which means 550,000-1,720,000 waves/second.

Signal Encoding

When you’re surfing the web on your laptop, your WiFi router and laptop are sending light waves back and forth to each other at a frequency of 2.4 or 5 GHz. When you’re driving in your car blasting “Party in the USA” on the radio, your car’s antenna is receiving light waves with a frequency of around 100 MHz sent by a radio tower.

When you’re talking on your cell phone, your cell phone and the cell tower are pinging light waves with a frequency around 750 MHz back and forth to each other to enable the conversation. How does this all work? How are these light waves actually carrying information?

Frequency and Amplitude Modulation (FM and AM)

Information in today’s day and age is stored as a bunch of numbers called bits: 0’s and 1’s. Every digitally stored song, all of the data on every website, the files on your computer. It’s all just long sequences of 0s and 1s like the following:

00001010010110111010100101010110010101110011011010100101010101

Thus, to send information via a light wave, we simply need to figure out how to convert a string of 0’s and 1’s, like the above, into a light wave. It turns out there are actually several ways of doing this.

Amplitude Modulation (AM)

What if we modified the amplitude, or height, of each wave to encode the 1’s and 0’s? We could say that if the height of the peak is above a certain amount, that indicates a 1. If the height of the peak is below a certain amount, that indicates a 0.

If whoever is receiving the light wave knows we’ve encoded it this way, they can just read the height of the waves to know whether we’re sending a 1 or a 0. Thus, with a continuous wave, we can send a sequence of 1’s and 0’s in this way. This is known as amplitude modulation.

Frequency Modulation (FM)

Alternatively, what if we slightly modified the frequency of each wave we sent? On average, the wave we send will still have a certain frequency, but we can push the peak of the wave to occur right before, or right after, a certain threshold. If the peak occurs before the threshold, it’s a 1, otherwise, it’s a 0. This is frequency modulation.

Signal encoding is more complicated than what was just described. Nonetheless, these basic strategies are at the root of how information is encoded via light waves. This is how everything from radio stations to wireless cell phones to WiFi works.

Wave Interference

We’ve discussed how light waves of varying frequencies are around us everywhere all the time: WiFi signals, radio waves, cell phone signals, visible light, etc. Furthermore, we identified that a light wave is simply the oscillation of the electric and magnetic field at a point in space and time.

This begs a natural question: how do all of these light waves not interfere with each other? Wave interference physics tells us that if multiple waves hit the same point at the same time, the amplitudes of the waves combine (aka add together).

If you were to drop two rocks into a pool and watch the waves emanate from rocks, eventually the wave patterns would hit each other. The result would be an interference pattern where the water height at a given point of the pool represents the sum of the individual wave heights.

Bringing this back to electromagnetism, consider an arbitrary point in the air in your house, like a point in the air right in front of your face right now.

Think about all of the different light waves hitting that point in space right now. Signals from radio towers, cell phone towers, your WiFi router, your neighbor’s WiFi router, television signals, perhaps UV light from the sun. Heck, even visible light is bouncing off everything, including your face, and hitting that point right now!

Given what we just learned about signal encoding, wouldn’t all of these waves interfere with each other and mess everything up? After all, decoding information from a light wave involves measuring the amplitude and/or frequency of the wave! If you have other waves mucking with the frequency and amplitude at a point in space and time, you wouldn’t be able to accurately understand the information anymore.

How do all these light waves not interfere with each other? Even if they’re at different frequencies, the waves themselves are still hitting the same point in space at the same time, so the wave patterns should interfere with each other.

How does anything work??? How do the light waves not interfere with each other???

Answer: they do interfere with each other. A lot. And everything works anyways.

How is that even possible?

Glad you asked. Fourier transforms.

Fourier Transforms

Fourier transforms are a mathematical technique named after their discoverer, French mathematician Joseph Fourier. A Fourier transform can decompose a real-world waveform into its individual constituent sinusoidal (wavelike) math terms added together. What the heck does that mean?

This means that a device can receive an electromagnetic signal that looks like a heaping pile of sh** formed from a bunch of light waves of different frequencies interfering with one another. It can run a Fourier transform to break apart that signal into a bunch of individual signals of different frequencies added together.

Once this is done, the device knows exactly which individual light waves were sent at which frequencies and what their amplitudes were at any given time!

For example, let’s suppose that the device picked up an electromagnetic signal that looked like this:

By running a Fourier transform, the device could determine that the only possible combination of individual waves that could result in this exact interference pattern is the combination of these three individual waves:

+

+

Note the varying frequencies and amplitudes of these individual waves. The first wave could be the electromagnetic signal getting sent from a cell phone tower at 900 MHz. The second wave could be the signal coming from your WiFi router at 2.4 GHz, while the third wave could be your local FM radio station broadcasting at 90 MHz.

All three of these waves interfere with each other to form this seemingly nonsensical combined signal:

Yet with a Fourier transform, we can determine the exact signal getting sent from each of the individual sources: the cell tower, the WiFi router, and the radio station.

Note that in practice, a Fourier transform does this same process with hundreds to thousands of individual light waves at different frequencies!

Why is this important? This means that it’s possible to glean out the signal sent at a specific frequency from the chaotic combination of interfering light waves. In other words, a device that runs a Fourier transform can listen to an input signal at a specific frequency, and filter everything else out!

I hope you see how mind-boggling this is. Forget about Hogwarts: this is pure magic.

I’ll illustrate with a practical analogy. Imagine you and your friends all peed together into the same cup. Along comes good ol Joe Fourier. He sees the cup, and asks you all if you’d like to separate your pee into individual cups.

“No!” you say. “What’s wrong with you, Joe!?”

Joe doesn’t care. He proceeds to pull out his magical Fourier transform device, sticks it in the cup, and out comes a bunch of nozzles, each pouring a specific person’s pee back out into an individual cup. This should not be possible to do, and yet it is, with the beauty of math! Thank you, Joseph Fourier.

Please don’t ask how Fourier transforms work mathematically. It involves advanced calculus and partial derivatives and is beyond the scope of this post (translation: I have no idea).

Fourier transforms are what allow your car to listen to a specific radio station. They allow your cell phone to communicate with the cell tower. They allow your laptop to talk to the WiFi router. Without Fourier transforms, the individual light waves would be forever lost in a sea of interference. Nothing as you know it would work.

In other words, the magic of Fourier transforms truly does make the world go round.

Appendix

Do Cell Phones Cause Cancer?

While researching this topic, another topic of interest kept surfacing again and again: do cell phones, especially 5G technology, cause cancer? After analyzing how electromagnetism actually works, I can confidently say the answer is a resounding no.

There are two potential ways that light waves from cell phone technology could cause cancer. The first is individual photons (particles of light) causing DNA damage. The second is damage from heat given off from the light wave.

We’ve previously discussed how the frequency of the light waves from your cell phone or cell towers is in the non-ionizing range. This means that individual photons do not have enough energy to cause DNA damage.

This is even true for the higher frequencies of 5G technology. 5G frequencies can get as high as 50 GHz, which sounds like a lot. But consider that the start of the “ionizing” band of the electromagnetic spectrum, ultraviolet light, is around 800,000 GHz, or 16,000 times the frequency of the highest 5G bands.

So the only way cell phone radiation would cause cancer is from damage due to excessive heat. Let’s investigate whether this would be possible.

Electromagnetic Heat Damage

A light wave can give off heat due to the rotation of polar molecules from the shifting electrical field. The amount of heat given off depends on the strength (or amplitude) of the wave.

A microwave oven, for example, gives off over 700 watts of power. Even though the frequency of the microwaves isn’t considerably high (2.4 GHz), because the waves have such a large amplitude focused on a specific area, this heats your food up.

Cell phone light waves, on the other hand, have a steady state power emission of about 0.02 watts, thousands of times less than the power of a microwave oven. Even if all of that radiation was absorbed by your entire body during a call, which it isn’t, it would take 20,900,000 seconds, or 241 days, to heat up your body by 1 degree celsius.

Even if all this occurred, your metabolism would likely be able to mitigate this process and cool you down. Thank you Inna Vishik for the detailed analysis and calculations.

The reality is that cell damage due to excessive heat from electromagnetic radiation is simply not possible in practice. And since we know that cell phone waves are also non-ionizing, there’s no realistic mechanism of action for cell phone radiation to cause cancer.

Same-Frequency Interference

You may have noticed one peculiarity from our discussion about Fourier transforms. Fourier transforms have the capability to separate signals sent at different frequencies, but what about two different signals sent at the same frequency?

Fourier transforms are quite magical, but they still cannot separate out the signals from independent sources if those sources broadcast signals at the same or very similar frequencies.

This is why radio stations are each assigned a specific frequency to broadcast at with a large enough frequency gap between the next closest frequency.

Furthermore, nearby cities ensure they do not have overlap between FM radio stations, otherwise people who are within range of both cities will not be able to pick up the signal from either station.

Cell phone towers communicate with each individual cell phone at a different frequency*, otherwise the signals getting sent would cancel each other out and be indiscernable, even by Joltin Joe Fourier.

Have you ever noticed your microwave oven interfering with your WiFi router? It sounds absurd, but most microwave ovens operate around the 2.4 GHz frequency. As it turns out, the two most popular frequency bands for a WiFi router are 5 GHz, and 2.4 GHz.

A microwave oven is designed to encapsulate its waves inside the microwave, but they aren’t always perfectly designed. Because the frequency is so similar to WiFi, this is another potential source that can confound even the magical Fourier transform.

* Cell towers also use a technology called TDMA, or Time Division Multiple Access, which allows multiple users to share the same frequency by splitting the signal into separate time slots and dividing the time slots among the multiple users.

Sources

1: Dangers of Visible Light: https://physics.stackexchange.com/questions/452592/if-visible-light-has-more-energy-than-microwaves-why-isnt-visible-light-danger

2: Signal Encoding: https://www.youtube.com/watch?v=FqSWpi8n1rY

3: Quora Question About Cell Phone Radiation: https://www.quora.com/Do-mobile-phones-cell-phones-emit-harmful-radiation-or-electromagnetic-fields

4: Inna Vishik’s Answer About Cell Phone Radiation: https://qr.ae/pGyLwa

5: Evolution Behind Why Humans See Visible Light Spectrum: https://www.quora.com/Why-do-we-only-see-light-in-a-particular-range-the-visible-light-spectrum-of-the-EM-spectrum-Is-there-a-reason-based-on-the-theory-of-evolution-behind-the-way-our-eyes-evolved-to-see-the-visible-light-spectrum

6: Christmas Lights WiFi Interference: https://www.pbs.org/newshour/science/war-on-christmas-lights-wifi-interference

7: NASA Electromagnetic Spectrum: https://imagine.gsfc.nasa.gov/science/toolbox/emspectrum1.html

8: Wavelength and Amplitude Photo: https://www.researchgate.net/figure/Wavelength-amplitude-and-frequency-of-an-electromagnetic_fig21_303377557

9: Fourier Transform: https://www.thefouriertransform.com/

10: 5G Frequency: https://www.sdxcentral.com/5g/definitions/what-is-5g-spectrum/

11: TDMA: https://en.wikipedia.org/wiki/Time-division_multiple_access

12: Featured Image: https://pixabay.com/illustrations/particles-wave-circle-color-1435363/

Published by Analytical Aspergian

I am an Aspergian who loves logically analyzing the world around me. On this blog, I analyze anything that interests me, from economic design to electromagnetism to sports nutrition and recovery.

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