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The article is interesting but some images have Error 400 so I can’t see them.
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Cool article; I’m still baffled though.
I've been studying for my amateur radio license recently, and this article is a great introduction to the basics.

But really, if you want to get your hands dirty with some practical electronics, and also want to be able to communicate without relying much on nearby infrastructure, amateur radio is a great hobby.

This reminds me of a beautiful book written by Paul Nahin, 'The Science of Radio'.
Bought it but never read it - is it worth pushing through?
To get an idea about radios, I made a crystal radio when I was in 7th grade, I only had few components. The only component I had difficulty in getting was the crystal oscillator (I was living in a rural town).

It was mind blowing when I first heard the audio through IEMs ! It felt magical that this contraption was working without any battery source.

Crystal radios don't normally contain crystal oscillators, which do require a battery source. Perhaps you mean a crystal rectifier.
Yeah perhaps I'm confused cuz this was long long back.
Never thought about the AM bandwidth thing before that is interesting and seems obvious now (from Fourier Transform point of view)
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I've been learning about radios for a while, and this article explained one of the key questions I had: why can't you turn on and off some single frequency waveform faster to transmit data faster? (answer: changing amplitude messes with the spectrum and makes it no longer a single frequency...)

I tend to prefer these visual and intuitive explanations to the mathematically based ones usually given in lectures. The "open capacitor" example was something I hadn't thought of before.

> In today’s article, I’m hoping to provide an introduction to radio that’s free of ham jargon and advanced math

…(scroll)…

> The identity for cos(α + β) can be trivially extended to cos(α - β), because subtraction is the same as adding a negative number:

…(scroll)…

> From the formula we derived earlier on, the result of this multiplication necessarily indistinguishable from the superposition of two symmetrical sinusoidal transmissions offset from a by ± b, so AM signals take up bandwidth just the same as any other modulation scheme.

EDIT: I am offended that you guys think my awesome explanation is from GenAI.

Imagine a circuit. Like a flashlight. The electrons flow from the battery to the lightbulb and back. It’s like a race track. They proceed in an orderly fashion.

There are some other electrical components. If you hook them up in just the right way, you get something called an LRC circuit. The electrons don’t flow in an orderly way now. They go back-and-forth. In spurts and fits. You’re making them wiggle. There are some very nice equations that allow you to specify exactly how much and how fast the wiggling is.

One cool thing about a circuit with wiggling electrons is that if you put some wires close by those electrons will also start wiggling.

This is called radio.

I was listening to story of QualComm in Acquired podcast. This is highly related.
This is a great review, but don't let theory interfere with practice. There's an art and feel to radio and antenna building. I've built antennas with scrap wire and baluns without double checking . Just a rough length of wire tweaked until you hear someone.

The theory seems overly complex (because it is). but for practical radio you can tune up nearly anything and chat -- and be way more productive with it.

Imagine obsessing over the mechanics of baseball or fishing without ever tossing a ball or casting a reel.

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If you dig deeper, you end up with I/Q signals. If you take the input from the antenna and delay it by 90 degrees, you then have something which can have a positive or even negative frequency. Using multiplication of complex terms, you can add or subtract frequencies without generating unwanted images.

You could also use signals from 2 antennas 90 degrees apart to get I and Q. This gives you the added ability in that signals from one side has a negative frequency. It's some really useful stuff.