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https://www.10stripe.com/articles/why-is-56k-the-fastest-dialup-modem-speed.php
If you've ever had dialup internet service, or still do, or just know someone that does, you have probably heard terms like "56k modem". "56k" has become almost synonymous with dialup Internet access. But it's such an arbitrary number. It's not divisible by ten, it's not a power of two... so why was it chosen as the fastest dialup speed? For the answer, we will have to travel back in time quite a while.
Our visitors from Google should be warned that this is not a "stripped down" explanation; it is intended for relatively technical readers. But if you really want to know where this magic number comes from, you need to understand some of the technical background. As we shall see, "56k" was not just pulled out of a hat.
[...] Anyone that has ever used a dialup modem knows full well that they don't actually get to connect at that speed, though. And that their connection speed varies each time they dial in. There are two factors at work here.
The first is the FCC. If you are in the United States, the FCC places a restriction on the power output of devices connected to the phone network. The result is that you will never be able to connect at a speed faster than 53.3 kbit/s.
The second is the overall complexity of the phone network. 56 kbit/s (or 53.3 kbit/s) requires very good operating conditions, as it is really operating beyond the paramaters of what the phone network is required to be capable of. Operating at these speeds requires that there only be one ADC between the user and their ISP (which is not guaranteed to be true, but typically is), and that the copper wiring in the user's "local loop" have very good electrical properties. Part of the dialup process that is used to initiate a connection is an evaluation of the overall quality of the connection; if it is determined to be lacking, the modem will automatically drop down to a lower data rate.
(Score: 2, Informative) by pTamok on Tuesday November 21 2023, @10:48AM
Guys,
you could read the V.92 standard. It is freely available. As is the V.90 standard it updates.
ITU-T: V.90 : A digital modem and analogue modem pair for use on the Public Switched Telephone Network (PSTN) at data signalling rates of up to 56 000 bit/s downstream and up to 33 600 bit/s upstream [itu.int]
ITU-T: V.92 : Enhancements to Recommendation V.90 [itu.int]
If the service provider was connected to the PSTN with a digital connection (ISDN or T- or E- carrier), then that end is a 'digital modem', and you can rely on the least significant bits of pulse-code modulation encoded signal being correct. This gives you a full 56 knit/s of capacity. The main reason it is 56 kbit/s and not 64 kbit/s is that fact that the PSTN in North America uses/used 'robbed-bit' signalling [wikipedia.org], that used the least-significant bit of a PCM encoded channel for in-band signalling (this is an oversimplification, but it'll do),
If your modem was connected to an analogue line (which it would be), then there was a conversion at the service provider using a modem/codec/DSP that sampled the incoming signal on the analogue line 8000 times a second and encoded it using pulse-code modulation (PCM [wikipedia.org]) in an 8-bit value, giving you a data rate of 64000 bits per second; otherwise written as 64 kilobits per second or a '64k channel'. It isn't a linear encoding, North America and Japan using a mapping function (compander) called 'ยต-law' [wikipedia.org] ('meuw-law') and the rest of the world using 'a-law' [wikipedia.org] (using the wrong companding function on companded data caused a noticeable drop in quality).
The FCC set a maximum power to be used on analogue phone lines in the USA in order to limit the effects of crosstalk between circuits, and that constraint reduces the set of points in the modulation constellation that can be used (certain combinations exceed the power levels), which means that fewer symbols can be used to carry the data*.
More background: Electronic Products: The standards behind V.90 and ADSL modems [electronicproducts.com]
More (mathematical details here: PDF: Voiceband modems and DSLs [nycu.edu.tw].
ADSL uses analogue POTS lines, but gets higher data rates because it operates at higher frequencies over the distance between the DSLAM and the end-ADSL modem. Also, many modems had data compression, so could get better effective throughput on compressible data.
Yes, I used to work in the industry, and have accumulated a lot of (now useless) knowledge. I didn't have to deal with understanding the mathematics behind convolutional and trellis codes: I 'just' used the products. And T-carriers, and E-carriers. And...
Now I'm just a dinosaur.
*A simple example - assume you have a signalling system that uses two tones Low and High. Each tone can be off or full power. The possible combinations that can be sent down the wire simultaneously are: Both off, one on, the other on, and both on. If there is a restriction on the maximum total power that can be sent, then you can't use the 'both on' combination, so you can use only three symbols (off,off), (off, on), and (on, off) because (on,on) is disallowed. Being able to use only three symbols instead of four cuts your data rate. The trellis code [wikipedia.org] used for V.92 has more possibilities, but some of the combinations would send too much power down the line, so can't be used in the USA, cutting the data rate.