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My understanding is that plain old telephone service passes frequencies between 300Hz and 3400Hz. How is it that a V.34 modem in the late 90s would be able to achieve data rates of 33,600bits/s with this amount of signal bandwidth? That's nearly 11 bits/hz which is a significantly higher spectral efficiency than many modern technologies (e.g. DOCSIS 3.1, 802.11AC, 4G LTE, etc).

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    $\begingroup$ Wow! Thems were the olden daze. i forgot. Didn't they get to be 56K modems? Before DSL? $\endgroup$ Aug 20 at 0:00
  • $\begingroup$ @robertbristow-johnson My understanding is that 56K speeds weren't achieved with the same end-to-end analog. I know that in the late 90s I had a V.92 modem but never got anything over 33.6K from my ISP. $\endgroup$
    – Chris_F
    Aug 20 at 0:04
  • $\begingroup$ It's all just a function bandwidth times SNR. DOCSIS 3.1 supports 4096 QAM which is 12 bits/symbol or 24 bits per Hz (theoretically) $\endgroup$
    – Hilmar
    Aug 20 at 0:13
  • $\begingroup$ @Hilmar My math could be wrong but I think that for a channel capacity of 33.6Kbit/s with a bandwidth of 3.1KHz would require a SNR of 66dB assuming your modulation was right at the Shannon-Hartley limit. That seems suspect. $\endgroup$
    – Chris_F
    Aug 20 at 4:51
  • $\begingroup$ How you get to the 66db? $\endgroup$
    – Moti
    Aug 20 at 5:41
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This article quotes the SNR of a phone line at 45 dB.

Combine this with a one sided bandwidth of about 3 kHz and you get a maximum channel capacity of about 45 kb/s.

45dB SNR is a perfectly reasonable assumption for a phone line, it's not great but certainly workable for speech. It corresponds roughly to the noise floor of a 7 bit A/D converter, so it's really not that outlandish.

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Phone lines presumably have higher SNR than your typical power/interference limited radio link. Plus, wired links tends to have less reflections and be more stable, thus the fundamental limits bandwidth and SNR in a Shannon sense was probably easier to achieve than todays variable and reflective radio channels (requiring modern tech).

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