# How is it that a dial-up modem can achieve 33.6Kbit/s with only 3100Hz of bandwidth?

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).

• Wow! Thems were the olden daze. i forgot. Didn't they get to be 56K modems? Before DSL? Commented Aug 20, 2021 at 0:00
• @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. Commented Aug 20, 2021 at 0:04
• 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) Commented Aug 20, 2021 at 0:13
• @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. Commented Aug 20, 2021 at 4:51
• yeah, you're using the "wrong" dB. it's $10^{x/10}$, not $/20$; SNR is a power, not an amplitude ratio. Commented Aug 20, 2021 at 6:42