# Mathematics behind dial up modem maximum data rate

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#### magnetra

##### Full Member level 5
Dial up modem

While connecting to the internet using dial up connection (i.e using copper pair) the max data rate is limited to 56kbps. Is there a mathematical approach to this limitation?

magnetra

#### techie

##### Advanced Member level 3
Dial up modem

I think it is partiallly maths and partially empirical. Since thw b/w of voice channnel is 8khz, you can only use a carrier of lower freq than that. and given a limited s/n ratio, you are limited as to how many bits you can encode.

#### throwaway18

##### Advanced Member level 2
Dial up modem

>Since thw b/w of voice channnel is 8khz

I don't think thats true for most telephone systems. Analog telephone conversations get digitised at the exchange.
8000samples/sec giving 4KHz max frequency is normal.

#### techie

##### Advanced Member level 3
Dial up modem

What i meant was that the limitation is due to the bandwidth and s/n ratio of the voice channel.

#### Nimer

##### Full Member level 3
Re: Dial up modem

hi
Non of the above is true , what makes the 56k limitation is the following :

in voice communiction telephones data rate of 64k are achievable , since we trnsmit 8k symbols per second and each one is coded with eight bits , so
8k*8 =64k

but indat commubication one bit of each sample is taken for control and synchronization purposes so we have 8k samples per second ,each represented by seven data bits and one control bit .....8k*7=56k which is the data rate .

#### techie

##### Advanced Member level 3
Dial up modem

Nimer, can you quote a reference to this "theory". I suspect that you are not right.

#### mehtesham

##### Junior Member level 3
Re: Dial up modem

i think nimer`s is correct. in fact each telephone line in europian standard (E-multiplexing) is 1/32 of a E1 line (2.048 Mbps) and this standard uses PCM standard[here 8 bit for each symbol,8k symbols per sec (because tele line is designed for voice,recieved signal is filterd to 8khz base band in communication centers)]. combine this with serial communication standard for PCs.

here is intro for E1.

#### techie

##### Advanced Member level 3
Dial up modem

I think you are mixing up digital and analog. Isn't encoding 32 voice channels on a digital E1 line different from encoding digital data on an analog telephone line. I am not an expert on this but from what I remember, the 56k modem uses more than 8 bits per sample. (I dont remember exactly how many).

##### Junior Member level 3
Re: Dial up modem

Nimer said:
hi
Non of the above is true , what makes the 56k limitation is the following :

in voice communiction telephones data rate of 64k are achievable , since we trnsmit 8k symbols per second and each one is coded with eight bits , so
8k*8 =64k

but indat commubication one bit of each sample is taken for control and synchronization purposes so we have 8k samples per second ,each represented by seven data bits and one control bit .....8k*7=56k which is the data rate .

i guess Nimer is correct considering the 64 k channel but what made it 56 as maximum is the quantization noise

#### changfa

##### Advanced Member level 4
Dial up modem

Shannon Theorem gives a hard limit of how fast you can transmit over a telephone line. Around 1945, shannon claimed that the maximum is around 30kbps based on his famous equation C=log2(1+SNR). Nowadays, we do have higher rates. The reason is not the theory is wrong, but we have better channel condition (high SNR than before).

#### techie

##### Advanced Member level 3
Dial up modem

Changfa is correct. This limit is due to the SN ratio of the analog line and has got nothing to do with the 8x7=56k. You are mixing up digital and analog worlds. How else would you get the 30k theoratical limit predicted by Shannon.

#### magnetra

##### Full Member level 5
Dial up modem

This is what Wikipedia had to say on thi issue,

A 56 kbit/s line is a digital connection (possibly a leased line, possibly switched) capable of carrying 56 kilobits per second (kbit/s), or 56,000 bit/s, the data rate of a normal single channel digital telephone line in North America. With the wide deployment of faster, cheaper, technologies such as ADSL and SDSL, 56 kbit/s lines are generally considered to be an obsolescent technology.

The figure of 56 kbit/s is derived from its implementation using the same digital infrastructure used since the 1960s for digital telephony in the PSTN, which uses a PCM sampling rate of 8,000 Hz used with 8-bit sample encoding to encode analogue signals into a digital stream of 64,000 bit/s.

However, in the T-carrier systems used in the U.S. and Canada, a technique called bit-robbing uses, in every sixth frame, the least significant bit in the time slot associated with the voice channel for Channel Associated Signaling. This effectively renders the lowest bit of the 8 speech bits unusable for data transmission, and so a 56 kbit/s line used only 7 of the 8 data bits in each sample period to send data, thus giving a data rate of 8000 Hz × 7 bits = 56 kbit/s.

This rate has nothing directly to do with modem rates over an analog telephone line, other than that they both represent a performance just short of 64 kbit/s: a 56 kbit/s line is a purely digital link.

#### throwaway18

##### Advanced Member level 2
Re: Dial up modem

techie said:
This limit is due to the SN ratio of the analog line and has got nothing to do with the 8x7=56k. You are mixing up digital and analog worlds.

This is obviously wrong. ADSL works at 500Kb/s or often faster on the same telephone lines you can only get 40-50K on using a modem.

How else would you get the 30k theoratical limit predicted by Shannon.

ADSL uses a wider frequency range.

#### sameerbabu

##### Member level 3
Re: Dial up modem

I only accept chagfa.
Can u please tell me how shannon got 30kbps limit?

#### qv1985

##### Newbie level 6
Re: Dial up modem

Acording to my teacher Dial up modem is a equipment which adaptive bit rate so the bit rate of dial up modem is
BR=56Kbps-(8/6)*n where n=0,1,2....

#### magnetra

##### Full Member level 5
Dial up modem

Do u have any document regarding this topic? If so, please post it in the board.

M

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