By J. H. Saltzer, February 28, 1997.

Here is a short topic to slip into a recitation that discusses the physical layer of computer networks: How can a 56 Kilobit/second modem possibly work over a telephone line? The answer, though requiring a bit of communications theory to understand completely, also has one exploring the telephone network as a system.

The puzzle

Everyone knows that an analog telephone line has a limited bandwidth, originally intended to carry only voice. (The actual passband is roughly 300-2400 Hz.)

And everyone also knows that Claude Shannon in 1948 showed that there is a limit to the data rate that one can transmit error-free over a bandwidth-limited channel. This data rate is proportional to the bandwidth available and to the logarithm of the Signal-to-Noise ratio of the channel. (When you do this calculation, using for the signal-to-noise level the loudest signal you are permitted to send down a telephone line and the specified maximum noise level of the telephone system, you get about 25,000 bits/second.)

And finally, everyone knows that inside the telephone system the analog signal is actually converted to digital for internal transmission, taking 7-bit samples eight thousand times per second, and the resulting stream of 56,000 bits per second is packaged in various ways for transmission within the telephone system. The sampling rate was chosen in accordance with the sampling theorem, which requires that if you wish to exactly reproduce an analog signal you must sample it at a rate more than twice the highest frequency in that signal. And the sample accuracy of 7 bits means that the voltage level is measured to one part in 128, which guarantees that there will be a certain minimum level of noise contributed by the digitizing process itself.

With constraints like these, how can a modem faster than, say 19,200 bits/second possibly work? More to the point, how could the recently announced X2 modem of U.S. Robotics, which sends data at 56,000 bits/second possibly work? Have Shannon's result and the sampling theorem been wrong all this time?

The solution

1. The actual noise levels on many telephone connections are not as high as the worst-case specification. The worst case happens when the local loop at each end is maximum length and the number of analog interconnecting links is at the maximum permitted.

2. To achieve a system-wide maximum noise level, different parts of the telephone network are assigned a noise "budget". The local loop from your modem to the telephone wire center is allowed to contribute a certain amount of noise, the telephone network is allowed to contribute a certain amount, and the local loop at the other end of the connection is allowed to contribute a certain amount. Each of these contributions must be substantially smaller than the total.

The second aspect is the one that the new 56 Kbit/second modems take advantage of. If one only had to deal with the (usually relatively small) noise contribution from the local loop between the telephone switch and your modem, it would be possible to send data at a much higher rate--if you are not far from the telephone switch and the line is reasonably quiet, one might be able to run 100,000 bits/second or more. (The telephone company is able to get 1.5 Mbits per second down a carefully chosen twisted pair if it is less than 6,000 feet long.) So the trick is to eliminate all of the rest of the noise budget.

For large customers, the telephone company must deal with the possibility that the customer owns an on-premise telephone switch. But this switch may include analog-to-digital conversion for customer-owned telephones, so the connection from the customer's switch to the telephone company may be an all-digital link. This all-digital interface from the customer to the telephone company opens up an interesting new possibility for data communications: Rather than starting with data, running it through a modem to create an analog signal, then converting the analog signal to digital, one can directly construct a stream of bits that, when converted from digital to analog by the telephone company at the local loop, creates an analog pattern carrying data at 56 Kbits/sec.

One could, of course, start by creating this same analog pattern and trying to send it all the way through the phone system, but the noise introduced by the originating loop and the a/d process would be too great and there would be many errors. And introducing an error-correcting code wouldn't help--56 Kbits/second is above the Shannon limit for this much noise.

Thus the 56 Kbit/second "modem" is really a relatively odd beast: there is a modem at the customer site that can receive an analog signal and extract 56 Kbits/second from it, but at the sending end there is no modem involved. And data flowing from the analog toward the digital end has a maximum possible data rate of only 33.6 Kb/sec. And since digital links are usually purchased in batches of 24, this arrangement makes most sense where one end of the connection is a large organization such as a university or an Internet Service Provider.