The first time I saw the word "modem" was (curiously enough) in a Sunday Tank McNamara strip in the late 1970s. A sports reporter had been sent to the boondocks to cover some local sports event. He was frantic about the lack of facilities at the stadium; he screamed "Where's my modem!!?!?!" in panic. One of the other 'casters held up a pigeon in a cage and said "Meet 'Modem' ".
When, in college, I got to actually use one of the thingies, modems were the strangest-looking things I'd ever seen. They were boxes about 9 inches long, 5 inches wide, and 3 inches high1, with these two round cups on top.

You guessed it, this was an acoustic modem. The modem would translate an acoustic signal, not an electric one. You dialed the number2, waited for the carrier signal to sound on the phone's handset, and then, you put the handset in the cups. You had to remember which cup to put which end of the handset in because it wouldn't work the other way.

My university had 300 baud modems and 1200 baud modems. The 300s would print (sometimes on a green screen, but often on HARDcopy, on a teletype), about as fast as you could type. The 1200s could print fast enough that you'd see a whole line appear at once! You could go down to the computing center and get on a hardwired 2400 baud terminal; that was really something. When I would complain about having to use a measly 300 baud modem instead of the snazzy 1200s, the old-timers would say "Consider youself lucky that you don't have to use a 110 baud modem anymore!"
120cm x 12cm x 5 cm.
2I remember holding the handset up to my TV speaker so that my VIC-20 could perform tone dialing, because that was faster than hand dialing on a rotary phone.
MODEM This is an acronym for modulator-demodulator. A modem is a device or program that enables a data transmission over telephone lines. While computer information is stored digitally, data sent over telephone lines is transmitted are in the form of analog waves. A modem converts from one form to the other. To faciliate compatibility between modems and computers, a standard interface, called RS-232, has been established so that any computer can be be connected with any modem adhering to the standard. Modems in the form an expansion board that are placed in a vacant expansion slot are called onboard or internal modems. There are a number of protocols for formatting the data transmitted through the modem,

While the modem interfaces are standardized, a number of different protocols exist for formatting data to be transmitted over telephone lines, like the CCITT1 V.34 and the X25 for high speeds. Usually a modem has built-in support for the most common protocols.

The characteristics of a modem, in addition to the formatting protocols, include:
bps
The modem transmission speed for sending and receiving data. For slow rates, the unit of measurement is called a baud. The slowest rate (and at one time the only one) is 300 baud. When higher speeds are attained, speed is measured in bits per second (bps). Even higher speeds are available when the data is compressed before transmission. In some areas telephone lines cannot reliably transmit at high speeds.
voice/data
Today most modems can switch between voice and data transmission. In voice mode, the modem acts like a telephone with onboard speaker and microphone. In data mode, the modem transmits only data.
auto-answer
Ths feature allows the computer to receive calls automatically.
compression
As mentioned before greater speeds of data transmission are obtained by compressing the data before it is sent. Obiously, the receiving modem must have the capability of decompressing the data, using the same rules.
Fax capability
Most modern modems can also be used for sending and receiving fax messages.
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1 = Comité Consultatif International Téléphonique et Télégraphique or International Telegraph and Telephone Consultative Comittee.

Necessity is the mother of invention. Modems are am ingenious hack that enable digital communication over an inherently analog medium, the existing phone network.

Modems are limited by Shannon's Law, which, on POTS lines in the USA, limits throughput to (8,000 samples per second) * (8 bits per sample) = 64 kilobits per second.

They are further limited (in the USA, at least) by the Federal Communications Commission (FCC) in that they cannot achieve connect rates of more than 53,000 bps due to FCC-imposed regulations regarding the maximum allowable voltage on telephone lines. Due to less-than-ideal line conditions, however, 56K modems generally connect from 42-52 kilobits per second (kbps).

There are three main types: the "classic" modem, the HCF modem, and the HSP modem.

The classic modem has three main components: a controller, a phone line interface, and a digital signal processor (DSP). They generally cost US $50 or more. I know US Robotics makes these.

A HCF modem, which is less expensive than a normal modem, gives the host CPU the job of controlling the modem's function, while the DSP is still handled by the modem's chipset. Often referred to as 'Winmodems' since the controller/driver software is generally only available for Windows or Mac. Probably the most common form of modem today, these are available in a variety of interfaces (PCI and USB) and can be used on different platforms, mostly PC or Macintosh These usually cost from $20-$50.

An HSP modem (also called softmodem), the cheapest of all, is the logical extension of the HCF modem. In this modem, the tasks of both the controller and the DSP are given to the host CPU, which are relatively insignifcant ones given the computing power of modern CPUs. All that remains on an HSP modem is the physical phone line interface. These are often (though incorrectly) referred to as Winmodems as well, since they have similar shortcomings. These cost $5-$20, and they usually obtain the same connect rates as the other two modem types, which is great, as long as you're using Windows (or if you're lucky enough to be able to make Linux/BSD work with it). These are generally sold as generic brands, but always(?) contain a PCTel chipset and are always PCI.

Also, from what I've seen, HSP modems never have transformers. If your modem has one, you can be pretty sure it's either a normal modem or an HCF modem.

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