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Okay, say you've got a flashlight you're shining at a buddy who's standing on top of the next hill. The beam of light hits him, shining steadily. That's the electricity going between sender and receiver. But there's no signal yet.
You both know Morse code, so you start blinking the flashlight on and off -- dots and dashes (zeroes and ones). Now you have a signal. In telecom-geek terms, the beam of light is the "carrier" and your busy thumb on the flashlight button is the "modulation." When modulation is applied to a carrier, the result is a signal. The signal transmits information.
I say "blinking" here to avoid confusion, because the term "switching" has a special sense in telecom: it means routing a signal from one to another of several possible circuits. This is different from the ordinary sense of switching a light on and off.
Depending on how agile your thumb is, you'll manage maybe eight blinks each second. That's your bandwidth, then -- 8 bits/second. Bandwidth is the amount of information that can be transmitted in a given amount of time. In one minute, you'd probably be able to send a short joke to your buddy.
With a fiber-optic cable connection, the laser "flashlight" blinks millions of times each second. The lowest class of fiber-optic does 45 megabits/second. In that same minute, it could easily transmit all of Shakespeare's works. So bandwidth is directly related to how fast the modulation is. When people talk about the "speed" of a connection, it's really just a loose way of referring to the amount of bandwidth.
Hope this helps...
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