Combining Multiple Signals in Data Transmission

Date: 02/18/2001 at 09:25:05
From: Ewout Bongers
Subject: Combining multiple signals

Hi,

My name is Ewout Bongers and I had this thought a couple of days ago. 
How can it be that there is so much information in our cables? I was 
thinking of coax here. After some assumptions I concluded that there 
should be something like 2-3 GB of data flowing through this cable 
every second (take 22 TV channels at a resolution of 740*600*16 (16-
bit color), including stereo sound, plus 20 cable radio (22kHz, 
stereo) and still space for cable Internet). 

So I figured that if you can make some software (I am a programmer 
myself) that would take any file as input and accept it as a wave, 
you could compress huge amounts of data into a relatively small file. 
Assuming some distortion you could use this very well for audio and 
video compression (I mean one distorted sample in 1000 is nothing to 
worry about). 

Now I was wondering, how do you combine one or more signals and pull 
them apart again? I asked my dad but he could only explain me how FM 
and AM worked and said I should look into Fourier analysis for this. 
I did. But I don't understand the first thing about it. (At school we 
haven't yet covered integration and that sort of stuff. We just 
recently learned logarithms.)

Any help would be appreciated.

Date: 02/19/2001 at 00:39:53
From: Doctor Douglas
Subject: Re: Combining multiple signals

Hello Ewout, and thanks for writing.

These are great questions! I'll give you an example of how one might 
do this in a communications system, since it's clear that you have 
some good background here. 

The key concept is *bandwidth*. Without access to Fourier 
representation, it's hard to be precise about what this means, but 
imagine this: suppose your signal is restricted to a single string of 
0's and 1's, and they have to occur at discrete, evenly spaced times 
(like a computer clock). Now, suppose that clock ticks only a few 
hundred times per second (say 300 bps). Then this is a very slow link, 
and cannot carry much information. But if the link is very fast, say 
10^9 bits per second, then we can send a huge data file quickly. It's 
up to us to use this bandwidth in whatever way we please - to have 
larger files, more resolution, better error correction, or, in the 
coaxial cable case, more signals.

The first method involves separating the signals in the "time-domain." 
Suppose each signal is sent as a series of small packets of bits. The 
carrier switches between the signals, but is always in use sending a 
long series of packets, i.e. a long sequence of bits at a high rate. 
The time axis is chopped up into segments that carry the different 
packets: AABBCCDDAABBCCDDAABBCCDDBBCCDDBBCCDDBBCCDDAA. Notice that 
signal A can be dropped for a short time, maybe because the cell phone 
signal got cut off. In this scheme, the segments between the signals 
A,B,C and D are separated in *time*.

The second method is where the signals are separated not in time, but 
in frequency. How does this work? I'm afraid that a real explanation 
will have to wait, but you can think of it this way. Suppose you have 
two radio stations that broadcast at frequencies f1 and f2. They can 
modulate their signals to encode information, using either AM or FM, 
or some other method. As long as f1 and f2 (plus the modulation 
effects) don't overlap each other, your radio can pick out one of the 
two frequencies while rejecting the other. The antenna in the radio 
doesn't do this - it receives everything - but there is a tuning 
circuit that oscillates at a particular frequency you can change by 
turning the tuning dial. If this tuning circuit oscillates at f1, then 
the radio station at f1 will excite the resonance of this circuit, and 
the signal at f1 will be picked out and amplified. The signal at f2 is 
either "averaged out" or is suppressed. In this way you can separate 
the signals by *frequency*.

This second method is actually at work in your ears: there is a 
membrane in the cochlea that vibrates. Because it is tapered, 
different parts of the membrane vibrate at different frequencies. So 
one part of the membrane will vibrate when you hear a bassoon, while 
another part may vibrate when you hear a piccolo. Each part is not 
all that sensitive to the frequencies that are not in its relatively 
narrow band of interest. Each part excites neurons that lead to your 
brain, and let you know whether or not the music (the combined signal) 
that you are listening to contains high frequencies (piccolo), low 
frequencies (bassoon), or both.
 
This separation of signals in the frequency domain is also done for
digitally clocked systems, such as cell phones. In this case the 
individual signals are usually carried along a single "channel." You 
can think of this channel as happening at a particular frequency 
(like the radio station example), but channels in real applications 
are a bit more complicated.

I hope this helps, and stimulates further thoughts and investigations 
on your part. Fourier analysis is a powerful and beautiful tool in 
mathematics, but I think it will have to wait until you have a bit 
more mathematical preparation.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/