Transmitting a time-slot over a small bandwidth

[EngIntoHW]

Hi,

We learned that if the signal is finit in the time-domain, then its infinite in the frequency domain.

So how come you can transmit a time-finit signal (transmit a time slot) over a small frequency-domain bandwidth?

If that signal is your voice, wouldn't it be distorted at the receiver end which passes only what's in a certain bandwidth?

 

[dknguyen]

Finally, a good question!

Many times we don't care about distortion as long as the meaning of the message isn't lost. Have you ever wondered why telephone audio quality sucks compared to CD quality? Telephone audio quality has been severely bandlimited in a very careful way to trade off quality for cheaper and easier transmission. However, the bandlimiting has been carefully designed for the human voice so legibility is not lost.

But yes, you're right in that we can't transmit a finite-time signal (which has infinite bandwidth) over a finite bandwidth (because this requires the signal to go on for infinite time). It's contradictory. Does that mean that only one of these kinds of signals can exist? That is actually the case.

One thing you slowly learn is that all these ideal mathematical principles you learn in class don't work out in real life when you put everything together. The math principles are ideal and rigid and like to treat the extreme cases of a single real-world phenomena as separate things (for ease of analysis). But in the real world, it's usually just one phenomena at work.

That means that one of our two signals is an idealization. One actually exists, and the other is the ideal approximation of an extreme case so we can analyze it more easily. So which can't exist? It shouldn't be too hard to see why an infinite time signal can't exist since no signal lasts forever. That would imply that no signal is truly bandwidth limited (which isn't so obvious). From that reasoning that means all signals are finite-time (which is pretty easy to understand) and infinite bandwidth (which can be rather puzzling).

So how does this all work then? Remember how filters can't be made that have perfect brick-wall roll offs. This is intimately tied into the fact that only infinite bandwidth signals exist while finite bandwidth signals do not. It means that all signal actually take up infinite bandwidth. It's just how fast the frequency components of the signal asymptotically decay towards zero. Using a filter will accelerate this. If a signal's frequency components decay fast enough, it won't interfere too much with other signals that mainly consist of other frequency components (which will be much stronger at those frequencies and overwhelm the residual frequency component of the first signal).

Saying that all real signals are finite-time (which any layman would agree with) is the same as saying all signals are infinite-bandwidth. It's also the same as saying that ideal brick-wall filters cannot exist. Are you able to wrap your head around that?

 

[EngIntoHW]

Thank you very much dknguyen!

It's a wonderful explanation and I got it.

It means that all signal actually take up infinite bandwidth. It's just how fast the frequency components of the signal asymptotically decay towards zero. Using a filter will accelerate this.

Is that why the channels with smaller bandwidth are more sensitive to interference?
Even though they have the filters you mentioned?

It relates to what you said below:

Have you ever wondered why telephone audio quality sucks compared to CD quality? Telephone audio quality has been severely bandlimited in a very careful way to trade off quality for cheaper and easier transmission. However, the bandlimiting has been carefully designed for the human voice so legibility is not lost.

Thank you

 

[dknguyen]

Is that why the channels with smaller bandwidth are more sensitive to interference?
Even though they have the filters you mentioned?

There are two types of interference. Interference from due to random noise noise (and other unrelated devices on different protocals). And interference from other users (other identical devices on different channels or other devices using the same protocal on different channels).

Low bandwidth signals are definately more vulnerable to interference that happens to fall within their frequency band. It's also harder to build high drop off filters so it takes more effort to get it to work as well (plus you also have less bandwidth to work with too).
----------------
Something analogous exists for code-division spread spectrum systems. In these system all users share the same bandwidth all the time and can transmit simultaneously or whenever they want. But before transmitting they encode their signals with "codes" or "keys". THese codes or keys are designed to be orthogonal to each other so mathematically you can extract a particular signal out of the RF jumble if you know the code. It turns out that this system is VERY good at rejecting random noise but has much more problem differentiating it's own signal between the signal of other users.

It actually would not be a problem to cram a crapload of users into the same bandwidth if all codes were perfectly orthogonal. But these are very few and far between. If we only used perfectly orthogonal codes we would end up with very few channels that worked very very well. Instead, we have opted to sacrifice quality for increased more channels for more capacity. So we also use codes that are "nearly orthogonal". This means that there is imperfect differentiation between signals. As you cram more users into the same bandwidth, you aren't making each user have an increasingly narrow channel bandwidth like in traditional systems. You are actually making use of more codes out of all possible codes, but the vast vast majority of these codes are nowhere near orthogonal to each other. As you cram more users in, you have less good codes to work with and as a result less perfect differentiation between signals which leads to higher error rates as you get more users.