So in this video we are now going to use the knowledge we have about how channels
behave, to develop what I'm going to call the basic channel model, that applies to
both wireless and wireline situations. As we know interference and noise are
going to be a problem with wireless channels.
Channels not so much with wireline channels.
That makes from a communication and engineering view point.
Wireline channels are fairly boring. They don't have much interest or design
flexibility. Whereas in wireless channels because of
the interference and the noise makes it a much more challenging and interesting
problem. I'm going to give an example on how to
apply this model of channels to what's called baseband communication, which is
perhaps the simplest kind of communication possible.
Baseband basically means you send the message directly.
But let's develop that basic channel model.
And as you know, this is our block diagram model of communications.
And we're concentrating, of course, on the channel.
And what we've seen that for both wireless and wire line situations.
This is a very good model for the channel. So what I'm doing is I'm putting the
channel in a box. Has x of t coming from the transmitter n.
And what we call r of t, the received signal coming out of the your channel.
It is delayed. The transmitted signal is delayed by some
amount, and in both situations. It may be attenuated, alpha's supposed to
represent a gain but usually alpha is less than one and certainly hopefully greater
than zero. For the wireline channel, basically alpha
is 1 but for the wireless channel, it can be significantly small, a very small
number indeed. Then to that are added the interference
which usually come from man-made sources and noise which is ubiquitous.
So a simple relationship between input and output is given by this expression.
And we're going to use this in our considerations of communication channels
and systems. So, for a wireline channel, to summarize.
The delay TAU is given by the distance between the transmitter and the receiver
divided by the speed of propagation which is I said is a fraction of a speed of
light in free space. Alpha's basically one and it interference
in the laser base is basically zero. So it makes it kind of uninteresting for
us. But in wireless channels, situations get
much more challenging. There is a delay which in many situations
can be small. But if you're talking about satellite
communications Or even ionospheric, communications.
That delay can become signifigant. What's much more important is the fact
that the attenuation is inversely related to the distance between transmitter and
receiver. The further away you are, the further away
that a receiver is from the transmitter. The smaller the signal you're going to
receive. And who knows what kind of interference
and noise are going to inhibit the reception of the signal.
Now we're going to basically assume that we can control the interference by
filtering. Basically is we're going to see as we go
through that in well designed communication systems the interference
from other communication signals is going to be out of band.
What that means is in a different bandwidth range of frequencies than x will
be. So we can just by floater and get rid of
it. Noise is another problem.
Noise we're going to consider as being wideband, technical term, which means it's
everywhere. And it's going to be out of band.
It's going to be in band. It's going to be in the same transmission
bandwidth, as we have, or as we're using and so we have to worry about it a lot.
But two big effects are going to be the attenuation and the noise.
So we need a way of describing noise in particular because it's going to be a new
signal for us. It has very interesting and special
properties. We're going to use a model called white
noise. The reason it's called white is because it
has powered all frequencies. Just like white, white has equal power
more or less in all the range of colors. Put all the colors together you get white.
So it's called white noise because it has power to all frequencies.
Random phase in amplitude. And here is an example of white noise that
I simulated on the computer. And it's very erratic, very wiggly.
Sometimes if you look very carefully it is not so wiggly.
And sometimes it has these really big values, which really is a random signal.
There one special property that noise signals have.
Is that when you add two noise signals together, where powers add nor their
amplitudes. So we're going to see this in more detail
just a second but this is very important property of noise signals, not so much for
regular, if you will. Non noise signals.
Suppose you add together two sign waves of the same frequency, well their amplitudes
more or less add, depends on the phase. But if you're in phase and at the same
frequency their amplitudes had. But their powers are, there's a greater
power than just the sum of their powers individually.
Not true for noise, for noise the powers add.
So we need to talk about, what's called the power spectrum because we're going to,
communication analyses relies a lot on understanding where signals have power.
And to make it convenient we like to talk about power in the frequency domain
through something called the power spectrum and very simply it is the
magnitude squared of the Fourier transform of the signal.
And that's what a power spectrum is. So the power spectrum for signal S is just
this magnitude squared. Well,as we know magnitude squared is S
times s conjugate of f, but this because of our properties, is just s of minus f.
And so when you multiply these to things together mathematically what you discover
is that power spectra are always even functions.
So, this is going to be important a little bit later.
So, it's an even function. It's just a mirror image about the,
vertical axis. For all power spectra have this property.
There are no exceptions. Okay, now next thing to talk about is this
power spectrum at the output of a filter, so we know that if we have a filter here,
which has x going in a transfer function of capital h.
And y coming out that the Fourier transform of the output is equal to the
transfer function times the Fourier transform of the input.
Well to find a power spectrum you just need the magnitude squared.
And the magnitude squared of a product is the product of the magnitude squared
that's Py, that's Px, and we get that relationship, pretty easy to see.
Not very difficult to derive at all. Now, because of Parseval's Theorem.
If you recall Parseval's theorem, lets say that the interval in time which we call
the power is equal to the interval of the 3 magnitude square root by tens form but
that's for us in other power spectrum. And because we now know that power spectra
are even we, the interval from minus infinity to infinity, is just twice the
interval from 0 to infinity. Because it's even, this interval is going
to be the same as that interval. So it's a little bit simpler for
calculational of purposes just to integrate it with a positive frequency
axis and multiply the answer by two. And finally, I want to point out this
little interesting result, that you can think about the power of the signal within
a given frequency range as being the interval of the power spectrum over that
frequency range. The reason that comes about is because of
that result. So, suppose capital H is a band pass
filter between f1 and f2. And so we know what that does is that the
output y only provide, only thing in the output is the frequency components of the
input in that frequency range. Well the power in the output is just given
by the interval. And so, when it is a band pass filter on
like this, magnitude at h of s squared is just 1 in that frequency range and that
interval simplifies to that. So you can interpret this band pass
filtering to be, I'm only measuring the power in the input signal over a given
frequency range. So, once you have the power spectrum for a
signal you can figure our how much power has in different ranges of frequencies,
simply by integrating over that frequency range.
It's very useful for us as you are going to see here is a second, especially as you
talk about noise. So going back to white noise, as we said
by definition, white noise signals have a constant power spectrum.
So your power spectrum using some sense the easiest, is just a constant.
And that amplitude of the constant it turns out as a not over 2 And we'll see
why there's this over tuned factor in just a second.
Now, what's the power in white noise? It's supposed to be in the inner row of
the power spectrum, what do you get for the power in white noise?
Of course, the answer is it's infinite. So white noise is a fiction.
It doesn't exist in the real world. But it's a very convenient fiction, very
useful for us, in analyzing communications problems.
And believe it or not we're going to assume in any channels the channel adds
white noise, infinite power noise, to our signal.
It's going to turn out it's easy to remove that noise because of filtering.
So again the power and noise signal we going to give in frequency range is just
going to be given that the spectral, so called spectral height times the
bandwidth. The range of frequencies of interest, and
that's called bandwidths. So the bandwidth is f 2 minus f 1.
So, again, using our result that we shown on the previous slide, the power spectrum
of the output of a linear filter, when white noise serves as the input Is n mod
over 2 the power spectrum of the input times magnitude of h of f squared.
So, in most situations when you have some sort of filter, like this, the power of
the output of that filter is now finite. Even though the power coming in is
infinite. So this is a convenient way of modeling
noise having any kind of spectral properties at all.
Is that you pass it through the appropriate kind of filter.
So, the beginnings that start with white noise because it's very, very easy to do.
Now, in analyzing the communication systems the bottom line measure equality
is the signal to noise ratio. So the idea is that, we're going to wind
up In almost all cases with a signal plus noise, our, our final signal that we
receive after going through the receiver, is going to be a signal relayed component
and a noise related component. We want a measured quality of how big the
signal is relative to the noise. And the signal to noise ratio is the
standard measure. It's the power in the signal part, divided
by the power in the noise part. And clearly, the bigger the SNR, the
better the quality it is, the smaller the noise is.
Well, given by what we've seen, seen in terms of power speculator, it's the ratio
of these 2 quantities. Power and the signal is given by that, and
the power of the noise is given by that. And also as we've seen SNRs frequently
expressed in decibels in DB. So here's the general lay of the land in
terms of our channel model. The channel introduces a delay, it
attenuates, adds interference and noise. We're, like I said, we're basically going
to assume interference is not there. And that gives us the final expression for
the signal to noise ratio coming out of the channel.
As being this quantity over here. So, certainly, the effects of attenuation
in a channel really hurt the numerator that makes can make it small in the
wireless situation as you get further and further away from the transmitter.
This number gets smaller and smaller and smaller, so the s and r goes down like the
square of that attenuation factor. And the noise part depends, of course, on
the spectral height. But also depends on the bandwidth over
which x exists. So I'm assuming that x is some signal in
the frequency domain. Which exists from frequency fl.
A lower frequency to some upper frequency. And the only thing we need to consider is
what's called the in-band noise. I'm about to show you how that works so
next talk about resign the transmitter and receiver.
For a channel, so we have now our channel model, and now we know what the
characteristics of the channel are. What we're going to do for all kinds of
reasons is simply transmit the message as it is.
The transmitter is going to provide some gain usually gain, the gain is much bigger
than 1. To bind some power to the signal, but it's
just going to send it as it is. Most message signals have a low pass
spectrum. They, they're support for their frequency,
transforms usually goes from 0 frequency up to some upper frequency.
For speech it goes up to like 6 or 7 kilohertz, audio music has a somewhat
higher frequency or range. Still, it's a low-pass spectrum.
So given that's the way we're going to do it, we're hopefully going to look at the
channel model and the transmitter design is going to be determined by how we think
we're going to send it and the characteristics of channel in that
frequency range. All as we know baseband communication does
not work well for wireless channels. They just low pass signals do not.
Propagate very well for wireless channels. So it's rarely, if ever used for wireless
channels. For wireline, there is also an attenuation
as we've seen. I mean a high frequency limit that we get
no attenuation. So it is used though for wireline channels
to for communication over limited distances that is definitely a case.
So let's see if we can figure out, the goal is going to be to figure out the
receiver that goes with this transmitter. And to figure out the signal to noise
ratio in the recovered message. There, so here's what's coming out of the
receiver, out of the channel rather, in the frequency domain.
So I'm assuming like I always do, that my message signal has this triangular
spectrum just a habit of mine, just to use a, a triangular looking spectrum.
So it has a highest frequency of w. It has a band width of w, its so called
band width is equal to W. And again bandwidth is defined to be that
portion of the positive frequency axis over which the signal is non zero.
So there's our good old message signal, like that.
It's contained in white noise. Which is very broad band.
I show this as a blue thing. It's everywhere.
Well, clearly, if I was to use that as my demodulated message devise.
If I used that, just sent that to the sync.
The signal to noise ratio would be 0. Because white noise has infinite power.
It's very clear we don't need this portion of the noise.
Why keep it? It's not what we call in band, in the
bandwidth, in the frequency band of the message.
So most receivers, the first thing they do, is remove what's called out of band
noise. It's out of the bandwith of, of the
transmitted signal. So in this case, since we are doing the
baseband with a low pass filter this is called front end filtering.
Virtually every receiver in the communications world, first thing it does
to the signal coming out of the channel is low cast filter.
And the base band communications situation, that is going to be our
message. Signal, it's going to be reconstructed, so
now the message in the frequency domain looks like this bandbase signal.
You have the message, part that transmitted message, comes through without
altering, being altered by the low pass filter, but the noise is certainly cut
down. Now, what is the signal to noise ratio?
It's going to be the attenuation squared introduced by the channel, times the gain
squared of our transmitter. Times the power in the message, whatever
that is, divided by n nought w. N nought times the bend with the noise,
which is just w. So you can see that the transmitter is
here as a gain to try to fight the attinuation introduced by the channel
basically. The bigger the noise is in the channel,
the smaller the SMR. That's pretty obvious.
I wouldn't worry too much about band width here, because the power in the message
will probably change if you change its band width.
This part of the SNR, basically, could be a constant, for most situations.
May or may not be, but it, I would not say that the wider the bandwidth, the smaller
the SNR has to be. That's certainly is not true.
Now, another thing I want to point out, is the those of you who are, who know what's
going on, who know that m hat, the, the signal part of the m hat is equal to alpha
times g times the message plus filtered noise, which I'll just call n tilde.
So do, don't we need to worry about alpha times g and in these situations we do not
demand that be 1. This corresponds to the knob on the radio.
Volume control if you will. Will make up for it by amplification a
little bit later. But if you amplify this entire signal
multiply it by a constant. You do not change the SNR.
So, we can study the SNR for this signal as it is.
And that will be a sufficiently characterized quality.
And we will fix the game problem, if you will, by further amplification.
So we now have designed for ourself a communication system.
The model that we've developed, for just the channel, describes the delay the, The
interference and the noise that the channels add.
They said nothing good happens in a channel, saying some get bigger and they
get delayed and sometimes attenuated by a lot especially in the wireless situation.
Interference gets added. Who knows what it is, depends on the
situation. And the noise is always prevalent.
You don't know really how bit it's going to be in any given situation.
And you have to do the best you can. To combat the presense of the noise and
the interference for that matter, most receivers from wireless, consist of a
front end filter. To remove out-of-band noise and
out-of-band interference, again out-of-band means not in the bandwidth of
the transmitter sig, not in the bandwidth of x.
Y keep the noise in any other signals floating around that are out of band when
I band pass. May be a low pass filter for, a base band
situation. A band passed over for modulated
situations can get rid of all other signals.
So, that's why most wireless receivers, right at the front, there's a fold.
And then further processing goes on, communications to yield the reconstructed.
Message. We'll talk about those kind of situations
in succeeding videos.