So lets get started. Any course that is concerned with the
fundamentals of electric engineering has to be concerned with information and
that's what we're going to talk about today to get started.
So terms that are really going to be important to us are Signals and Systems.
So the first thing I'm going to do is to define a signal, turns out this
definition is very simple. You already know what a signal is, it
turns out and we'll get, get through that very quickly.
What's going to be more important are the various kinds of signals.
Now when electrical engineers think about signals, the different ways in which we
think about and categorize them. And then I'm finally going to link
information to signals, and it turns out this link may be a bit tighter than you
think. I would say some provocative things, I
hope. Finally we'll talk about systems, which
are. The way you operate on signals.
And we're going to put systems together to create what is known as the
fundamental model of communication. This model is very, very important.
We're going to return to it throughout this course.
Think of all this, not only will. In the process of understanding it, when
we understand how real communication systems work, we will also, when we put
the various components together, we'll be able to apply the things that we learned
in this course. So, what's a signal? So, the definition
of a signal is quite easy. It's a function.
So, anything you've learned about functions in calculus, you've already
learned about signals. They're exactly the same thing, there's
no difference. So I can plot a signal.
So lets say we have a signal as a function of t, which I, take to be time.
And I'm going to call my signal s of t, and, there it is.
That's a signal. That's all there is to it.
I don't, there's nothing more complicated about a signal than it's just a function.
And, before we get too far, I want to talk about electrical signals.
And there's some interesting aspects of electric signals that have proven to be
very important in the information age. As you know, electrical signals are
voltages and currents, and electromagnetic waves.
but what's very important to appreciate is that they carry two different
quantities. First of all, they carry power You would
plug into the socket in the wall, that's an electrical signal, that's carrying
power, which powers your computer, your refrigerator in the home, the laser
printer in the office, all kinds of things.
Very important. There's also signals that convey
information. For example a wireless signal.
What you use with your cell phone to call one person.
It's carrying information. It's carrying what you're saying.
Now it turns out we're not going to talk much about power in this course.
it's not our major concern. What is our concern, of course, is
information. But you cannot design an information
system without having some consideration about the power.
How much power am I using, does it use, to send, to call somebody over my cell
phone? How efficiently am I expending that power? These are going to be issues
that are very important to the design of modern information systems, so we've
gotta talk about power a little bit. But we won't certainly focus on it.
it will come up. So, the various categories of signals
that we think about. Well, there's two broad categories,
Analog and Digital. A analog signal is a function of a
continuous variable. So here's my good old s of t again.
And it's a function of time and certain times continuous variable, and we can
plot it, looks soemthing like that, very crudely The sinusoid, the sine wave is
governed by 3 parameters versus the amplitude.
Which determines how big the signal is. The frequency at zero, determines how
often the peaks occur. If I were to whistle, [SOUND], that turns
out to be an acoustic sine wave, at least, approximately.
The frequency of my whistle, the pitch of my whistle.
Corresponds with the frequency f 0. Phase has to do with how the sine wave
begins at the origin. So here we have a very good analog signal
as a sine wave as an explicit formula controlled by three different parameters.
Now the next analog signal I want to talk about though, is a little different, and
it's different in a very interesting way. So here's a plot of a segment of speech.
That's me saying the vowel e, and what I want to point out is that there's no
format. The sine wave has a formula.
There is no explicit analytic formula for speech.
However, we are going to want to operate on signals to understand the structure of
signals, whether there is a formula or not.
The structure of the signal with a formula, like I said, is covered by 3
parameters. Here, we're going to have to think about,
what, and for the speech signal, what the structure of it is.
It kind of looks like a sine wave. But I think you can see that the
succeedant peak values don't look the same.
The peaks of these wave forms don't look the same.
And wherever the peaks of every other, thing, tend to look the same.
This is even more apparent at the bottom, where there's the big guys, very negative
peaks, and then the ones that are smaller, that's negative.
And they kind of interlace with each other.
Well, that's a very tedious way to try to get at the structure of this signal.
We're going to have to develop much more interesting ways and much more
informative ways to under, appreciate the structure of a signal.
But we will have signals that don't have any signal, formula.
We'll have Signals that do have formulas. And we had to develop methods by which we
can appreciate the structuring process, the signals.
Whether or not there is a formula. It's going to be very interesting for us.
So, the next kind of signal is a digital signal.
An additional signal is a function of the integers.
It's a function of the discrete valued independent variable, which we're just
going to take to be the integers. So here's an example of a digital signal,
and it's a sequence of numbers. So I intend for these numbers to be read
from left to right and here's a plot of it.
Just for fun, so this goes across like that and that corresponds to these
sequence of values that we see across here.
So from left to right to top to bottom you see the values of my Digital signal,
So, I don't know about you, but I don't see much structure there.
I don't quite see what's so interesting about this signal.
I do want to point out, before we get too far along, about how I plotted it.
This is what's known as a stem plot And every value is drawn by, with a little
line, with a little bubble at the top to indicate the value.
So this indicates this function only exists at the integers.
The value at 29 1/2 doesn't exist. It's not even defined.
This is a function only advantage. That's what a digital signal is.
So, what is this thing? Well, it turns out, this is another digital signal.
This is text and, and you may say that's a signal.
Well, it turns out, if you take position of the character along the line.
The value, of the signal, is a member of the alphabet.
So, s of 1, is capital T. Well that's a digital signal, it's a
function of the integers. What's even more interesting about this
example, is that these 2 are the same thing.
So I typed this text into my computer, inside the computer what the computer did
was turn those symbols, the type characters into numbers.
So a capital T is 84, a lower case h is a 104 etc.
So. They're exactly the same.
So I think you'll agree that the text version, you can understand the
structure, you know it's English text. For example, it's not something random.
It actually makes sense. Whereas you look at a sequence of
numbers, It's not quite apparent. And certainly if you plot it, it's even
less apparent. What the structure is.
The structure's signal may depend on how it's presented, kind of interesting thing
to think about. So, let's talk about some other kinds of
signals. another way of categorizing and I want to
talk first about images. So the speech signal we already talked
about is a function of time. Images are functions of space.
So, they're a function of X and Y. So, I can give you an example of a black
and white image. Here's a picture of a woman.
And you say, why are you showing a picture of a.
A beautiful woman. Well it turns out this image has been
used in the image processing field for decades as a test image.
And here's why. This one image contains a curve, contains
some straight lines, contains a border, contains a highly complicated area.
Something we call texture. Contains a smooth surface with a gradient
of shading and it certainly contains a face, a very important thing.
So, if you're trying to develop an image algorithm or image processing procedure.
You want an image to test out how well it works.
Well, you may discover that your procedure works very well on straight
lines and curves, but doesn't work too well on texture.
And so you want an image to test it with. Rather than having a textured image and a
straight line image, etc., it's a bit more convenient to just have one image
you can run through and see how well it works on that.
That's why electrical engineers develop such test images so this one has been
used for decades because it has all the right things in it that you need to test
out images. But this is a function.
Well I'm going to plot it. So here it is as a function.
And so, here is the x and y. And, this is what's known as a heat map.
This is a way of applying what's called 2, what we call dimensional signals, so
you can readily see what's, big and what's small.
So big values are indicated in red, and big in the image turns out corresponds to
white, so in terms of that corresponds to that little area right there.
Blue values are small and that corresponds to black.
So the gray scale values in between are somewhere In between.
This is called a heat map, because the big values are hot, they're red.
The small values are cold, they're black. So, this is just a very convenient way,
of looking, at a plot, of such a signal. However, here we have the same situation
we just had and with the digital signal. Looking at the image, the plot of the
function, I have a hard time appreciating what it really is.
It's a lot easier for me as a human to look at the image as opposed to a plot of
it. Again this is how the computer sees it.
There's absolutely no difference between the 2 and we're going to develop more
insight into the presentation of information as we go through the course.
Now another kind of signal that's interesting.
This video, because you know video is a sequence of images occurring in time so
that makes it a, function of three variables.
So space and time, so the jargon is that an image is a 2D signal because it
depends on two variables. This is a 3D signal.
Cause it depends on three variables. and this is common terminology.
So, again I one dimensional signal, for example was for speech, that we say
earlier. 'kay.
But it can be a little bit more complicated than that.
Because, if it's color video. Actually is 3, 3 D signals rolled up into
one. Consider it as a signal, signal so our
color video actually consists of a red, a green, and a blue signal each of which is
a function of space and time. So this is a multivalued signal.
So this image up here is in color, as an image.
Not as a plot but as an image. And it's still considered a 2D image.
So again, the dimension of a signal has to do with the number of variables upon
which it depends, not how many values it may have.
So this 2D image here Is red, green and blue components, each of which is a 2D
signal. Now let's bring in information.
Where's the information? So obviously something controversial here.
That information does not exist without a signal.
To represent it. So, lets think about this for a second.
I've already shown you that text, is a signal.
So, all books, contain signals from 1 viewpoint.
Certainly there's more to it than just that, it's the structure of that signal,
that conveys Information. You may say, well information exists
without there being a signal. For example, a memory that I have in my
head. Well, it turns out that memory is
mediated and brought to fore, in your brain, by electrical signals.
And we'll talk about that a little bit. So It makes, there is no such thing
information just existing in and of itself without there being a signal to
represent it. So, signals and information are linked
together in very important ways. Not all signals can be, contain
information, but information doesn't exist without there being a signal.
So, we talk about information being encoded in a signal by modifying its
structure. That's why I was talking earlier about
the structure of a signal, cause that's where the information is.
It varies the structure of the signal, that's how you think about it.
So. What we're going to do is look at this
sine wave, so if I take this signal, which we've already talked about as 3
parameters, highlighted in blue here, if I change the amplitude of that signal in
such a way that the amplitude varies according to the speech signal that I'm
producing now, that turns out to be What's called amplitude modulation, and
that underlies AM radio. If I change the frequency, according to
what I say, that gives you frequency modulation, or FM radio.
There is something called phased modulation, but it turns out it's not
used very often, because it's been shown that frequency modulation has better
technical characteristics than phase modulation, but anyway, this, that's why
I wanted to find out what the structure of the components of a sine wave were, so
that I can change those. Components according to the information I
want to encode in. So, what does the system do? What does
the system that operates on signals to produce a modified signal so the system
might do the encoding of the information, and it also.
It can do the extraction. It can also use a thing that extracts
information and the way we indicate, systems is with a what's called a block
diagram kind of thing. So here is the idea.
The input of my system is x. It's always in most.
Drawings comes in from the left side of this block, box that denotes the system.
And the output is y, usually comes out from the right.
And so the arrow indicates the inputs, and the arrow coming out, indicates the
output. So what this system does, is it Operates
on the signal x, does something, who knows what, and produced an output y.
So we could go down the mathematical path to talk about functions that are
functions of functions So, that, that turns out to be overly com-, complicated.
We won't need to go down that path. We'll just appre-, just learn and talk
about systems as things that take, as an input, a signal, and produce another
signal. So here's that Fundamental Model of
Communication that I keep, was saying earlier is so important.
It's a sequence of systems. I've labeled important signals.
The source The idea is there is a source that has a message signal.
And the goal of communication system is to get that signal over to its
destination, which technically is called the sink.
I do not know where the term sine comes from.
Its just the terminologies used. We'll just go withe flow and use the
terminology. The, the idea of three occasions to get
something that contains information from one place to another.
Now it turns out, and the easy if it wasn't for this funky, cloud-like thing
in the middle, the channel. That's another system, but I've drawn it
like this because this basically refers to phenomena we have no control over.
And let me guarantee you at the channel nothing good happens to a signal in a
channel.What comes in, is not what comes out at all.
An in fact the channel can do Things the signals that you may not know in great
detail. So trying to send the message in directly
through this channel is a big mistake, in most cases, it won't get through.
So here's what we're going to do. We're going to take our message signal
in, and put it into a transmitter. And what the transmitter does, is,
creates a signal, which is better suited, to go through the channel, then the
message m would be. This is called engineering design.
This is what electrical engineers, and communication engineers do.
Try to get signals through the channel. So this is termed the modulated message.
That's where the transmitted signal is. Transmitter is a box which takes the
message m as input and produces x as an output, that goes through the channel.
Where nothing good happens, and produces the received signal r, which is, trust
me, it's a corrupted modulated message. it's going to do all kinds of things, and
nothing good happens in a channel. The receiver's job, is to try to figure
out What that message was that was being transmitted, despite the fact that it's
been corrected, so I put this hat on top of the, symbol here, because it means
that it's not exactly equal to m, so it's an approximation.
So and half basically equals the message M plus some error.
And one of the things you wanted to do when you design communication systems is
try to fight the channel so you can minimize that error.
And it's this. Message in hat that goes into the sink.
Goes to the destination. That's what you get.
It's known as the demodulated message. So, here are all the labeled pieces of
our fundamental model. We are going to talk about this again and
again. We're going to worry about how
transmitters are designed. How receivers are designed.
Turns out. You cannot design these without designing
them together. and we will talk about that as we go
through the course. So this is the fundamental model.
You now know what a signal is, you know what a system is generically, and now
we're ready to get serious and talk about things to get.