In this video, we're going to talk about the Signals that are the most important
for this course. we're going to see these signals over,
and over, and over again. one you've already seen already, is the
Sinusoid, and signals that are basically related to the sinusoid in some way.
Well, today we're going to talk about a very different class of signals, they're
pulse-like signals. These, you haven't seen in your calculus
course. These are going to be interesting signals
as you'll see. But probably the most important thing in
this video, is the segment on how to think about signals.
How to build them and how to think about them in a simpler Parts.
That's going to be a very very important thing we're going to talk about.
So here's our sinusoid, and we've seen it before, and as we've said it has an
amplitude A which determines how big it is, so this negative Right down here as a
size of -A, goes down to -A. The frequency that turns how often this
reappears. This is a periodic signal.
Which means it repeats, and it's certainly clear And if you hop over to
the next peak, your going to get the same signal back again.
What occurred here, is going to re-occur here, etc.
Well the period we call capital T. And just a symbol that we'll use often
for, what the period of a signal is. How is it related to f naut? And if you
think about it for a second, the sine wave repeats, whenever f naut times t, is
an integer. Which means that t has to be 1 over f0.
So f0 again is the frequency in Hertz, and this formula gives t in seconds.
So, hertz has units of inverse seconds if you will.
And then finally there is the phase, which basically defines how this starts
at the origin. So that value right there.
Is A sine of T. So if T is 0, this thing starts at 0 of
the origin and, therefore there's no, what's called, phase offset.
Now I should point out, that if you have cosine, 2pie f naut t It's another
sinusoid. That's equal to a phase shifted sine
wave. When it's shifted by 90 degrees, pi over
2. You should verify that formula is
correct. I hope I didn't make an error.
I think it's right. and therefore, a sine wave can be written
in terms of a phase shifted cosine, also. So.
There's really no difference between this cosine and a sine, they're phase-shifted
versions of each other. So this is what phase does, it moves
things around in time. Now the next signal we're going to talk
about is the complex exponential Yes, complex valued signals.
So j here is the square root of -1 and this signal is incredibly important for
us. It's incredibly important for all
electrical engineers. This is another periodic signal.
Its period is the same as the sinusoid we saw before and its period = 1 / f not or
1 / frequency. Now you can plot this in the complex
plane. So this is the real part.
This is the imaginary part. And it turns out, that it goes around in
a circle. So, if you,
The angle at any one time, is 2pi f0t, for everytime.
So, as time increases, this angle increases, And this, complex exponential,
looks like an arrow that goes around, and around, and around.
Unit circle. And the time it takes it, to go, around
once, is the period. That makes it periodic.
Again I think it's pretty clear, again, that that period has to be such F nought
times T is an integer. That makes the period one over F nought.
Now I hope you all know Oiler's relationship Which says that a complex
exponential actually consists of a cosine and a sine, so the cosine is the real
part of the complex exponential. The sine is the imaginary part of the
complex exponential. I cannot stress how important these
signals are in our course. So in particular, the complex value
nature of it is very very important to understand.
So, if you're a little unfamiliar with complex numbers Oiler's formula.
I suggest you look at the complex numbers video, which I've made.
I hope it will help you, develop a better mathematical appreciation of complex
numbers and complex valued signals. Okay.
Let's go on to our pulses. pulses are going to be a little different
that what you've seen in Calculus courses, because most of them are
discountinuous. You probably shied away from
discontinuous functions in your calculus course but pulse signals occur all the
time. And perhaps the simplest signal is what
we call the unit step = 0 for t less than 0, and then = 1 for t greater than 0.
So it looks like a step, a step up, where it gets its name.
Now you probably noticed right away, I didn't define it at the origin.
And that's just the way it is. Its value at the origin is undefined.
Turns out it doesn't matter what its value at the origin is.
And you may not like that, but get used to it.
It turns out it's going to occur all the time, and it turns out we don't really
care what the value is at the origin. It just plain doesn't matter.
And so another signal that has discontinuities is what we call the Unit
pulse. By the way, the word unit here refers to
the fact that the amplitude of this pulse, that now it jumps.
The size of its continuity is 1. So the amplitude of this pulse, is 1,
because it's a unit pulse. So, As the formula says, it's 0, for t
less than 0. It's then 1 in between 0 and delta, and
delta is called the pulse width. And, then it's zero again afterwards.
So, it it pops up, stays constant then goes down and continues on, so that's a
single pulse. We're going to discover the pulses used
very frequently in digital communications.
It's used to represent. It's, that's what makes it extremely
important. And the next signal we're going to talk
about is the square wave. So, a square wave is a periodic signal,
it has a period of capital T as I've labeled it here, because you can see it
repeats every capital T. Goes on forever.
the amplitude of a square wave is, there is no convention for that, there is no,
we don't really talk about the unit square wave.
You explicitly have to give the amplitude, which I've just labeled here
as A. So it has discontinuities and The origin
T over 2, T, 3 T over 2. So it's discontinuous ever T over 2.
And, what it's value is at those points, it doesn't matter, it's discontinuous.
Just have to appreciate that as we go through the course.
Alright, here comes the good stuff. Here comes the part that's very, very
important. We have to build signals, what do we mean
by that? The thing we have to first talk about, before we get into building
signals, is what's called a signal delay. So, here, the delay, is tao, and I claim
this signal, is delayed in time, by tao, so lets see how that works.
The example I like to use is a unit step. Because this point of discontinuity gives
us an handle, to latch on to where the origin is, in the signal.
So, I'm going to pretend that my signal s is a unit step.
So here it is, just u(t), just like we've seen in, earlier in this video.
If I delay it, I claim it looks like this.
So, let's see if we can figure this out. The discontinuity occurs where the
argument of u is zero. So argument u is still a function.
It's now 0 when T equals tau. That's where the discontinuity is.
So this means that this formula right now, corresponds to a, to a step that's
been delayed, shifted over in time, to tau.
So U. Of, t plus tao, is called a signal
advance. That means, this discontinuity, now
occurs at -tao, wherever this argument is 0.
So, we usually think of tao as being a positive number.
So that the sign in the argument here tells you if it's the signal delay or an
advance. Signal delay is going to be a very
important thing in building signals as you're about to see.
So let's have a little game. Let's see if you can tell me what this
signal is. So we have here a unit step and a delayed
unit step. It's delayed by delta.
And I subtracted the two signals. What does it look like? Okay, well, let's
try to plot it. We'll plot the pieces.
So the unit step course looks like that. The delayed unit step is just as I shown
you already. It looks like that, occurring at delta.
But we have a minus sign over here, so what does that do? Well that inverts it,
flips it over, so now it looks like that. So what's the sum of the red curve, and
the, and the green curve? It's zero here, and all we get is the red guy up here,
and then when they subtract, the red and the green cancel each other, and we get
that. Ta-da.
What we have, is the unit pulse. So, actually, you can think of the pulse
As being a sum of the unit step and a delayed unit step by sum as the same as
the differece here, there's a difference between a unit step and a delayed unit
step and now decompose, The signal into what I claim are simpler pieces.
I actually look at this pulse, and I see those two components when I look at it.
Let's try an example here to see how this works for a little bit more complicated
case. So This thinking about signals as a sum
or difference of simpler signals is called signal superposition.
This word superposition is going to occur again and again here, so superposition
means just to add them up. 'Kay? Here's our example.
How would you build this? How would you deconstruct it? How would you think? What
are the pieces that you see in this signal? Okay.
Well, what I see, is I see that big discontinuity.
Right away, the first thing that jumps out is that it's discontinue.
That tells me that's gotta be a unit step in here somewhere.
And if we put in a unit step, negative unit step because it, the step is going
down. If the origin, of course that's not the
right place. We need to shift it over to 1, well,
we've seen already. That's the play.
So we just move it over. And now we get a discontinuity at the
right place. Alright, that's good.
But now, we want to, talk about this earlier part of the wave form, it looks
like it goes up. Linearly, with a slope of one.
How do we describe that? Well, we can describe that by what's called the ramp
function. that's why I called it r(t) and I want to
point out it's the integral of the unit step.
So the ramp function is 0, 4t less than 0, and then it goes up with a slope of 1
forever and that's the same as this interval.
Okay. So, now, what do we got? If we add
together the green curve and the red curve? let's see we got this, in this
range, always going to be 0 back there. There's no, red curve 0, so we get the
ramp, and then you get a discontinuity, because of the, In steps which jumps
down. The ramp is continuous, the unit step is
discontinuous that causes the discontinuity when you add'em up.
But it keeps on going forever because of the ramp.
I now that's not quite what we want, we want something that cancels This ramp
part, in this area, only in that area, what would you do? Think about that for a
second. What signal could you add out there, or
subtract, to make that part disappear? And I think you know what the answer is
Hence it is a delayed ramp. So we now have R, loose signal is that
delayed ramp and when you add them all together you get the black curve.
And so our final expression for S of T. Is, that it's a sum, a superposition of L
and 3g signals. It has a ramp, a ramp collade, and it
also has a unit step. This is what I mean by signal
decomposition. Signal super position is building
signals, decomposition is thinking about the pieces.
I would also point out that this ramp, is built from a unit step.
It's the integral of it. So basically this signal s, although it
looks a little screwy Turns out it's actually built, the pieces that bfuilt it
are integrals and the, of the unit steps and the unit step itself.
So the unit step is the key thing because in this signal, even though this early
part of the signal doesn't look anything like the unit step and it turns out it
lurch underneath inside and. That's kind of cool.
We're going to use this over and over, and over again as we go through this
course. Now okay, so let's review the important
signals, are sinusoids and more importantly is the complex exponential it
turns out. And the real and imaginary parts of the
complex exponential, corresponding to the sinusoid.
We've talked about the unit step. Discontinuous signals, are going to occur
over and over again, both theoretically, and in practice.
Pulses are used, to communicate bits, and as we've seen, a pulse can be considered
as a superposition of unit step. Of unit step functions.
So this construction in deconstructing signals is a sum of simpler signals.
Oh, is this important. This is going to make your life easy in
doing calculations. And I can't stress, it also helps you
understand what is going on. We'll see this in the succeeding videos.