Okay, in this video we're going to talk 
about the basic ideas behind systems. 
We'll first talk about some very simple 
systems, systems that the operations they 
perform in the input signal to boost an 
output are quite simple and 
straightforward. 
We're then going to talk about how you 
can book systems together, to make more 
complicated input, output relationships. 
The relationship between the input and 
the output . 
Then finally we're going to talk about 
linear systems, and time invariant 
systems. 
And the ones that are both linear. 
And time invariant are going to be 
extremely important to us, as we're going 
to see. 
Alright, so let's talk about some simple 
systems, ones that are pretty easy to 
understand. And the first one is quite 
simple. 
It's called a gain, and what the system 
does is, the output is equal to the input 
times a gain. 
A number G, a scalar. 
So I think that's pretty easy to 
understand it. 
It just takes the input, multiplies it 
times a number and that's its So, we can 
use the unit step as a example input to 
see how this works. 
So the input here, is I've drawn it as a 
unit step. 
And then the output is the same unit 
step, except it's now got an amplitude of 
G instead of 1, whatever that is. 
In this special case that I've drawn, 
I've indicated here of an amplifier has 
to do with the fact that if g is bigger 
than 1, the terminology is, that it is an 
amplifier. 
It makes things bigger. 
It amplifies. 
There are cases in which g is less than 1 
in which the case it serves as an 
attenuator. 
It makes things smaller. 
And finally, a somewhat confusing 
terminology, if G is negative, it's 
called an inverter. 
So it does not mean take the reciprocal. 
So if we use our unit step input for 
example here. 
So, we bring in new step, and we look at 
what the output is going to be. 
If it G was negative, the output, would 
be like that. 
So, it doesn't mean take the reciprocal 
it means change the sign. 
And I have no idea where the terminology 
inverter came from, but that's what it 
is. 
Okay, let's talk about some more systems, 
and this one we've seen already. 
The time delay, the output is equal to 
the input a little bit later, it's been 
delayed, and again using unit step as our 
example, you delay the output is also a 
unit step that is now delayed. 
It seems like I said, already talking 
about signals, pretty easy to understand 
what it does. 
Turns out is a more complicated to build. 
So, delay boxes turn out to not be easy. 
We do want to point out that they do 
occur naturally. 
Light, in fact, all electrical signals 
travel at a finite speed. 
They don't travel from one place or 
another instant. 
So, if something happens at the sun, a 
change in the brightness of the sun, it 
won't reach the earth for about eight and 
a half minutes. 
So that's a delay. 
That's a physical delay. 
Just run an electrical signal through a 
long piece of wire. 
Introduces the delay so in some cases 
it's pretty easy, it's very hard to have 
a controllable delay but that could be a 
bit more complicated. 
So like I said last time if you have tau 
greater than 0 that's called a time 
delay, if tau is negative it's a time 
advance that occurs earlier in time. 
turns out such systems are pretty hard to 
build, because, again using arguments 
step as an example, if time were 
negative, the output would come out 
before the change in the input occured. 
So if this is, for example, -1, this 
output at time 1, -1 is going to have to 
occur before it occurs at the input. 
So, somehow the system would have to know 
the future value, the future behavior of 
the input signal. 
Well, that's a little weird. 
And so, time advances are great 
mathematical things, but building them, 
is, more than just a little complicated. 
And another example that's along similar 
lines, something that's easy to write, 
but hard to build, and that's called time 
reversal. 
So here, the, using it again our unit 
step input, you put in the unit step, and 
it comes out, delayed. 
Not delayed, I'm sorry, my mistake it 
comes out time-reverse. 
So notice, that again, the value at, lets 
say time-1 is going to be the value of 
the input at time 1. 
The system would have to know what was 
going to happen 2 seconds ahead in that 
case to produce an output. 
Well it gets even worse. 
Let's clean this out, what happened at 
time 10 would have to come out at the 
output at time minus 10. 
Which means they would have to be able to 
predict way ahead in the future, in fact, 
have to predict the entire waveform into 
the infinite future, and then put it out 
at the infinite past, so time reverse 
systems are very weird, very hard to 
build. 
however, they have a very simple 
mathematical description. 
not all simple systems can be built. 
Well, let's talk about some systems that 
are maybe not so simple, but easy to 
understand, and that's the derivative 
system. 
So, what this system does is simply take 
the input and evaluate its derivative, 
and that produces the output. 
So, easy, understanding actually 
derivative systems are actually easy to 
build. 
We can also talk about, systems that 
integrate, and I'm going to point out 
something about this, again these are 
easy to build There's a convention in 
this course that's important and that is 
there are no things such as indefinite 
integrals in this course. 
Every integral is a definite integral. 
And most of our integrals will start at 
minus infinity and go up to time t. 
I tend to use Greek letters for variables 
of integration. 
It's just a convention I have to make it 
clear that it is a variable of 
integration. 
This will come up very frequently. 
So, the output y is equal to the integral 
over all previous time of the input x. 
This [INAUDIBLE] and like I said there is 
no trouble building this system. 
Okay. 
Those were our simple systems. 
Let's put them together. 
So, let's talk about some simple 
structures. 
Perhaps the simplest is called the 
cascade structure, and essentially, all 
it has is one system after another. 
We've already seen this in the 
fundamental model communication. 
It is a cascade of several systems, in 
fact five of them. 
So the input to the entire chain is x, 
goes through s1, produces w and then w in 
turn produces y. 
So we can think about this as one system, 
which is constructed as a cascade of two 
simpler systems. 
In fact, that's exactly what cascade 
structures are for. 
For example, suppose you wanted a system 
that delayed and inverted. 
And I could do that by building an 
inverter let's say here. 
Then passing that, the output of inverter 
into delay.And then the overall 
relationship between x and y would be an 
inverted delay. 
So, that's the way you get something a 
bit more complicated from simpler. 
Systems. 
Another way you can hook systems together 
to do interesting things is what's called 
a parallel connection. 
And I think it's pretty easy to see why 
they're in parallel. 
You have two systems here that are in 
parallel to each other. 
There are a couple of conventions here we 
need to talk about. 
I want you to notice here that I have x 
coming in, and it gets split. 
At least that's what the diagram would 
seem to indicate. 
In system theory, the convention is that 
when such as, a split occurs, it doesn't 
mean split the signal in two or anything. 
What it means is the same signal is 
applied to both systems after the split, 
so just a convention. 
That's why I labeled this very carefully 
x(t). 
When we get to talking about this in 
special cases, you're not going to see me 
write this. 
It will be understood that the input to 
s1 and s2 is x(t). 
That's what this diagram Diagram means. 
We also have a new system over here. 
We have a two input one output system. 
And I think it's pretty obvious what it 
is, it's an adder, so if we were to call 
this y 1, the operative of that system, 
and that y2 and y = y1 + y2. 
So, and that's what an adder does, I 
think it's pretty self-explanatory in 
terms of this what's called block 
diagram. 
So a parallel system. 
And there's two systems in parallel each 
having the same input x and you just add 
up their outputs to produce the total 
output. 
So for example suppose you wanted to 
model an echo chamber. 
So in that case the system s1, we let's 
say do nothing. 
It's output is equal to the input. 
The output of s2 or rather s2 would be a 
delay box. 
It would delay it's input. 
After you add them together, you get the 
same signal, plus it's self delayed. 
That's what happens when you have an 
echo. 
You get the same, you get a signal, plus 
its delayed version. 
So I could build, construct something out 
of very simple systems, it's a bit more 
complicated. 
Okay, finally, we have a the most 
complicated structure that we're going to 
encounter the feedback. 
And I'll explain how it's used in a 
second, but let's go through this. 
So we have the input and here's the 
output. 
The input is going to go through an adder 
and the other input to the adder comes 
from something derived from y. 
So, the way we think about this is that 
this output of this adder is an error 
signal. 
Because you can see this little minus 
sign sitting here. 
And that's the standard convention which 
means that the e is equal to x minus the 
output of S2. 
So that little minus sign, right there 
corresponds to that minus sign. 
So that's how you indicate a difference 
instead of a sum. 
So what this feedback configuration does, 
is it takes e passes it through S1 to 
produce the output. 
But that output gets sent back through S2 
to occur at the input. 
That gets subtracted from x, and produces 
the air signal, which goes around, and 
round, and round, and round. 
So this is the way a cruise control works 
on a car. 
You may choose the setting for the 
velocity which you want the car to go 
and, if that is different than it's 
current velocity, that produces an error 
signal. 
That causes the engine to rub. 
Let's clean that to describe the engine. 
Causes and output velocity y. 
Well that goes through the control system 
as two to produce the air signal. 
And finally when the air goes to zero, 
the cruise control kind of settles down 
and doesn't accelerate the car any more. 
So, that's a great example of feedback 
system. 
I want to point out that everything I 
just said in that example wasn't 
electrical. 
Turns out systems can describe a very 
general class of signal. 
The fundamental reason is, they're just 
mathematical constructs. 
And system theory and signal theory for 
that matter applies to electrical signals 
and beyond, so It's interesting that you 
can use these kind of models to describe 
much more general things. 
Alright, let's talk about special cases, 
special systems, and perhaps one of the 
most important, or linear systems. 
So let's look at this. 
So we have a system S. 
If it is linear, if you take an input 
that consists of a sum of inputs, sum of 
the signals. 
So x1 is a signal, x2 is a signal, and a2 
and a1 are scalars, constants. 
The output of that sum, is the sum of the 
outputs to the simple sequence. 
So, if you put in x1 by itself in S, you 
get that for an output, put x2 by itself 
you get that. 
You add those two together with the 
waiting cost and that's the same. 
As if the system saw this sum, presented 
to it as an input. 
So this is a very important property of 
linear systems. 
It's called the principle of 
superposition. 
We're going to use this a lot. 
We've already seen when we talked about 
signals that is very convenient, in many 
ways, to think about signals as a sum of 
simpler signals. 
So, for example, that triangular signal 
we saw in the signals video. 
We wrote as a sum of three signals. 
Well, to find out what the output to sum 
system is to that signal, all I have to 
do is figure out the output to a step, 
and to a ramp, and a delayed ramp, add 
them all up, and that's the going to be 
output to our complicated signal. 
So, this principle of superposition is 
very, very important. 
It helps decompose problems, once you can 
decompose a signal into simpler parts, 
you now can find the output because you 
probably can easily find the output to a 
simple signal. 
So, here's the some special cases of the 
linear system. 
Let's assume that the linear system here 
as is the linear. 
If you multiply the input by a number, it 
changes the input too, now the output is 
going to be the output you had before 
multiplied by the same gain factor and 
there was a way, almost sloppy way of 
saying this, you double the input, you 
double the output. 
If that happens, then the system could be 
linear. 
If that does not happen, if doubling the 
input does not double the output, then 
the system can't be linear. 
It violates the rules. 
That's a special case of what I wrote up 
here Because, here we have a 2 0, for the 
case that I'm showing down here. 
Okay, then the other case is, just, as I 
described in some detail, if you just, 
decompose a signal into a sum, it's 
output Of the system, the system's output 
to that sum is the sum of the output's to 
each, so that, goes just into detail of 
what we've said before. 
But, this result turns out has a very 
interesting consequence, so let's 
consider a special case. 
A2 is the negative of a1. 
And let's say that x1 and x2 are the same 
thing. 
We're just going to call that x. 
So, a1 and a2 are negatives of each 
other, and these two signals has the same 
thing. 
Well, what that gives us, is that the, 
input to our system is a1x- a1x. 
Well that's called 0. 
But now, let's look at what the linear 
system equation says in that special 
case. 
Now, since x1 and x2 are the same, we 
have S(x) and S(x), and they're weighted 
by the same a1, but they're subtracted to 
get 0. 
So, if you have. 
The system is linear. 
You stick in 0, identically 0, into a 
linear system, what you get out is 0. 
Any system which doesn't obey that rule, 
cannot be linear. 
It's sort of a very simple test, but it's 
not what's called, suffiecient, it's a 
neccessary property, but not suffiecient. 
The real test of wheter something is 
linear or not, goes back to the original 
definetion. 
And by the way, in detail, this 
Principles super position has to apply 
for all signals to XY and X2, and all 
constants A1 and A2. 
So, it's a pretty, demanding requirement, 
that, when you have a linear system, it's 
very, very nice. 
Another special. 
A class of systems or so-called timing 
variance systems. 
So what these kind of systems are are is 
that if you come into the lab, let's say, 
and you build a device and you put in an 
input and you measure the output end, you 
leave the system alone, you come back the 
next day, you put in the same signal And, 
what you should get out is the, exactly 
the same output, as you had the previous 
day. 
In that scenario, I just described, means 
that, I delay the input to the next day, 
and what happened? I got out that day, 
the same thing as what I got the previous 
day, but delayed. 
[INAUDIBLE] day, you have the 
information. 
So systems that do not change their 
behavior with time, are called 
Time-Invariant Systems. 
That does not mean, signals can't vary 
with time, that's certainly not the case. 
What it does mean, the system doesn't 
change its behavior, with time, stays the 
same. 
So, let's go through a little, set of 
examples, and see if we can, figure out 
how to classify signals according to 
whether they're linear, or 
time-invariant. 
So, here we have the game box. 
Multiply the signal the, and amplifier, 
let's say g is bigger than 1 and 
positive, it's an amplifier. 
Is that linear? Well, if I change x to x1 
+ x2, do I get G * x1 + G * x2? 
I certainly do. 
It's linear. 
Is it time invariant? Well, I think it's 
pretty obvious that it is but I just 
change T to T minus tau. 
What I get is the previous expression for 
Y, if I stick in T minus tau there. 
So this is linear and time invariant. 
No problem with that. 
Okay, another example. 
How about the derivative box. 
Something that takes the derivative. 
Is that linear? If you take the 
derivative, of a sum, is that the sum of 
the derivatives? And I think you will 
agree, yeah sure, that's true. 
So that's, linear. 
How about time-invariant? Once the 
derivative of x / t minus tau, well, 
that's, in calculus you learn that's the 
derivative of x evaluated at t minus tau. 
Well that makes it time invariant. 
So, that's we got two special cases here 
of linear and time invariant systems. 
How about this system? How about this 
squaring system? So the output equals to 
the input squared. 
While that is certainly not linear. 
Alright, because the square of X 1 plus X 
2 is not equal to X 1 squared plus X 2 
squared. 
How about time-invariant? If I, replace 
t, by t minus tau, is that the same 
formula I'd get if I replaced t minus tau 
in this, in this relationship, 
input/output relationship, and the answer 
is, it certainly is. 
This is an example of a non-linear, but 
Time-Invariant System. 
And finally, how about this system, which 
is called a modulator? So, what we have 
is our input is here, and what happens is 
that the system multiplies it by cosine 2 
pi f ct to produce an output. 
Well, is it linear? So if I have x1 + x2, 
will I get out, cosine x1 + cosine x2? 
And I certainly do. 
So, that's linear. 
How about time invariant? So, the thing 
to notice is that the delay applies only 
to x. 
So if I replace x here by t minus tau. 
You don't replace it over here because 
that's inside the system. 
That's not the input delay. 
Well that's not the same expression that 
you get, if you replace t minus tau over 
here in this formula, you have to replace 
the t in both places by t minus tau. 
So this system is not time-invariant, due 
to delay of the input you do not get the 
outputs we had before, delayed because 
this co sign is changing behavior with 
time. 
So we have the cases where both of these 
examples you going to have systems also 
that are not linear and not time 
invariant of course. 
And the ones that are really going to be 
special are the ones that are linear and 
And time invariant. 
We're going to have to deal with this 
one. 
But we're clearly can't use the linear 
time invariant system theory to talk 
about it, because it is not time 
invariant. 
We'll see that in a second. 
Well okay, we now know a lot about 
systems. 
What systems do, is they operate on input 
to produce an output for some reason. 
You might want to amplify it. 
You might want to delay the signal. 
Take it's derivative, whatever. 
That's what systems do. 
They can do all kinds of things to 
signals that are interesting and by 
building up structures in simple systems, 
you can make something that's a lot more 
complicated than any of the components. 
And that's basically the way electrical 
engineers do a lot of design. 
Breakdwon a complicated input and output 
relationship into a simpler set that I 
can realize by cascade, parallel, and 
feedback structures. 
And I keep emphasizing that linear 
time-invariant systems are very 
important. 
What we're going to call LTI. 
LTI systems really are important. 
They occur a lot, and as what we see when 
we start talking about circuits, we're 
going to talk about linear time invariant 
circuits. 
When we get to signal processing we'll 
talk about linear time-invariant signal 
processing. 
Those special systems are very important 
because you can do a lot of very 
interesting things and have a very good 
theory for them when we start talking 
about them. 
That comes very soon.