In this video, we're going to learn about 
how to process signals on a computer. 
we'll venture into the world known as 
digital signal processing. It turns out 
that my research area is DSP, you're 
going to hear me talk about DSP a lot. 
And so, we're going to to start just like 
we did for analog signal. 
We're going to start talking about 
fundamental signals and the basic 
systems. 
And one of the points of this video is 
that the, in the digital world and in the 
analog world, there are a lot of 
similarities about how you think about 
signals and their spectra. 
So we'll see that you're going to already 
know a lot of what [UNKNOWN] say but 
there's a few little things a little bit 
different. 
Alright. 
And the first thing, of course, is we're 
going to start with the complex 
exponential. 
Our friend, 
this is our most important signal that we 
have just about. 
And if you recall, a discrete-time signal 
is a function only of the integers. 
So we call this a digital signal. 
I'm being a little careful here about 
saying discrete-time. 
Digital signals usually refer to discrete 
in time, functions of the integers, and 
discrete in amplitude. 
Well, I'm going to just talk about 
discrete-time signals pretending that the 
amplitude is a continuos value variable, 
and it could, it could be any real 
number. 
It turns out that's not really true. 
In a computer, you can't represent all 
numbers exactly. 
Everything is discrete-valued but we're 
going to hide that detail in the, in the 
way we do our theory. 
Alright. 
The first thing to note is in this 
complex exponential here, 
the frequency variable has to 
dimensionless. 
Has no dimensions, because n has no 
dimensions, and you cannot exponentiate 
anything that has units. 
So, f is dimensionless, and furthermore, 
here's a very interesting property. 
Suppose I consider, the frequency f plus 
an integer, so l is another integer. 
And if I write this as a product, e^j 2pi 
times an integer is one. 
So, you get that this is equal to that. 
That's the definition of a periodic 
function. 
So, f is periodic for any integer and, of 
course, one with the smallest period. 
So, this is periodic, the period of one. 
That's very important. 
So, every function that frequency that 
we're going to talk about has to be 
periodic with period one. 
Because of, of the dimensionalist nature 
of f and the fact that n is an integer. 
We didn't have this for the analog 
complex exponential because t was any 
real number and f had to have units of 
inverse seconds in order for the exponent 
to be dimensionless. 
n here is dimensionless that makes f 
dimensionless. 
Alright. 
But this periodic behavior has a 
interesting consequence so f is periodic 
with period one. 
And it turns out, let's consider this 
complex exponential which has a frequency 
of 1-f and if you go through all the math 
again, that's one and that's the same as 
the complex exponential at negative 
frequency. 
So, here's our f-axis. 
And here's 1/2, -1/2, and 1. 
Let's extend it out to -1. 
So, if you have some it's got to be 
periodic with period one so that means 
this part looks like that. 
It turns out that this part, the part 
greater than a half, has to be the same 
as this part because it's periodic. 
So, this is the same as that. 
So, a frequency at 0.6, let's say, a 
little bit bigger than a half, according 
to this formula, well, that is 0.6 is 
that makes this f 0.4, 0.4 which means 
that f is got to be the same of the same 
value as -0.4. 
So, the highest frequency you can have, 
it turns out, is a 1/2 because once you 
go above a 1/2, that winds you up in 
negative frequency. 
So, here's the name of the game. 
The lowest frequency sinusoid occurs at 
f=0, the highest frequency complex 
exponential whatever is at frequency of 
0.5, 
that's the highest frequency that makes 
any sense. 
You got little bit higher than that, you 
wind up back at may be frequency. 
so, in this highest frequency complex 
exponential, it turns out to be an 
alternating between plus and -1 forever. 
So, when we talk about signals, there are 
functions of quantities, there are 
functions of frequency. 
We're either going to define them over 
-1/2 to 1/2. 
That's a period. 
Or we can talk about them going as a 
function from 0 to 1 and realize that 
this part is the same as that part. 
This is positive frequency, this part is 
negative frequency and then there, that's 
the one difference, a very important 
difference between discrete-time signals 
and analog signals. 
Alright. 
Let's then extend this to a sinusoid. 
the formula looks the same and the 
difference is, is that n is an integer, 
which again, makes f have no no 
dimensions. 
And we're usually going to pick f around, 
in that range. 
So, the highest value of f makes any 
sense to talk about is f. 
And you get what looks a sinusoid but, of 
course, it's only defined with integers 
so you get a stem plot. 
And what doesn't change, of course, is 
Euler's formula. 
So, this sinusoid can be written as the 
real part of Ae to the j phi times 
e^j2pifn. 
So again, that's the complex amplitude. 
That all falls through nothing really 
changes in that regard. 
So, let's talk about some other 
fundamental signals. 
And here's our friend, the unit step. 
And I want to point out that I'm now 
defining it everywhere. 
[LAUGH] Be happy I guess. 
So, here's the n-axis and I'll plot u of 
n. 
And it's zero with, and this, zero pops 
up to one at the origin and there's one 
forevermore, 
okay? 
So, that's the unit step. 
it's a familiar quantity we've seen 
before. 
Here's a new basic signal that we didn't 
have in the [UNKNOWN] called the unit 
sample. 
And this one is very simple. 
It is 0 basically everywhere except at 
the original where it's 1. 
That's why it's called a unit, and it's 
called a unit sample, because it looks 
like it's one value, okay, and is being 
rewritten by delta(n). 
Okay. 
Now, here's the reason we talk about the 
unit sample. 
Let's sketch out here some signal which I 
am just going to call S(n) and let's 
assume that this is the origin n=1, n=2, 
etc. 
along that scale, the horizontal scale 
there. 
The value of the signal at the origin is, 
of course, S(0). 
What is the signal at the origin? Well, 
it's a unit sample whose amplitude is 
S(0), 
okay? 
S(0) is just a number. 
So, what the signal is at the orgin is a 
unit sample. 
This signal, the value, the signal at n=1 
is S(1) times a unit sample which is a 
delayed version of the unit sample. 
It's delta(n)-1, alright? Because this 
signal is a delayed version of that and 
it's delayed by 1 unit, 
etc. 
This is S(2), S(3), S(4), S(5), etc. 
because it's just a 5 down there. 
So, what we get is a somewhat confusing 
but very important formula. 
That a signal, any signal can be 
expressed as a superposition of unit 
samples. 
In fact, that's the definition of a 
discrete-time signal. 
It is a superposition of unit samples 
where we use amplitude at every sample 
expresses whatever the value the signal 
has at that lay. 
So this may look a little funny here but 
the idea here is that the real signal is 
the part that depends on n. 
Here's the only part that depends on the 
n in the formula. 
We're going to find this expression to be 
extremely handy a little bit later. 
Alright. 
Now, let's move on to some simple systems 
and we've seen this one before. 
The simple amplifier where G is the gain, 
of course. 
Now we had to use operational amplifiers, 
op amps, when we talk about analog 
signals that amplified. 
Well, now an amplifier is easy, which 
corresponds to a multiply. 
So, you just kind of do it. 
So, I've written this in kind of a 
computer like code thing, because that's 
exactly the way it works. 
Multiplies are very simple to do. 
So, are delays 
and time delay is simply delaying, of 
course, only by an integer because all 
discrete-time signals are defined only if 
the integer makes no sense to delay by f, 
for example, because must have an integer 
argument signals. 
So, and a delay is also easy to implement 
in a computer. 
no, no real dificulty thre. 
Also, the definition of linear system is 
the same basically as it was for analog 
systems. 
Notice the change in terminology here. 
We say shift-invariant because the delay 
can only be in integers. 
So, the word shift is supposed to mean 
only delaying by an integer, so it was 
just a slight terminology change. So, 
linear time invariant systems basically 
refers to an analog system and your 
shift-invariant system refers to a 
discrete-time system. 
So, the linear part of the definition is 
the same as it was for analog, 
superposition applies. 
So, whenever you express an input as a 
sum of simpler signals, 
the output is also, is equal to the super 
position of the outputs that you got when 
you put each analog. 
So, nothing changes at all. 
And shift-invariant is the same, is the 
same basic form as time invariant. 
If you delay the input by some amount, 
the output that you get is the original 
input to this, this, when the signal 
wasn't delayed. 
And then, you just delay that and that's 
what the output is. 
No matter what delay you pick, the system 
doesn't change with time. 
And all of the examples that we 
encountered for the linear and time 
invariance systems apply here, through 
this special case of discrete-time. 
All these examples follow through so 
there's really no difference here. 
So, let's summarize the basics that here, 
digital signals, discrete-time signals 
are functions of the integers that has a 
very important consequence. 
We're talking about the complex 
exponential, things that are functions of 
frequency. 
Frequency is dimensionless and is only 
defined uniquely over unit-length 
intervals. 
Usually it's going to be 0 to 1 or -1/2 
to 1/2 and it's the -1/2 to 1/2 is the 
one we're going to use a lot wehn we're 
doing theory. 
It's going to turn out that in a 
practical, if you do the calculation, 
the, it's convenient to take frequency 
going over zero to 1. 
I'll point this out when we get to it. 
The most intriguing thing and the 
simplest thing from your viewpoint, is 
that linear signal in systems theory is 
exactly the same as it is for analog 
signals. 
There is no difference, you don't have to 
learn anything new. 
So, we're on our way and now we're going 
to start talking about some more details, 
in particular, the frequency domain in 
discrete-time.