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In this video, we're going to learn about 
how to process signals on a computer. 

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we'll venture into the world known as 
digital signal processing. It turns out 

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that my research area is DSP, you're 
going to hear me talk about DSP a lot. 

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And so, we're going to to start just like 
we did for analog signal. 

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We're going to start talking about 
fundamental signals and the basic 

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systems. 
And one of the points of this video is 

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that the, in the digital world and in the 
analog world, there are a lot of 

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similarities about how you think about 
signals and their spectra. 

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So we'll see that you're going to already 
know a lot of what [UNKNOWN] say but 

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there's a few little things a little bit 
different. 

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Alright. 
And the first thing, of course, is we're 

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going to start with the complex 
exponential. 

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Our friend, 
this is our most important signal that we 

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have just about. 
And if you recall, a discrete-time signal 

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is a function only of the integers. 
So we call this a digital signal. 

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I'm being a little careful here about 
saying discrete-time. 

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Digital signals usually refer to discrete 
in time, functions of the integers, and 

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discrete in amplitude. 
Well, I'm going to just talk about 

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discrete-time signals pretending that the 
amplitude is a continuos value variable, 

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and it could, it could be any real 
number. 

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It turns out that's not really true. 
In a computer, you can't represent all 

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numbers exactly. 
Everything is discrete-valued but we're 

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going to hide that detail in the, in the 
way we do our theory. 

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Alright. 
The first thing to note is in this 

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complex exponential here, 
the frequency variable has to 

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dimensionless. 
Has no dimensions, because n has no 

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dimensions, and you cannot exponentiate 
anything that has units. 

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So, f is dimensionless, and furthermore, 
here's a very interesting property. 

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Suppose I consider, the frequency f plus 
an integer, so l is another integer. 

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And if I write this as a product, e^j 2pi 
times an integer is one. 

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So, you get that this is equal to that. 
That's the definition of a periodic 

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function. 
So, f is periodic for any integer and, of 

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course, one with the smallest period. 
So, this is periodic, the period of one. 

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That's very important. 
So, every function that frequency that 

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we're going to talk about has to be 
periodic with period one. 

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Because of, of the dimensionalist nature 
of f and the fact that n is an integer. 

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We didn't have this for the analog 
complex exponential because t was any 

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real number and f had to have units of 
inverse seconds in order for the exponent 

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to be dimensionless. 
n here is dimensionless that makes f 

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dimensionless. 
Alright. 

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But this periodic behavior has a 
interesting consequence so f is periodic 

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with period one. 
And it turns out, let's consider this 

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complex exponential which has a frequency 
of 1-f and if you go through all the math 

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again, that's one and that's the same as 
the complex exponential at negative 

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frequency. 
So, here's our f-axis. 

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And here's 1/2, -1/2, and 1. 
Let's extend it out to -1. 

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So, if you have some it's got to be 
periodic with period one so that means 

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this part looks like that. 
It turns out that this part, the part 

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greater than a half, has to be the same 
as this part because it's periodic. 

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So, this is the same as that. 
So, a frequency at 0.6, let's say, a 

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little bit bigger than a half, according 
to this formula, well, that is 0.6 is 

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that makes this f 0.4, 0.4 which means 
that f is got to be the same of the same 

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value as -0.4. 
So, the highest frequency you can have, 

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it turns out, is a 1/2 because once you 
go above a 1/2, that winds you up in 

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negative frequency. 
So, here's the name of the game. 

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The lowest frequency sinusoid occurs at 
f=0, the highest frequency complex 

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exponential whatever is at frequency of 
0.5, 

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that's the highest frequency that makes 
any sense. 

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You got little bit higher than that, you 
wind up back at may be frequency. 

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so, in this highest frequency complex 
exponential, it turns out to be an 

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alternating between plus and -1 forever. 
So, when we talk about signals, there are 

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functions of quantities, there are 
functions of frequency. 

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We're either going to define them over 
-1/2 to 1/2. 

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That's a period. 
Or we can talk about them going as a 

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function from 0 to 1 and realize that 
this part is the same as that part. 

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This is positive frequency, this part is 
negative frequency and then there, that's 

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the one difference, a very important 
difference between discrete-time signals 

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and analog signals. 
Alright. 

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Let's then extend this to a sinusoid. 
the formula looks the same and the 

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difference is, is that n is an integer, 
which again, makes f have no no 

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dimensions. 
And we're usually going to pick f around, 

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in that range. 
So, the highest value of f makes any 

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sense to talk about is f. 
And you get what looks a sinusoid but, of 

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course, it's only defined with integers 
so you get a stem plot. 

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And what doesn't change, of course, is 
Euler's formula. 

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So, this sinusoid can be written as the 
real part of Ae to the j phi times 

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e^j2pifn. 
So again, that's the complex amplitude. 

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That all falls through nothing really 
changes in that regard. 

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So, let's talk about some other 
fundamental signals. 

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And here's our friend, the unit step. 
And I want to point out that I'm now 

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defining it everywhere. 
[LAUGH] Be happy I guess. 

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So, here's the n-axis and I'll plot u of 
n. 

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And it's zero with, and this, zero pops 
up to one at the origin and there's one 

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forevermore, 
okay? 

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So, that's the unit step. 
it's a familiar quantity we've seen 

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before. 
Here's a new basic signal that we didn't 

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have in the [UNKNOWN] called the unit 
sample. 

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And this one is very simple. 
It is 0 basically everywhere except at 

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the original where it's 1. 
That's why it's called a unit, and it's 

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called a unit sample, because it looks 
like it's one value, okay, and is being 

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rewritten by delta(n). 
Okay. 

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Now, here's the reason we talk about the 
unit sample. 

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Let's sketch out here some signal which I 
am just going to call S(n) and let's 

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assume that this is the origin n=1, n=2, 
etc. 

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along that scale, the horizontal scale 
there. 

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The value of the signal at the origin is, 
of course, S(0). 

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What is the signal at the origin? Well, 
it's a unit sample whose amplitude is 

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S(0), 
okay? 

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S(0) is just a number. 
So, what the signal is at the orgin is a 

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unit sample. 
This signal, the value, the signal at n=1 

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is S(1) times a unit sample which is a 
delayed version of the unit sample. 

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It's delta(n)-1, alright? Because this 
signal is a delayed version of that and 

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it's delayed by 1 unit, 
etc. 

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This is S(2), S(3), S(4), S(5), etc. 
because it's just a 5 down there. 

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So, what we get is a somewhat confusing 
but very important formula. 

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That a signal, any signal can be 
expressed as a superposition of unit 

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samples. 
In fact, that's the definition of a 

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discrete-time signal. 
It is a superposition of unit samples 

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where we use amplitude at every sample 
expresses whatever the value the signal 

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has at that lay. 
So this may look a little funny here but 

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the idea here is that the real signal is 
the part that depends on n. 

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Here's the only part that depends on the 
n in the formula. 

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We're going to find this expression to be 
extremely handy a little bit later. 

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Alright. 
Now, let's move on to some simple systems 

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and we've seen this one before. 
The simple amplifier where G is the gain, 

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of course. 
Now we had to use operational amplifiers, 

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op amps, when we talk about analog 
signals that amplified. 

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Well, now an amplifier is easy, which 
corresponds to a multiply. 

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So, you just kind of do it. 
So, I've written this in kind of a 

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computer like code thing, because that's 
exactly the way it works. 

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Multiplies are very simple to do. 
So, are delays 

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and time delay is simply delaying, of 
course, only by an integer because all 

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discrete-time signals are defined only if 
the integer makes no sense to delay by f, 

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for example, because must have an integer 
argument signals. 

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So, and a delay is also easy to implement 
in a computer. 

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no, no real dificulty thre. 
Also, the definition of linear system is 

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the same basically as it was for analog 
systems. 

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Notice the change in terminology here. 
We say shift-invariant because the delay 

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can only be in integers. 
So, the word shift is supposed to mean 

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only delaying by an integer, so it was 
just a slight terminology change. So, 

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linear time invariant systems basically 
refers to an analog system and your 

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shift-invariant system refers to a 
discrete-time system. 

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So, the linear part of the definition is 
the same as it was for analog, 

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superposition applies. 
So, whenever you express an input as a 

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sum of simpler signals, 
the output is also, is equal to the super 

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position of the outputs that you got when 
you put each analog. 

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So, nothing changes at all. 
And shift-invariant is the same, is the 

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same basic form as time invariant. 
If you delay the input by some amount, 

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the output that you get is the original 
input to this, this, when the signal 

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wasn't delayed. 
And then, you just delay that and that's 

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what the output is. 
No matter what delay you pick, the system 

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doesn't change with time. 
And all of the examples that we 

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encountered for the linear and time 
invariance systems apply here, through 

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this special case of discrete-time. 
All these examples follow through so 

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there's really no difference here. 
So, let's summarize the basics that here, 

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digital signals, discrete-time signals 
are functions of the integers that has a 

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very important consequence. 
We're talking about the complex 

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exponential, things that are functions of 
frequency. 

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Frequency is dimensionless and is only 
defined uniquely over unit-length 

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intervals. 
Usually it's going to be 0 to 1 or -1/2 

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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 

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doing theory. 
It's going to turn out that in a 

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practical, if you do the calculation, 
the, it's convenient to take frequency 

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going over zero to 1. 
I'll point this out when we get to it. 

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The most intriguing thing and the 
simplest thing from your viewpoint, is 

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that linear signal in systems theory is 
exactly the same as it is for analog 

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signals. 
There is no difference, you don't have to 

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learn anything new. 
So, we're on our way and now we're going 

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to start talking about some more details, 
in particular, the frequency domain in 

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discrete-time.