In this video, we're going to define the 
Fourier Transform for digital signals. 
It's known by the complicated name of a 
discrete-time Fourier Transform. 
it's so long-winded, everybody calls it 
the DTFT. 
We're going to see that it looks very 
similar to what we've already encountered 
and we're going to talk about its 
properties. 
Many of which are shared with the Fourier 
Transform we've already defined for 
analog signals. 
So this is a fairly easy leap to go into 
the discreet time world. 
So, let's define it and like I said the 
definition looks very similar What we 
have for analog signals. 
the big difference is, like I keep 
saying, is. 
That it's a function of f, which has no 
units. 
And f is in the range minus a half to a 
half or zero to one. 
is the case may be we'll figure this out 
a little bit later. 
You want to point out the notational 
change this is written, the DTFT is 
written as S(e^jt2pif) the variable is f 
but this helps us distinguish it from the 
Fourier Transform of an analog c So, this 
is the notation we're going to use for 
the DTFT, and when I, when you see a 
s(f), you'll know that means an analog 
Fourier Transform. 
And f in that case has units of hertz. 
Okay, so again the all spectra are 
periodic with period 1. 
And as we know, that comes from the fact 
that this is periodic. 
with a period of 1 as a function of f so 
it has to be periodic and I'm going to 
show you an example to show you how that 
works. 
So, here's a little simple example. 
It's a exponential signal. 
And let's plot it, so we understand it. 
So, a^n u(n). 
Okay, 
the first thing to note is that the unit 
step multiplies the a^n. 
So unit step, you recall Use 0 for the 
negative and it's 1 for n = 0 and 
positive. 
And that's going to the part that's back 
here for -n is going to set to 0 the 
exponential, so the only thing we get Is 
the exponential for positive, index 
values. 
Positive time values. 
And it may look something like that. 
If A is bigger than 1 this thing could 
take off. 
It's still 1 at the origin. 
But it could very big very quickly if A 
is bigger than 1. 
So, what I have done here is some example 
here with a [UNKNOWN]. 
Ok, well let's compute spectrum and way 
that proceeds the best referred I am just 
going to Plug in for s and I'll plug in 
what my signal is. 
There it is. 
And, again what the unit step does is 
that it sets to zero anything for n 
negative so I can change the limits of 
the sum and therefore get rid of the u(n) 
and get a much simpler for where I can 
play with. 
And I'd like to point out that the 
complex exponential here can be thought 
of as e to the minus j, 2pi f, all raised 
to the nth. 
And that means, I can merge those 2. 
Exponential's in the formula.This is 
going to be the key step for getting a 
analytic expression from the spectrum. 
So here's what I did same thing. 
And now we have to recall the geometric 
series, very important in digital signal 
processing to know this property. 
You sum up a exponential in alpha, in 
zero to infinity. 
As long as the absolute value of alpha. 
The magnitude of alpha is less than 1. 
It's hidden by this very concise formula. 
So I'm going to use that formula. 
Because that's exactly what I have here. 
And this formula works, even if the alpha 
is complex valued. 
The only thing that matters is that its 
magnitude is be, is less than 1. 
Which it certainly is, in this case. 
If I assume that the absolute value of A 
is less than 1. 
Alright. 
So we can use it. 
Very simple formula pops up. 
There it is. 
And so, that is the spectrum of the 
exponential, signal. 
let's plot it. 
And for a specific value of a, which we 
have a half. 
And here you see the periodic, nature of 
it, you can go between -1/2 and 1/2 or 0 
to 1, whichever you want. 
I want to, show you how I plotted those 
curves for the magnitude and phase. 
you'll notice I didn't actually give you 
a formula for the magnitude and the 
phase. 
it turns out that for, discreet time 
spectra, those expressions can be fairly 
unwieldy. 
So, just recall here's what the transfer 
function was for our example. 
The exponential signal. 
And what I want to show you, is that by, 
in using modern tools, like MATLAB Our 
octave. 
It turns out you can basically type this 
expression in as it stands. 
And get all the information about the 
magnitude and phase that you want. 
So, let's see how that works. 
So here, I have the MATLAB command 
window. 
And I'm going to type in. 
[SOUND] A range of frequencies going 
between minus 2 and 2. 
that's sort of what was shown in the 
previous figure. 
And I'm going to say that A is a half, 
which is what we had. 
And I'm going to type in the formula for 
h. 
[SOUND] And you notice I'm literally 
typing it in [SOUND] J and all. 
Turns out the default in MATLAB and 
Octave is that J is equal to the square 
root of minus 1. 
And the only, and the only thing that's 
interesting here is, notice the dot 
slash. 
What that means is term by term divide. 
Because f is a vector. 
And the exponential here of the, 
of an expression containing an f, is also 
a vector. 
And so, when you want turn by turn 
division, which is exactly what I want, 
this is the expression you'll want. 
You'll want to put that in. 
Okay. 
That's that. 
I'm going to create a subplot, it's 
called, in my plot window here. 
And, I'm just going to plot it. 
Plot f. 
Now, we want to plot the magnitude and in 
MATLAB an octave it's ab, abs function 
for absolute value. 
And there it is. 
Turn on the grid. 
So, the [UNKNOWN] range is between -1/2 
and 1/2 that's the range of which the 
frequencies are really an interest, the 
rest is periodic so you can get the 
Everything else is a periodic repetition 
just like we've been talking about. 
So that's negative frequency, that's 
positive frequency, negative, positive, 
negative etc. 
Notice how easy it was to apply, just 
type in the expression for the spectrum 
and there it is. 
Alright, 
so now I'm going to do the angle. 
[SOUND] [INAUDIBLE] plot, 
and I'm going to go back and change that 
to angle. 
That's the function you want. 
Now, it turns out, this is going to 
return an angle in radians, and I'm not 
you, but I find degrees a little bit 
easier to understand, so the way to do 
that is multiply your angle by 180/pi. 
And also, I should point out that 
[INAUDIBLE] octave know what pi is, too. 
That's a predefined variable. 
I didn't have to define, to very many 
digits. 
And there's the [INAUDIBLE]. 
turn on the grid abd now we can see that 
the 
angle never really gets beyond about 30 
degrees. 
That's kind of interesting. 
you can prove that the phase at the 
origin, 0 frequency, is either going to 
be 0 or pi. 
It has to be real valued, and that's 
going to be the same at a half. 
Now, the the half may seems a little odd, 
until you realize what the complex 
exponential is? 
So, if you look at e to the minus j(2pif) 
and then evaluated at f equal a half. 
And you've already done that, 
that's minus 1 to the n. 
So when you calculate your spectrum, your 
DTFT at that frequency. 
It only contains real numbers. 
So again, the phase has to be either zero 
or pi. 
And so phase tends to have this rather 
complicated appearance to it. 
But also, we need to point out that the 
phase is odd and the magnitude is even, 
just like they should be and both are 
periodic because these are the J(2pi) of 
n's. 
But the real point of this segment is. 
How easy it is to plot very complicated 
transfer functions, once you have a 
formula for the, transfer function or the 
spectrum that you want. 
Okay, 
what's the inverse DTFT? Well, we are 
going to use under the [UNKNOWN] 
relationship that we used long time ago 
for [UNKNOWN] series and we show that the 
[UNKNOWN] for two different industries 
here was equal to this expression. 
[UNKNOWN] we can write as a uni Much more 
concisely than writing it like that. 
You know, a same kind of thing follows. 
We are going to multiply, our expression 
for the spectrum by the complex 
exponential integrate. 
And that will give us back our signal. 
So, driving the inverse. 
DTFT is very easy and again just like we 
had for analog signals, we can go back 
and find the time domain and the 
frequency domain, using these formulas. 
We go from time domain to the frequency 
domain using the DTFT and the inverse 
that So, just like we had before, we can 
think about a signal either way depending 
on the application. 
So, let's explore some of the properties 
of the DTFT. 
None of these should be terribly 
surprising of course, the DTFT is linear 
because it's just a A sum of terms, you 
can sum, a sum this way, or do it 
directly. 
So, it's linear, very important property 
it has conjugate symmetry, which is easy 
to show, so it negative frequency's the 
same Hence, the conjugated positive 
frequencies. 
And they also show through the properties 
of the complex exponential. 
Because it's periodic with period 1, it's 
also, what's at negative frequency also 
occurs at 1-f. 
So, the old time delay formula still 
works if you delay. 
A signal by, n zero, values. 
And in-, indices samples, if you will. 
That's the same as multiplying the 
spectrum by a complex exponential, having 
it delayed. 
But there, and the other way around, if 
you multiply by a complex, 
Exponential in the time domain is the 
same as shifting the spectrum in the 
frequency domain by the frequency 
[UNKNOWN] of the complex modulation. 
And of course our old friend Parseval's 
theorems still applies. 
Now in a computer, um,the power in the 
signal is not That role, not that 
important. 
So, this is definitely an abstraction 
about calling that a power, but what else 
we want to call it. 
And of course you can find it in the 
frequency domain or in the time domain, 
whichever is easier using Parseval's 
Theorem. 
The proof of this is just like it was Of 
all of the previous examples we've gone 
through. 
So, the DTFT shares many of the 
properties of the Fourier Transform we've 
already encountered for analog signals. 
It's very important to note all spectra 
are periodic and the reason for that is 
Just because of the way the math works 
out. 
Tthat's one way of looking at it. 
And because it has to be periodic, the 
highest possible frequency it can have in 
the Discrete-Time world is 1/2. 
There is no higher frequency in math 
because it has to be periodic. 
We'll get used to that. 
Now What we've talked about with the DTFT 
is how to, compute it. 
We've defined it, 
Remember back when we talked about 
signals? There are some signals for which 
there are no formulas. 
In particular, speech. 
How in the world did I show you that 
speech spectrogram several videos ago? 
How do they do it, I don't have a formula 
so it's going to be a little difficult to 
compute this directly and how we do that 
is the subject of the next video.