The Discrete Fourier Transform (DFT)
Summary
TLDRThis video introduces the Discrete Fourier Transform (DFT), explaining its role in converting discrete data into Fourier coefficients to analyze signals. The speaker clarifies how DFT differs from the continuous Fourier Transform and emphasizes the importance of the Fast Fourier Transform (FFT), a highly efficient algorithm widely used in fields like audio compression, image processing, and scientific computing. The mathematical concepts behind DFT are explored, with the speaker highlighting how it can be written as a matrix multiplication and how FFT enables faster computation of large datasets.
Takeaways
- 🧮 The Discrete Fourier Transform (DFT) is key for computing Fourier series on discrete data.
- 📉 DFT approximates a Fourier series for a finite, periodic interval of data rather than an infinite Fourier transform integral.
- 🖥️ The Fast Fourier Transform (FFT) is a computationally efficient way to compute the DFT, especially for large datasets.
- 🎶 DFT is used to decompose data into sine and cosine components, which is valuable for tasks like signal processing and diagnostics.
- 🔢 Input data for DFT is represented as a vector of discrete values, which may be sampled from continuous functions.
- 📊 DFT transforms this discrete data into Fourier coefficients, revealing the frequencies present in the data.
- 🧩 The transformation process involves a matrix multiplication, where the DFT is defined mathematically using exponential terms.
- 🔄 You can also reconstruct the original data from the Fourier coefficients using an inverse DFT formula.
- ⚙️ The FFT allows efficient computation of the DFT without explicitly performing large matrix operations.
- 🌀 The result of the DFT includes complex numbers, with the magnitude indicating the frequency's strength and the phase indicating its alignment.
Q & A
What is the Discrete Fourier Transform (DFT)?
-The Discrete Fourier Transform (DFT) is a mathematical transformation that approximates a Fourier series on a finite interval where the function is periodic. It allows one to convert data points from the time domain into frequency components, revealing the sine and cosine components that make up the signal.
Why is the name 'Discrete Fourier Transform' considered misleading?
-The name 'Discrete Fourier Transform' is considered misleading because it more accurately represents a Fourier series approximation rather than a true Fourier transform, which involves an integral over infinite intervals. However, due to historical reasons, it is called DFT.
What is the relationship between the DFT and the Fast Fourier Transform (FFT)?
-The FFT is a computationally efficient algorithm to compute the DFT. While the DFT can be calculated through direct matrix multiplication, the FFT reduces the computation time significantly, making it feasible to process large datasets.
What are some applications of the Fast Fourier Transform (FFT)?
-The FFT is widely used in areas such as image compression, audio compression, signal processing, and high-performance scientific computing. It's considered one of the most powerful and important algorithms of the 20th century.
What is the basic idea behind using Fourier series to approximate functions?
-Fourier series approximates a function by breaking it down into a sum of sine and cosine terms. This is particularly useful for periodic functions, where the series captures the frequency components that make up the signal.
How is the DFT applied to experimental or simulation data?
-The DFT is applied to discrete data points obtained from experiments or simulations, assuming an underlying continuous function. The DFT converts these discrete data points into frequency components that describe the signal.
What is the mathematical representation of the DFT?
-The DFT of a dataset is represented as a sum over data points, where each data point is multiplied by a complex exponential term. This sum results in a set of Fourier coefficients, representing different frequency components in the data.
What is the inverse DFT, and how does it work?
-The inverse DFT allows one to reconstruct the original data from its Fourier coefficients. It involves summing over the frequency components, using a similar formula to the DFT but with a positive exponential term and a normalization factor of 1/N.
What is the significance of the complex exponential terms in the DFT?
-The complex exponential terms in the DFT represent the sine and cosine components of different frequencies. They allow the transformation of time-domain data into frequency-domain data by capturing both the magnitude and phase of each frequency component.
How is the DFT represented as a matrix operation?
-The DFT can be written as a matrix operation, where the data vector is multiplied by a matrix of complex exponential terms (DFT matrix) to obtain the Fourier coefficients. This matrix approach simplifies the computation, especially for large datasets.
Why is the FFT preferred over directly computing the DFT?
-The FFT is preferred because it provides a highly efficient way to compute the DFT without constructing and multiplying large matrices. The FFT significantly reduces the computation time, making it practical for real-time applications and large datasets.
Outlines
📊 Introduction to Discrete Fourier Transform (DFT)
The speaker begins by discussing the Fourier series and Fourier transforms, emphasizing the importance of the Discrete Fourier Transform (DFT) in computing. They humorously suggest that DFT should be named the Discrete Fourier Series because it approximates the Fourier series on a finite interval. The DFT is crucial for applications like image and audio compression, signal processing, and scientific computing. The Fast Fourier Transform (FFT), an efficient algorithm for computing the DFT, is highlighted as one of the most significant algorithms of all time. The lecture aims to explain the DFT in terms of matrix multiplication and its application in transforming data vectors into their sine and cosine components, which can be useful for various applications, including diagnosing car issues through audio signal analysis.
🔍 Computing Fourier Coefficients
The speaker explains the process of computing Fourier coefficients from data points. They describe how to transform a data vector into its frequency components using the DFT. The Fourier coefficients are obtained by summing over all data points, multiplied by a complex exponential term. The inverse DFT is also discussed, which allows reconstruction of the original data from its Fourier coefficients. The concept of a fundamental frequency, denoted as Omega_n, is introduced as a key element in the computation of the DFT matrix. The speaker clarifies that the Fourier transform and its inverse involve complex exponentials with positive and negative exponents, respectively, and both are normalized differently.
🧮 Matrix Representation of DFT
The speaker illustrates how the DFT can be represented as a matrix operation, where the data vector is multiplied by the DFT matrix to obtain the Fourier transform vector. They discuss the tediousness of computing the DFT using a direct summation approach and instead propose a matrix representation that simplifies the computation. The matrix is constructed using the fundamental frequency Omega_n, and the speaker provides an example of how the matrix looks for different values of K. The first row of the matrix is all ones, representing the constant value or zeroth frequency, while subsequent rows represent higher frequencies. The speaker emphasizes that the DFT matrix is complex and that each entry in the Fourier coefficients vector is a complex number, indicating both the magnitude and phase of the corresponding frequency component.
🚀 Efficiency through Fast Fourier Transform (FFT)
The speaker concludes by acknowledging the computational expense of directly computing the DFT using the matrix approach. They hint at the upcoming discussion on the Fast Fourier Transform (FFT), an algorithm that efficiently computes the DFT without constructing the entire matrix. The FFT is a crucial tool for handling large datasets and is fundamental to many modern computational applications. The speaker assures that the mathematical definition of the DFT will be practically applied through the FFT in subsequent lectures.
Mindmap
Keywords
💡Fourier Transform
💡Discrete Fourier Transform (DFT)
💡Fast Fourier Transform (FFT)
💡Fourier Series
💡Data Vector
💡Frequency Components
💡Fourier Coefficients
💡Matrix Multiplication
💡Exponential Function
💡Phase and Magnitude
Highlights
Introduction to the Discrete Fourier Transform (DFT) and its significance in computing.
The DFT is often misnamed and should be called the Discrete Fourier Series.
Explanation of the DFT as an approximation of the Fourier series on a finite interval.
The importance of the Fast Fourier Transform (FFT) as a powerful algorithm.
Applications of the FFT in various fields like image and audio compression, signal processing, and scientific computing.
The mathematical transformation of the DFT can be represented as a matrix multiplication.
The FFT provides a computationally efficient way to compute the DFT for large datasets.
The process of approximating periodic functions using sums of sines and cosines.
The common scenario of having discrete data points rather than an analytic function.
The concept of transforming a data vector into its sine and cosine components through DFT.
Practical use case: Diagnosing car issues through audio signal analysis using DFT.
The DFT can also be used to compute or approximate derivatives of data.
Explanation of the formula for computing the DFT and obtaining Fourier coefficients.
The inverse DFT formula for reconstructing data from Fourier coefficients.
The significance of the fundamental frequency in the context of DFT.
The DFT matrix and how it can be used to transform data into Fourier coefficients.
The complex nature of Fourier coefficients and their interpretation in terms of magnitude and phase.
The computational expense of directly computing the DFT matrix and the need for the FFT.
Upcoming lectures will cover the efficient computation of the DFT using the FFT.
Transcripts
[Music]
welcome back so we're talking about the
Fourier series Fourier transforms and
now I'm going to tell you about how you
compute these in a computer using the
discrete Fourier transform okay so this
is really really important the discrete
fourier transform and I think this is
actually I always joke with my students
that this is kind of a terrible name
really this should be called a discrete
Fourier series because what we're
actually doing is we're approximating
that Fourier series approximation on a
finite interval where your function is
periodic that's what we're actually
approximating is the finite series and
not the infinite Fourier transform
integral but through some joke of
history it is now called the discrete
Fourier transform or the DFT and this is
an extremely important concept which
will eventually lead to the fast Fourier
transform the FFT which is probably the
most powerful algorithm of the last
century maybe of all time it's one of
the most important algorithms today it's
used for image compression audio
compression signal processing
high-performance scientific computing
you name it fast Fourier transform is
probably at the heart of some of these
computations and what I want to get
across right now is that the discrete
Fourier transform is a mathematical
transformation that can be written in
terms of a big matrix multiplication the
fast Fourier transform is a
computationally efficient way of
computing the DFT
that scales to very very large datasets
so in some sense these are kind of
synonymous the FFT is how we compute the
DFT okay but today I'm going to tell you
about the DFT the discrete Fourier
transform and soon I'll show you how to
actually can numerically implement this
via the FFT okay so we talked about
approximating functions that are
periodic using infinite sums of sines
and cosines let me show that this is
this is possible but in many many cases
most of the time I don't
have an analytic function I have
measurement data from an experiment or
from a simulation okay so what I what I
often have is some F defined at discrete
locations x1 x2 x3 all the way up to
some X n so I have a discrete vector of
data some function and maybe we believe
that there's a continuous function
underlying this data but I'm only
sampling it at these discrete locations
x1 and x2 x3 and so on and so forth up
to X n okay so what I have is a vector
of data in fact I'm just gonna literally
write this out I have a vector of data
f1 f2 f3 dot dot dot all the way down to
F and okay this is gonna be my I'm just
gonna call this you know some some
vector of data and what I'm gonna want
to do is Fourier transform compute the
discrete Fourier series of that data
vector okay so I also want to break this
data up into a sum of sines and cosines
and this can be very useful you can
figure out if this is audio data you can
figure out what are kind of the the
tones that add up to make that audio
data I always tell my students you know
what I want to do is build an iPhone app
so you put it on the roof of your car or
the hood of your car and it listens to
the audio signal it computes the Fourier
transform and based on the frequency
content maybe it can diagnose if
something's wrong with your car like you
have a belt slipping or some kind of
vibration that shouldn't be there okay
so so turning a data vector into its
sine and cosine components through this
discrete Fourier transform can be very
very useful it's also nice to compute
derivatives to approximate derivatives
of of data using using these Fourier
transform derivative properties okay so
I'm gonna walk you through what the
Fourier transform the discrete Fourier
transform is how you can write it as a
big matrix multiplication but remember
it's basically just a Fourier series on
data
instead of on contingent on functions
analytic functions okay so let's just
get started so I'm just gonna write down
what the the 48 discrete Fourier
transform is and then I'm going to show
you some ways of interpreting this okay
so for each of these data points FK what
we're going to be able to do is compute
a as you might imagine what we're going
to try to get out of this is some vector
of Fourier coefficients f1 hat f2 hat f3
hats all the way down to F and hat so
just like we took our function f and we
turned it into these coefficients that
multiplied our cosine and sines that's
what we're doing here we're taking our
data F and we're going to obtain this
Fourier transform vector F hat of
frequency components so f1 hat will tell
me how much of that first low frequency
is in the data f2 hat will tell me how
much of that next higher frequency is in
the data and so on and so forth where FN
hat is the highest frequency possible
with n data points if it was going you
know as fast as possible oscillating
over n data points and so the way that
we get that F hat vector we say that F
hat K maybe I'll write it up here F hat
K the caithe Fourier coefficient is a
sum over all of my all of my data to n
minus 1 of my data F J the J element of
my data times e to the minus I 2 pi J K
over N this is kind of a pain I'll walk
you through all of these terms here ok
so the case Fourier coefficient is
obtained by taking the sum over all J
data points at the jet frequency times
the caithe frequency divided by N and
I'll tell you what this kind of e to the
I 2 pi JK over n means in a minute ok
but this is how you compute all of those
case Fourier coefficients and similarly
just like an ax for you
transform or a Fourier series if I have
my data I can get my Fourier transform
but if I have my Fourier transform I
should be able to go back and
reconstruct my data it's all write down
what that is and it's again it's pretty
similar so FK once I had all of these
blue hats the FK is again a sum over all
frequencies now of J equals 0 to n minus
1 of my F J hat ok my J Fourier
coefficient times now e to the positive
I 2 pi JK over N so this is very very
similar to the Fourier transform pair
where this is a negative exponent and
this is a positive exponent and this
whole thing is times 1 over N remember
just like in the Fourier transform this
had a 1 over I think pi or 2 pi this
also has to a slightly different
normalization ok and so what this means
is that if I have if I have my data kind
of F 1 F 2 dot dot
to F n I can run it through this Fourier
transform this DFT this discrete Fourier
transform here this is my DFT and I can
get my Fourier frequencies f1 hat f2 hat
dot dot F and hat ok and these literally
are the f1 f2 F and hats are literally
telling me how much of each each
frequency I need to add up to
reconstruct this data in in F ok and
there will be some subtleties I'll tell
you about but to compute this thing this
is kind of interesting is that notice
that all of these these these terms in
this series are multiplying these
Exponential's which are some integer
multiple of e to the I to PI over N ok
so we have this kind of e to the minus 2
pi I over N I is the the complex the
imaginary
i square root of one I'll just write I
equals square root of negative one okay
this is I and this defines a fundamental
frequency that's defined some Omega n
okay which is some fundamental frequency
that is related to two what kinds of
sines and cosines I can approximate with
n discreet values in a domain X okay so
this is kind of the fundamental
frequency we get to work with if I have
an interval with n data points and all
of these Fourier transforms are adding
up integer multiples of that fundamental
frequency times my data and the same
thing with my inverse Fourier transform
okay so we're gonna be able to use this
fundamental frequency Omega n to compute
a matrix that would multiply my data and
give me my Fourier transform so that's
really really important and I'm gonna
write this out here okay okay good so so
we have this this sum over the data to
compute the Fourier transform and the
inverse Fourier transform but I don't
want to go you know for K 1 for K 2 for
K 3 4 K 4 and compute this whole sum
this seems very tedious this would be
kind of too big for loops if I wanted to
actually code that up and I don't want
to do that so instead what we're going
to do is we're going to try to write
what this sum would be for each of these
k's in terms of a matrix operation where
i take my data vector and i multiply it
by a matrix to get my Fourier transform
vector and I'm gonna write it in terms
of these fundamental frequencies Omega
ends okay and I'll point out you can
absolutely go back to your definition of
the Fourier series the complex Fourier
series where you had it written in terms
of the complex exponential and you can
show that this is exactly what you would
get if you took your your Fourier series
and approximated it with N and discrete
values in the middle this is exactly
analogous to the cosines and sines of
those discrete frequencies in the
Fourier series okay good so we have this
fundamental frequency and we're going to
walk through and sometimes what I like
to do is just think if K
was one what would this series look like
okay so if K was one I would take all of
my f J's and multiply them by some the
first row vector and that would tell me
the coefficient on each of those f JS
okay now if K was yeah I probably should
have k zero yeah sorry I'm being pretty
sloppy here let's say that I actually
have a 0th index here let's say I have
an x0
and that this is f0 f1 f2 all the way up
to FN and this is f0 f1 f2 all the way
up to FN this will make a lot more sense
if J my data has to be indexed from 0 to
n minus 1 okay so the 0th this is the
0th frequency kind of the constant value
if I had K equals 0 here then all of my
exponents would be e to the I times 0
because K is 0 and e to the 0 is 1 no
matter what J is if K is 0 this whole
exponent is 1 is that is this whole
exponent is 0 so e to the 0 is 1 and so
the coefficient on every single data
point here is 1 and I'm adding up F 0
plus F 1 plus F 2 plus F 3 all the way
to F n that will give me the first the
zeroth F hat so this row is 1 maybe I'll
make it in yellow this is is 1 1 1 dot
dot dot 1 okay this entire first row is
just ones now for the second row now
let's figure out what happens for F 1
hat okay so in K equals 1 now all of
these coefficients multiplying my data F
for J equals 0 I get e to the 0 is 1 ok
for J equals 1 I get e to the minus 2 pi
divided by n remember K is 1 so that's
just Omega when J equals 2 I get
basically Omega squared I get e to the
minus 2 pi I times 2 divided by n this
is Omega N squared
dot dot dot and eventually at the very
end I'll have Omega n to the power n
minus 1 ok and I can keep going down if
K was 2 then I would have everything you
know multiplying 2 in the exponent so I
would still have for J equals 0 a 1 here
I would have an Omega N squared because
my K is 2 I'd have Omega N to the 4th
dot dot dot Omega and to the 2 n minus 1
and so on and so forth you can just go
down the list I'll always have ones on
the first row and the first column down
here I'm gonna have something like Omega
N to the N minus 1 Omega n to the 2 n
minus 1 dot dot dot and eventually this
last entry is going to be Omega n to the
N minus 1 squared ok so this is
something you can you can readily kind
of convince yourself that every row of
this is one value of K here and you can
figure out what the coefficients are of
the data that will be the row of this
matrix DFT here okay so long story short
what I'm trying to convince you of is
that if you have a data vector instead
of an analytic function so you just have
data in these these pink dots here you
can still compute something like a
fourier series like a fourier transform
but it's the discrete Fourier transform
okay and the form of the the discrete
Fourier transform pair to go from data
to Fourier coefficients or to go from
Fourier coefficients back to your data
looks very much like a Fourier series
but because this is defined on discrete
data I can write it as a big matrix
operation okay so this here is the DFT
matrix this is the DFT the discrete
Fourier transform matrix and if you had
your your vector of data you multiply it
by this matrix you get your Fourier
transform out very very useful okay and
these Fourier coefficients you'll notice
that this is a complex number okay this
is e to the this has a cosine plus an I
sine part this is a complex number so
this is a complex matrix and these
Fourier coefficients these blue Fourier
coefficients are complex valued and so
these somehow tell you how much of that
0th mode cosine and sine mode there are
but it also tells you the phase okay so
the magnitude of this complex number
tells you how much magnitude of that
first mode you have and then the cut the
the phase angle of that complex number
tells you you know how much cosine or
how much sine or what phase in between
your your sine and cosine is okay so
every single one of these values is a
complex number and the magnitude tells
you how much of that set that that
second frequency sine and cosine you
have and the phase tells you what is the
the phase between cosine and sine and it
gives you exactly the sines and cosines
you need to add up to approximate your
data perfectly okay so this is the
discrete Fourier transform this is
pretty expensive to actually compute
this matrix and multiply your data to
compute this Fourier transform and so
what I'm going to show you in the next
few lectures is how you compute this
efficiently using the fast Fourier
transform so you never actually have to
build this matrix and do this
multiplication this is how its
mathematically defined but
computationally we're always going to
compute the DFT
using the fast Fourier transform
algorithm okay that's all coming up soon
thank you
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