# 1.8 Dft as a matrix operation

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This module introduces linear algebra, DFT, FFT, matrix and vector.

## Matrix review

Recall:

• Vectors in $\mathbb{R}^{N}$ : $\forall {x}_{i}, {x}_{i}\in \mathbb{R}\colon x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N-1}\\ \end{pmatrix}$
• Vectors in $\mathbb{C}^{N}$ : $\forall {x}_{i}, {x}_{i}\in \mathbb{C}\colon x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N-1}\\ \end{pmatrix}$
• Transposition:
• transpose: $x^T=\begin{pmatrix}{x}_{0} & {x}_{1} & & {x}_{N-1}\\ \end{pmatrix}$
• conjugate: $(x)=\begin{pmatrix}\overline{{x}_{0}} & \overline{{x}_{1}} & & \overline{{x}_{N-1}}\\ \end{pmatrix}$
• Inner product :
• real: $x^Ty=\sum_{i=0}^{N-1} {x}_{i}{y}_{i}$
• complex: $(x)y=\sum_{i=0}^{N-1} \overline{{x}_{n}}{y}_{n}$
• Matrix Multiplication: $Ax=\begin{pmatrix}{a}_{00} & {a}_{01} & & {a}_{0,N-1}\\ {a}_{10} & {a}_{11} & & {a}_{1,N-1}\\ & & & \\ {a}_{N-1,0} & {a}_{N-1,1} & & {a}_{N-1,N-1}\\ \end{pmatrix}\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N-1}\\ \end{pmatrix}=\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ \\ {y}_{N-1}\\ \end{pmatrix}$ ${y}_{k}=\sum_{n=0}^{N-1} {a}_{kn}{x}_{n}$
• Matrix Transposition: $A^T=\begin{pmatrix}{a}_{00} & {a}_{10} & & {a}_{N-1,0}\\ {a}_{01} & {a}_{11} & & {a}_{N-1,1}\\ & & & \\ {a}_{0,N-1} & {a}_{1,N-1} & & {a}_{N-1,N-1}\\ \end{pmatrix}$ Matrix transposition involved simply swapping the rows with columns. $(A)=\overline{A^T}$ The above equation is Hermitian transpose. $A^T_{k, n}=A_{n, k}$ $(A)_{k, n}=\overline{A}_{n, k}$

## Representing dft as matrix operation

Now let's represent the DFT in vector-matrix notation. $x=\begin{pmatrix}x(0)\\ x(1)\\ \\ x(N-1)\\ \end{pmatrix}$ $X=\begin{pmatrix}X(0)\\ X(1)\\ \\ X(N-1)\\ \end{pmatrix}\in \mathbb{C}^{N}$ Here $x$ is the vector of time samples and $X$ is the vector of DFT coefficients. How are $x$ and $X$ related: $X(k)=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi }{N}kn)}$ where ${a}_{kn}=e^{-(i\frac{2\pi }{N})}^{(kn)}={W}_{N}^{(kn)}$ so $X=Wx$ where $X$ is the DFT vector, $W$ is the matrix and $x$ the time domain vector. $W_{k, n}=e^{-(i\frac{2\pi }{N})}^{(kn)}$ $X=W\left(\begin{array}{c}x(0)\\ x(1)\\ \\ x(N-1)\end{array}\right)$ IDFT: $x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi }{N}}^{(nk)}$ where $e^{i\frac{2\pi }{N}}^{(nk)}=\overline{{W}_{N}^{(nk)}}$ $\overline{{W}_{N}^{(nk)}}$ is the matrix Hermitian transpose. So, $x=\frac{1}{N}(W)X$ where $x$ is the time vector, $\frac{1}{N}(W)$ is the inverse DFT matrix, and $X$ is the DFT vector.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
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