# 1.10 Filtering with the dft

 Page 1 / 1
The ideas of using the DFT to filter a signal and recover a signal from a noisy transmission are addressed based on the ideas of the DFT and convolution.

## Introduction

$y(n)=(x(n), h(n))=\sum$ x k h n k
$Y()=X()H()$
Assume that $H()$ is specified.

How can we implement $X()H()$ in a computer?

Discretize (sample) $X()$ and $H()$ . In order to do this, we should take the DFTs of $x(n)$ and $h(n)$ to get $X(k)$ and $X(k)$ . Then we will compute $\stackrel{~}{y}(n)=\mathrm{IDFT}(X(k)H(k))$ Does $\stackrel{~}{y}(n)=y(n)$ ?

Recall that the DFT treats $N$ -point sequences as if they are periodically extended ( ):

## Compute idft of y[k]

$\stackrel{~}{y}(n)=\frac{1}{N}\sum_{k=0}^{N-1} Y(k)e^{i\times 2\pi \frac{k}{N}n}=\frac{1}{N}\sum_{k=0}^{N-1} X(k)H(k)e^{i\times 2\pi \frac{k}{N}n}=\frac{1}{N}\sum_{k=0}^{N-1} \sum_{m=0}^{N-1} x(m)e^{-(i\times 2\pi \frac{k}{N}m)}H(k)e^{i\times 2\pi \frac{k}{N}n}=\sum_{m=0}^{N-1} x(m)\frac{1}{N}\sum_{k=0}^{N-1} H(k)e^{i\times 2\pi \frac{k}{N}(n-m)}=\sum_{m=0}^{N-1} x(m)h({\left(\left(nm\right)\right)}_{N})$
And the IDFT periodically extends $h(n)$ : $\stackrel{~}{h}(n-m)=h({\left(\left(nm\right)\right)}_{N})$ This computes as shown in :

$\stackrel{~}{y}(n)=\sum_{m=0}^{N-1} x(m)h({\left(\left(nm\right)\right)}_{N})$
is called circular convolution and is denoted by .

## Dft pair

Note that in general:

## Regular vs. circular convolution

To begin with, we are given the following two length-3 signals: $x(n)=\{1, 2, 3\}$ $h(n)=\{1, 0, 2\}$ We can zero-pad these signals so that we have the following discrete sequences: $x(n)=\{, 0, 1, 2, 3, 0, \}$ $h(n)=\{, 0, 1, 0, 2, 0, \}$ where $x(0)=1$ and $h(0)=1$ .

• Regular Convolution:
$y(n)=\sum_{m=0}^{2} x(m)h(n-m)$
Using the above convolution formula (refer to the link if you need a review of convolution ), we can calculate the resulting value for $y(0)$ to $y(4)$ . Recall that because we have two length-3 signals, our convolved signal will be length-5.
• $n=0$ $\{, 0, 0, 0, 1, 2, 3, 0, \}$ $\{, 0, 2, 0, 1, 0, 0, 0, \}$
$y(0)=1\times 1+2\times 0+3\times 0=1$
• $n=1$ $\{, 0, 0, 1, 2, 3, 0, \}$ $\{, 0, 2, 0, 1, 0, 0, \}$
$y(1)=1\times 0+2\times 1+3\times 0=2$
• $n=2$ $\{, 0, 1, 2, 3, 0, \}$ $\{, 0, 2, 0, 1, 0, \}$
$y(2)=1\times 2+2\times 0+3\times 1=5$
• $n=3$
$y(3)=4$
• $n=4$
$y(4)=6$

• Circular Convolution:
$\stackrel{~}{y}(n)=\sum_{m=0}^{2} x(m)h({\left(\left(nm\right)\right)}_{N})$
And now with circular convolution our $h(n)$ changes and becomes a periodically extended signal:
$h({\left(\left(n\right)\right)}_{N})=\{, 1, 0, 2, 1, 0, 2, 1, 0, 2, \}$
• $n=0$ $\{, 0, 0, 0, 1, 2, 3, 0, \}$ $\{, 1, 2, 0, 1, 2, 0, 1, \}$
$\stackrel{~}{y}(0)=1\times 1+2\times 2+3\times 0=5$
• $n=1$ $\{, 0, 0, 0, 1, 2, 3, 0, \}$ $\{, 0, 1, 2, 0, 1, 2, 0, \}$
$\stackrel{~}{y}(1)=1\times 1+2\times 1+3\times 2=8$
• $n=2$
$\stackrel{~}{y}(2)=5$
• $n=3$
$\stackrel{~}{y}(3)=5$
• $n=4$
$\stackrel{~}{y}(4)=8$

illustrates the relationship between circular convolution and regularconvolution using the previous two figures:

## Regular convolution from periodic convolution

• "Zero-pad" $x(n)$ and $h(n)$ to avoid the overlap (wrap-around) effect. We will zero-pad the two signals to a length-5 signal (5being the duration of the regular convolution result): $x(n)=\{1, 2, 3, 0, 0\}$ $h(n)=\{1, 0, 2, 0, 0\}$
• Now take the DFTs of the zero-padded signals:
$\stackrel{~}{y}(n)=\frac{1}{N}\sum_{k=0}^{4} X(k)H(k)e^{i\times 2\pi \frac{k}{5}n}=\sum_{m=0}^{4} x(m)h({\left(\left(nm\right)\right)}_{5})$
Now we can plot this result ( ):

We can compute the regular convolution result of a convolution of an $M$ -point signal $x(n)$ with an $N$ -point signal $h(n)$ by padding each signal with zeros to obtain two $M+N-1$ length sequences and computing the circular convolution (or equivalently computing the IDFT of $H(k)X(k)$ , the product of the DFTs of the zero-padded signals) ( ).

## Dsp system

• Sample finite duration continuous-time input $x(t)$ to get $x(n)$ where $n=\{0, , M-1\}$ .
• Zero-pad $x(n)$ and $h(n)$ to length $M+N-1$ .
• Compute DFTs $X(k)$ and $H(k)$
• Compute IDFTs of $X(k)H(k)$ $y(n)=\stackrel{~}{y}(n)$ where $n=\{0, , M+N-1\}$ .
• Reconstruct $y(t)$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!