<< Chapter < Page Chapter >> Page >

Given a desired frequency response, the frequency sampling design method designs a filter with a frequency response exactly equal to the desired response at a particular set of frequencies k .


k k
    o 1 N 1
H d k n M 1 0 h n k n
Desired Response must incluce linear phase shift (if linear phase is desired)

What is H d for an ideal lowpass filter, cotoff at c ?

M 1 2 c c 0 c c

Got questions? Get instant answers now!
This set of linear equations can be written in matrix form
H d k n M 1 0 h n k n
H d 0 H d 1 H d N - 1 0 0 0 1 0 M 1 1 0 1 1 1 M 1 M - 1 0 M - 1 1 M - 1 M 1 h 0 h 1 h M 1

or H d W h So

h W H d
W is a square matrix for N M , and invertible as long as i j 2 l , i j

Important special case

What if the frequencies are equally spaced between 0 and 2 , i.e. k 2 k M

Then H d k n M 1 0 h n 2 k n M n n M 1 0 h n n 2 k n M DFT! so h n n 1 M k M 1 0 H d k 2 n k M or h n n M k M 1 0 H d k 2 n k M n IDFT H d k

Important special case #2

h n symmetric, linear phase, and has real coefficients. Since h n h M n 1 , there are only M 2 degrees of freedom, and only M 2 linear equations are required.

H k n M 1 0 h n k n n M 2 1 0 h n k n k M n 1 M even n M 3 2 0 h n k n k M n 1 h M 1 2 k M 1 2 M odd k M 1 2 2 n M 2 1 0 h n k M 1 2 n M even k M 1 2 2 n M 3 2 0 h n k M 1 2 n h M 1 2 M odd

Removing linear phase from both sides yields A k 2 n M 2 1 0 h n k M 1 2 n M even 2 n M 3 2 0 h n k M 1 2 n h M 1 2 M odd Due to symmetry of response for real coefficients, only M 2 k on 0 need be specified, with the frequencies k thereby being implicitly defined also. Thus we have M 2 real-valued simultaneous linear equations to solve for h n .

Special case 2a

h n symmetric, odd length, linear phase, real coefficients, and k equally spaced: k 0 k M 1 k n k M

h n IDFT H d k 1 M k M 1 0 A k 2 k M M 1 2 2 n k M 1 M k M 1 0 A k 2 k M n M 1 2

To yield real coefficients, A mus be symmetric A A A k A M k

h n 1 M A 0 k M 1 2 1 A k 2 k M n M 1 2 2 k n M 1 2 1 M A 0 2 k M 1 2 1 A k 2 k M n M 1 2 1 M A 0 2 k M 1 2 1 A k 1 k 2 k M n 1 2

Simlar equations exist for even lengths, anti-symmetric, and 1 2 filter forms.

Comments on frequency-sampled design

This method is simple conceptually and very efficient for equally spaced samples, since h n can be computed using the IDFT.

H for a frequency sampled design goes exactly through the sample points, but it may be very far off from the desired response for k . This is the main problem with frequency sampled design.

Possible solution to this problem: specify more frequency samples than degrees of freedom, and minimize the total errorin the frequency response at all of these samples.

Extended frequency sample design

For the samples H k where 0 k M 1 and N M , find h n , where 0 n M 1 minimizing H d k H k

For l norm, this becomes a linear programming problem (standard packages availble!)

Here we will consider the 2 l norm.

To minimize the 2 l norm; that is, n N 1 0 H d k H k , we have an overdetermined set of linear equations: 0 0 0 M 1 N - 1 0 N - 1 M 1 h H d 0 H d 1 H d N - 1 or W h H d

The minimum error norm solution is well known to be h W W W H d ; W W W is well known as the pseudo-inverse matrix.

Extended frequency sampled design discourages radical behavior of the frequency response between samples forsufficiently closely spaced samples. However, the actual frequency response may no longer pass exactly through any of the H d k .

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Digital filter design. OpenStax CNX. Jun 09, 2005 Download for free at http://cnx.org/content/col10285/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital filter design' conversation and receive update notifications?