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Notice from [link] for N even, A(ω) is an even function around ω=0 , an odd function around ω=π , and is periodic with period . This requires A(π)=0 .

For the case in [link] where K 1 = π / 2 , an odd symmetry is required of the form

h ( n ) = - h ( N - n - 1 )

which, for N odd, gives

H ( ω ) = j A ( ω ) e j M ω


A ( ω ) = n = 0 M - 1 2 h ( n ) sin ω ( M - n )

and for N even

A ( ω ) = n = 0 N / 2 - 1 2 h ( n ) sin ω ( M - n )

To calculate the frequency or amplitude response numerically, one must consider samples of the continuous frequency response function above. L samples of the general complex frequency response H ( ω ) in [link] are calculated from

H ( ω k ) = n = 0 N - 1 h ( n ) e - j ω k n .

for k = 0 , 1 , 2 , , L - 1 . This can be written with matrix notation as

H = F h

where H is an L by 1 vector of the samples of the complex frequency response, F is the L by N matrix of complex exponentials from [link] , and h is the N by 1 vector of real filter coefficients.

These equations are possibly redundant for equally spaced samples since A ( ω ) is an even function and, if the phase response is linear, h ( n ) is symmetric. These redundancies are removed by sampling [link] over 0 ω k π and by using a defined in [link] rather than h . This can be written

A = C a

where A is an L by 1 vector of the samples of the real valued amplitude frequency response, C is the L by M real matrix of cosines from [link] , and a is the M by 1 vector of filter coefficients related to the impulse response by [link] . A similar set of equations can be written from [link] for N odd or from [link] for N even.

This formulation becomes a filter design method by giving the samples of a desired amplitude response as A d ( k ) and solving [link] for the filter coefficients a ( n ) . If the number of independent frequency samples is equal to the number of independent filter coefficients and if C is not singular, this is the frequency sampling filter design method and the frequency response of the designed filter will interpolate thespecified samples. If the number of frequency samples L is larger than the number of filter coefficients N , [link] may be solved approximately by minimizing the norm A ( ω ) - A d ( ω ) .

The discrete time fourier transform with normalization

The discrete time Fourier transform of the impulse response of a digital filter is its frequency response, therefore, it is an important tool.When the symmetry conditions of linear phase are incorporated into the DTFT, it becomes similar to the discrete cosine or sine transform(DCT or DST). It also has an arbitrary normalization possible for the odd length that needs to be understood.

The discrete time Fourier transform (DTFT) is defined in [link] which, with the conditions of an odd length-N symmetrical signal, becomes

A ( ω ) = n = 1 M a ( n ) cos ( ω n ) + K a ( 0 )

where M = ( N - 1 ) / 2 . Its inverse as

a ( n ) = 2 π 0 π A ( ω ) cos ( ω n ) d ω

for n = 1 , 2 , , M and

a ( 0 ) = 1 K π 0 π A ( ω ) d ω

where K is a parameter of normalization for the a ( 0 ) term with 0 < K < . If K = 1 , the expansion equation [link] is one summation and doesn't have to have the separate term for a ( 0 ) . If K = 1 / 2 , the equation for the coefficients [link] will also calculate the a ( 0 ) term and the separate equation [link] is not needed. If K = 1 / 2 , a symmetry results which simplifies equations later in the notes.

Four types of linear-phase fir filters

From the previous discussion, it is seen that there are four possible types of FIR filters [link] that lead to the linear phase of [link] . These are summarized in [link] .

Questions & Answers

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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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