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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

with

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

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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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