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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket 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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