# 2.2 Least squared error design of fir filters  (Page 11/13)

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If there are $F$ distinct band edges ${\omega }_{k}$ , the first and last are ${\omega }_{1}=0$ and ${\omega }_{F}=\pi$ . This means part of the first term in the sum of [link] is always zero and part of the last is zero except when $n=m=0$ where it is $\pi$ . Using these facts allows [link] to be written

${c}_{w}\left(m,n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{\pi }\sum _{k=1}^{F-2}\left({W}_{k}-{W}_{k+1}\right)\left[\frac{sin\left(n-m\right){\omega }_{k+1}}{n-m},+,\frac{sin\left(n+m\right){\omega }_{k+1}}{n+m}\right]$

which, together with appropriately modified [link] and [link] , are good forms for programming. The Matlab program in the appendix containsthe details.

${a}_{w}\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{\pi }\sum _{k}\left[{W}_{k},{\int }_{{\omega }_{k}}^{{\omega }_{k+1}},{A}_{{d}_{k}},\left(\omega \right),\phantom{\rule{0.166667em}{0ex}},cos,\left(\omega n\right),\phantom{\rule{0.166667em}{0ex}},d,\omega \right].$

These integrals have been evaluated for the four basic filter types - constant gain passband lowpass filter, linear gain passband lowpass filter,differentiator plus lowpass filter, and Hilbert transformer plus lowpass filter - giving simple design formulas in [link] , [link] , [link] , and [link] .

Each basic filter type plus the effects of a transition band can be calculated and combined according to [link] . An example low pass filter with a weight of ${W}_{1}$ in the passband and ${W}_{2}$ in the transition band is given for odd $N$ gives for the intermediate coefficients ${\stackrel{^}{h}}_{w}\left(n\right)$ from [link] are

${a}_{w}\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{W}_{1}\left[\frac{sin\left({\omega }_{2}n\right)-sin\left({\omega }_{1}n\right)}{\pi n}\right]+{W}_{2}\left[\frac{sin\left({\omega }_{0}n\right)}{\pi n},{\left(\frac{sin\left(\Delta n/p\right)}{\Delta n/p}\right)}^{p},-,\frac{sin\left({\omega }_{1}n\right)}{\pi n}\right]$

A similar expression can be derived for even $N$ using Equation 8 from Constrained Approximation and Mixed Criteria .

This means the left hand vector in [link] can be calculated as a weighted sum of inverse DTFTs such as in [link] if the ideal desired amplitude can be constructed from the four basic types in FIR Digital Filters , each with optimal transition bands.

If one or more of the integrals in [link] has no analytical solution, ${a}_{w}\left(n\right)$ can be calculated numerically using a truncated weighted sum of inverse DFTs of a dense sampling of ${A}_{{d}_{k}}\left(\omega \right)$ or made up of the passbands calculated from inverse DFTs and the transition bands addedby multiplying appropriately by sinc functions since constructing an optimal spline transition function to be sampled would not be easy.

This gives a very powerful design method that allows multi band weighted least squares design of FIR filters. The calculation of the matrix ${\mathbf{C}}_{\mathbf{w}}$ in [link] is always possible using [link] . Because using atrue “don't care" transition band with a weight of zero might causes ill conditioning of [link] for $\left({f}_{s}-{f}_{p}\right)N>12$ as discussed in [link] , one can add a spline transition function in ${A}_{d}\left(\omega \right)$ to the definition in [link] as done in [link] and [link] . A very small weight used in the transition bands together with a splinetransition function will improve the conditioning of [link] with minor degradation of the optimality. This point needs further evaluation.

By using an inverse FFT perhaps plus a sinc induced transition function to calculate the components of [link] , this method can be used to design arbitrary shaped passbands. It can also be used for complex approximationby applying it to the real and imaginary parts of the desired ${H}_{d}\left(\omega \right)$ separately and using the full, nonsymmetric $h\left(n\right)$ .

The form of the simultaneous equations [link] that must be solved to design a filter by this method is interesting. If the weights in allpass, stop, and transition bands are unity, the ${\mathbf{C}}_{\mathbf{w}}$ matrix is the identity matrix and ${\stackrel{^}{\mathbf{h}}}_{\mathbf{w}}$ contains the filter coefficients. As the weights become less and less uniform or equal, the ${\mathbf{C}}_{\mathbf{w}}$ matrix becomes poorer conditioned. If the weights for the transition bands are zero, it is the smallest eigenvalues of ${\mathbf{C}}_{\mathbf{w}}$ that control the actual amplitude response $A\left(\omega \right)$ in the transition bands. This explains why numerical errors in solving [link] show up primarily in the transition bands. It also suggests this effect can be reduced by allowinga small weight in the transition bands. Indeed, one can design long filters by using spline transition functions with a small weight whichthen allows different pass and stopband weights.

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it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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