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Section conclusions

This section has derived the four basic ideal lowpass filters: the constant gain passband lowpass filter, the linearly increasing gain passbandlowpass filter, the differentiator with a lowpass filter, and the Hilbert transformer with a lowpass filter. It is shown that each of these can bemodified to allow a spline transition function by a simple weighting function.

Because of using an L 2 approximation error criterion and because of the orthogonality of the basis functions of the Fourier transform, it is shownthat an optimal multiband filter can be built from the linear combination of these optimal building blocks. This new filter design method has theflexibility of the Parks-McClellan algorithm but the simplicity of the windowing methods. It is extremely fast and has no numerical problems.Unlike the windowing methods, the new method allows explicit independent control of multiple transition band edges and gives an optimal design.Its only limitation is not allowing error weighting.

We then derived a second method that likewise allowed multiple pass, stop, and transition bands with arbitrary band edges, but also allowedindependent weighting of each frequency band. There are two limitations on this method. For long filters with wide transition bands with zeroweights and where N ( f p - f s ) > 12 , the equations that must be solved are ill conditioned. This can be partially addressed using optimal splinefunctions with small weights in the transition bands. The second problem is that solving a large number of simultaneous equations can be slow andrequire considerable memory. These problems might be addressed by using special Toeplitz or Toeplitz plus Hankel algorithms [link] or some iterative method.

When should these methods be used? The second method which minimizes the weighted integral squared error should be used anytime the originalproblem dictates a squared error criterion and the product of the length and transition band width is less than twelve, N ( f p - f s ) < 12 . These conditions are often met because the squared error is a measure of thesignal or noise energy and one seldom wants a long filter and a wide transition band. Even though this method requires solution of a set ofsimultaneous equations and is, therefore, slower than the spline transition function method, it executes in a few seconds on a PC orworkstation and allows independent weighting of different frequency bands.

The first method which uses spline functions in the ideal response transition bands will design essentially arbitrarily long filters veryquickly but it will not allow any error weighting. Although artificial transition functions are used in the ideal response, the optimized splinefunctions are very close to the response actually obtained by the second method with zero weighting in the transition band. This means the optimalapproximation to the ideal response with spline functions transition bands is close to that obtained using the second numerical method. Comparisonsof these effects for a single band can be found in [link] . If a Chebyshev approximation is desired, the Parks-McClellan method should beused although it too has numerical problems for long filters with wide transition bands. If different error measures are wanted in differentbands, the iterative reweighted least squares (IRLS) algorithm [link] should be used. Recent research suggest that for many practical signal specifications, a mixture of Chebyshev and least squaresis appropriate with no explicit transition bands [link] .

If the equations that must be solved to obtain the optimal filter coefficients are ill-conditioned, an orthogonalization procedure can beused to improve the conditioning [link] .

Characteristics of optimalFilters

Gibbs phenomenon, transition band, pole-zero plots, etc.

ComplexAnd minimum phase approximation

Here we talk about which methods also solve the complex approximation problem. We talk about the minimum phase filter.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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