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Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 5:48 AM

As I understand it, the ideal frequency response for a filter is that we want maximum gain in pass band, a vertical or instant cutoff and 0 in the stop band, however this is ideal and can never be achieved in real life.

Along came Butterworth and found a relationship / formula for choosing the right combination of inductors and capacitors to be able to create filters that matched a certain spec and resembled the frequency response that we want - but not ideal.
With the exact same filter structure along came Chebyshev and he worked out an even better formula to get an even tighter / more tuned relationship to give us a filter that was closer to idea than Butterworths relationship, for example a much more steeper transition band than the same design using Butterworths.
Then again, along came Cauer with his own method of calculating the relationships and values needed to get an even better and closer approximation to an idea filter. With the steepest transition yet.
With all these, they haven't discovered a new "structure" or found that by making a different connection you get a better filter, but instead each one devised a better and more accurate formula for choosing the values of your components to get your filter close and closer to ideal.
So my question is this.
Lets say we are dealing with a 5 order low pass filter. (The type doesn't really matter). With Butterworth we have a good filter, then still using a 5th order and same structure Chebyshev was able to "tune" it even more to create and even better response, and then the same with Cauer.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
And then when someone is able to devise or work out a relationship linking all these values together and with the spec thats required it would then be a classified its own filter type such as Butterworth etc?
My second question is then,
Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
Similar to saying, if NP = P was proven it would mean we know for sure that there are solution to the problems but we haven't worked out yet how to get there / havnt worked out an algorithm for it. (Sorry if thats a bad example or its wrong but its the best analogy I can think of for now)
See the datasheet of the filters at http://www.kynix.com/Parts/2686543/5487BP15C675.html.

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#1

Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 7:14 AM

I think a better way to phase the ideal frequency would be that we want 0 dB gain in the pass band and infinite attenuation in the stop band. Remember 0 dB gain means output is equal to the input.

Or to put it another way, we want the pass band output to match the input to the filter and we want no output in the stop band.

And let's not forget about the phase. Minimal (0?) phase lag in the pass band.

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#2

Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 7:50 AM

most curios question?

I was fully unaware that current filter design left much to be desired.

nevertheless the knowledge you express for a topic that most would find rather dry is noteworthy.

For just about everything... graphene seems like a possible future..

if you can build a circuit at the atomic level there's not a lot of wiggle room for failure.

good luck in finding your answer.

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#3

Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 9:13 AM

Filter design is a trade off. You will always have some ripple in the pass band, some finite rate of fall off in the transition band, and some finite rejection in the stop band. Depending on your application, you may prefer to optimize one of these qualities at the detriment of the others, and different filter designs do this. So "best" depends on your application.

Keep in mind that the closer to ideal you achieve in the frequency domain, the farther from ideal the results will be in the time domain. For example, a filter that has a sharp drop off will have a poor response to a step or an inpulse function.

http://www.analog.com/media/en/training-seminars/design-handbooks/Basic-Linear-Design/Chapter8.pdf?doc=ADA4661-2.pdf

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Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 11:33 AM

You have a lot of misunderstandings on filters and linear filter design. Then again many people misunderstand linear filter design.

The physicist/engineer Stephan Butterworth found a mathematical approach to accurate, repeatable linear filter design. One of the key things he identified in his work is there are many different quantify able and measurable attributes to any linear filter. In this work he proved that one and only one family of filter designs will have a maximally flat (no ripple) pass band response. This family of filters was then named the Butterworth filter. His mathematical approach then allowed the theoretical mathematical work of Gauss, Chebyshev, Bessel or any other desired mathematically described linear transfer. So if one can accept a certain pass band ripple but needs a large amount of attenuation at an interfering frequency then a Chebyshev filter with a "zero" at the interfering frequency can be designed with far less poles than other filters. (A Butterworth filter has no "zero" in the stop band. Some out of band levels always get through.)

Then there's the problem of trying to realize an ideal mathematical equation with less than ideal but real components that one can obtain. The advent of high open loop gain operational amplifiers introduced multiple active elements that prevented both circuit loading complications and minimizing parasitic complications. The single op-amp Sallen-Key topology is but one example.

My point is that there is no single "best" filter configuration. Every mathematical and real circuit topology has limitations.

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#5

Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 3:35 PM

You might want to get a hold of Don Lancaster's book Active Filter Cookbook.

It's an excellent reference for Sallen-Key filters.

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Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 9:19 PM

Sallen-Key is a good topology when only one op-amp is available to make a bi-quad filter. IMHO the multiple op-amp topologies of a gyrator, Tow-Thomas, Ackerberg-Mossberg and a few others allow for better independent control.

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Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 3:48 PM
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Re: Can Anyone Answer My Hypothetical Question About Filters?

10/20/2016 5:55 PM

Hi zxcvb,

Let me start by establishing the theory of electronic filters. Your questions will be answered at the end of this message.

You are right the ideal filters frequency7 response is as you say. Here you have the response of the four main IDEAL filters:

As you can see the passband has a maximum value and the stopband has a magnitude of zero. The transition from passband to stopband is instantaneous. This is an impossibility. A more realist response is the following (I call these responses real-ideal):

Here the transition is not vertical but it has a slop (roll-off). The response of these filter is not real either.

It happens that a real-world filter response may have ripples in the passband or/and the stopband. The next figure is a typical frequency response of a low-pass general filter with all the possible behaviors in each section:

Here you have the definitions for each part in th e figure:

Passband. The range of frequencies where the output have a gain.

Stopband. The range of frequencies where the output is zero or very small.

Passband ripple. This is the variations or oscillations in the bandpass, also called error band. In a typical filter, these oscillations occur around the nominal value of 1.0, or at 0 dB, if the amplitude;is expresses in decibels. the ripple value is 2a₁, where a₁is a parameter dependent of the circuit components, as we will see later.

Stopband ripple. This represents the variations in the stopband region. The ripple is determined by the values of the circuit components.

Critical frequency, f_c. This is the frequency at which the response leaves the passband ripple. Normally, for certain type of filters (Butterworth filter, for instance), at this frequency the amplitude of the response is 1/√2 of the maximum amplitude. If the nominal amplitude is A_nom we can determine the value of the amplitude at f_c by using the following equation:

f=f_c⇒ A_C=1/√2 A_nom=0.707A_nom

A_c=20 log⁡(0.707A_nom )=-3 dB+(A_nom )_dB

Stopband frequency, f_s. This is the frequency at which the maximum stopband ripple (a₂) occurs

Transition band. This represents the range (f_s-f_c ) of frequencies between the critical and cutoff frequencies.

The slope or steepness of the transition region is related to the number of poles in the transfer function of the response (next section), also known as the order of the filter.

•A pole is a root of the denominator of the transfer function. For a standard Butterworth filter every pole adds -20 dB/decade or -6 dB/octave to the slope of the response. The slope of the line is called the roll-off of the transition. Also, a pole represents one RC stage in the circuit, as shown below .

•A filter that has only one RC network is called a single-pole or order-one filter, one with two RC circuits we call it a 2-pole or second order filter, and so on.

Now, we have to consider the following:

•Not all filters have these attributes.

•A pole (zero) is a root of the numerator of the transfer function. Every pole adds -20 dB/decade or --6 dB/octave to the roll-off •A filter circuit not only affect the magnitude of the signal, but also its phase, as we will see later.

Amplitude response. As we have seen, the amplitude response is the plot of the filter amplitude versus frequency.

•The amplitude is the ratio of output voltage to input voltage. Generally the amplitude is in decibels and the frequency axis is a logarithmic scale.

Roll-off. This is the steepness of the transition band slope. The roll-off increases as the number of poles increases. •The next figure shows the amplitude response of a low-pass Butterworth filter with 2, 4, 6, and 8 poles. In this figure the amplitude is in decibels (dB)

Frequency response types: There are several characteristics of filters depending on the shape of the response. Some considerations:

•The frequency response of any filter (LPF,HPF,PBF,BRF) can be designed by properly selecting the circuit components. •The characteristics of filters is defined by the shape of the frequency response curve.

•The most important response shapes are named after a researcher who studied the particular filters. •There are filter of type Butterworth, Chebyshev (types I and type II) , Elliptic (or Cauer), and Bessel, to mention the most important.

•These filter types are named after the British researcher Stephen Butterworth, the Russian mathematician Pafnuty Chebyshev, the German scientist Wilhelm Cauer, and the German mathematician Friedrich Bessel, repetitively.

•Each one of these filter types has a particular advantage in certain applications. The figure shows the characteristics of four low-pass filters, each one of four-poles and cutoff frequency of 10.

Notice the following:

1)Butterworth filter: flat passband and stopband. See figure:

2) Chebyshev Type I: Passband has ripples, no ripples in the stopband transition steeper than Buttterworth. See figure of comparison with Butterworth:

3) Chebyshev Type II: flat passband, ripples in the stopband, transition steeper than Buttterworth. See figure:

4) Elliptic or Cauer: ripples in the passband and stopband. transition is the steepest of all. See the following figure. Shows how many poles you need in a Butterworth and Chebyshev filters to chieve the same roll-off for a 5-pole Elliptic filter. •Number of poles to achieve a 5-pole elliptic filter:

–Butterworth: 36 poles

–Chebyshev Type I and Type II: 12 poles

Now your questions:

Question 1) Lets say we are dealing with a 5 order low pass filter. (The type doesn't really matter). With Butterworth we have a good filter, then still using a 5th order and same structure Chebyshev was able to "tune" it even more to create and even better response, and then the same with Cauer.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
And then when someone is able to devise or work out a relationship linking all these values together and with the spec thats required it would then be a classified its own filter type such as Butterworth etc?

Answer: If you have, say, a 5-pole Butterworth filter and optimize the parameters (resistors, capacitors, inductors, etc.) by using an optimization algorithm, it is possible to get the best "Butterworth" filter.You are not going to get a different type of filter and call it the"Joe" filter, because it will still be a Butterworth, even if it is the best one. The same goes if you use a 5-pole Chebyshev or elliptic filter. By optimizing the parameter you do not get a "new" type of filter. In other words, if you are optimizing a Butterworkth, the result cannot be called differently but Butterworth.

Question 2) Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
Similar to saying, if NP = P was proven it would mean we know for sure that there are solution to the problems but we haven't worked out yet how to get there / have not worked out an algorithm for it. (Sorry if that's a bad example or its wrong but its the best analogy I can think of for now)

Answer: The equations and design techniques for the type of filters (Butterworth, Cauer, Chebyshev, Multiple feedback and others) are very well known. So, in general, if one of these filters is designed using the proper equations and algorithms, then it is always possible to build it. I repeat: this is if you can use parameter values that are real and existent. If you come up with a best design for filter where the solution to the filter equation tells you that you need a capacitor, for example, of 1 mega Farads, then this is impossible to build (there is no way you can have 1 MF capacitor that fits on the surface of the Earth!). So the final answer is: if the result of the design model produces parameters values that are real and existent, then the filter will always be able to manufacture.

I hope I answered your question. Let me know.

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#9

Re: Can Anyone Answer My Hypothetical Question About Filters?

10/21/2016 10:20 PM

Back in the '70s, I had to simulate a dial up phone line using active filters. I used 2nd order low pass and high pass sections and possibly band pass section to get the proper amplitude response, but I had to add ALL PASS sections to get the proper phase response. In the end, I had around a 14th order filter, but it was very close to the response, both amplitude and phase, of a dial up phone line.

An All pass filter has constant amplitude across the frequencies of interest, but it provides different phase delay at various frequencies (like the other filter types: LPF, HPF, BPF, & BRF).

I don't have the actual working design handy. It is probably in a lab book in storage.

Sincerely

Bill

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