Third Order High Pass Filter (Report)

[Whiley]

Homework Statement

[/B]
The filters of FIGURE 4(a) to (c) where designed using a commercial software package (http://focus.ti.com/docs/toolsw/folders/print/filterpro.html)

Write a report (of approximately 750 words) that discusses and compares the design of each filter in terms of their performance, circuit topology, application and any other relevant characteristics. You may for example attempt to relate the choice of component values to the filters performance

Homework Equations

The Attempt at a Solution

So far I have found that each of the circuits contain both a 1st order and second order filter cascaded to form a 3rd order filter. I am now looking at the graphs shown to analyse the data such as frequency response / phase response of each circuit.

 

[Whiley]

I have some questions relating to chebychev vs butterworth. Can I determine which of these by simply the frequency response. I see that (a) is chebychev (b) is butterworth and (c) possibly butterworth or bessel..

Is it ok to determine these in this way or I am I totaly off the mark?

 

[rude man]

I have some questions relating to chebychev vs butterworth. Can I determine which of these by simply the frequency response. I see that (a) is chebychev (b) is butterworth and (c) possibly butterworth or bessel..

Is it ok to determine these in this way or I am I totally off the mark?

In the frequency domain the Bessel never has ripple in either the passband or stopband regions but the Butterworth can look almost as flat as the Bessel yet still have some overshoot. The Chebyshev always has passband ripple, usually pronounced.

The other differentiator is phase reponse, just as important as frequency response.

The different filter types are tradeoffs between rapidity of rolloff (chebyshev best), flatness (Bessel is best) and phase linearity (Bessel is best). Butterworth is a compromise between Bessel and Chebyshev - little ripple and acceptable phase linearity.And then there are still other response scenarios, e.g elliptic.
cf. http://www.analog.com/media/en/training-seminars/design-handbooks/.../Chapter8.pdf

Phase linearity is important if pulse integrity is to be maintained. A pulse consists of a wide (theoretically infinitely wide) band of frequencies. If phase shift is a linear function of frequency then the pulse shape will be retained (see appendix below). The Bessel best approximates this but it has the worst frequency rolloff characteristic.

(Appendix: A pure (undistorted) delay in the frequency domain is exp(-jωT) = exp(-jθ) where T is delay and θ is phase.. So clearly dθ/dω = T = constant must hold.)

 

[Whiley]

In the frequency domain the Bessel never has ripple in either the passband or stopband regions but the Butterworth can look almost as flat as the Bessel yet still have some overshoot. The Chebyshev always has passband ripple, usually pronounced.

The other differentiator is phase reponse, just as important as frequency response.

The different filter types are tradeoffs between rapidity of rolloff (chebyshev best), flatness (Bessel is best) and phase linearity (Bessel is best). Butterworth is a compromise between Bessel and Chebyshev - little ripple and acceptable phase linearity.And then there are still other response scenarios, e.g elliptic.
cf. http://www.analog.com/media/en/training-seminars/design-handbooks/.../Chapter8.pdf

Phase linearity is important if pulse integrity is to be maintained. A pulse consists of a wide (theoretically infinitely wide) band of frequencies. If phase shift is a linear function of frequency then the pulse shape will be retained (see appendix below). The Bessel best approximates this but it has the worst frequency rolloff characteristic.

(Appendix: A pure (undistorted) delay in the frequency domain is exp(-jωT) = exp(-jθ) where T is delay and θ is phase.. So clearly dθ/dω = T = constant must hold.)

Thanks for your response, that is very helpful. I can speak about the frequency response quite a bit but I don't know how to explain the impact of the phase response or what it means when differentiating between the 3 responses given. Do you have any more insight as to what is either causing the change in phase or the impact that the change has on the response please.

 

[rude man]

Do you have any more insight as to what is either causing the change in phase or the impact that the change has on the response please.

EDIT: I edited out the original first paragraph because I'm not prepared to discuss it further if asked to.

The best way to appreciate the effect on a pulse shape by the various filter transfer functions is to take the Fourier integral of the pulse, multiply by each of the Fourier transforms of your various filter transfer functions, and inverse-Fourier transform the outputs. (You can do this with Laplace transforms also, in which case jω is replaced by s in the filter transfer function).

You can also take a simple transfer functions like 1/(1+jωτ), input the transformed pulse, and inverse-transform the output, varying τ while keeping the pulse duration constant. You know the phase of this network is φ = -tan^-1(ωτ) so dφ/dω = -τ/(ω²τ² +1) so you can see how dφ/dω is low if ωτ << 1 and also you know separately that this 1st order low-pass filter will have minimum pulse distortion for this condition since then it approaches a constant gain without frequency effects. So there is an obvious correlation between phase response and pulse distortion.

 

[LvW]

The different filter types are tradeoffs between rapidity of rolloff (chebyshev best), flatness (Bessel is best) and phase linearity (Bessel is best). Butterworth is a compromise between Bessel and Chebyshev

I think, I should mention some corrections:
It is not quite correct that the Bessel response is best - as far as "flatness" is concerned.
Another name for Butterworth response is "maximum flat" response.
That means: Butterworth is between Bessel and Chebyshev because it has a maximum flat response WITHOUT any peaking !

 

[Whiley]

Thanks guys, I am getting there now, The only thing (which is not at all discussed in any lessons) that I have to discuss is delay. Am I correct to to state that the greater the change in delay over a frequency range the greater the distorsion of the signal? and therefor a flat graph of delay is better. If this is the case I would say that graph (a) is the best in terms of this as its generally flat(ish) compared to the others except for a spike between approx 1khz-3khz.

 

[LvW]

Thanks guys, I am getting there now, The only thing (which is not at all discussed in any lessons) that I have to discuss is delay. Am I correct to to state that the greater the change in delay over a frequency range the greater the distorsion of the signal? and therefor a flat graph of delay is better. If this is the case I would say that graph (a) is the best in terms of this as its generally flat(ish) compared to the others except for a spike between approx 1khz-3khz.

Speaking about distortions, you must discriminate between linear and non-linear distortions (caused by non-linear amplifiers).
Non-linear effects are outside of the scope of this discussion.
Now, speaking of linear distortions only - we have
(a) linear amplitude distortions (due to attenuation or damping), and
(b) phase distortions (cause by a non-linear phase characteristics); note that group delay is defined as the negative diff. quotient d(phi)/d(w).

 

[rude man]

I think, I should mention some corrections:
It is not quite correct that the Bessel response is best - as far as "flatness" is concerned.
Another name for Butterworth response is "maximum flat" response.
That means: Butterworth is between Bessel and Chebyshev because it has a maximum flat response WITHOUT any peaking !

True. I should have said that Bessel has no ripple wheras Butterworth usually does. But in any case that's not the OP's focus.

 

[rude man]

Thanks guys, I am getting there now, The only thing (which is not at all discussed in any lessons) that I have to discuss is delay. Am I correct to to state that the greater the change in delay over a frequency range the greater the distorsion of the signal? and therefor a flat graph of delay is better. If this is the case I would say that graph (a) is the best in terms of this as its generally flat(ish) compared to the others except for a spike between approx 1khz-3khz.

Actually, the answer is (b). You have to concentrate on the constancy of the passband group delay which is to the left in your delay graphs. It's not the amount of delay but its lack of variation (flatness) with frequency that is of import. So the delay in (a) varies more than that in (b) even at low frequencies, making (b) the closest to a Bessel response.

What is called "delay" in your graphs is "group delay" which is dφ/dω and so has the units of time. So you can also see that phase φ itself should increase linearly (in a negative directon) with frequency ω to attain a constant group delay. The integrity of a pulse correlates directly with the constancy of group delay with ω.

 

[Whiley]

Yes, I was looking at the graphs comparing them as if they had the same time scale but I see now that (a) varies more in time. I had it backwards sorry, so (b) is the best case which compromises in all 3 aspects being frequency roll off, flatness in phase response and group delay to provide the best case in terms of minimizing distortion?

 

[rude man]

Yes, I was looking at the graphs comparing them as if they had the same time scale but I see now that (a) varies more in time. I had it backwards sorry, so (b) is the best case which compromises in all 3 aspects being frequency roll off, flatness in phase response and group delay to provide the best case in terms of minimizing distortion?

Right. Flat delay (vs. frequency) = low distortion (vs. time).

Try this link. If you can open it it's a great app note from a top integrated cicuits mfr:
http://www.analog.com/media/en/training-seminars/design-handbooks/.../Chapter8.pdf
Look at pp. 8.37 and 8.38 to see what really great group delay and temporal response characteristics look like!

 

[LvW]

Right. Flat delay (vs. frequency) = low distortion (vs. time).

To Whiley: "Distortions" is an ugly word - and everybody its trying to avoid this.
HOWEVER: Speaking about constant delay or a flat delay function , we are referring to PHASE DISTORTIONS only, which in many cases are no problem.
Not to confuse it with non-linear AMPLITUDE distortions which create additional frequency components!

 

[rude man]

To Whiley: "Distortions" is an ugly word - and everybody its trying to avoid this.
HOWEVER: Speaking about constant delay or a flat delay function , we are referring to PHASE DISTORTIONS only, which in many cases are no problem.
Not to confuse it with non-linear AMPLITUDE distortions which create additional frequency components!

Non-constant (non-flat) delay DOES cause amplitude distortion. A rectangular pulse sent into a LINEAR network with non-constant delay w/r/t frequency will distort IN AMPLITUDE as we all know.

The networks the OP is talking about are all linear. You might confuse the OP by even mentioning non-linear networks.

 

[LvW]

@rude man, I cannot agree with you.

(1) At first, the OP has asked for distortion properties of the various alternatives.
It was the primary intention of my contribution to DEFINE the term "distortion", because I was not sure if the OP was aware of the correct use of this term.
In this context, I think it is very important to point to the fact that non-linear amplitude distortions (better known as "harmonic distortions") are caused by non-linear circuits only.
And I clearly have stated that these non-linear effects are NOT subject of this discussion.
So - I cannot see why the OP should be "confused" by mentioning non-linear effects.

(2) You write: "A rectangular pulse sent into a LINEAR network with non-constant delay w/r/t frequency will distort IN AMPLITUDE as we all know."
I am afraid that THIS sentence could creae confusion because it uses wrong terms.
Here comes the definition for "phase distortion" (www.merriam-webster.com):

Definition of phase distortion
change of wave form of a composite wave due to change of relative phase of its component harmonics

Hence, you must not use the term "amplitude distortion" for describing a change of the waveform caused by a non-linear phase response.
The term "Amplitude distortion" (better: "Harmonic Distortion") refers to unequal amplification or attenuation of the various frequency components of the signal only.

 

[The Electrician]

True. I should have said that Bessel has no ripple wheras Butterworth usually does. But in any case that's not the OP's focus.

Butterworth filters have no ripple.

From: https://en.wikipedia.org/wiki/Butterworth_filter: "The frequency response of the Butterworth filter is maximally flat (i.e. has no ripples) in the passband and rolls off towards zero in the stopband."

 

[The Electrician]

Thanks for your response, that is very helpful. I can speak about the frequency response quite a bit but I don't know how to explain the impact of the phase response or what it means when differentiating between the 3 responses given. Do you have any more insight as to what is either causing the change in phase or the impact that the change has on the response please.

The change in phase is caused by the change in frequency response. If you know the frequency response of a filter which has no singularities in the right half plane, the phase response is completely determined by that frequency response.

 

[rude man]

Butterworth filters have no ripple.

From: https://en.wikipedia.org/wiki/Butterworth_filter: "The frequency response of the Butterworth filter is maximally flat (i.e. has no ripples) in the passband and rolls off towards zero in the stopband."

Butterworth is not necessarily montonically non-increasing with frequency (for a low-pass). whereas Bessel is. I considered Butterworth "ripple" but better is "over-shoot".

The change in phase is caused by the change in frequency response. If you know the frequency response of a filter which has no singularities in the right half plane, the phase response is completely determined by that frequency response.

View attachment 209604

The above does not hold for an all-pass network which has zeros in the right-hand plane. It does for the three networks the OP initially posted.

 

[The Electrician]

Butterworth is not necessarily montonically non-increasing with frequency (for a low-pass). whereas Bessel is. I considered Butterworth "ripple" but better is "over-shoot".

A little past halfway down the page under the heading "Maximal flatness" at: https://en.wikipedia.org/wiki/Butterworth_filter

is found "the derivative of the gain with respect to frequency can be shown to be...monotonically decreasing for all ω since the gain G is always positive", which is even stronger than "non-increasing".

The above does not hold for an all-pass network which has zeros in the right-hand plane.

Why would anyone suppose that it does, since it excludes networks with singularities in the right half plane?

 

[rude man]

A little past halfway down the page under the heading "Maximal flatness" at: https://en.wikipedia.org/wiki/Butterworth_filter
is found "the derivative of the gain with respect to frequency can be shown to be...monotonically decreasing for all ω since the gain G is always positive", which is even stronger than "non-increasing".

Look at my attached link from Analog Devices for yourself. "The gain is always positive" does not preclude a bump in the G-w characteristic anyway. If it had said "dG/dw is always negative" that would be different. I've worked with A/D for 30+ years and they're a pretty fair bunch of EE's.

 

[The Electrician]

Look at my attached link from Analog Devices for yourself. "The gain is always positive" does not preclude a bump in the G-w characteristic anyway. If it had said "dG/dw is always negative" that would be different. I've worked with A/D for 30+ years and they're a pretty fair bunch of EE's.

Are you referring to the link in post #3? That link doesn't work, but I think I found what it should be: http://www.analog.com/media/en/trai...-Linear-Design/Chapter8.pdf?doc=ADA4661-2.pdf

On page 21 under the heading "Butterworth", it says: "The Butterworth filter is the best compromise between attenuation and phase response. It has no ripple in the pass band or the stop band, and because of this is sometimes called a maximally flat filter."

The Wikipedia page I linked to: https://en.wikipedia.org/wiki/Butterworth_filter, under the heading "Maximal flatness",shows an expression for dG/dw. The fact that the gain is always positive means that dG/dw is always negative, even though that is not explicitly stated. The next sentence concludes from the properties of dG/dw that "The gain function of the Butterworth filter therefore has no ripple.".

 

[LvW]

I considered Butterworth "ripple" but better is "over-shoot".
.

The term "ripple" always refers to the frequency response (Butterworth is maximally flat without any ripple), whereas the term "overshoot" must be used for the step response only (time domain).

 

[NascentOxygen]

I have some questions relating to chebychev vs butterworth. Can I determine which of these by simply the frequency response. I see that (a) is chebychev (b) is butterworth and (c) possibly butterworth or bessel..

Is it ok to determine these in this way or I am I totally off the mark?

If you say (b) is a Butterworth, then you can't say (c) is also a Butterworth, because Butterworths have a fixed shape and you can see that (c) is not as sharp at the corner. So (b) may well be a Butterworth, but it then follows that (c) cannot be a Butterworth. Sure, suggest (c) to be a Bessel.

 

[NascentOxygen]

The different filter types are tradeoffs between rapidity of rolloff (chebyshev best), flatness (Bessel is best) and phase linearity (Bessel is best). Butterworth is a compromise between Bessel and Chebyshev - little ripple and acceptable phase linearity.And then there are still other response scenarios, e.g elliptic.
cf. http://www.analog.com/media/en/training-seminars/design-handbooks/.../Chapter8.pdf

Phase linearity is important if pulse integrity is to be maintained. A pulse consists of a wide (theoretically infinitely wide) band of frequencies. If phase shift is a linear function of frequency then the pulse shape will be retained (see appendix below). The Bessel best approximates this but it has the worst frequency rolloff characteristic.

You are listing low-pass characteristics, but is any of this relevant to the high-pass filters shown? Apart from near the band edge, they all have around 0 group delay. And near the band edge they all distort in their own way (group delay varies) and I can't see the group delay curve of one filter being significantly better than another.

 

[NascentOxygen]

The Butterworth is maximally flat over both passband and stopband. There is only one set of Butterworth polynomials. It is easy enough to show the Bode Plot to be monotonic throughout—simply count the sign changes in the first derivative.

Designers are free to do what they like, but if they change the polynomial it's no longer a Butterworth. Convention says it would be termed a modified Butterworth, at the very least. Wikipedia indicates it is possible to design for an even flatter passband if willing to accept a bit of ripple in the stop band. It would seem logical for a marketing department to list such filters in their Butterworth Filter pages, because that's where buyers would be looking for flat passband filters, even though these I mentioned are technically a type of Chebyshev design.

 

[rude man]

The fact that the gain is always positive means that dG/dw is always negative, even though that is not explicitly stated.

Not explicitly stated because wrong.
Looky here: see the bump on the Butterworth frequency response? Have to look carefully maybe ...

 

[NascentOxygen]

Not explicitly stated because wrong.
Looky here: see the bump on the Butterworth frequency response? Have to look carefully maybe ...

What is the significance of the small rectangles at points along each of your plots?

 

[rude man]

What is the significance of the small rectangles at points along each of your plots?

Your guess is as good as mine. Look like data points on empirically derived data but everything in this 144-page app note is seemingly computer-derived. And I see them only on the Chebyshev, not the Butterworth or Bessel curves.

I know my link wouldn't open so I suggest googling "analog filters" and finding "Chapter 8: Analog Filters - Analog Devices" which is how I found the paper. All you ever wanted to know about analog filters and then some.

 

[rude man]

The term "ripple" always refers to the frequency response (Butterworth is maximally flat without any ripple), whereas the term "overshoot" must be used for the step response only (time domain).

There is also "overshoot" on frequency plots. Let's not get pedantic. See my post 26.
As I said already, all this is of no relevance to the OP who simply wondered about the relationship between pulse distortion and dφ/df flatness (cf. his last posts, ##7 and 11).

 

[rude man]

You are listing low-pass characteristics, but is any of this relevant to the high-pass filters shown?

It is.
Consider simple illustration: a rectangular pulse of duration T into 1st order low-pass and high-pass networks, cutoff radian frequency = a:
Input is (1/s)(1 - exp(-sT).
Low-pass is a/(s+a) and hi-pass is s/(s+a).
Group delay is the same for both: dφ/dω = -τ/(ω²τ² +1) with τ = 1/a as I derived earlier.
Output for the low-pass is 1 - exp(-at) - [1 - U(t-T)exp(-a(t-T))]. This is a distorted pulse with slow rise and fall times but a flat top (for T >> 1/a).
Output for the high-pass is exp(-at) - U(t-T)exp(-a(t-T)) which is also a distorted pulse with sharp rise and fall times but a drooping top (for T >> 1/a).
Totally complementary situation. Both filters distort the pulse, but in different ways.

Further: dφ/dω of the input pulse = -T/2.
So the total delay of this pulse into a network with linear phase over all ω (in which φ = -kω so dφ/dω = -k) would be -T/2 - k
which proves that a network with linear phase over all ω would not distort the pulse anywhere.