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LEGO Block Filters

Started by Safwan 1 year ago12 replieslatest reply 1 year ago298 views

Hello All,

Considering the availability of mature and and advanced linear phase FIR filter design techniques, many might look sideways at "novel claims" for filter design approaches. What possible benefit could come from such  endeavours.  Well, it is not my goal (or even hope) to convince anyone here to keep an open mind  (scientists and engineers always do, by the way, being led by the philosophical principle of falsifiability). 

So, allow me to introduce a novel method for linear phase filter design. It is still in its infancy, and there is much room for contribution (for those interested). What is new about it is that it is purely an algebraic method. There is no explicit application of the DTFT . A pulse shaped function in the frequency response (Dirichlet function) is developed to constitute a basis set from which all possible transfer functions may be constructed (as limited by symmetric impulse responses of specified length). An assembly of these pulses are stacked together, kind of like kids  LEGO brick, the sum of which gives the transfer function of the filter. 

Let me demonstrate, consider this low pass differentiator (LPD), where the black line is the filter magnitude response is obtained by summing the blue pulses.  The impulse response of the filter is also the sum of the individual impulse responses of the pulses.

                                              xxxxx_47298.png

Look also at this stepped filter  stepped filter.jpg

Dear readers, I apologise for not pursuing this development in the professional venue of peer-reviewed publishing (don't have enough energy). But the rigorous theory and details for this method are in this paper here

I hope you will find value in it.

Safwan Arekat








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Reply by Rick LyonsMay 22, 2023

Hello Safwan Arekat.

Regarding your paper titled: “Dirichlet Pulse Filter Design (LEGO Block Filters)“, it appears to me that your FIR filter design method is the same as a design method that creates what are called “frequency sampling filters.” The frequency sampling filter design method has been known since the late 1960’s.

● Linear phase FIR frequency sampling filters comprise a comb filter whose output drives a bank of resonators. (The comb filter forces the impulse responses of the resonators to be finite in duration.) Each resonator has a different resonant frequency, and a single resonator has frequency response equal to that sin(x)x response curve you show in the second figure of your paper.

● In frequency sampling filters the output of each resonator is multiplied by a user-defined coefficient which I believe is the same as the A[k} coefficient on the seventh page of your paper. (Safwan it’s difficult to refer to the contents of your paper because you didn’t number your paper’s equations, figures, or pages.)

● Many decades ago the design of frequency sampling filters was one of only two known methods for designing linear phase FIR filters. However, the advent of the powerful Parks-McClellan tapped-delay line FIR filter design method has driven the frequency sampling filter design method to near obscurity. Thus in the 1970s frequency sampling filters lost favor to the point where their coverage in today’s DSP classrooms and textbooks ranges from very brief to nonexistent.

● I cover frequency sampling filters is great detail in Chapter 7 of my “Understanding Digital Signal Processing” textbook. I did so because for some linear phase FIR narrowband filter applications frequency sampling filters are far more computationally-efficient than Parks-McClellan designed FIR filters.

● Safwajn, if I could see the block diagram, or the z-domain transfer function, of your “LEGO block filter” then I could be sure whether or not your design method is equivalent to the 50-year old frequency sampling filter design method.

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Reply by SafwanMay 22, 2023

Rick Lyons

Dear Sir,

Firstly, I am honoured you saw fit to reply to my post. I have learned much from your expertise over the past couple of years, and from your engaging style and keen insight. Your papers were instrumental in getting me back on track in signal processing, after decades of doing physics.

I am quite certain the Dirichlet Pulse Method in my paper is not the same as the “Frequency Sampling Filter” in Ch7 of your book. The fact that it raises this doubt from a professional such as yourself tells me I could have done a much better job at presenting my idea. For one thing, I did not give the block diagram or the z-domain transfer function because I assumed that the mere mention of FIR filter was a sufficient classification. Apparently, I was wrong. I should have been more specific in classifying it as a non-recursive linear phase FIR. The FSF you refer to in your post is a recursive linear phase FIR, utilising feedback loops to create either real or complex resonators, whose unit circle poles cancel some unit circle zeros of the comb filter, thus producing an effective transfer function with a finite impulse response and linear phase, (or passband linear if the poles and zeroes are migrated slightly inside the unit circle).

The block diagram I should have provided is the standard non-recursive tapped delay line FIR filter, just like the Parks-McCLellan circuit, where the z-domain transfer function of a z^-n zeroes-only polynomial is implemented.

The method in the paper is a novel one, it (theoretically) allows obtaining any transfer function achievable from symmetric IR of length M, even the Parks-McCLellan transfer function itself – with properly set design criteria and proper iteration of the A[k] coefficients. It is also an algebraic method, just add the Dirichlet pulses, and you have your transfer function. Just add the corresponding cosine impulse responses, and you have your filter coefficients. There is no need to take any transforms in the design process.

Dear Rick Lyons, I am certain that The Dirichlet Pulse Method is not the same as the FSF.

So sure, that it hurts !!!

Thank you so much

Safwan Arekat

P.S. Sorry for not paginating or equation numbering. I will strive for better.

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Reply by kazMay 22, 2023

Hi Safwan,

Without deep analysis I feel your algorithm is different from current popular methods. It seems based on impulse breakdown instead of Fourier. This breaking down concept itself is not new but utilising it for a filter is.

According to my experience with tools like Matlab. They got at least three main categories of FIR filter design algorithms: windowed sinc (fir1), frequency sampling method (fir2) and iterative methods like PM (Remez). Apart from some special filter functions like raised cosine, halfband, etc.

Personally I view windowed sinc as a case of frequency sampling method as we target a given frequency shaping but instead of using iDFT (fir2) it uses the pre-known sinc function which is related to Fourier anyway.

So in all cases they are dependant on DFT/iDFT, not impulse breakdown.

Kaz


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Reply by SafwanMay 22, 2023

Hi Kaz,

Yes Kaz, I agree. All the cases you mentioned in Matlab  depend on the Fourier theorems. Other bases like using, for example, the discrete Walsh functions and transform to represent and analyse signals have never been developed to threaten Fourier decomposition, which is dependent on sinusoidal functions.   All the cases you mentioned depend to the core on Fourier analysis. It is just a matter of algorithm and representation.

b.t.w.  I was able to define a new mathematical function which I call the Cine function. THIS function might threaten the throne of Sine and Cosine - if ever more researchers became interested enough to follow where it might lead. 

Safwan


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Reply by Rick LyonsMay 22, 2023

Hello Safwan Arekat.

Thank you for your gracious and clearly written reply.

The text of your above reply’s second paragraph is a good description of a frequency sampling filter (FSF), and I’m now convinced you’re quite familiar with FSFs.

I want to tell you that one item of interest on the sixth page of your paper was when you wrote, “(Note: 𝐴[0] and 𝐵[0] must be reduced by a factor of 2 in the synthesis equation above).” That sentence “caught my eye” because the A[0] coefficient in an FSF must *also* be reduced by a factor of 2!

OK, ...your above reply’s third paragraph makes it perfectly clear that your Dirichlet Pulse Method designed filter is definitely not an FSF. So now, in my opinion, your paper becomes even more interesting.

Safwan, I’m wondering how your Dirichlet Pulse design method is related to the so-called “Frequency-Sampling” design method described on page 630 of the 3rd edition of the Proakis and Manolakis DSP textbook (book title: “Digital Signal Processing: Principles, Algorithms, and Applications”).

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Reply by SafwanMay 22, 2023

Hello Rick Lyons,

I gave it some more thought since last night. I must have stumbled upon a version of the Sampling Theorem, one that is applied to periodic transforms in the frequency domain. Just like any aperiodic continuous time signal can be reconstructed out of sinc functions modulated by signal samples (limited by the sampling frequency of course), my paper seems to demonstrate that any periodic transform in frequency space can be reconstructed out of Dirichlet pulses modulated by transform samples (limited by the number of time impulses) . But this is too broad a thing to have been overlooked by mathematicians. Maybe I should look if there are any versions of the sampling theorem for periodic signals. I bet the Dirichlet makes a prominent presence there.

So this must tie in with the frequency sampling method. Again they could give identical results, even though the mathematical representation is different.


Safwan

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Reply by SafwanMay 22, 2023

Hello Rick Lyons,

Thank you for your reply. Your question about the "Frequency Sampling Method" has much merit to it. I have not read the book, but I remember the method applies the inverse DFT to a discrete set of frequency samples to calculate the time domain IR. I will review the details and get back. Who knows, it might be one and the same. I say this because after reading your question, I glanced again at the demonstrations above, and noticed that the filter magnitude response (black line) always touches the peaks of the pulses! Can't be a coincidence.

But it 10 PM in Bahrain, so later please.

Safwan

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Reply by CedronMay 22, 2023
"Dear readers, I apologise for not pursuing this development in the professional venue of peer-reviewed publishing (don't have enough energy)."


Don't apologize for that.  This is quite an appropriate venue, perhaps the most appropriate venue, to introduce new DSP formulas to the community.  Academia is only a subset of it.  Here you will find more experienced practitioners.

To reach a broader, more diluted audience (e.g. more novices and students), there is also the DSP stack exchange.  Many members here also contribute there.

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Reply by Rick LyonsMay 22, 2023

Hi Cedron

I agree with you. Too many DSP publications allow authors to present signal processing operations to be described by a long sequence of equations without explaining what the equations mean or why the equations are important. That makes the authors look smart to their university colleagues, but many of the author's readers don't understand what the authors are saying.

Too many DSP authors take the attitude of "I understand what I'm saying, why don't you?



 

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Reply by SafwanMay 22, 2023

Hello Cedron,

Yeah.. I guess I am still influenced by academic prejudice. But your opinion is taken to heart, and so is Rick's.

Still, there ought to be "authoritative venues" to build scientific and engineering consensus. It is just that most published research - even in these - do not fit the bill for a whole range of reasons (watch Derek), and I would add, for self-importance and conceit. Unfortunately, sometimes this enhances unearned longevity in academia.


Thanks

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Reply by CedronMay 22, 2023

Post deleted by author

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Reply by CedronMay 22, 2023
The second part is about Academic publishing.  This is much more important to me.

There are basically three steps to get an idea to be the currently accepted best answer:

1) Presentation
2) Validation
3) Dissemination

You're perspective is definitely Academically skewed by what you say.  The video highlights many of the problems quite well.  They are also well known.  My opinion is the whole system needs to be abolished.  I make my recommendations in an editorial in another forum:  "Opinion Piece on Academia"

Journals offer all three steps in one swoop:  You submit, they peer review, then publish.

Have a look at the "Academic Acceptance" section of my overview article on this site.

Besides the problems already mentioned, the volume has gotten way too large that the system as is simply can't keep up.  This also crosses over to Rick's point about the lack of mathematical literacy among many of the readers.  Not all "experts" can read math.  However, the source document of an innovation should definitely show all the math and not "leave it as an exercise for the reader".  With a Journal submission, 1 and 3 above are the same, then 3 gets replicated in "dumbed down" or "demathified" form in the popular press.  There is no need for the version in 1 to be used in 3, as long as 3 references 1.

To me, items in 1 for Academics should be hosted by their own institutions.  Since the institution now has responsibility, it should decide what criteria or level of internal validation should occur before something is posted on its site.  Once posted, you not only have the ability for peer review, but also miscellaneous expert review.  If any flaws are found, they can be relayed directly to the author and the flaw fixed, or the item retracted.

Dissemination need not be anything more than a press release to various publications.  Having a site like Arxiv, for full articles or digests, is also good.  If a publication wants to stay relevant, it needs to keep up and select articles to highlight, either in original form, or a reworked interpretation.

In general publishing, when an author pays to be published it is called "Vanity Press".  Academics have to for their careers, hobbyists don't.  Why should a hobbyist pay to give a gift?

This one won't.  My blog is full of novel formulas, with derivations, that the IEEE has failed to pick up on.  To me, they are functionally a book club with a captive base not worthy of tax exempt status.  A legalized extortion racket.  I've told them so and also told them to make me an offer if they want to publish my stuff.

Not surprisingly, they have not replied.  Then again, I was intentionally rude.  So, they have to get over that, or they don't have state of the art.

Appealing for a "Central Authority" is just plain wrong.  Every one of those will ultimately go self-centered and corrupt.  Who is going to stand up to them?

This was kind of a problem during the pandemic and the ill advised vaccines.  Academia failed society and cowed to the suppression of dissent.