Thanks again.
Dave

I’ll restate the motivation: Imagine you make a set of B bandpass filters, and then set the EQ so it’s all level (e.g. the weight for each band that you use to sum into the signal is at 1). Now ask: will running this “null equalization” on an arbitrary signal always give you the original signal back, or will the different characteristics of the filters cause interference and noise to appear?

I’m not aware of a way to design a set of bandpass filters that would not introduce some noise. Especially when you’re trying to capture different frequencies covering logarithmic ranges, and thus possibly using different size filters…! If you know how to do it, I’d be interested to know.

What I came up with is trickier because it was trying to take the principles of Perfect Reconstruction and apply them to a graphic equalizer. In addition to acting like you’d expect an equalizer to act when you have the weights set at different values, the process of all this turmoil spits out the exact original bytes you put in when the weights are 1! I believed I’d proven it to work, and my TAs in college did too, so maybe it’s for real…but we could have been wrong.

The Remez code I found is by Egil Kvaleberg, ported from Fortran. I will email it to you.

The decreasing of array widths is a conceptual thing…and really the data can all be processed in place. But to more clearly explicitly show what’s happening and what all the bookkeeping needs to do, I tried to “pull it apart”. Here I’ll badly hack up the “next step” if you want more than 3 bands in your EQ…and you’ll see why I preferred to stop after 2 steps.

So imagine that if you have a low pass Remez filter that basically passes everything below 2kHz, and you want to pass everything below 1kHz instead. To get an equivalent to a low pass 1kHz Remez filter, just use the 2kHz design on the even samples, the odd samples, and then interleave them back together in time. (The idea of taking a 4kHz Remez filter and dropping it to 1kHz by taking every 1st, 2nd, 3rd, and 4th sample… running the process…and then putting the samples back in their sequence post-filtering would be expected if this were true.)

This process is what I’m referring to as a “logarithmic drop”. I almost certainly can’t prove myself today that process would give you another perfect reconstruction. That must have been something I read or learned at the time, because I don’t know how I’d have just taken that for granted unless the professor said I could! Still—it would be very interesting to me if someone could recreate the effect (or debunk it). Please let me know how it goes…

(P.S. I found an old C program that generates Remez filter coefficients if you don’t have Matlab or something like that available.)

Would be interesting to have someone reproduce it! I actually had coded this for a Texas Instruments DSP chip. Lost the source code but still had the writeup and diagrams, so I figured…well, might as well post them…

Not sure I understand your question, but I’ll try and answer. (Note that this algorithm isn’t suitable for processing real-time data; you must know how long your input signal is…let’s call that length L).

This is a recursive process. Each time you apply “the process” you will get two signals of length L. One of those is the time-domain representation of a higher pass band which you’re done filtering. It’s suitable for weighting and adding together to get a band in the overall signal. The other you have a choice: (1) you can take it as it is as a band in its own right, or (2) you can run the algorithm again on it to get another two signals of length L.

So if you refer to the “Implementation Diagram”, it does two steps on your data to produce two completed series of length L (the black ones: labeled band 1 and band 2). Then after the second step you’ve got a band on the left that you may choose to run the process on again, or consider yourself done and use it as band 3.

Make sense? Each time you get a completed band it needs to be the same size as the original. Then you multiply the samples in that signal by your EQ setting for that band, and add them all up.

If you write a modern reference implementation I’d appreciate you sharing your findings or tools used. I looked into tweaking Audacity to use the method but it looked like more of a project than I wanted at the time…

I still remember when my 6th-grade science teacher told us that the Sun emitted light at all frequencies equally (which theoretically requires infinite energy). I think that’s when I popped. Now I’m more careful in questioning Dirac’s constant.

My undestanding, last I looked at these issues, was that the more accurate a frequency manipulation you desired, the more latency needed to be introduced, the bane of live performance, and the irritant of synchronized-sound applications (A/V players).

I also understand that almost all software frequency analyzers and most hardware ones use various methods to cheat to make their tasks easier, figuring that no one will ever notice their lack of frequency response fidelity.