Algorithm Challenge

[Skippy]

It's a shame you put this 'competition' on when I have very limited internet access.
So I'm a late entrant, I guess.
To calculate the magnitude of your vector:
Split the 0 to 180 degrees angle into 45 degree segments, and use an approximation that's good for 0 to 45 degrees.
Swap the X and R values over for 45 to 90 degrees, for example.
An approximation based on a parabola works well, amplitude errors of less than 1.03% can be achieved.
Summary:

Amplitude = x + 0.428676 * y² / x

Alternatives are A = x + (3/7) y² / x ... very good

and A = x + 110 y² / ( 256 * x ) ... pretty good, maybe useful for computer maths.

Attached explanation:

[Skippy]

Regarding the arctan function:
Is arctan ideally what you want, or are you simply after a decent TID generator?

I quite like what Minelab did with the Equinox 800 TID scale.
I reverse-engineered it and posted my observations over on Steve's Detector Prospector Forum.
They have made a very accurate logarithmic scale, so each ID value step represents a fixed ratio change of target frequency. This seems to get around the issues of stretching/squashing to get a practical useable scale.

My analysis shows ML chose a mid-point "20" to represent a 25 microsecs / 6366 Hz target. The frequency step ratio is 1.147203 so that 8 numbers on the scale represents a 3:1 frequency change.
The full "1" to "39" scale then covers about 80 kHz through to 500 Hz. This is targets about 13 times higher in freq than the 6366 Hz, and 13 times lower at the 'top' end.

Mathematically, the TID value can be expressed as:

TID = 20 + log10 ( f_d / f_t ) / 0.05964

or 20 + 16.7673 * log10 ( f_d / f_t )

where f_d is target frequency and f_d is detector frequency.

The thread on Detector Prospector:

https://www.detectorprospector.com/t...bration-table/


The TID table is attached here.


How to do a log scale?
Thanks to "Abramowitz & Stegun" , there is a simple and decently accurate log10 approximation, see attached image.



Source: ( see page 68 )
Scanned Version of Abramowitz and Stegun, Version 1.1 (bham.ac.uk)

It's good for a full decade, from 0.316 to 3.16, and for our more practical purposes can be squeeked out to cover (1 / 3.5) to 3.5, a 1:12 range, and maybe a bit more.
So it's ideal for covering the 0 to 45 degree range of the Equinox scale. Swapping the X and R components easily allows the 45 to 90 end of the range to be calculated.

The practical details:
Our x value we want to log covers the range 0 to 1.00, or more practically, 0.07 to 1.00.
So pre-scaling this x up by a factor or 3.5 puts it in the log10 approximator's range. The resulting log10 then needs re-scaling back down, with a subtraction of log10 (3.5) = 0.54407

because:
log ( c * d ) = log (c) + log (d)

So for example if we have x = 0.4; then 3.5 * x = 1.40.
log10 (1.4) = 0.1461
subtracting log10(3.5) gives ( 0.1461 - 0.54407 ) = - 0.3979
which is the correct value for log10 (0.4)

This can then be put into the TID formula above, to create a TID value.

I ran a simulation in QuickBasic, and it works impressively well, putting out TID values like "10.98" for a "11" target.

The maths can also be simplified a lot to speed it up, only one division is needed.

The log10 approximation takes in one number; we have a division of two numbers, ( m/n ) as our source.
But it's not necessary to calculate ( m/n ) and then do a second division to find t.
They can be combined:

t = (m/n - 1 )/ ( m/n + 1 ) = ( m - n ) / ( m + n )

Likewise, many of the scaling constants can be integrated / combined, reducing the maths complexity.
See the attached screenshot of a QuickBasic routine for 0 to 45 degrees ( TID 1 to 20 )
Example:
A TID "10" target, 25.1 kHz frequency.
Enter 'top' of the fraction as '33'; enter 'bottom' as '130'
and the result is a "10" give or take a tiny bit.

For fun, I also simulated a 'revised Equinox scale' ; scaled from 1 to 79, with "40" at centre-scale, and frequency ratio steps half what the were. It all worked great over the full 80 point range, really rather impressive.

[moodz]

After wrestling the archane STM32 build platform for a few days I was able to compile ARMRADIO which is an SDR radio into the STM32f407 chip.

Its on a STM32F407 diso board however this build uses only the CPU ... none of the other bling chips on the board are used.
The signal comes into one of the 12 bit ADCs is processed and the result comes out of the DAC.

ARMRADIO has all the DSP goodies already baked in to provide a solid framework for a DSP based processing system for metal detector use.

As it stands it can process from 9 khz to about 900 khz.

This project was released some years ago ... hence the wrestling to update to current tools ( STM32 cube )

you can read about it in the attached PDF. Its open source so google it.

[Carl-NC]

What process do you use to attach files & pics? Because they occasionally don't show up. I need to chase this down.

[moodz]

What process do you use to attach files & pics? Because they occasionally don't show up. I need to chase this down.

err the "usual" way from the post editor. Some where png format ... it seemed to work for me. I could see them but nobody else ??
The last post I thought I attached a PDF also .. I cant see it.

here it is.

[Carl-NC]

There are different ways even from the editor. I simply drag-and-drop items right into the editor window. Or use Window's snip tool to do a screen capture, then paste in the editor window.

[moodz]

The main blocks for an advanced VLF IB metal detector will require ...

1. AIB ( Active Induction Balance )
2. LLIA (Log Lock In Amplifier )
3 PAD ( Phase Detector )


AIB will largely counter magnetic grounds. Amplitude vs frequency ( khz )



TX1 is the main TX coil. TX2 is the bucking coil.


ARM radios software is rearranged


to form a VDI detector ..



and all we need is the MAGIC log amplifier for sine waves

as we cant use an ordinary "LOG" amplifier as the phase information is lost ( must maintain phase and log of amplitude gain)

We cant use this ... the phase info is lost and the log amplifier is wideband .... it will log all the input noise and as well.



We need the response below ... The noise is removed. The phase is intact. The sine wave RMS value is a log value but the waveform is not log distorted.



... should then have a motion / non motion hi spec detector ( adding batteries / cases / LCD bling etc etc etc ).

moodz

[moodz]

Hi Carl ...on my last post the image thumbnails are not visible ( except the last image ) Note ( the thumbnails not visible have spaces in the image filename ). If I log in I can see all the images. ??

[Carl-NC]

For me, only the last image shows, the rest are just squares like before. I have a help request posted to vBulletin, I found that someone else sees the exact same problem.

One thing I wonder is, after you click [Post Reply] whether you are then jumping somewhere else before the files have finished processing. This could vary with Internet speed which might explain why it happens only sometimes.

Otherwise, I'm intrigued. I worked on SDR some 20+ years ago at Analog Devices along with Siemens in Germany. This was for GSM/Edge base stations. I was working on the ADC which ran 14-bit 100MSps. Today, that is no big deal, but 20 years ago it was.

[moodz]

...hmm the only thing I could see different was that the last image is called inout.png ... the other images are called similiar to screenshot from date.png ...therer are spaces in the names.

I have a 100mbit connection here ... and it is very reliable but it could be something. I will try posting stuff from a windows box .. the usual one is linux.

The cutting edge for RF conversion is Rydberg Quantum Radios that convert DC to 20 Ghz simultaneously .... https://phys.org/news/2021-02-quantu...-spectrum.html

[Qiaozhi]

For me, only the last image shows, the rest are just squares like before.

Same here.

[Skippy]

Seeing as this 'challenge' seems to still be open ( no response from Carl ) , I spent some time over the Christmas holiday working on an Arctan approximation. I'm quite pleased with the result.

I started with the simple approximation:

arctan (x) ~ 90 * x/(x+1)

This was discussed in the previous 'arctan' thread:
Arc tangent function - Geotech Forums (geotech1.com)

It's correct at 0, 45 and 90 degrees, but can be over 4 degrees in error elsewhere.
The error term resembles a cubic equation, so I worked out a cubic correction term and computer-optimised it.
The resulting new approximation has an error of only +/-0.35 degrees over the full 90 degree range.

In summary:

(1) calculate t = x/(x+1) ... or t = opposite/(opposite+adjacent) if you have I and Q values.
(2) calculate f = -0.92735 * t * (t * (t - 1.5) + 0.5)
(3) Revised arctan approximation = 90 * (t + f)

Attached is a pdf explaining my calculations. There's also an Excel file with a full example (when I work out how ..) and the resulting error plot for the simple approximation, and the revised 'cubic' version.

I originally did all the calculations in QuickBasic, so I've attached screengrabs of example code and results. The .XLS file is probably easier to play with, though.




IMG3_20240104_0001.pdf

[Carl-NC]

This was discussed in the previous 'arctan' thread:
Arc tangent function - Geotech Forums (geotech1.com)

That thread is only a year old and I had completely forgotten about it. Gettin' old. Yes, 90*R/(X+R) is a shortcut that works well enough. Because you will almost always have some sort of look-up table to correlate the computed phase with a VDI number, you can easily adjust for the 4° error in the estimation. If you need more accuracy, then certainly your correction method can be used. But so far I like the CORDIC approach the best, which doesn't require any multiplies and uses a very small LUT.

For a magnitude short-cut, instead of I would propose |X| + |R|. The question now becomes, how much calculated depth error will that impose and is it reasonable?

There is no right or wrong answer in any of this, it's just a challenge to spur discussion, see where it goes.

[Skippy]

Quote: "For a magnitude short-cut, instead of I would propose |X| + |R|."

That's pretty 'rough'. If 'simple + dirty' is needed, X + 0.41 * R is better. Use it for 0 to 45 degrees. If R >X , then swap them over and apply the formula to 90 to 45 degrees.

Quote: "Because you will almost always have some sort of look-up table to correlate the computed phase with a VDI number"

So you WERE wanting a VDI scale, and not actually an arctan calculation? I did ask in my ( rather good ) reply detailing the logarithmic scaling, but it was completely looked over while you were consumed with attachment problems.

[Carl-NC]

So you WERE wanting a VDI scale, and not actually an arctan calculation?

What we want is a simple way to come up with some kind of phase representation of the target response. Whether it's for displaying a VDI or doing discrimination or ground tracking. In all cases, having extreme precision is not really necessary, but consistency is. If you are writing a ground tracking algorithm, you can write it whether the phase calculation is perfect, or is off by a few percent, as long as a particular phase is always off by the same amount. Same with Disc, same with VDI. Obviously a phase calculation that is closer is better than one that is not, as you will not need to make as many fudges to various end-use algorithms. Detectors using crappy micros fake this stuff all the time, and it's good enough that no one seems to notice. One of the best ground tracking designs is in the MXT, which uses a lowly PIC16C76 clocked at 8MHz. That what this thread is about, getting good-enough results on a crappy micro.