LabRadar and Signal-to-Noise Ratio

[WobblyHand]

For this equation:

Code: [Select]

Cd = -2*Mass*LN(V2/V1)/(Rho/32.174)/Area/(D2-D1)what is 32.174?  Is that g the acceleration due to gravity?  So g has units of ft/sec²?  Is LN the natural logarithm? 

For Ron's data (the only complete data set I have), what is the mass of the projectile?  Please include the units, be they grains, grams, or slugs.  Also the projectile frontal area would be helpful.  As well as the air temperature.  I'd like to play around with the data and knowing this would help.  One thing I did note was that the velocity is significantly noisier past 125 feet.  I plotted the range and velocity difference (adjacent points)  The range difference stays at the same noise level, but the velocity difference clearly gets worse with increasing range (or time).

Likewise if one does the division, V2/V1, the behavior is the same.
As well as ln(V2/V1)/(D2-D1)

[WobblyHand]

One could apply a median filter to the data to knock down some of the fliers.  It seems to improve performance out to about 250 feet.

[rsterne]

Yes, 32.174 ft/sec^2 is the value for "g" to convert lbs. to slugs for the air density.... and LN is the natural logarithm....

You can clearly see what the problem is from your plots.... Past a certain range (ie below a certain SNR) you start getting V2 > V1, which is impossible.... This yields a negative Cd.... This is almost always followed by a large swing in the other direction, and the two extreme values tend to average out.... Of greater concern is what Ron showed in his first chart, where the Cd tends to increase drastically below about Mach 0.65 with the LR unit at the muzzle.... However, moving the LR out to 50 yards totally changes the Cd values to what is expected....

I don't know what a "Median Filter" is.... 

Incidently, here is the same chart (all the data points for 10 shots, .177 cal) as the bottom one in Reply #16, but with the vertical scale changed....



This is more typical of what we might expect for a Cd curve for a pellet.... Note that Excel uses all the data points (as shown in the previous chart) to calculate the trendline equation (it is the same), even though some of the points are not shown because they exceed the vertical scale limits....

Bob

[WobblyHand]

Of greater concern is what Ron showed in his first chart, where the Cd tends to increase drastically below about Mach 0.65 with the LR unit at the muzzle.... However, moving the LR out to 50 yards totally changes the Cd values to what is expected....

I don't know what a "Median Filter" is.... 

This kind of makes sense to me.  What is happening is that the SNR is too low for these lower velocities.  The low velocities are occuring at the furthest distances from the LR.  Parameter estimation error dramatically changes with SNR loss.  If the LR is moved down range, and the measurement done again, the longer range segments, with their lower velocities will now have higher SNR.  This is because the LR is physically closer to the projectile. 

The power received by the LR is given by:

P_r= P_tG_tG_rlambda²sigma / ( (4pi)³R⁴)

The noise in the LR is given by

P_n = F kT B

The SNR is just P_r/P_n.  So SNR is proportional to 1/R⁴.  The LR doesn't care where the shooter is, only the relative position of the projectile to the radar.  So moving the radar down range from the shooter will result in better SNR for the lower velocities (at longer range from the muzzle).

Median filters are just another tool in a signal processing engineers toolbox.  A brief explanation can be found at https://en.wikipedia.org/wiki/Median_filter.  It is commonly used to remove salt and pepper noise in images.  I have found it can be useful to reduce impulsive noise.  The median of a sequence is the value when 50% of the sequence is above the that value.  Some simple examples.  x = {1,2,3},  med(x) = 2.  x = {1,3,2,5,7,4,9}, med(x) = 4.  You see the order doesn't matter.  In the last case, there are 7 elements.  If we reordered them we would have y = {1,2,3,4,5,7,9}.  4 is the element which is in the middle, there are 3 elements above it and 3 below it.  4 is the median.  The median filter I used is 15 elements long.  This filter slides along the sequence.  The first output is the median of the first 15 elements of the sequence.  The second output is the median of samples 2-16, the third output is the median of samples 3-17, etc.

I have an idea...  Let me try something.  Be back in a while.

[rgb1]

In order to illustrate the effects that surroundings have on SNR, I've attached
three pics from work last summer. They're not to the same scale and I apologize for
that.......at the time no thought was given to the possibility of sharing them in the
future. The pellet is Crosman 10.5 gr at 900 ft/s. Numerous shots were made and
analyzed, and an example was saved from each scenario.
Pic1 represents what we would see from a normal setup........shooter at a bench with
the unit three feet from the ground and shooting parallel to the ground. In this instance,
as well as the following two, the ground is my front yard (flat). For this shot there was
a small shrub at 70 yards off to the side approximately 15 feet.
Pic2 is with the shooting bench on an elevated porch. The unit is now above the ground
7  feet with the pellet traveling parallel to the ground. Nothing else is in the radars
"field of view".
Pic takes it a step further. Still shooting from the porch, the aiming point is now a
small, nearly stationary cloud about five degrees above the horizon which effectively
places the ground further out of the "field of view". As you can see, each change makes
an improvement.


                                                                                Ron

[rsterne]

Thanks, Bruce.... So a Median filter is like a moving average, except it takes the "center" value, rather than the average value.... We have played with moving averages of various breadths (eg. MA=5, MA=10, MA=20), and it certainly smooths the data.... Unfortunately, you lose a number of data points from each end (from already scarce data)…. For example, if you are using a 10 point moving average of 200 data points, the first available MA would be points 1-10, then 2-11, etc. and the last being 191-200.... That reduces the number of plottable points by 5 at each end (190 points instead of 200)…. Doubling the MA to 20 points brings you down to 180 points.... The same thing would apply to a Median filter, would it not?.... (I see how you can fudge the data by repeating end windows, but that is probably what caused the sudden upturn at the end of your curve)….

The moving average curves I have done look very similar to your Median filter curves.... I do see, however, that the Median filter would discard high and low data points.... They tend to occur near each other in opposing pairs, so a lot of the time a moving average "averages" them out, but not always, because at some point the sliding window for the MA picks up one (eg. a high) but not the other (eg. a low)…. The Median filter should discard both extremes all the time....

Bob

[WobblyHand]

Bob, sounds like you understand it.  Moving average can reduce your data size - it's the price one pays.  A sliding median filter can have the same loss, but some people a lot smarter than me learned that if you loop your data (wrap it around) the penalty in 'bad' points is not as bad.  As you noted, in this dataset, that caused the upturn.  I did play around with the data again, and aggressively median filtered both the velocity and the range to smooth it and reprocessed the data.  I came up with a pseudo Cd vs Mach #.  (At the time I didn't have values for all the constants in the equation).  I have to say the result is not satisfying.  Disregard the y-axis, all that is important is the shape.  You can see below Mach 0.6 things look ugly.

Unfortunately, (most of the time) we can't save all the data points.  Another technique that might work (at the expense of end points) is what is called forward backward filtering.  Matlab calls this filtfilt.  Maybe I can get a better result.  That will take a while, got to remember how to do it.  (Need to search old files where I did it.) 

[WobblyHand]

Ron, something I just thought of.  I seemed to remember somewhere (maybe in the xarchiv paper I linked?) that people had to play with the pointing angle (azimuth and elevation) to get the best SNR.  The paper stated they had to first set up on large projectiles before they could measure smaller ones.  They started with 9mm.  This is because real packaged antenna patterns are not as smooth as we would like them to be.  They are in fact lumpy.  I've measured automotive radar antennas at sensor level and found (depending on how good a job we did) the pattern deviated a bit from a cosⁿ(theta) shape.  The pattern ripple could be many dB (1-5dB).  I have also seen evidence of squinted antennas where the peak of the pattern is not where we expect it to be.  This all depends on how well the packaged system was 3D electromagnetically modeled and simulated.  This is considered to be large scale modeling, servers need to be used for this.  Since this product is considered to be 'consumer class', I don't know how detailed the simulations were.  At some point when doing consumer goods one has to say, "that's good enough".

So the fact that minor changes in angle are affecting your results could be explained by the above.  If I am interpreting your pics correctly, there is about a 4-5dB change in SNR due to the tilt of the antenna (elevation change). 

Your pics do show no evidence of multipath, so that is good.  Multipath would show up as huge dips in the SNR in a pseudo-periodic fashion (high frequency in close range, lower frequencies in further range).

I also don't see significant evidence of tumbling or large yaw.  When I get a chance, I'll run the velocity data through an FFT to see if there is anything there.  Nope.  Nothing there.

[WobblyHand]

Here is the filtfilt version of pseudo Cd and SNR vs Mach Number.  Filtfilt smoothed out some of the trash, down to 0.6.  Used an 8 pole Butterworth filter Wn=0.125.  How much do I trust this?  Wouldn't bet the farm on it.  Seems like the data from 25 dB and higher is ok.  From a practical standpoint, I'd recommend only retaining data with > 25dB SNR.  The low SNR data just plain isn't dependable.

[WobblyHand]

Ron, what is the weight and diameter of the projectile for the dataset that you posted?  My Cd is off by a factor of 20.  You seem to have numbers of ~ 0.4, I'm getting numbers like 8.   

I guessed the weight was 8.5 grains, and diameter of 0.177"
Used a conversion factor of 0.000004440136 to change grains to slugs
For rho I have 0.002373 lb/ft^3
Computed frontal area simply by pi*((D^2)/2)

Is this sensible?

[rgb1]

I read the Courtney paper while still trying to decide whether or not
to buy an LR. Their struggle with range vs projectile size was personally
discouraging but I had no such experience and was able to do much
better right out of the box. Since the half power beam width is only seven
degrees wide aiming is critical. I equipped mine with a mechanical sight.......
a short piece of tubing attached with Elmers. The unit was laid face down on
a surface plate and a small v block provided alignment until dry. Of course this
makes the assumption the radiation pattern isn't skewed from a line normal to
the case, as you suggest it could be.
In use, the gun and unit are always aimed at the same point. In my lay opinion,
a comparison of 2 and 3 demonstrate, more than anything, the reflectivity of
grass and the resulting degradation of SNR. I could, if you feel it worthwhile,
deliberately miss aim the two and see what happens.
For the sample shot...... use 15.7 grains, .22 cal, .002376 for rho, 58 F.......don't be
too concerned if you come up with slightly different numbers, there are a
variety of reasons.


                                                           Ron

[WobblyHand]

LR might have made improvements since the paper was written. 

Thanks for providing the HPBW.  Is this the total HPBW, or just the transmitter HPBW?   I haven't checked the specs yet.

Sometimes, as I noted, there can be a squint of the beam.  I'm pleased that you have some sort of sight to help get consistent measurements.  From what I see, just a few dB in SNR is greatly changing the results, (which is normal) so it might be worthwhile to check if there is any squint. 

I'm not recommending this, as it's not intrinsically safe, but if you were in an unpopulated area and could point LR to the sky and shoot safely, it might be instructive to see if the SNR is improved.  Please don't do this if you can't guarantee everyone's safety!!!  It's kind of a rhetorical thought question.  The experiment might answer the question if ground return actually is suppressing the SNR.  It's not a perfect test, because the radiation leaks out of the back side of the unit, but it should give an indication.

I had the wrong diameter and weight.  That might help get the values closer.  Alas no, I have an error in my program.    Hey, it's just a bug.  It will be fixed.

How do you determine frontal area, just pi r²?  Or something more sophisticated?

-Bruce

[rsterne]

Bruce, not sure I think wrapping the data around for the Median filter is a good idea.... It doubles the number of points at each end, right?.... I would sooner use a set of 9, and then discard 4 datasets on each end.... Additionally, since the range data is much smoother than the velocity, would it be better to use a (small) moving average for the range and a median for the velocity?.... Just a thought....

Yeah, your Pseudo Cd and Filtfilt Cd curves are pretty much unusable.... BTW, are you watching your units?.... In Imperial units the frontal area was in sq.ft…. I would think in Metric you might need to use sq.metres?....

I don't think Ron's improvement in SNR is due to the angle change, I think it is simply a matter of removing the ground clutter.... To a lesser degree, that is why the 2nd dataset is better than the first, because it is higher above the ground....

Bob

[WobblyHand]

Bob, the wrapping is done with just enough points to do the median filter.  In other words, if the median filter is of length 15, only 15 samples were wrapped and used.  The extra processing is minimal.  No doubling of points.  The range data is smoother than the velocity, perhaps a moving average would be ok.  All the processing is being done on a PC, so it doesn't matters much.  If this was to be implemented on an embedded platform, I'd think about stripping the processing to the bare essentials. 

I thought I was watching my units!  Everything was converted to Imperial, just to be consistent with your values.  Apparently there is a problem - eventually it will get sorted out.  I'll do a dimensional analysis, with all the units and see what comes up.  Usually that's a very good way to resolve this sort of thing.  Probably was a little sloppy somewhere, as the bulk of the code was written in a day.  (About 170 lines) 

My curve was derived from only 1 dataset, a single shot.  That's all the data that is available to me right now.  Using more data helps improve the SNR, which in turn improves the results.  Using aggregate data can be beneficial.

Ron's improvement in SNR could come from a several places.  The best way to find out is to construct an experiment or two to find out.  A sky shot, if safely done, could eliminate ground clutter.  If the LR was on a good tripod with accurate azimuth and elevation markings, and one did a sky shot, one could show if there is ripple or squint in the pattern.  At the moment, we can't rule it out.  To make it easier, one could scan only one dimension to see if it mattered, say in azimuth.

There's theory and there's data.  Data (in my opinion) rules.  Sure, there can be bad experiments, or technique, but once one has good technique, and a sensible experiment, data is reality.  Given the choice, I'd rather measure things rather than guess. 

[rsterne]

I thought Ron's 3rd chart (actually first in the order shown) was a "sky shot", he said he was shooting at a cloud about 5 deg. above the horizon....

Bob

[WobblyHand]

The antenna pattern isn't as abrupt as one might think.  Say the beam is 7 degrees wide.  What this really means is the half power beamwidth is 7 degrees.  That means the power is down by 1/2, or 3dB.  If the beam is pointed 5 degrees off the horizon into the sky, a significant part of the beam is still in the ground.  Here is a figure to show what I mean.

Positive angles are toward the sky.  Negative angles (including zero) are painting the ground.  The red line is the half power points.  The peak gain is 22dB.  At the ground the antenna gain is 16 dB, or -6dB peak.  There still is a lot of energy (clutter) coming in from the ground, especially from up close.  If you recall (from my previous ramblings) the receive power falls as 1/R^4.  If the radar is 3 feet off the ground, and pointed 5 degrees up from the horizon, the -6dB point hits the ground at 34 feet.  r = 3 ft/ sin(5 degrees).  The -10dB point hits the ground sooner.  If you want insignificant amounts of ground clutter in the experiment, one needs to point the beam higher. As a SWAG, about 15 degrees up off the horizon or more.  The pattern that was shown was idealized, the real one is worse. 

[rsterne]

Good to know.... so elevating the "Target" helps, how much needs to be determined by experiment.... Now all I need is a solid target backstop hanging 500 ft. below a blimp hovering at 1000 ft.... and never miss it, of course.... One problem is that gravity will be slowing the bullet if it is shot upwards, decreasing the velocity.... but I guess a correction could be done in the math to allow for that?....

Quick question, how much noise would you get from a forest of trees at 150 yards from the LR unit?.... How much would it degrade the SNR at 50 yards?.... 75 yards?.... 100 yards?.... Could I aim the LR at the top of the tallest tree, which is on a hill (say 15 deg. angle) and use that as a POA?.... How much better would that be than shooting level at 4 feet above a grassy field, also at 150 yards from the same forest?.... What would happen to the SNR if I backed up to 300 yards from the forest (noise level of 1/16th)?....

Bob

[WobblyHand]

Hah, hah.  What one really needs to worry about is what is called close in clutter.  So one way of "fixing" this is to shoot over a valley or dip in the land.  The ground should fall away, like a V.  If the beam intercepts the ground at say 30 yards away, rather than 10 yards, the clutter power would be reduced by a factor around 10.  That should be noticable.  The target could be on the hill.  Outdoor antenna ranges often are designed to have a dip in the land between the the Transmitter and the test antenna.  Or, the Transmitter is put up in a tower and the test antenna is on its own tower.

A forest 150 yards away will have very little effect on the results.  It's the stuff in close that can desensitize the radar. 

Could I aim the LR at the top of the tallest tree, which is on a hill (say 15 deg. angle) and use that as a POA?....  Yes.  Or shoot from hillock to hillock.  One doesn't need a mountain or cliff.  How much better would that be than shooting level at 4 feet above a grassy field, also at 150 yards from the same forest?....   Well, at least a dB, since that's what Ron's data shows.  Beyond that, try measuring it.  It's tough to estimate much more that this from a distance of thousands of miles away. 

Getting the correct answer would take a lot of modeling and simulation, which at the moment, doesn't make sense to do.  There's not that much public published papers on low angle ground clutter at 24 GHz, to my knowledge.  Not that it's impossible, but it's not a casual effort, especially if one doesn't already have code in place to modify.  Based on past experiences, it's like a couple thousand lines of code.  That takes a while to write, and far longer, to validate.  People have made careers out of this.  Heck, at the moment, I'm not sure of what their waveform is.  To do the modeling, the waveform is a prerequisite, as well as a terrain map.  It's a lot faster to measure it than simulate.

That being said, I just found the LR FCC site, which does have User Manuals and a copy of their test report.  https://fccid.io/PDG-LABRADAR  Going to take a look there...  Might learn something there.

[rsterne]

Bruce, thanks for your help.... I'm not trying to create a make-work project here.... *LOL*…. Pardon some of the rookie questions, but when it comes to radar, I don't even know what I don't know....

Bob

[WobblyHand]

Learned something from the internal pictures.  The transmitter is a 4x8 patch array, it has about 22 dBi gain, according to the User Manual.  LR is using a giant receiver array, it is 12x16.  I'll estimate the gain in a little while.  The chip that is in common to both the transmitter (TX) and receiver (RX) appears to have a similar footprint to the BGT24MTR, which is a 24 GHz transceiver chip.  From looking up the part, I can confirm the small array is the TX.  This makes sense, because the total TX power is capped according to regulatory agencies, like the FCC and IC.  The RX array is connected to a low noise amplifier - which is good design, and then to the BGT24MTR.  Now I understand how they get their sensitivity. 

From basic antenna theory, we find the HPBW ~ 0.89 x lambda/D [radians], where lambda is the wavelength (c/fc), and D is the aperture.  So the bigger the antenna, the higher the gain, and the narrower the beamwidth.  In the LR, we see that the D_horizontal ~ (16-1)*n*lambda, and D_vertical ~ (12-1)*n*lambda.  Typical antenna element spacing is a little wider than 0.5*lambda, but let's measure it since there is a scale on the drawing.  It seems the actual spacing for the TX is about 0.72 lambda.  If the spacing is greater, the beams will be narrower.  So lets put in the correct values.

HPBW_RX_h = 0.89 * lambda/( 15*0.72*lambda) = 0.0824 rads = 4.7 degrees, give or take.
HPBW_RX_v = 0.89 * lambda/( 11*0.72*lambda) = 0.1124 rads = 6.4 degrees

I guess the design choice was made to have a narrower beam in azimuth than in elevation.  As you can see, the RX beam is narrower than the TX.  This does help, as the total gain is the product of the TX with the RX.  So roughly speaking, the HPBW of the RX is about 1/2 of the transmitter in azimuth.  This makes sense, since the array is twice the size.  In elevation the beam is about 1/3 the width.

So... pointing angles or squint of the RX seem to be much more important than of the TX. 

Also the specification of the antenna beamwidth is 7.6 degrees in azimuth, and 18.5 degrees in elevation, at least if I have the correct orientation of the RF board.  So it seems the TX beam is painting a lot more of the ground than I expected.  But it seems the limiting factor is the RX beam, and it's elevation beamwidth is about 7 degrees.

I'm still thinking that going from hillock to hillock is a good test.  This was a good exercise.

Sorry, I can't quite post the image - it's 1.6MB, and the limit is 1.2.  I tried.  It's in the Internal Photos document at the fcc url.