AWR1642BOOST: + DCA1000 EVM ; Calculation of RCS from raw ADC Data ( by using PSD and/or SNR data)

I may not answer in the order in which you raised the questions but I will attempt to see if I can explain you in a way that it makes sense so that if the basics are more clear, you will be able to extrapolate to what you intend to do. I have to admit though that I am no expert in this so take what I say with a grain of salt. From the materials I have come across so far, the RCS estimation does not look like an exact science. It depends on what is your end goal in terms of accuracy of estimation. I don't know if your are doing this an an academic exercise or it is towards some commercial RCS measuring device, because that would be an unconventional use of our sensors, would be useful if you can share what if your goal in terms of application. I guess if yours is a commercial application, you will probably do a calibration against a known RCS device [such as a corner reflector] to take out most of the non-changing quantities in the estimation equation.

Below I will quote the relevant estimator arithmetic [mostly same in visualizer also] from https://dev.ti.com/gallery/view/1792614/mmWaveSensingEstimator/ver/1.3.0/app/input.js

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var non_coherent_combining_loss = function(num_virtual_rx) {
if (num_virtual_rx >= 8) {
return 3;
} else if (num_virtual_rx == 4) {
return 2;
} else if (num_virtual_rx == 2) {
return 1;
} else {
return 0;
}
};

var combined_factor_in_dB = function(tx_power, tx_gain, rx_gain, non_coherent_combining_loss,
detection_loss, system_loss, implementation_margin, detection_SNR, noise_figure) {
return tx_power+tx_gain+rx_gain-non_coherent_combining_loss-
detection_loss-system_loss-implementation_margin-detection_SNR-noise_figure;
};

var combined_factor_linear = function(combined_factor_in_dB) {
return Math.pow(10, combined_factor_in_dB/10);
};

var max_range_for_typical_detectable_object = function(rcs_value, combined_factor_linear,
lambda, num_virtual_rx, chirp_time, min_num_of_chirp_loops, cube_4pi, kB, ambient_temperature) {
return Math.sqrt(Math.sqrt((0.001*rcs_value*combined_factor_linear*Math.pow(lambda,2)*num_virtual_rx*chirp_time*min_num_of_chirp_loops)/(0.9*cube_4pi*kB*ambient_temperature*1e12)))
};

var min_rcs_detectable_at_max_range = function(maximum_detectable_range, cube_4pi, kB,
ambient_temperature, combined_factor_linear, lambda, num_virtual_rx, chirp_time, min_num_of_chirp_loops) {
return (0.9*Math.pow(maximum_detectable_range, 4)*
cube_4pi*kB*ambient_temperature*1e12)/(0.001*combined_factor_linear*lambda*lambda*num_virtual_rx*chirp_time*min_num_of_chirp_loops);
};

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The code syntax above isn't too complicated to make sense of the arithmetic it is displaying, so I hope you will be able to digest it at the raw level despite the syntax.

I want to point to another thread https://e2e.ti.com/support/sensors/f/1023/p/852046/3163323#3163323 the purpose of which is for you to appreciate why we have the 0.9 factor in the estimator calculation above. Although it may be confusing to see the term Beff as 1/fs [later fc correction] which in the estimator is replaced by the time term described in more detail further below.

The above equation in the estimator assumes you are doing a TDM-MIMO chirping and your processing is basically our oob processing chain. In this chain, we do 1D (range) FFT + 2D (doppler) FFT and then do sum of log magnitudes across the virtual antennas (numTx * numRx). This sum, which is a matrix of dimension number of range bins x number of doppler bins is then fed to the CFAR detection algorithm, which will detect objects if the SNR exceeds the detection SNR set in the CFAR algorithm, all objects above the detection SNR will be detected, the detection SNR is indirectly representing (along with the variant of the CFAR algorithm used) desired probability of detection (Pd in the literature) and probability of false alarm (Pfa). The purpose of the estimator was, for a given detection SNR setting in CFAR [expressing indirectly user intended Pd and Pfa], to let you find the max range for an object of given RCS [beyond the range the SNR from that object would be just below the detection SNR] or to find the minimum RCS required of an object at a given range to be detectable. My understanding is that what you and others who aspire to measure RCS of an object are basically doing is using the same equation to measure the RCS of an object [that may not necessarily be the minimum RCS at a given range where you are placing the object or at the maximum range detectable for this object's RCS] by measuring the SNR itself so the detection SNR is not a threshold of CFAR but the measurement of the SNR itself. The measurement of SNR can be done from the detection matrix just like how the CFAR is measuring before it compares against the detection SNR. If your experiment is controlled in that your object is stationary and nothing else in the scene is moving at the max velocity, this can be done by seeing the range profile and noise profile in the visualizer display - these are nothing but the detection matrix [along the range dimension] at 0th doppler and max doppler bin. So if your object were static, then the level at the range position of the object in the range profile will be the signal power and the noise power will be simply the [mostly flat looking] level of the noise profile and the subtraction [when display in dB] would give the SNR in dB. The range line at max doppler gives a good approximation of receive noise floor unless something in your scene is moving at near the max velocity when it will get messed up.

One thing to note in above is that the so called object energy in the range profile at the range bin that you see lighted up corresponding to your object is an aggregation of all energies reflected from all objects located at that range bin, there may be several or just clutter at different angles [in elevation and azimuth] which will all appear concentrated in that single bin. If you want more discrimination, you have to go do the angle processing within its limits of resolution. On the other hand, the windowing in the FFTs may broaden the main lobe to spill the energy from the main object bin to neighboring bins [in range, doppler, angle] so you may need to add some neighbors for a more accurate estimation of signal power that is truly representing your actual (point-like) object.

Related to your question of what you do when you have multiple Tx/Rx, in the basic radar equation, the time term represents the total integration time which in the case of TDM-MIMO involves all the physical chirps and all the Rx antennas. So this total time is either represented by number of virtual rx [=numTx*numRx] * physical chirp time * number of chirp loops, a chirp loop representing the number of chirps related to a single transmit antenna [the dimension involved in doppler FFT], or equivalently it is numRx (typically 4) * physical chirp time * physical number of chirps. The physical number of chirps = numTx * number of chirp loops. Bottomline is that you are considering the full signal integration time that is involved in your signal processing. In radar texts, some of these terms may be represented as "signal processing gain" - when you do an FFT, you get a gain of 10log10(N) because signal is getting concentrated by this amount while noise is not. The gain in fast time (range dimension) is represented here by the physical chirp time term and similarly, the gain in doppler dimension [10*log10(Num doppler bins)] is represented by the slow time dimension i.e number of chirp loops and the numRx similarly represents the multiple rx channel aggregation, all these are enhancing the SNR. So your measured SNR has already been enhanced by these multiplying terms in the time factor and therefore you can see the RCS calculation divides this (enhanced) measured SNR by the enhancement [to give you an intuitive sense], otherwise it would be grossly overestimated.

While the range and doppler dimension aggregation is coherent, the summing the magnitudes in the (virtual) antenna (angle) dimension is a non-coherent combining operation and so compared to coherent combining, there is a loss which is accounted by the non-coherent combining loss term, this is an empirical coding from some radar text. You can also have a coherent combining processing chain [with much more computational complexity] in which case you don't need this loss [your gain will be fully the numRx factor, you don't have to discount it] but we don't do coherent processing in oob [you may in Matlab].

Note also that the antenna gains are directional on most of our EVMs [antenna patterns are published in EVM UGs] so dependent on the angle of placement of the object, you may have to account for gain dependence on angle. An automated measurement would probably store measured antenna pattern in a table and pick the value of the gain based on the measured angle [elevation and azimuth] for the estimation of RCS and may need to make it temp dependent for more accuracy.

With the above understanding, you will be able to make sense of what to do when you have a different chirping scheme than TDM-MIMO, or if you do calculations in Matlab with some assumptions about your scene. For example if you activate all numTx in the same chirp at the same phase say, then you are doing beamforming with max energy at boresight and there will be nulls depending on the Tx spacing [typically 2*lambda on our EVMs between consecutive Tx-es] so for example if your object is at a null, it may see very poor SNR if it is even detectable in the noise/clutter and your RCS measurement will not make sense. In this simultaneous Tx case, you are not really integrating several physical chirp times (corresponding to same Tx like in TDM, you are integrating along the doppler dimension though) but here you will still see enhancement because your transmit power is higher so you will need to multiply your Gtx (in linear domain) by the simultaneous Tx-es you are activating.

In your simplistic case where you are using only one chirp i.e you don't have doppler information, your reliance on using non-lighted bins to estimate noise power assumes that you have no physical reflections coming from anywhere in your scene except the range bin where the object is placed, this may only be seen in an idealistic anechoic chamber. Using the doppler dimension processing is a more reliable way to see the receiver noise floor as it eliminates all reflections unless any were moving at the max speed. Some customers have also attempted to measure noise by exploring to activate only the receiver while sending nothing on the transmitter [there are some e2e posts on this but I don't recall if this was doable on our sensors].

PSD is o.k to use, note you don't need to worry bout scale factors [like Nfft] as SNR is relative so fixed scales will cancel out. As usual, when you calculate magnitude square, you do dB as 10*log10(|.|^2), if you estimate magnitude then you do 20*log10(|.|). Averaging noise levels is o.k but generally you will not need to average, you see a flat noise floor when you look at the noise profile in the visualizer. If you wanted to measure the actual dBm level from the ADC samples using FFT, you would need to worry about scaling.

Range index to meters is based on oob doxygen of SDK 2.1 release. In SDK 3.x, it will be embedded in the code itself (you can study its doxygen and locate the code) as we give out the point cloud x,y,z in meters directly.

Regarding your last questions, I want to first make sure we are on same page in terms of what is the noise we are talking about - this is the noise generated in the receiver electronics, mostly thermal noise. Basically the receiver output is reflections from the scene + receiver noise. Note the noise, being generated in the receiver is independent of anything in the scene [the noise does depend on the sensor temp though]. In order to estimate noise, you must guaranteed to not have any reflections or have reflections that are much lower than the noise floor. When you only do range processing i.e have one chirp only and do range FFT, each range bin represents energy reflected from all objects that are at the radial distance represented by that range bin, and there may be multiple such objects at different angles (in 3 dimensions) along the range position and furthermore they could be at any velocity. So unless you have a scene in which the only reflection is coming from the object of your interest (at the range where it is placed) and rest of the range bins are therefore noise, you will not estimate noise correctly. In an office environment for example, you may get reflections from ground and walls [some may be multi-path and may be constructive] at different ranges and this will tend to overwhelm the noise so you will be lucky if you see a single bin showing no reflections [and just the noise] and you will not be able to find out easily even if one exists [if you want you can see what I am saying in the range profile display of visualizer which is like your single chirp range processing as it is the display of range bins at 0 velocity]. The undesired reflections are called clutter, the clutter degrades with range so at higher ranges, the character of receiver output starts to look more like noise. So the way to measure receiver noise reliably is to eliminate all clutter or measure receiver output when there is no transmission [and hence no reflection] i.e the Tx antennas need to be silent while Rxes are enabled [I don't recall our devices can do this].

The convenience of doppler processing in this case is that it is easy to ensure that in your experimental scene nothing is moving at max velocity during such measurements, whereas it is much harder to control physical clutter because you have to have a idealistic anechoic chamber for that. When something moves at max velocity, its reflection will appear in the max velocity bin at the range at which the object resides and so you cannot use that for noise estimation. You could still see the noise if you knew the velocity [max or otherwise] at which the object moved and so  you would know which doppler bin it will light up [while others still show noise]. If you were to run visualizer and observe the noise profile in a static scene and if you wave your hand fast enough in front of the sensor, you will see the stable noise floor gets disturbed [goes higher] because your motion causes energy from reflection of your hand to appear in the max velocity bin.

I understand you are trying to learn from first principles but I encourage you to just run the out of box demo with visualizer if you haven't done it as it runs out of flash on power up and visualizer is easy to launch and the range and noise profiles are easy to see, you can look at it to help your learning without getting deep into how the demo or visualizer is implemented, you may be able to appreciate some of the dynamics mentioned above.