too many unknowns: speed and range of sawtooth, distance and radar cross section of targets…

No, speed and range of the sawtooth is presented in the question. Distance and radar cross sections has obviously been ruled out as the problem

OK; so the good news is that of course, signals of single-digit kHz frequencies don't randomly "leak". The question is really how well the your oscillator can follow these sawtooths. In other words, what his control loop bandwidth is, compared to your 1 kHz, and how much frequency shift your 0.6 V to 3 V variation is supposed to effect. Do you happen to have a way to verify the emitted waveform?

Have you simulated your circuit? If not why not?

@Allfadern Have you confirmed you are actually producing a frequency sweep in the chirp? What is the sweep rate? Also, when you say "chirp with a frequency", are you referring to the carrier (or center) frequency?

@MarcusMüller According to the datasheet of the VCO, it should provide a relatively linear tuning range from 0.5 - 4.5 volts. The frequency ranges from 2.36 - 2.59 GHz. Sadly i don't have any way to measure that the emitted waveform as of now.

@Andyaka No, I am not aware of any free RF simulation tools that would be able to simulate the system.

@Envidia Yes, my way of testing this was to use another device using the same frequency band as the system. Putting the device infront of the Tx antenna clearly jams the communication of the device, however if no FM is used the device does not get jammed which tells me that there is at least some frequnecy sweep. My sweep rate (of the sawtooth) is roughly 586uS, so when i referred to the frequency of the chirp i was referring to the sawtooth supplied to the VCO

I have now included a additional figure in the question, showing the IF output and the chirp on my oscilloscope

@Allfadern do I presume right that the yellow trace is thicker because the oscillations at beat frequency happen at a much higher frequency, so that you'd need to zoom in a lot to see them? Because otherwise, I really don't understand what I'm looking at, I'm afraid.

@marcusMüller No, the yellow trace is thicker because of the amplitude setting (100mV for yellow and 500mV for purple). The yellow and purple trace have the same frequency ~1.71 kHz so the "horizontal" zoom is the same.

but we expect the IF to be a beat frequency, so to do multiple oscillations during your one sawtooth period, right?

@Marcus - I agree, something doesn't look right with the waveforms shown. The RF sweep should be ~150 MHz with a chirp input range of 2.4 V. Something about the yellow trace doesn't make sense.

If you have a fixed target at zero range, your IF output should be close to DC. As you move the fixed target further away, the IF output should increase, proportional to the 2-way range and your sweep rate, MHz per us, or whatever.

@Allfadern can you show us a capture of the IF signal when you use a constant control voltage, with a target at a known distance, in a zoom where you can see the IF in the signal? then do the same zoom while using the ramp.

I did a quick back of the envelope estimation. Your chip is 138 MHz (2.4V) in 770 us (1.3 KHz). If you put a target 5' away, the round trip time is ~ 10 ns (1 ns/foot). So the IF output should be 1.8 KHz (the frequency change in your signal over 10 ns). Is this what you measure?

@SteveSh Yes something seems strange. The chirp is actually 1.71 kHz as seen in the oscilloscope figure, the "1.3 kHz" was en estimate. When the mesurment with the oscilloscope was captured the antennas where pointed at a wall roughly 2 meters (6.56 feet) away, calculations indicate that the beat frequency should be around 3 kHz. Capturing the IF output and creating a spectrogram image shows no "moving frequency" in the image even when i move a large metal platen infront of the antennas, but the doppler shift is very apparent.

@MarcusMüller I've calculated the approximate beat frequency of an object 2 meter in front of the antennas and it should be somewhere around 3 kHz

@Allfadern I'm still a bit concerned about the o-scope capture. What are you actually capturing when you say you measure the chirp in purple? Seems like you are measuring the input voltage to the VCO. What would provide more information is if you could probe the output of the VCO, so we can see the actually LFM sweep.

@Envidia - Yes, I think the purple trace is the voltage input to the VCO.

Looking at the 'scope traces, they make some sense to me now. The yellow that overlays the purple is 1 cycle (actually, a smidgen less) of the 1.3 KHz IF output. So if you move the target, you should see the IF output (yellow) change in frequency. That is, see more or less "cycles of the IF output during the chirp.

@Envidia Yes the purple trace is the voltage tuning input of the VCO, sadly my scope has a max sampling rate of 1 GSa/s and the frequency of the VCO output is roughly 2.36 - 2.59 GHz. The only way for me at the time to confirm that there is any frequency modulation is to observe if other "2.4 ghz devices" in close vicinity to the Tx antenna get affected, which they only do if i supply a variable voltage to the VCO tune.

If you change the chirp time - increase or decrease the slope of the VCO input - and keep everything else the same, you should see the IF frequency change accordingly.

@SteveSh Yes if I change the chirp time the IF frequnecy changes accordingly. But all the peaks in the IF FFT are related to the chrip frequency. Example: I use chirp freq 1.7 kHz (sweep time 586 uS) the first peak in the FFT shows up at 1.7 kHz, second at 3.4 kHz and so on... (harmonics). There are no other peaks that correlate to any known targets

@Allfadern Because the sweep is only about 230 MHz from what you've said, you can still capture the signal sampling at 1 GHz. Remember that the Nyquist sampling criterion is concerned with the bandwidth of the signal, not necessary its absolute frequencies. Look up "bandpass sampling".

@Envidia You are absolutly right, I never though of that. Will be testing it soon and post a screenshot of the scope.

@Allfadern The signal won't be centered in frequency where you expect and might likely fold around DC, so the o-scope might show you something a bit different regarding the frequency content, but the sweep nature should be apparent.

Here's a potential source of your problem. Look at your yellow trace, the 1.7 KHz (?) If output. Note that it is not continuous chip to chirp. That is, the end point of chirp n does not match up with the starting point of chirp n+1. That discontinuity will show up in any frequency analysis - like a FFT - you do of the IF.

@Envidia I probed the Tx output, and it looked very much like the output from the IF which is strange... could be something with the mixer, but then the doppler shift should not be possible to see

@Allfadern Do your troubleshooting with stationary targets (no Doppler) first so you can at least get the ranging correctly without the additional frequency shift. It does seem like you've found a problem to investigate!