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I was wondering why if the AF (Ambiguity Function) has a ridge like form it has low Doppler resolution.

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If you examine a contour plot of the Ambiguity function - the lines connect equal values of the Ambiguity function. This means that if you travel along one of these lines (i.e. keeping the same value of the function) then your system will not be able to distinguish between the signal that have this combination of Doppler and time delay. Note that the values for time/Doppler resolution depend on what value of the Ambiguity function you choose i.e. which contour line you decide on. I'm saying just pick one for the purposes of illustration.

If you look the Ambiguity function for a CW pulse of 2 secs duration: Ambiguity function for CW pulse

You see that if you choose the lowest contour value, that we would be unable to distinguish between time delay of -1.5 secs and 1.5 sec, and our Doppler resolution ranges from -0.5 Hz to 0.5 Hz. If you were use a pulse of longer duration the Ambiguity function would be wider (left to right) and shorter (top to bottom). This means your time resolution would be worse, but your Doppler resolution would be better. The opposite is true if you used a shorter pulse.

If we examine the Ambiguity function for a LFM chirp pulse: enter image description here

Now we see that the ambiguity exists for a combination of Dopplers and time delays (chose one of the contours that is essentially an ellipse).Previously, for the CW pulse, across a range of time delays the ambiguity was mostly constant for the same Doppler value. The advantage of the LFM pulse over a CW pulse, is that I can use the same pulse length but achieve better time resolution. You could also use a longer pulse and achieve better time resolution. The longer pulse means you can get more energy on the target and improve the signal to noise ratio.

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Not really sure why ambiguity function with wide frequency response is interpreted as a loss of doppler resolution. My interpretation is: wide frequency response means the signal pulse compression will not be degraded (as much) with doppler shift. LFM has this characteristic. Which means there's still a nice output from the matched filter. For doppler resolution, it's a function of the slow-time (pulse to pulse) FFT, so that seems like an (almost) completely different matter.

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