I was wondering why is the ideal AF has zero value in all the function except in (0, 0).
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2$\begingroup$ Because if there is just one peak, there are no ambiguities. Any additional peak in the AF could be mistaken for an additional target. $\endgroup$– FlorianCommented Sep 5, 2019 at 16:31
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1$\begingroup$ It is the adjective ideal that is the trouble-maker here. One has to define what an ideal ACF is, and anything other than a function that is zero everywhere except at $(0,0)$ can be disparaged by others on various well-chosen grounds ("my ideal AF is better than your ideal AF because it has ...."). With 0 you have sunk as low as it is possible everywhere. $\endgroup$– Dilip SarwateCommented Sep 6, 2019 at 3:49
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An ideal ambiguity function is by definition a Dirac Delta function with an impulse response at 0 delay and 0 frequency with zero value elsewhere. It only takes on a non-zero value when the delay and frequency are matched between the filter and the signal.
That is of course not the whole story as that doesn't address why this is a useful definition. This definition is chosen so that there is no ambiguity as to whether the signal and filter are matched. If they are not matched then the response is zero.
As the Wikipedia link above states, this is not usually desired in practice as there will usually be some mismatch between the filter and signal:
This is not usually desirable (if a target has any Doppler shift from an unknown velocity it will disappear from the radar picture)...
In practice one often wants an ambiguity function that is close to the idea, but not exactly to handle mismatches in the delay or frequency offset.
answered Jan 10, 2022 at 13:22