What is the importance of signal bandwidth in radar systems for velocity and localization estimation
Bandwidth affects the &quot;gate/cell width&quot; of your measurements. If you think about an LFM waveform reflecting back off an object and going through a matched filter to determine the range, a signal with a very low bandwidth is going to have a more ambiguous return time (range) than a more aggressive chirp. The rough calculation for the smallest range resolution for a certain bandwidth is $\frac{c}{2B}$.
I imagine the relationship is reversed for doppler processing. A signal with low bandwidth would provide a lot of doppler resolution because there isn't a great deal of ambiguity as to the return's center frequency, so the doppler gate width should increase proportional to bandwidth. That is a guess, though.
Radar and other signal processing disciplines involve these tradeoffs a lot. Certainty in frequency versus certainty in time and vice versa, maximum  unambiguous range versus improvement factor from pulse repetition frequency, etc.