Prove that cuts in the ambiguity function are always defined by an ellipse. Hint: approximate the ambiguity function using a Taylor series expansion about the values $(\tau ,f_d)=(0,0)$; use only the first three terms in the Taylor series expansion.
The radar uncertainty principle establishes a lower bound for the time bandwidth product. More specifically, if the radar effective duration is ${\tau }_e$, and its effective bandwidth is $B_e$, show that $B_e^2{\tau }_e^2(1-{\rho }_{RDC}^2)\ge {\pi }^2$, where ${\rho }_{RDC}$ is the range-Doppler coupling coefficient defined in Chapter 4.