Consider a sonar system with range resolution \(\Delta R=4 \mathrm{~cm}\). 

(a) A sinusoidal pulse at frequency \(f_{0}=100 \mathrm{kHz}\) is transmitted. What is the pulse width, and what is the bandwidth? 

(b) By using an up-chirp LFM centered at \(f_{0}\), one can increase the pulse width for the same range resolution. If you want to increase the transmitted energy by a factor of 20, give an expression for the transmitted pulse. (c) Give an expression for the causal filter matched to the LFM pulse in part (b).